View source for Physics:Entropy

{{short description|Property of a thermodynamic system}}
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{{Infobox physical quantity
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| unit =joules per kelvin (J⋅K<sup>−1</sup>)
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| symbols = ''S''
| baseunits = kg⋅m<sup>2</sup>⋅s<sup>−2</sup>⋅K<sup>−1</sup>
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{{Modern physics}}
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'''Entropy''' is a [[Science|scientific]] concept that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the microscopic description of nature in [[Physics:Statistical physics|statistical physics]], and to the principles of [[Information theory|information theory]]. It has found far-ranging applications in [[Chemistry:Chemistry|chemistry]] and [[Physics:Physics|physics]], in biological systems and their relation to life, in [[Astronomy:Cosmology|cosmology]], economics, sociology, [[Physics:Atmospheric science|weather science]], [[Earth:Climate change|climate change]], and [[Information system|information system]]s including the transmission of information in telecommunication.<ref>{{Cite journal|last=Wehrl|first=Alfred|date=1 April 1978|title=General properties of entropy|url=https://link.aps.org/doi/10.1103/RevModPhys.50.221|journal=Reviews of Modern Physics|volume=50|issue=2|pages=221–260|doi=10.1103/RevModPhys.50.221|bibcode=1978RvMP...50..221W}}</ref>

Entropy is central to the [[Physics:Second law of thermodynamics|second law of thermodynamics]], which states that the entropy of an isolated system left to spontaneous evolution cannot decrease with time. As a result, isolated systems evolve toward [[Physics:Thermodynamic equilibrium|thermodynamic equilibrium]], where the entropy is highest. A consequence of the second law of thermodynamics is that certain processes are [[Physics:Irreversible process|irreversible]]. 

The thermodynamic concept was referred to by Scottish scientist and engineer [[Biography:William Rankine|William Rankine]] in 1850 with the names ''thermodynamic function'' and ''heat-potential''.<ref>{{cite book |last=Truesdell |first=C. |date=1980 |title=The Tragicomical History of Thermodynamics, 1822–1854 |url=https://archive.org/details/tragicomicalhist0000unse |url-access=registration |location=New York |publisher=Springer-Verlag |isbn=0387904034 |page=215 |via=[[Internet Archive]]}}</ref> In 1865, German physicist [[Biography:Rudolf Clausius|Rudolf Clausius]], one of the leading founders of the field of thermodynamics, defined it as the quotient of an infinitesimal amount of heat to the instantaneous temperature. He initially described it as ''transformation-content'', in German ''Verwandlungsinhalt'', and later coined the term ''entropy'' from a Greek word for ''transformation''.<ref name=brush1976>[[Biography:Stephen G. Brush|Brush, S.G.]] (1976). ''The Kind of Motion We Call Heat: a History of the Kinetic Theory of Gases in the 19th Century, Book 2, Statistical Physics and Irreversible Processes'', Elsevier, Amsterdam, {{ISBN|0-444-87009-1}}, pp. 576–577.</ref>

Austrian physicist [[Biography:Ludwig Boltzmann|Ludwig Boltzmann]] explained entropy as the measure of the number of possible microscopic arrangements or states of individual atoms and molecules of a system that comply with the macroscopic condition of the system. He thereby introduced the concept of statistical disorder and [[Probability distribution|probability distribution]]s into a new field of thermodynamics, called [[Physics:Statistical mechanics|statistical mechanics]], and found the link between the microscopic interactions, which fluctuate about an average configuration, to the macroscopically observable behavior, in form of a simple [[Logarithm|logarithm]]ic law, with a [[Proportionality (mathematics)|proportionality constant]], the [[Physics:Boltzmann constant|Boltzmann constant]], that has become one of the defining universal constants for the modern [[International System of Units]] (SI).

== History ==
[[File:Clausius.jpg|thumb|upright|Rudolf Clausius (1822–1888), originator of the concept of entropy]]
{{Main|Physics:History of entropy}}

In his 1803 paper ''Fundamental Principles of Equilibrium and Movement'', the French mathematician [[Biography:Lazare Carnot|Lazare Carnot]] proposed that in any machine, the accelerations and shocks of the moving parts represent losses of ''moment of activity''; in any natural process there exists an inherent tendency towards the dissipation of useful energy. In 1824, building on that work, Lazare's son, [[Biography:Nicolas Léonard Sadi Carnot|Sadi Carnot]], published ''[[Physics:Reflections on the Motive Power of Fire|Reflections on the Motive Power of Fire]]'', which posited that in all heat-engines, whenever "[[Physics:Caloric theory|caloric]]" (what is now known as heat) falls through a temperature difference, work or [[Physics:Work|motive power]] can be produced from the actions of its fall from a hot to cold body. He used an analogy with how water falls in a [[Engineering:Water wheel|water wheel]]. That was an early insight into the [[Physics:Second law of thermodynamics|second law of thermodynamics]].<ref>{{cite web |url=http://scienceworld.wolfram.com/biography/CarnotSadi.html |title=Carnot, Sadi (1796–1832) |publisher=Wolfram Research |year=2007 |access-date=24 February 2010}}</ref> Carnot based his views of heat partially on the early 18th-century "Newtonian hypothesis" that both heat and light were types of indestructible forms of matter, which are attracted and repelled by other matter, and partially on the contemporary views of Count Rumford, who showed in 1789 that heat could be created by friction, as when cannon bores are machined.<ref>{{Cite book |last=McCulloch |first=Richard, S. |title=Treatise on the Mechanical Theory of Heat and its Applications to the Steam-Engine, etc. |publisher=D. Van Nostrand |year=1876}}</ref> Carnot reasoned that if the body of the working substance, such as a body of steam, is returned to its original state at the end of a complete engine cycle, "no change occurs in the condition of the working body".

The [[Physics:First law of thermodynamics|first law of thermodynamics]], deduced from the heat-friction experiments of James Joule in 1843, expresses the concept of energy and its [[Physics:Conservation of energy|conservation]] in all processes; the first law, however, is unsuitable to separately quantify the effects of [[Physics:Friction|friction]] and [[Dissipation|dissipation]].{{cn|date=May 2023}}

In the 1850s and 1860s, German physicist [[Biography:Rudolf Clausius|Rudolf Clausius]] objected to the supposition that no change occurs in the working body, and gave that change a mathematical interpretation, by questioning the nature of the inherent loss of usable heat when work is done, e.g., heat produced by friction.<ref name="Clausius">{{Cite journal |last=Clausius |first=Rudolf |title=Über die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen |language=de |year=1850 |doi=10.1002/andp.18501550306 |journal=Annalen der Physik |volume=155 |issue=3 |hdl=2027/uc1.$b242250 |bibcode=1850AnP...155..368C |pages=368–397 |hdl-access=free}} [On the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat] : Poggendorff's ''Annalen der Physik und Chemie''.</ref> He described his observations as a dissipative use of energy, resulting in a ''transformation-content'' ({{lang|de|Verwandlungsinhalt}} in German), of a [[Physics:Thermodynamic system|thermodynamic system]] or working body of [[Chemistry:Chemical species|chemical species]] during a change of [[Physics:Thermodynamic state|state]].<ref name="Clausius" /> That was in contrast to earlier views, based on the theories of [[Biography:Isaac Newton|Isaac Newton]], that heat was an indestructible particle that had mass. Clausius discovered that the non-usable energy increases as steam proceeds from inlet to exhaust in a steam engine. From the prefix ''en-'', as in 'energy', and from the Greek word {{lang|el|τροπή}} [tropē], which is translated in an established lexicon as ''turning'' or ''change''<ref>Liddell, H. G., Scott, R. (1843/1978). A Greek–English Lexicon, revised and augmented edition, Oxford University Press, Oxford UK, {{ISBN|0198642148}}, pp.&nbsp;1826–1827.</ref> and that he rendered in German as {{lang|de|Verwandlung}}, a word often translated into English as ''transformation'', in 1865 Clausius coined the name of that property as ''entropy''.<ref name = "Clausius German">{{cite journal |last1=Clausius |first1=Rudolf |year=1865 |title=Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie (Vorgetragen in der naturforsch. Gesellschaft zu Zürich den 24. April 1865) |language=de |journal=Annalen der Physik und Chemie |volume=125 |issue=7 |pages=353–400 |doi=10.1002/andp.18652010702 |bibcode=1865AnP...201..353C |url=https://zenodo.org/record/1423700 |quote=Sucht man für ''S'' einen bezeichnenden Namen, so könnte man, ähnlich wie von der Gröſse ''U'' gesagt ist, sie sey der ''Wärme- und Werkinhalt'' des Körpers, von der Gröſse ''S'' sagen, sie sey der ''Verwandlungsinhalt'' des Körpers. Da ich es aber für besser halte, die Namen derartiger für die Wissenschaft wichtiger Gröſsen aus den alten Sprachen zu entnehmen, damit sie unverändert in allen neuen Sprachen angewandt werden können, so schlage ich vor, die Gröſse ''S'' nach dem griechischen Worte ἡ τροπή, die Verwandlung, die ''Entropie'' des Körpers zu nennen. Das Wort ''Entropie'' habei ich absichtlich dem Worte ''Energie'' möglichst ähnlich gebildet, denn die beiden Gröſsen, welche durch diese Worte benannt werden sollen, sind ihren physikalischen Bedeutungen nach einander so nahe verwandt, daſs eine gewisse Gleichartigkeit in der Benennung mir zweckmäſsig zu seyn scheint. |quote-page=390}}</ref> The word was adopted into the English language in 1868.

Later, scientists such as [[Biography:Ludwig Boltzmann|Ludwig Boltzmann]], [[Biography:Josiah Willard Gibbs|Josiah Willard Gibbs]], and [[Biography:James Clerk Maxwell|James Clerk Maxwell]] gave entropy a statistical basis. In 1877, Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of [[Physics:Ideal gas|ideal gas]] particles, in which he defined entropy as proportional to the [[Natural logarithm|natural logarithm]] of the number of microstates such a gas could occupy. The [[Proportionality (mathematics)|proportionality constant]] in this definition, called the [[Physics:Boltzmann constant|Boltzmann constant]], has become one of the defining universal constants for the modern [[International System of Units]] (SI). Henceforth, the essential problem in statistical thermodynamics has been to determine the distribution of a given amount of energy ''E'' over ''N'' identical systems. [[Biography:Constantin Carathéodory|Constantin Carathéodory]], a Greek mathematician, linked entropy with a mathematical definition of irreversibility, in terms of trajectories and integrability.

== Etymology ==
In 1865, Clausius named the concept of "the differential of a quantity which depends on the configuration of the system", ''entropy'' ({{lang|de|Entropie}}) after the Greek word for 'transformation'.<ref name="Gil399">{{cite book |last=Gillispie |first=Charles Coulston |title=The Edge of Objectivity: An Essay in the History of Scientific Ideas |url=https://archive.org/details/edgeofobjectivit0000gill |url-access=registration |year=1960 |publisher=Princeton University Press |isbn=0-691-02350-6 |page=[https://archive.org/details/edgeofobjectivit0000gill/page/399 399] |quote=Clausius coined the word entropy for <math>S</math>: "I prefer going to the ancient languages for the names of important scientific quantities, so that they may mean the same thing in all living tongues. I propose, accordingly, to call <math>S</math> the entropy of a body, after the Greek word 'transformation'. I have designedly coined the word entropy to be similar to 'energy', for these two quantities are so analogous in their physical significance, that an analogy of denomination seemed to me helpful."}}</ref> He gave "transformational content" ({{lang|de|Verwandlungsinhalt}}) as a synonym, paralleling his "thermal and ergonal content" ({{lang|de|Wärme- und Werkinhalt}}) as the name of [[Physics:Internal energy|<math>U</math>]], but preferring the term ''entropy'' as a close parallel of the word ''energy'', as he found the concepts nearly "analogous in their physical significance".<ref name=Gil399/> This term was formed by replacing the root of {{lang|grc|ἔργον}} ('ergon', 'work') by that of {{lang|grc|τροπή}} ('tropy', 'transformation').<ref name = "Clausius German"/>

In more detail, Clausius explained his choice of "entropy" as a name as follows:<ref name="Cooper">{{cite book |last1=Cooper |first1=Leon N. |title=An Introduction to the Meaning and Structure of Physics |date=1968 |publisher=Harper |page=331}}</ref>
<blockquote>
I prefer going to the ancient languages for the names of important scientific quantities, so that they may mean the same thing in all living tongues. I propose, therefore, to call ''S'' the ''entropy'' of a body, after the Greek word "transformation". I have designedly coined the word ''entropy'' to be similar to energy, for these two quantities are so analogous in their physical significance, that an analogy of denominations seems to me helpful.
</blockquote>
[[Biography:Leon Cooper|Leon Cooper]] added that in this way "he succeeded in coining a word that meant the same thing to everybody: nothing".<ref name = "Cooper" />

== Definitions and descriptions ==
{{quote box|width=30em|quote=Any method involving the notion of entropy, the very existence of which depends on the second law of thermodynamics, will doubtless seem to many far-fetched, and may repel beginners as obscure and difficult of comprehension.|source=Willard Gibbs, ''Graphical Methods in the Thermodynamics of Fluids''<ref>{{cite book|title=The scientific papers of J. Willard Gibbs in Two Volumes|url=https://archive.org/stream/scientificpapers01gibbuoft#page/11/mode/1up |volume=1 |year=1906 |publisher=Longmans, Green, and Co. |page=11 |access-date=2011-02-26}}</ref>}}

The concept of entropy is described by two principal approaches, the macroscopic perspective of classical thermodynamics, and the microscopic description central to [[Physics:Statistical mechanics|statistical mechanics]]. The classical approach defines entropy in terms of macroscopically measurable physical properties, such as bulk mass, volume, pressure, and temperature. The statistical definition of entropy defines it in terms of the statistics of the motions of the microscopic constituents of a system – modeled at first classically, e.g. Newtonian particles constituting a gas, and later quantum-mechanically (photons, phonons, spins, etc.). The two approaches form a consistent, unified view of the same phenomenon as expressed in the second law of thermodynamics, which has found universal applicability to physical processes.

=== State variables and functions of state ===
Many thermodynamic properties are defined by physical variables that define a state of [[Physics:Thermodynamic equilibrium|thermodynamic equilibrium]]; these are ''state variables''. State variables depend only on the equilibrium condition, not on the path evolution to that state. State variables can be functions of state, also called ''state functions'', in a sense that one state variable is a [[Function (mathematics)|mathematical function]] of other state variables. Often, if some properties of a system are determined, they are sufficient to determine the state of the system and thus other properties' values. For example, temperature and pressure of a given quantity of gas determine its state, and thus also its volume via the [[Physics:Ideal gas law|ideal gas law]]. A system composed of a pure substance of a single [[Physics:Phase (matter)|phase]] at a particular uniform temperature and pressure is determined, and is thus a particular state, and has not only a particular volume but also a specific entropy.<ref>J. A. McGovern,{{cite web|url=http://theory.phy.umist.ac.uk/~judith/stat_therm/node29.html |title=2.5 Entropy |access-date=5 February 2013 |url-status=dead |archive-url=https://web.archive.org/web/20120923080936/http://theory.phy.umist.ac.uk/~judith/stat_therm/node29.html |archive-date=23 September 2012 }}</ref> The fact that entropy is a function of state makes it useful. In the Carnot cycle, the working fluid returns to the same state that it had at the start of the cycle, hence the change or [[Line integral|line integral]] of any state function, such as entropy, over this reversible cycle is zero.

=== Reversible process ===
Total entropy may be conserved during a [[Physics:Reversible process (thermodynamics)|reversible process]]. The entropy change ''<math display="inline">d S</math>'' of the system (not including the surroundings) is well-defined as heat ''<math display="inline">\delta Q_\text{rev}</math>'' transferred to the system divided by the system temperature ''<math display="inline">T</math>'', <math display="inline">d S = \frac{\delta Q_\text{rev}}{T}</math>. A reversible process is a [[Physics:Quasistatic process|quasistatic]] one that deviates only infinitesimally from thermodynamic equilibrium and avoids friction or other dissipation. Any process that happens quickly enough to deviate from thermal equilibrium cannot be reversible, total entropy increases, and the potential for maximum work to be done in the process is also lost.<ref>{{cite web|title=6.5 Irreversibility, Entropy Changes, and ''Lost Work''|url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node48.html|website=web.mit.edu|access-date=21 May 2016}}</ref> For example, in the [[Physics:Carnot cycle|Carnot cycle]], while the heat flow from the hot reservoir to the cold reservoir represents an increase in entropy in the cold reservoir, the work output, if reversibly and perfectly stored in some energy storage mechanism, represents a decrease in entropy that could be used to operate the heat engine in '''reverse''' and return to the previous state; thus the ''total'' entropy change may still be zero at all times if the entire process is reversible. An irreversible process increases the total entropy of the system and surroundings.<ref>{{cite web|last1=Lower|first1=Stephen|title=What is entropy?|url=http://www.chem1.com/acad/webtext/thermeq/TE2.html|website=chem1.com|access-date=21 May 2016}}</ref>

=== Carnot cycle ===
The concept of entropy arose from [[Biography:Rudolf Clausius|Rudolf Clausius]]'s study of the [[Physics:Carnot cycle|Carnot cycle]] that is a [[Physics:Thermodynamic cycle|thermodynamic cycle]] performed by a Carnot heat engine as a reversible heat engine.<ref>{{cite book| last1=Lavenda| first1=Bernard H.| title=A new perspective on thermodynamics| date=2010|publisher=Springer|location=New York|isbn=978-1-4419-1430-9|edition=Online-Ausg.|chapter=2.3.4}}</ref> In a Carnot cycle, heat {{math|''Q''<sub>H</sub>}} is absorbed isothermally at temperature {{math|''T''<sub>H</sub>}} from a 'hot' reservoir (in the isothermal expansion stage) and given up isothermally as heat {{math|''Q''<sub>C</sub>}} to a 'cold' reservoir at {{math|''T''<sub>C</sub>}} (in the isothermal compression stage). According to [[Physics:Carnot's theorem (thermodynamics)|Carnot's principle or theorem]], [[Physics:Work|work]] from a heat engine with two thermal reservoirs can be produced only when there is a temperature difference between these reservoirs, and for reversible engines which are mostly and equally efficient among all heat engines for a given thermal reservoir pair, the work is a function of the reservoir temperatures and the heat absorbed to the engine {{math|''Q''<sub>H</sub>}} (heat engine work output = [[Physics:Heat engine#Efficiency|heat engine efficiency]] × heat to the engine, where the efficiency is a function of the reservoir temperatures for reversible heat engines). Carnot did not distinguish between {{math|''Q''<sub>H</sub>}} and {{math|''Q''<sub>C</sub>}}, since he was using the incorrect hypothesis that [[Physics:Caloric theory|caloric theory]] was valid, and hence heat was conserved (the incorrect assumption that {{math|''Q''<sub>H</sub>}} and {{math|''Q''<sub>C</sub>}} were equal in magnitude) when, in fact, the magnitude of {{math|''Q''<sub>H</sub>}} is greater than the magnitude of {{math|''Q''<sub>C</sub>}}.<ref>{{cite book|last1=Carnot|first1=Sadi Carnot|editor1-last=Fox|editor1-first=Robert |title=Reflexions on the motive power of fire|url=https://archive.org/details/reflexionsonmoti0000carn|url-access=registration| date=1986|publisher=Lilian Barber Press| location=New York|isbn=978-0-936508-16-0|pages=[https://archive.org/details/reflexionsonmoti0000carn/page/26 26]}}</ref><ref>{{cite book|last1=Truesdell|first1=C.|title=The tragicomical history of thermodynamics 1822–1854|url=https://archive.org/details/tragicomicalhist18221854iiic|url-access=limited|date=1980|publisher=Springer|location=New York|isbn=978-0-387-90403-0|pages=[https://archive.org/details/tragicomicalhist18221854iiic/page/n85 78]–85}}</ref> Through the efforts of Clausius and [[Biography:Lord Kelvin|Kelvin]], it is now known that the work done by a reversible heat engine is the product of the Carnot efficiency (it is the efficiency of all reversible heat engines with the same thermal reservoir pairs according to the [[Physics:Carnot's theorem (thermodynamics)|Carnot's theorem]]) and the heat absorbed from the hot reservoir:
{{NumBlk||<math display="block">W = \left(\frac{T_\text{H}-T_\text{C}}{T_\text{H}}\right)Q_\text{H} = \left(1-\frac{T_\text{C}}{T_\text{H}} \right) Q_\text{H}</math>|{{EquationRef|1}}}}

Here <math>W</math> is work done by the Carnot heat engine, <math>Q_\text{H}</math> is heat to the engine from the hot reservoir, and  <math>-\frac{T_\text{C}}{T_\text{H}}Q_\text{H}</math> is heat to the cold reservoir from the engine. To derive the ''Carnot efficiency'', which is {{math|1 − ''T''<sub>C</sub>/''T''<sub>H</sub>}} (a number less than one), Kelvin had to evaluate the ratio of the work output to the heat absorbed during the isothermal expansion with the help of the Carnot–Clapeyron equation, which contained an unknown function called the Carnot function. The possibility that the Carnot function could be the temperature as measured from a zero point of temperature was suggested by [[Biography:James Prescott Joule|Joule]] in a letter to Kelvin. This allowed Kelvin to establish his absolute temperature scale.<ref>{{cite book |last1=Clerk Maxwell|first1=James|editor1-last=Pesic|editor1-first=Peter|title=Theory of heat|date=2001|publisher=Dover Publications |location=Mineola|isbn=978-0-486-41735-6|pages=115–158}}</ref> It is also known that the net work {{math|''W''}} produced by the system in one cycle is the net heat absorbed, which is the sum (or difference of the magnitudes) of the heat {{math|''Q''<sub>H</sub>}} > 0 absorbed from the hot reservoir and the waste heat {{math|''Q''<sub>C</sub>}} < 0 given off to the cold reservoir:<ref name="PlanckBook">{{cite book |last=Planck |first=M. |title=Treatise on Thermodynamics |page=§90 & §137|quote=eqs.(39), (40), & (65) |publisher=Dover Publications |year=1945}}.</ref>
{{NumBlk||<math display="block">W=Q_\text{H}+Q_\text{C}</math>|{{EquationRef|2}}}}

Since the latter is valid over the entire cycle, this gave Clausius the hint that at each stage of the cycle, work and heat would not be equal, but rather their difference would be the change of a state function that would vanish upon completion of the cycle. The state function was called the [[Physics:Internal energy|internal energy]], that is central to the [[Physics:First law of thermodynamics|first law of thermodynamics]].<ref name="Clausius1867">{{cite book |author=Rudolf Clausius |title=The Mechanical Theory of Heat: With Its Applications to the Steam-engine and to the Physical Properties of Bodies|url=https://books.google.com/books?id=8LIEAAAAYAAJ |year=1867 |publisher=J. Van Voorst |isbn=978-1-4981-6733-8|page=28}}</ref>

Now equating ({{EquationNote|1}}) and ({{EquationNote|2}}) gives, for the engine per Carnot cycle,<ref name="FermiBook">{{cite book |last=Fermi |first=E. |title=Thermodynamics |page=48 |quote=eq.(64) |publisher=Dover Publications (still in print) |year=1956}}.</ref><ref name="PlanckBook"/>

{{NumBlk||<math display="block">Q_\text{H} / T_\text{H}+Q_\text{C} / T_\text{C} = 0</math>|{{EquationRef|3}}}}

This implies that there is a function of state whose change is {{math|''Q''/''T''}} and this state function is conserved over a complete Carnot cycle, like other state function such as the internal energy. Clausius called this state function ''entropy''. One can see that entropy was discovered through mathematics rather than through laboratory experimental results.{{citation needed|date=November 2022}} It is a mathematical construct and has no easy physical analogy.{{citation needed|date=November 2022}} This makes the concept somewhat obscure or abstract, akin to how the concept of energy arose. This equation shows an entropy change per Carnot cycle is zero. In fact, an entropy change in the both thermal reservoirs per Carnot cycle is also zero since that change is simply expressed by reverting the sign of each term in the equation ({{EquationNote|3}}) according to the fact that, for example, for heat transfer from the hot reservoir to the engine, the engine receives the heat while the hot reservoir loses the same amount of the heat;

{{NumBlk||<math display="block"> \Delta S_\text{r,H}+  \Delta S_\text{r,C} = - Q_\text{H} / T_\text{H} - Q_\text{C} / T_\text{C} = 0</math>|{{EquationRef|4}}}}

where we denote an entropy change for a thermal reservoir by {{math|1=Δ''S''<sub>''r,i''</sub> = - ''Q''<sub>''i''</sub>/''T''<sub>''i''</sub>}}, for ''i'' as either H (Hot reservoir) or C (Cold reservoir), by considering the above-mentioned signal convention of heat for the engine.

Clausius then asked what would happen if less work is produced by the system than that predicted by Carnot's principle for the same thermal reservoir pair and the same heat transfer from the hot reservoir to the engine {{math|''Q''<sub>H</sub>}}. In this case, the right-hand side of the equation ({{EquationNote|1}}) would be the upper bound of the work output by the system, and the equation would now be converted into an inequality<math display="block"> W < \left(1 - \frac{T_\text{C}}{T_\text{H}}\right) Q_\text{H}</math>When the equation ({{EquationNote|2}}) is used to express the work as a net or total heat exchanged in a cycle, we get<math display="block"> Q_\text{H}+Q_\text{C}<\left(1-\frac{T_\text{C}}{T_\text{H}}\right)Q_\text{H}</math>or<math display="block"> |Q_\text{C}|>\frac{T_\text{C}}{T_\text{H}}Q_\text{H}</math>by considering the sign convention of heat where {{math|''Q''<sub>H</sub>}} > 0 is heat that is from the hot reservoir and is absorbed by the engine and {{math|''Q''<sub>C</sub>}} < 0 is the waste heat given off to the cold reservoir from the engine. So, more heat is given up to the cold reservoir than in the Carnot cycle. The above inequality <math> Q_\text{H}+Q_\text{C}<\left(1-\frac{T_\text{C}}{T_\text{H}}\right)Q_\text{H}</math> can be written as<math display="block">\frac{Q_\text{H}}{T_\text{H}} + \frac{Q_\text{C}}{T_\text{C}} < 0.</math>If we, again, denote an entropy change for a thermal reservoir by {{math|1=Δ''S''<sub>''r,i''</sub> = - ''Q''<sub>''i''</sub>/''T''<sub>''i''</sub>}}, for ''i'' as either H (Hot reservoir) or C (Cold reservoir), by considering the abovementioned signal convention of heat for the engine, then<math display="block">\Delta S_\text{r,H}+  \Delta S_\text{r,C} > 0</math>or
{{NumBlk||<math display="block"> \Delta S_\text{r,C} > - \Delta S_\text{r,H} = \left \vert \Delta S_\text{r,H} \right \vert</math>|{{EquationRef|5}}}}telling that the magnitude of the entropy earned by the cold reservoir is greater than the entropy lost by the hot reservoir. The net entropy change in the engine per its thermodynamic cycle is zero, so the net entropy change in the engine and both the thermal reservoirs per cycle increases if work produced by the engine is less than the work achieved by a Carnot engine in the equation ({{EquationNote|1}}).

The [[Physics:Carnot cycle|Carnot cycle]] and Carnot efficiency as shown in the equation ({{EquationNote|1}}) are useful because they define the upper bound of the possible work output and the efficiency of any classical thermodynamic heat engine. Other cycles, such as the [[Physics:Otto cycle|Otto cycle]], [[Physics:Diesel cycle|Diesel cycle]] and [[Physics:Brayton cycle|Brayton cycle]], can be analyzed from the standpoint of the Carnot cycle. Any machine or cyclic process that converts heat to work and is claimed to produce an efficiency greater than the Carnot efficiency is not viable because it violates [[Physics:Second law of thermodynamics|the second law of thermodynamics]].

For very small numbers of particles in the system, statistical thermodynamics must be used. The efficiency of devices such as photovoltaic cells requires an analysis from the standpoint of quantum mechanics.

=== Classical thermodynamics ===
{{Main|Physics:Entropy (classical thermodynamics)}} 
{{Conjugate variables (thermodynamics)}}

The thermodynamic definition of entropy was developed in the early 1850s by [[Biography:Rudolf Clausius|Rudolf Clausius]] and essentially describes how to measure the entropy of an [[Physics:Isolated system|isolated system]] in [[Physics:Thermodynamic equilibrium|thermodynamic equilibrium]] with its parts. Clausius created the term entropy as an [[Physics:Intensive and extensive properties|extensive]] thermodynamic variable that was shown to be useful in characterizing the [[Physics:Carnot cycle|Carnot cycle]]. Heat transfer in the isotherm steps (isothermal expansion and isothermal compression) of the Carnot cycle was found to be proportional to the temperature of a system (known as its [[Physics:Absolute temperature|absolute temperature]]). This relationship was expressed in an increment of entropy that is equal to incremental heat transfer divided by temperature. Entropy was found to vary in the thermodynamic cycle but eventually returned to the same value at the end of every cycle. Thus it was found to be a function of state, specifically a thermodynamic state of the system.

While Clausius based his definition on a reversible process, there are also irreversible processes that change entropy. Following the [[Physics:Second law of thermodynamics|second law of thermodynamics]], entropy of an isolated [[Physics:Thermodynamic system|system]] always increases for irreversible processes. The difference between an isolated system and closed system is that energy may ''not'' flow to and from an isolated system, but energy flow to and from a closed system is possible. Nevertheless, for both closed and isolated systems, and indeed, also in open systems, irreversible thermodynamics processes may occur.

According to the [[Physics:Clausius theorem|Clausius equality]], for a reversible cyclic process: <math display="inline">\oint \frac{\delta Q_\text{rev}}{T} = 0</math>. This means the line integral <math display="inline">\int_L \frac{\delta Q_\text{rev}}{T}</math> is [[Physics:State function|path-independent]].

So we can define a state function {{mvar|S}} called entropy, which satisfies <math display="inline">d S = \frac{\delta Q_\text{rev}}{T}</math>.

To find the entropy difference between any two states of a system, the integral must be evaluated for some reversible path between the initial and final states.<ref>{{Cite book|last=Atkins|first=Peter|author2=Julio De Paula|title=Physical Chemistry, 8th ed.|publisher=Oxford University Press|year=2006|page=79|isbn=978-0-19-870072-2}}</ref> Since entropy is a state function, the entropy change of the system for an irreversible path is the same as for a reversible path between the same two states.<ref>{{Cite book|last=Engel|first=Thomas|author2=Philip Reid|title=Physical Chemistry|publisher=Pearson Benjamin Cummings|year=2006|page=86|isbn=978-0-8053-3842-3}}</ref> However, the heat transferred to or from, and the entropy change of, the surroundings is different.

We can only obtain the change of entropy by integrating the above formula. To obtain the absolute value of the entropy, we need the [[Physics:Third law of thermodynamics|third law of thermodynamics]], which states that ''S'' = 0 at [[Physics:Absolute zero|absolute zero]] for perfect crystals.

From a macroscopic perspective, in classical thermodynamics the entropy is interpreted as a [[Physics:State function|state function]] of a [[Physics:Thermodynamic system|thermodynamic system]]: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. In any process where the system gives up energy Δ''E'', and its entropy falls by Δ''S'', a quantity at least ''T''<sub>R</sub> Δ''S'' of that energy must be given up to the system's surroundings as heat (''T''<sub>R</sub> is the temperature of the system's external surroundings). Otherwise the process cannot go forward. In classical thermodynamics, the entropy of a system is defined only if it is in physical [[Physics:Thermodynamic equilibrium|thermodynamic equilibrium]]. (But chemical equilibrium is not required: the entropy of a mixture of two moles of hydrogen and one mole of oxygen at 1 bar pressure and 298 K is well-defined.)

=== Statistical mechanics ===
{{main|Physics:Entropy (statistical thermodynamics)}}
The statistical definition was developed by [[Biography:Ludwig Boltzmann|Ludwig Boltzmann]] in the 1870s by analyzing the statistical behavior of the microscopic components of the system. Boltzmann showed that this definition of entropy was equivalent to the thermodynamic entropy to within a constant factor—known as the [[Physics:Boltzmann constant|Boltzmann constant]]. In short, the thermodynamic definition of entropy provides the experimental verification of entropy, while the statistical definition of entropy extends the concept, providing an explanation and a deeper understanding of its nature.

The [[Physics:Entropy (statistical thermodynamics)|interpretation of entropy in statistical mechanics]] is the measure of uncertainty, disorder, or ''mixedupness'' in the phrase of [[Biography:Josiah Willard Gibbs|Gibbs]], which remains about a system after its observable macroscopic properties, such as temperature, pressure and volume, have been taken into account. For a given set of macroscopic variables, the entropy measures the degree to which the probability of the system is spread out over different possible [[Physics:Microstate (statistical mechanics)|microstates]]. In contrast to the macrostate, which characterizes plainly observable average quantities, a microstate specifies all molecular details about the system including the position and velocity of every molecule. The more such states are available to the system with appreciable probability, the greater the entropy. In statistical mechanics, entropy is a measure of the number of ways a system can be arranged, often taken to be a measure of "disorder" (the higher the entropy, the higher the disorder).<ref name=McH>{{cite book |last1=Licker |first1=Mark D. |title=McGraw-Hill concise encyclopedia of chemistry |date=2004 |publisher=McGraw-Hill Professional |location=New York |isbn=978-0-07-143953-4}}</ref><ref name="Sethna78" /><ref>{{cite book |last1=Clark |first1=John O. E. |title=The essential dictionary of science |date=2004 |publisher=Barnes & Noble |location=New York |isbn=978-0-7607-4616-5}}</ref> This definition describes the entropy as being proportional to the natural logarithm of the number of possible microscopic configurations of the individual atoms and molecules of the system ([[Physics:Microstate (statistical mechanics)|microstates]]) that could cause the observed macroscopic state (macrostate) of the system. The constant of proportionality is the [[Physics:Boltzmann constant|Boltzmann constant]].

The Boltzmann constant, and therefore entropy, have dimensions of energy divided by temperature, which has a unit of [[Joule|joule]]s per [[Kelvin|kelvin]] (J⋅K<sup>−1</sup>) in the [[International System of Units]] (or kg⋅m<sup>2</sup>⋅s<sup>−2</sup>⋅K<sup>−1</sup> in terms of base units). The entropy of a substance is usually given as an [[Physics:Intensive and extensive properties#Intensive properties|intensive property]]{{snd}}either entropy per unit [[Physics:Mass|mass]] (SI unit: J⋅K<sup>−1</sup>⋅kg<sup>−1</sup>) or entropy per unit [[Physics:Amount of substance|amount of substance]] (SI unit: J⋅K<sup>−1</sup>⋅mol<sup>−1</sup>).

Specifically, entropy is a [[Logarithmic scale|logarithmic]] measure of the number of system states with significant probability of being occupied:

:<math>S = -k_\text{B} \sum_i p_i \ln p_i,</math>

(<math>p_i</math> is the probability that the system is in <math>i</math>th state, usually given by the [[Physics:Boltzmann distribution|Boltzmann distribution]]; if states are defined in a continuous manner, the summation is replaced by an [[Integral|integral]] over all possible states) or, equivalently, the expected value of [[Entropy (information theory)#Rationale|the logarithm of the probability]] that a microstate is occupied

:<math>S = -k_\text{B} \langle\ln p\rangle,</math>

where ''k''<sub>B</sub> is the [[Physics:Boltzmann constant|Boltzmann constant]], equal to {{val|1.38065|e=-23|u=J/K}}.
The summation is over all the possible microstates of the system, and ''p<sub>i</sub>'' is the probability that the system is in the ''i''-th microstate.<ref name="Perplexed">[http://charlottewerndl.net/Entropy_Guide.pdf Frigg, R. and Werndl, C. "Entropy – A Guide for the Perplexed"]. In ''Probabilities in Physics''; Beisbart C. and Hartmann, S. (eds.); Oxford University Press, Oxford, 2010.</ref> This definition assumes that the basis set of states has been picked so that there is no information on their relative phases. In a different basis set, the more general expression is

:<math>S = -k_{\mathrm{B}} \operatorname{Tr}\big(\widehat{\rho} \ln(\widehat{\rho})\big),</math>

where <math>\widehat{\rho}</math> is the [[Density matrix|density matrix]], <math>\operatorname{Tr}</math> is [[Trace (linear algebra)|trace]], and <math>\ln</math> is the matrix logarithm. This density matrix formulation is not needed in cases of thermal equilibrium so long as the basis states are chosen to be energy eigenstates. For most practical purposes, this can be taken as the fundamental definition of entropy since all other formulas for ''S'' can be mathematically derived from it, but not vice versa.

In what has been called ''the fundamental assumption of statistical thermodynamics'' or ''the fundamental postulate in statistical mechanics'', among system microstates of the same energy ([[Physics:Degenerate energy levels|degenerate]] microstates) each microstate is assumed to be populated with equal probability; this assumption is usually justified for an isolated system in equilibrium.<ref>{{cite book|last1=Schroeder|first1=Daniel V.|title=An introduction to thermal physics|url=https://archive.org/details/introductiontoth00schr_817|url-access=limited|date=2000|publisher=Addison Wesley|location=San Francisco, CA |isbn=978-0-201-38027-9|page=[https://archive.org/details/introductiontoth00schr_817/page/n68 57]}}</ref> Then for an isolated system ''p''<sub>''i''</sub> = 1/Ω, where Ω is the number of microstates whose energy equals the system's energy, and the previous equation reduces to
:<math>S = k_\text{B} \ln \Omega.</math>

In thermodynamics, such a system is one in which the volume, number of molecules, and internal energy are fixed (the [[Physics:Microcanonical ensemble|microcanonical ensemble]]).

For a given thermodynamic system, the ''excess entropy'' is defined as the entropy minus that of an ideal gas at the same density and temperature, a quantity that is always negative because an ideal gas is maximally disordered.<ref>{{Cite journal |last1=Allen |first1=Michael P. |last2=Tildesley |first2=Dominic J. |date=23 November 2017 |title=Computer Simulation of Liquids |journal=Oxford Scholarship Online |doi=10.1093/oso/9780198803195.001.0001 |isbn=9780198803195}}</ref> This concept plays an important role in liquid-state theory. For instance, Rosenfeld's excess-entropy scaling principle<ref>{{Cite journal |last=Rosenfeld |first=Yaakov |date=1 June 1977 |title=Relation between the transport coefficients and the internal entropy of simple systems |journal=Physical Review A |language=en |volume=15 |issue=6 |pages=2545–2549 |doi=10.1103/PhysRevA.15.2545 |bibcode=1977PhRvA..15.2545R |issn=0556-2791}}</ref><ref>{{Cite journal |last=Dyre |first=Jeppe C. |year=2018 |title=Perspective: Excess-entropy scaling |journal=The Journal of Chemical Physics |language=en |volume=149 |issue=21 |pages=210901 |doi=10.1063/1.5055064 |pmid=30525736 |bibcode=2018JChPh.149u0901D |issn=0021-9606 |doi-access=free}}</ref> states that reduced transport coefficients throughout the two-dimensional phase diagram are functions uniquely determined by the excess entropy.<ref>{{Cite journal |last=Bell |first=Ian H. |year=2019 |title=Probing the link between residual entropy and viscosity of molecular fluids and model potentials |journal=Proceedings of the National Academy of Sciences |language=en |volume=116 |issue=10 |pages=4070–4079 |doi=10.1073/pnas.1815943116 |issn=0027-8424 |pmc=6410835 |pmid=30770449 |arxiv=1809.05682 |bibcode=2019PNAS..116.4070B |doi-access=free}}</ref><ref>{{Cite journal |last1=Bell |first1=Ian H. |last2=Dyre |first2=Jeppe C. |last3=Ingebrigtsen |first3=Trond S. |year=2020 |title=Excess-entropy scaling in supercooled binary mixtures |journal=Nature Communications |language=en |volume=11 |issue=1 |pages=4300 |doi=10.1038/s41467-020-17948-1 |issn=2041-1723 |pmc=7453028 |pmid=32855393 |bibcode=2020NatCo..11.4300B}}</ref>

The most general interpretation of entropy is as a measure of the extent of uncertainty about a system. The equilibrium state of a system maximizes the entropy because it does not reflect all information about the initial conditions, except for the conserved variables. This uncertainty is not of the everyday subjective kind, but rather the uncertainty inherent to the experimental method and interpretative model.<ref>{{Cite journal|last=Jaynes|first=E. T.|date=1 May 1965|title=Gibbs vs Boltzmann Entropies|url=https://aapt.scitation.org/doi/10.1119/1.1971557|journal=American Journal of Physics|volume=33|issue=5|pages=391–398|doi=10.1119/1.1971557|bibcode=1965AmJPh..33..391J|issn=0002-9505}}</ref>

The interpretative model has a central role in determining entropy. The qualifier "for a given set of macroscopic variables" above has deep implications: if two observers use different sets of macroscopic variables, they see different entropies. For example, if observer A uses the variables ''U'', ''V'' and ''W'', and observer B uses ''U'', ''V'', ''W'', ''X'', then, by changing ''X'', observer B can cause an effect that looks like a violation of the second law of thermodynamics to observer A. In other words: the set of macroscopic variables one chooses must include everything that may change in the experiment, otherwise one might see decreasing entropy.<ref>{{cite book |url=http://www.mdpi.org/lin/entropy/cgibbs.pdf |author=Jaynes, E. T. |chapter=The Gibbs Paradox |title=Maximum Entropy and Bayesian Methods |editor1=Smith, C. R. |editor2=Erickson, G. J. |editor3=Neudorfer, P. O. |publisher=Kluwer Academic: Dordrecht |year=1992 |pages=1–22 |access-date=17 August 2012}}</ref>

Entropy can be defined for any Markov processes with reversible dynamics and the [[Detailed balance|detailed balance]] property.

In Boltzmann's 1896 ''Lectures on Gas Theory'', he showed that this expression gives a measure of entropy for systems of atoms and molecules in the gas phase, thus providing a measure for the entropy of classical thermodynamics.

=== Entropy of a system ===
[[File:system boundary.svg|thumb|A thermodynamic system]]

[[File:Temperature-entropy chart for steam, imperial units.svg|thumb|A [[Physics:Temperature–entropy diagram|temperature–entropy diagram]] for steam. The vertical axis represents uniform temperature, and the horizontal axis represents specific entropy. Each dark line on the graph represents constant pressure, and these form a mesh with light gray lines of constant volume. (Dark-blue is liquid water, light-blue is liquid-steam mixture, and faint-blue is steam. Grey-blue represents supercritical liquid water.)]]

Entropy arises directly from the [[Physics:Carnot cycle|Carnot cycle]]. It can also be described as the reversible heat divided by temperature. Entropy is a fundamental function of state.

In a [[Physics:Thermodynamic system|thermodynamic system]], pressure and temperature tend to become uniform over time because the equilibrium state has higher [[Probability|probability]] (more possible [[Combination|combination]]s of [[Physics:Microstate (statistical mechanics)|microstates]]) than any other state.

As an example, for a glass of ice water in air at [[Physics:Room temperature|room temperature]], the difference in temperature between the warm room (the surroundings) and the cold glass of ice and water (the system and not part of the room) decreases as portions of the [[Physics:Thermal energy|thermal energy]] from the warm surroundings spread to the cooler system of ice and water. Over time the temperature of the glass and its contents and the temperature of the room become equal. In other words, the entropy of the room has decreased as some of its energy has been dispersed to the ice and water, of which the entropy has increased.

However, as calculated in the example, the entropy of the system of ice and water has increased more than the entropy of the surrounding room has decreased. In an [[Physics:Isolated system|isolated system]] such as the room and ice water taken together, the dispersal of energy from warmer to cooler always results in a net increase in entropy. Thus, when the "universe" of the room and ice water system has reached a temperature equilibrium, the entropy change from the initial state is at a maximum. The entropy of the [[Physics:Thermodynamic system|thermodynamic system]] is a measure of how far the equalization has progressed.

Thermodynamic entropy is a non-conserved [[Physics:State function|state function]] that is of great importance in the sciences of [[Physics:Physics|physics]] and [[Chemistry:Chemistry|chemistry]].<ref name="McH" /><ref name="Wiley91">{{cite book|last1=Sandler|first1=Stanley I.|title=Chemical, biochemical, and engineering thermodynamics|url=https://archive.org/details/chemicalbiochemi00sand|url-access=limited|date=2006|publisher=John Wiley & Sons|location=New York|isbn=978-0-471-66174-0|page=[https://archive.org/details/chemicalbiochemi00sand/page/n104 91]|edition=4th}}</ref> Historically, the concept of entropy evolved to explain why some processes (permitted by conservation laws) occur spontaneously while their [[Physics:T-symmetry|time reversals]] (also permitted by conservation laws) do not; systems tend to progress in the direction of increasing entropy.<ref name="McQuarrie817">{{cite book|last1=Simon|first1= John D. |first2=Donald A. |last2=McQuarrie |title=Physical chemistry : a molecular approach|date=1997|publisher=Univ. Science Books|location=Sausalito, Calif.|isbn=978-0-935702-99-6|page=817|edition=Rev.}}</ref><ref>{{Cite book|last=Haynie|first=Donald T.|title=Biological Thermodynamics|publisher={{wipe|Cambridge University Press}}|year=2001|isbn=978-0-521-79165-6}}</ref> For [[Physics:Isolated system|isolated system]]s, entropy never decreases.<ref name="Wiley91" /> This fact has several important consequences in science: first, it prohibits "[[Unsolved:Perpetual motion|perpetual motion]]" machines; and second, it implies the [[Physics:Entropy (arrow of time)|arrow of entropy]] has the same direction as the [[Physics:Arrow of time|arrow of time]]. Increases in the total entropy of system and surroundings correspond to irreversible changes, because some energy is expended as waste heat, limiting the amount of work a system can do.<ref name="McH" /><ref name="Sethna78">{{cite book|last1=Sethna|first1=James P.|title=Statistical mechanics : entropy, order parameters, and complexity.|url=https://archive.org/details/statisticalmecha00seth_912|url-access=limited|date=2006|publisher=Oxford University Press|location=Oxford|isbn=978-0-19-856677-9|page=[https://archive.org/details/statisticalmecha00seth_912/page/n97 78]|edition=[Online-Ausg.]}}</ref><ref name="OxSci">{{cite book|last1=Daintith|first1=John|title=A dictionary of science|date=2005|publisher=Oxford University Press|location=Oxford|isbn=978-0-19-280641-3|edition=5th}}</ref><ref>{{Cite book|last=de Rosnay|first=Joel|title=The Macroscope&nbsp;– a New World View (written by an M.I.T.-trained biochemist)|publisher=Harper & Row, Publishers|year=1979|isbn=978-0-06-011029-1|title-link=M.I.T.}}</ref>

Unlike many other functions of state, entropy cannot be directly observed but must be calculated. Absolute [[Chemistry:Standard molar entropy|standard molar entropy]] of a substance can be calculated from the measured temperature dependence of its [[Physics:Heat capacity|heat capacity]]. The molar entropy of ions is obtained as a difference in entropy from a reference state defined as zero entropy. The [[Physics:Second law of thermodynamics|second law of thermodynamics]] states that the entropy of an [[Physics:Isolated system|isolated system]] must increase or remain constant. Therefore, entropy is not a conserved quantity: for example, in an isolated system with non-uniform temperature, heat might irreversibly flow and the temperature become more uniform such that entropy increases.<ref>{{cite web|last=McGovern|first=J. A.|title=Heat Capacities|url=http://theory.phy.umist.ac.uk/~judith/stat_therm/node50.html|url-status=dead|archive-url=https://web.archive.org/web/20120819175243/http://theory.phy.umist.ac.uk/~judith/stat_therm/node50.html|archive-date=19 August 2012|access-date=27 January 2013}}</ref> Chemical reactions cause changes in entropy and system entropy, in conjunction with [[Physics:Enthalpy|enthalpy]], plays an important role in determining in which direction a chemical reaction spontaneously proceeds.

One dictionary definition of entropy is that it is "a measure of thermal energy per unit temperature that is not available for useful work" in a cyclic process. For instance, a substance at uniform temperature is at maximum entropy and cannot drive a heat engine. A substance at non-uniform temperature is at a lower entropy (than if the heat distribution is allowed to even out) and some of the thermal energy can drive a heat engine.

A special case of entropy increase, the [[Physics:Entropy of mixing|entropy of mixing]], occurs when two or more different substances are mixed. If the substances are at the same temperature and pressure, there is no net exchange of heat or work – the entropy change is entirely due to the mixing of the different substances. At a statistical mechanical level, this results due to the change in available volume per particle with mixing.<ref>{{cite journal|last1=Ben-Naim|first1=Arieh|title=On the So-Called Gibbs Paradox, and on the Real Paradox|journal=Entropy|date=21 September 2007|volume=9|issue=3|pages=132–136|doi=10.3390/e9030133|url=http://www.mdpi.org/entropy/papers/e9030132.pdf|bibcode=2007Entrp...9..132B|doi-access=free}}</ref>

=== Equivalence of definitions ===
Proofs of equivalence between the definition of entropy in statistical mechanics (the [[Physics:Entropy (statistical thermodynamics)#Gibbs entropy formula|Gibbs entropy formula]] <math display="inline">S = -k_{\mathrm{B}}\sum_i p_i \log p_i</math>) and in classical thermodynamics (<math display="inline">d S = \frac{\delta Q_\text{rev}}{T}</math> together with the [[Physics:Fundamental thermodynamic relation|fundamental thermodynamic relation]]) are known for the [[Physics:Microcanonical ensemble|microcanonical ensemble]], the [[Physics:Canonical ensemble|canonical ensemble]], the [[Physics:Grand canonical ensemble|grand canonical ensemble]], and the [[Physics:Isothermal–isobaric ensemble|isothermal–isobaric ensemble]]. These proofs are based on the probability density of microstates of the generalized [[Physics:Boltzmann distribution|Boltzmann distribution]] and the identification of the thermodynamic internal energy as the ensemble average <math>U=\left\langle E_{i}\right\rangle </math>.<ref>{{cite book |last= Callen|first= Herbert|date= 2001|title= Thermodynamics and an Introduction to Thermostatistics (2nd ed.)|publisher= John Wiley and Sons|isbn= 978-0-471-86256-7}}</ref> Thermodynamic relations are then employed to derive the well-known Gibbs entropy formula. However, the equivalence between the Gibbs entropy formula and the thermodynamic definition of entropy is not a fundamental thermodynamic relation but rather a consequence of the form of the [[Physics:Boltzmann distribution#Generalized Boltzmann distribution|generalized Boltzmann distribution]].<ref>{{cite journal |last1= Gao |first1= Xiang |last2= Gallicchio |first2= Emilio |first3= Adrian |last3= Roitberg |year= 2019 |title= The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy |journal= The Journal of Chemical Physics|volume= 151|issue= 3|pages= 034113|doi= 10.1063/1.5111333|pmid= 31325924 |arxiv= 1903.02121 |bibcode= 2019JChPh.151c4113G |s2cid= 118981017 }}</ref>

Furthermore, it has been shown that the definitions of entropy in statistical mechanics is the only entropy that is equivalent to the classical thermodynamics entropy under the following postulates:<ref name="Gao2022">{{cite journal |last1= Gao |first1= Xiang |date= March 2022 |title= The Mathematics of the Ensemble Theory |journal= Results in Physics|volume= 34|pages= 105230|doi= 10.1016/j.rinp.2022.105230 |bibcode= 2022ResPh..3405230G |s2cid= 221978379 |doi-access= free |arxiv= 2006.00485 }}</ref>
{{ordered list
| The probability density function is proportional to some function of the ensemble parameters and random variables.
| Thermodynamic state functions are described by ensemble averages of random variables.
| At infinite temperature, all the microstates have the same probability.
}}

== Second law of thermodynamics ==
The [[Physics:Second law of thermodynamics|second law of thermodynamics]] requires that, in general, the total entropy of any system does not decrease other than by increasing the entropy of some other system. Hence, in a system isolated from its environment, the entropy of that system tends not to decrease. It follows that heat cannot flow from a colder body to a hotter body without the application of work to the colder body. Secondly, it is impossible for any device operating on a cycle to produce net work from a single temperature reservoir; the production of net work requires flow of heat from a hotter reservoir to a colder reservoir, or a single expanding reservoir undergoing adiabatic cooling, which performs [[Physics:Adiabatic process|adiabatic work]]. As a result, there is no possibility of a [[Unsolved:Perpetual motion|perpetual motion]] machine. It follows that a reduction in the increase of entropy in a specified process, such as a [[Chemistry:Chemical reaction|chemical reaction]], means that it is energetically more efficient.

It follows from the second law of thermodynamics that the entropy of a system that is not isolated may decrease. An air conditioner, for example, may cool the air in a room, thus reducing the entropy of the air of that system. The heat expelled from the room (the system), which the air conditioner transports and discharges to the outside air, always makes a bigger contribution to the entropy of the environment than the decrease of the entropy of the air of that system. Thus, the total of entropy of the room plus the entropy of the environment increases, in agreement with the second law of thermodynamics.

In mechanics, the second law in conjunction with the [[Physics:Fundamental thermodynamic relation|fundamental thermodynamic relation]] places limits on a system's ability to do [[Physics:Work (thermodynamics)|useful work]].<ref name="Daintith">{{Cite book|last=Daintith| first=John|title=Oxford Dictionary of Physics|publisher=Oxford University Press|year=2005|isbn=978-0-19-280628-4}}</ref> The entropy change of a system at temperature <math display="inline">T</math> absorbing an infinitesimal amount of heat <math display="inline">\delta q</math> in a reversible way, is given by <math display="inline">\delta q / T</math>. More explicitly, an energy <math display="inline">T_R S</math> is not available to do useful work, where <math display="inline">T_R</math> is the temperature of the coldest accessible reservoir or heat sink external to the system. For further discussion, see ''[[Physics:Exergy|Exergy]]''.

Statistical mechanics demonstrates that entropy is governed by probability, thus allowing for a decrease in disorder even in an isolated system. Although this is possible, such an event has a small probability of occurring, making it unlikely.<ref>{{Cite journal |title=Entropy production theorems and some consequences|pages=1–10 |journal=Physical Review E |volume=80 |issue=1 |doi=10.1103/PhysRevE.80.011117 |pmid=19658663 |year=2009 |last1=Saha |first1=Arnab |last2=Lahiri |first2=Sourabh |last3=Jayannavar |first3=A. M. |bibcode=2009PhRvE..80a1117S |arxiv=0903.4147 |s2cid=22204063 }}</ref>

The applicability of a second law of thermodynamics is limited to systems in or sufficiently near [[Physics:Thermodynamic equilibrium|equilibrium state]], so that they have defined entropy.<ref>{{cite journal|last1=Martyushev|first1=L. M.|last2=Seleznev|first2=V. D.|title=The restrictions of the maximum entropy production principle|journal=Physica A: Statistical Mechanics and Its Applications|year=2014|volume=410|doi=10.1016/j.physa.2014.05.014|pages=17–21|arxiv=1311.2068|bibcode=2014PhyA..410...17M|s2cid=119224112}}</ref> Some inhomogeneous systems out of thermodynamic equilibrium still satisfy the hypothesis of [[Physics:Thermodynamic equilibrium#local and global equilibrium|local thermodynamic equilibrium]], so that entropy density is locally defined as an intensive quantity. For such systems, there may apply a principle of maximum time rate of entropy production.<ref>{{cite book|last1=Ziegler|first1=H.|title=An Introduction to Thermomechanics|date=1983|location=North Holland, Amsterdam.}}</ref><ref>{{cite journal|last1=Onsager|first1=Lars|title=Reciprocal Relations in Irreversible Processes|journal=Phys. Rev. |volume=37|issue=4|page=405|year=1931|doi=10.1103/PhysRev.37.405|bibcode=1931PhRv...37..405O|doi-access=free}}</ref> It states that such a system may evolve to a steady state that maximizes its time rate of entropy production. This does not mean that such a system is necessarily always in a condition of maximum time rate of entropy production; it means that it may evolve to such a steady state.<ref>{{cite book|last1=Kleidon|first1=A.|last2=et.|first2=al.|title=Non-equilibrium Thermodynamics and the Production of Entropy|date=2005|publisher=Springer|location=Heidelberg}}</ref><ref>{{cite journal|last1=Belkin|first1=Andrey|last2=et.|first2=al.|title=Self-assembled wiggling nano-structures and the principle of maximum entropy production|journal=Scientific Reports |year=2015|doi=10.1038/srep08323|pmid=25662746|pmc=4321171|volume=5|pages=8323|bibcode=2015NatSR...5E8323B}}</ref>

== Applications ==
=== The fundamental thermodynamic relation ===
{{Main|Physics:Fundamental thermodynamic relation}}
The entropy of a system depends on its internal energy and its external parameters, such as its volume. In the thermodynamic limit, this fact leads to an equation relating the change in the internal energy <math>U</math> to changes in the entropy and the external parameters. This relation is known as the ''fundamental thermodynamic relation''. If external pressure <math>p</math> bears on the volume <math>V</math> as the only external parameter, this relation is:

: <math>dU = T \, dS - p \, dV</math>

Since both internal energy and entropy are monotonic functions of temperature <math>T</math>, implying that the internal energy is fixed when one specifies the entropy and the volume, this relation is valid even if the change from one state of thermal equilibrium to another with infinitesimally larger entropy and volume happens in a non-quasistatic way (so during this change the system may be very far out of thermal equilibrium and then the whole-system entropy, pressure, and temperature may not exist).

The fundamental thermodynamic relation implies many thermodynamic identities that are valid in general, independent of the microscopic details of the system. Important examples are the [[Physics:Maxwell relations|Maxwell relations]] and the [[Physics:Relations between heat capacities|relations between heat capacities]].

=== Entropy in chemical thermodynamics ===
Thermodynamic entropy is central in [[Chemistry:Chemical thermodynamics|chemical thermodynamics]], enabling changes to be quantified and the outcome of reactions predicted. The [[Physics:Second law of thermodynamics|second law of thermodynamics]] states that entropy in an [[Physics:Isolated system|isolated system]] – the combination of a subsystem under study and its surroundings – increases during all spontaneous chemical and physical processes. The Clausius equation of <math>\delta q_{\text{rev}} / T = \Delta S</math> introduces the measurement of entropy change, <math>\Delta S</math>. Entropy change describes the direction and quantifies the magnitude of simple changes such as heat transfer between systems – always from hotter to cooler spontaneously.

The thermodynamic entropy therefore has the dimension of energy divided by temperature, and the unit [[Joule|joule]] per [[Kelvin|kelvin]] (J/K) in the International System of Units (SI).

Thermodynamic entropy is an [[Physics:Intensive and extensive properties|extensive]] property, meaning that it scales with the size or extent of a system. In many processes it is useful to specify the entropy as an [[Physics:Intensive and extensive properties|intensive property]] independent of the size, as a specific entropy characteristic of the type of system studied. Specific entropy may be expressed relative to a unit of mass, typically the kilogram (unit: J⋅kg<sup>−1</sup>⋅K<sup>−1</sup>). Alternatively, in chemistry, it is also referred to one [[Mole (unit)|mole]] of substance, in which case it is called the ''molar entropy'' with a unit of J⋅mol<sup>−1</sup>⋅K<sup>−1</sup>.

Thus, when one mole of substance at about {{val|0|u=K}} is warmed by its surroundings to {{val|298|u=K}}, the sum of the incremental values of <math display="inline">q_{\text{rev}} / T</math> constitute each element's or compound's standard molar entropy, an indicator of the amount of energy stored by a substance at {{val|298|u=K}}.<ref name="ctms">{{Cite book|last=Moore|first=J. W.|author2=C. L. Stanistski|author3=P. C. Jurs|title=Chemistry, The Molecular Science|publisher=Brooks Cole|year=2005|isbn=978-0-534-42201-1|url-access=registration|url=https://archive.org/details/chemistrymolecul0000moor}}</ref><ref name="Jungermann">{{cite journal|last1=Jungermann|first1=A.H.|s2cid=18081336|year=2006|title=Entropy and the Shelf Model: A Quantum Physical Approach to a Physical Property|journal=Journal of Chemical Education|volume=83|issue=11|pages=1686–1694|doi=10.1021/ed083p1686|bibcode = 2006JChEd..83.1686J}}</ref> Entropy change also measures the mixing of substances as a summation of their relative quantities in the final mixture.<ref>{{Cite book|last=Levine|first=I. N.|title=Physical Chemistry, 5th ed.|url=https://archive.org/details/physicalchemistr00levi_1|url-access=registration|publisher=McGraw-Hill|year=2002|isbn=978-0-07-231808-1}}</ref>

Entropy is equally essential in predicting the extent and direction of complex chemical reactions. For such applications, <math>\Delta S</math> must be incorporated in an expression that includes both the system and its surroundings, <math>\Delta S_{\text{universe}} = \Delta S_{\text{surroundings}} + \Delta S_{\text{system}}</math>. This expression becomes, via some steps, the [[Physics:Gibbs free energy|Gibbs free energy]] equation for reactants and products in the system: <math>\Delta G</math> [the Gibbs free energy change of the system] <math>= \Delta H</math> [the enthalpy change] <math>- T\,\Delta S</math> [the entropy change].<ref name="ctms" />

=== World's technological capacity to store and communicate entropic information ===
A 2011 study in Science (journal) estimated the world's technological capacity to store and communicate optimally compressed information normalized on the most effective compression algorithms available in the year 2007, therefore estimating the entropy of the technologically available sources.<ref name="HilbertLopez2011">{{Cite journal|last1=Hilbert|first1=Martin|last2=López|first2=Priscila|date=2011-02-11|title=The World's Technological Capacity to Store, Communicate, and Compute Information|journal=Science|language=en|volume=332|issue=6025|pages=60–65|doi=10.1126/science.1200970|pmid=21310967 |bibcode=2011Sci...332...60H |s2cid=206531385 |issn=0036-8075|doi-access=free}}</ref> The author's estimate that human kind's technological capacity to store information grew from 2.6 (entropically compressed) exabytes in 1986 to 295 (entropically compressed) exabytes in 2007. The world's technological capacity to receive information through one-way broadcast networks was 432 exabytes of (entropically compressed) information in 1986, to 1.9 zettabytes in 2007. The world's effective capacity to exchange information through two-way telecommunication networks was 281 petabytes of (entropically compressed) information in 1986, to 65 (entropically compressed) exabytes in 2007.<ref name="HilbertLopez2011"/>

=== Entropy balance equation for open systems ===
[[File:First law open system.svg|thumb|upright=1.4|During steady-state continuous operation, an entropy balance applied to an open system accounts for system entropy changes related to heat flow and mass flow across the system boundary.]]
In [[Chemistry:Chemical engineering|chemical engineering]], the principles of thermodynamics are commonly applied to "[[Open system (systems theory)|open systems]]", i.e. those in which heat, [[Physics:Work (thermodynamics)|work]], and [[Physics:Mass|mass]] flow across the system boundary. Flows of both heat (<math>\dot{Q}</math>) and work, i.e. <math>\dot{W}_\text{S}</math> ([[Physics:Work (thermodynamics)|shaft work]]) and <math>P(dV/dt)</math> (pressure-volume work), across the system boundaries, in general cause changes in the entropy of the system. Transfer as heat entails entropy transfer <math>\dot{Q}/T</math>, where <math>T</math> is the absolute [[Physics:Thermodynamic temperature|thermodynamic temperature]] of the system at the point of the heat flow. If there are mass flows across the system boundaries, they also influence the total entropy of the system. This account, in terms of heat and work, is valid only for cases in which the work and heat transfers are by paths physically distinct from the paths of entry and exit of matter from the system.<ref>{{cite book|author=Late Nobel Laureate Max Born|title=Natural Philosophy of Cause and Chance|url=https://books.google.com/books?id=er85jgEACAAJ|date=8 August 2015|publisher=BiblioLife|isbn=978-1-298-49740-6|pages=44, 146–147}}</ref><ref>{{cite book|last1=Haase|first1=R.|title=Thermodynamics|date=1971|publisher=Academic Press|location=New York|isbn=978-0-12-245601-5|pages=1–97}}</ref>

To derive a generalized entropy balanced equation, we start with the general balance equation for the change in any extensive quantity <math>\theta</math> in a [[Physics:Thermodynamic system|thermodynamic system]], a quantity that may be either conserved, such as energy, or non-conserved, such as entropy. The basic generic balance expression states that <math>d\theta/dt</math>, i.e. the rate of change of <math>\theta</math> in the system, equals the rate at which <math>\theta</math> enters the system at the boundaries, minus the rate at which <math>\theta</math> leaves the system across the system boundaries, plus the rate at which <math>\theta</math> is generated within the system. For an open thermodynamic system in which heat and work are transferred by paths separate from the paths for transfer of matter, using this generic balance equation, with respect to the rate of change with time <math>t</math> of the extensive quantity entropy <math>S</math>, the entropy balance equation is:<ref name="Pokrovskii 2020">{{Cite book|url=|title= Thermodynamics of Complex Systems: Principles and applications. |last= Pokrovskii |first=Vladimir|language=English | publisher= IOP Publishing, Bristol, UK.|year=2020|isbn=|pages=|bibcode= 2020tcsp.book.....P }}</ref><ref>{{Cite book|last=Sandler|first=Stanley, I.|title=Chemical and Engineering Thermodynamics|publisher=John Wiley & Sons|year=1989|isbn=978-0-471-83050-4}}</ref><ref group="note" name=overdot>The overdots represent derivatives of the quantities with respect to time.</ref>

:<math>\frac{dS}{dt} = \sum_{k=1}^K  \dot{M}_k \hat{S}_k  + \frac{\dot{Q}}{T} + \dot{S}_\text{gen}</math>

where
* <math display="inline">\sum_{k=1}^K  \dot{M}_k  \hat{S}_k</math> is the net rate of entropy flow due to the flows of mass into and out of the system (where <math>\hat{S}</math> is entropy per unit mass).
* <math display="inline">\frac{\dot{Q}}{T}</math> is the rate of entropy flow due to the flow of heat across the system boundary.
* <math display="inline">\dot{S}_\text{gen}</math> is the rate of [[Physics:Entropy production|entropy production]] within the system. This entropy production arises from processes within the system, including chemical reactions, internal matter diffusion, internal heat transfer, and frictional effects such as viscosity occurring within the system from mechanical work transfer to or from the system.

If there are multiple heat flows, the term <math>\dot{Q}/T</math> is replaced by <math display="inline">\sum \dot{Q}_j/T_j,</math> where <math>\dot{Q}_j</math> is the heat flow and <math>T_j</math> is the temperature at the <math>j</math>th heat flow port into the system.

The nomenclature "entropy balance" is misleading and often deemed inappropriate because entropy is not a conserved quantity. In other words, the term <math>\dot{S}_\text{gen}</math> is never a known quantity but always a derived one based on the expression above. Therefore, the open system version of the second law is more appropriately described as the "entropy generation equation" since it specifies that <math>\dot{S}_\text{gen} \ge 0</math>, with zero for reversible processes or greater than zero for irreversible ones.

== Entropy change formulas for simple processes ==
For certain simple transformations in systems of constant composition, the entropy changes are given by simple formulas.<ref>{{cite web |url=http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/Mathematical_Thinking/ideal_gases_under_constant.htm |title=GRC.nasa.gov |publisher=GRC.nasa.gov |date=27 March 2000 |access-date=17 August 2012 |archive-url=https://web.archive.org/web/20110821135844/http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/Mathematical_Thinking/ideal_gases_under_constant.htm |archive-date=21 August 2011 |url-status=dead }}</ref>

=== Isothermal expansion or compression of an ideal gas ===
For the expansion (or compression) of an [[Physics:Ideal gas|ideal gas]] from an initial volume <math>V_0</math> and pressure <math>P_0</math> to a final volume <math>V</math> and pressure <math>P</math> at any constant temperature, the change in entropy is given by:
:<math>\Delta S = n R \ln \frac{V}{V_0} = - n R \ln \frac{P}{P_0} .</math>

Here <math>n</math> is the amount of gas (in [[Mole (unit)|moles]]) and <math>R</math> is the [[Physics:Ideal gas constant|ideal gas constant]]. These equations also apply for expansion into a finite vacuum or a throttling process, where the temperature, internal energy and enthalpy for an ideal gas remain constant.

=== Cooling and heating ===
For pure heating or cooling of any system (gas, liquid or solid) at constant pressure from an initial temperature <math>T_0</math> to a final temperature <math>T</math>, the entropy change is
:<math>\Delta S = n C_P \ln \frac{T}{T_0}.</math>
provided that the constant-pressure molar [[Physics:Heat capacity|heat capacity]] (or specific heat) ''C''<sub>''P''</sub> is constant and that no [[Physics:Phase transition|phase transition]] occurs in this temperature interval.

Similarly at constant volume, the entropy change is
:<math>\Delta S = n C_v \ln \frac{T}{T_0},</math>
where the constant-volume molar heat capacity ''C''<sub>v</sub> is constant and there is no phase change.

At low temperatures near absolute zero, heat capacities of solids quickly drop off to near zero, so the assumption of constant heat capacity does not apply.<ref>{{cite web|last1=Franzen|first1=Stefan|title=Third Law|url=http://www4.ncsu.edu/~franzen/public_html/CH433/lecture/Third_Law.pdf|publisher=ncsu.edu|archive-url = https://web.archive.org/web/20170709093839/http://www4.ncsu.edu:80/~franzen/public_html/CH433/lecture/Third_Law.pdf |archive-date = 9 July 2017}}</ref>

Since entropy is a state function, the entropy change of any process in which temperature and volume both vary is the same as for a path divided into two steps – heating at constant volume and expansion at constant temperature. For an ideal gas, the total entropy change is<ref>{{cite web|url=http://www.grc.nasa.gov/WWW/K-12/airplane/entropy.html |title=GRC.nasa.gov |publisher=GRC.nasa.gov |date=11 July 2008 |access-date=17 August 2012}}</ref>
:<math>\Delta S = nC_v \ln \frac{T}{T_0} + nR \ln \frac{V}{V_0}.</math>

Similarly if the temperature and pressure of an ideal gas both vary,
:<math>\Delta S = nC_P \ln \frac{T}{T_0} - nR \ln \frac{P}{P_0}.</math>

=== Phase transitions ===
Reversible [[Physics:Phase transition|phase transition]]s occur at constant temperature and pressure. The reversible heat is the enthalpy change for the transition, and the entropy change is the enthalpy change divided by the thermodynamic temperature.<ref>{{cite book|first=Michael E.|last=Starzak|date=2010|chapter=Phase Equilibria & Colligative Properties|chapter-url=https://books.google.com/books?id=cw0QV7l559kC&pg=PA138|title=Energy & Entropy: Equilibrium to Stationary States|pages=138–140|isbn=978-1489983671|publisher=Springer Science+Business Media|access-date=5 September 2019}}</ref> For fusion ([[Physics:Melting|melting]]) of a solid to a liquid at the melting point ''T''<sub>m</sub>, the [[Physics:Entropy of fusion|entropy of fusion]] is
:<math>\Delta S_\text{fus} = \frac{\Delta H_\text{fus}}{T_\text{m}}.</math>
Similarly, for [[Physics:Vaporization|vaporization]] of a liquid to a gas at the boiling point ''T''<sub>b</sub>, the [[Physics:Entropy of vaporization|entropy of vaporization]] is
:<math>\Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{b}}.</math>

== Approaches to understanding entropy ==
As a fundamental aspect of thermodynamics and physics, several different approaches to entropy beyond that of Clausius and Boltzmann are valid.

=== Standard textbook definitions ===
The following is a list of additional definitions of entropy from a collection of textbooks:
* a measure of energy dispersal at a specific temperature.
* a measure of disorder in the universe or of the availability of the energy in a system to do work.<ref>{{cite book|last1=Gribbin|first1=John|editor1-last=Gribbin|editor1-first=Mary|title=Q is for quantum : an encyclopedia of particle physics|date=1999|publisher=Free Press|location=New York|isbn=978-0-684-85578-3|url=https://archive.org/details/qisforquantumenc00grib}}</ref>
* a measure of a system's [[Physics:Thermal energy|thermal energy]] per unit temperature that is unavailable for doing useful [[Physics:Work (thermodynamics)|work]].<ref>{{cite web|title=Entropy: Definition and Equation|url=https://www.britannica.com/EBchecked/topic/189035/entropy|website=Encyclopædia Britannica|access-date=22 May 2016}}</ref>

In Boltzmann's analysis in terms of constituent particles, entropy is a measure of the number of possible microscopic states (or microstates) of a system in thermodynamic equilibrium.

=== Order and disorder ===
{{Main|Physics:Entropy (order and disorder)}}
Entropy is often loosely associated with the amount of order or [[Randomness|disorder]], or of [[Chaos theory|chaos]], in a [[Physics:Thermodynamic system|thermodynamic system]]. The traditional qualitative description of entropy is that it refers to changes in the status quo of the system and is a measure of "molecular disorder" and the amount of wasted energy in a dynamical energy transformation from one state or form to another. In this direction, several recent authors have derived exact entropy formulas to account for and measure disorder and order in atomic and molecular assemblies.<ref name="Brooks">{{cite book|last1=Brooks|first1=Daniel R.|last2=Wiley|first2=E. O.|title=Evolution as entropy : toward a unified theory of biology|date=1988|publisher=University of Chicago Press|location=Chicago [etc.]|isbn=978-0-226-07574-7|edition=2nd}}</ref><ref name="Landsberg-A">{{cite journal | last1 = Landsberg | first1 = P.T. | year = 1984 | title = Is Equilibrium always an Entropy Maximum? | journal = J. Stat. Physics | volume = 35 | issue = 1–2| pages = 159–169 | doi=10.1007/bf01017372|bibcode = 1984JSP....35..159L | s2cid = 122424225 }}</ref><ref name="Landsberg-B">{{cite journal | last1 = Landsberg | first1 = P.T. | year = 1984 | title = Can Entropy and "Order" Increase Together? | journal = Physics Letters | volume = 102A | issue = 4| pages = 171–173 | doi=10.1016/0375-9601(84)90934-4|bibcode = 1984PhLA..102..171L }}</ref> One of the simpler entropy order/disorder formulas is that derived in 1984 by thermodynamic physicist Peter Landsberg, based on a combination of [[Physics:Thermodynamics|thermodynamics]] and [[Information theory|information theory]] arguments. He argues that when constraints operate on a system, such that it is prevented from entering one or more of its possible or permitted states, as contrasted with its forbidden states, the measure of the total amount of "disorder" in the system is given by:<ref name="Landsberg-A" /><ref name="Landsberg-B" />
:<math>\text{Disorder}={C_\text{D}\over C_\text{I}}.\,</math>

Similarly, the total amount of "order" in the system is given by:
:<math>\text{Order}=1-{C_\text{O}\over C_\text{I}}.\,</math>

In which ''C''<sub>D</sub> is the "disorder" capacity of the system, which is the entropy of the parts contained in the permitted ensemble, ''C''<sub>I</sub> is the "information" capacity of the system, an expression similar to Shannon's [[Engineering:Channel capacity|channel capacity]], and ''C''<sub>O</sub> is the "order" capacity of the system.<ref name="Brooks" />

=== Energy dispersal ===
{{Main|Physics:Entropy (energy dispersal)}}
[[File:Ultra slow-motion video of glass tea cup smashed on concrete floor.webm|thumb|thumbtime=0:04|Slow motion video of a glass cup smashing on a concrete floor. In the very short time period of the breaking process, the entropy of the mass making up the glass cup rises sharply, as the matter and energy of the glass disperse.]]
The concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature.<ref>{{cite web|last1=Lambert |first1=Frank L. |title=A Student's Approach to the Second Law and Entropy |url=http://franklambert.net/entropysite.com/students_approach.html }}</ref> Similar terms have been in use from early in the history of classical thermodynamics, and with the development of statistical thermodynamics and [[Physics:Quantum mechanics|quantum theory]], entropy changes have been described in terms of the mixing or "spreading" of the total energy of each constituent of a system over its particular quantized energy levels.

Ambiguities in the terms ''disorder'' and ''chaos'', which usually have meanings directly opposed to equilibrium, contribute to widespread confusion and hamper comprehension of entropy for most students.<ref>{{cite journal|last1=Watson|first1=J.R.|last2=Carson|first2=E.M.|title=Undergraduate students' understandings of entropy and Gibbs free energy.|journal=University Chemistry Education|date=May 2002|volume=6|issue=1|page=4|url=http://www.rsc.org/images/Vol_6_No1_tcm18-7042.pdf|issn=1369-5614}}</ref> As the [[Physics:Second law of thermodynamics|second law of thermodynamics]] shows, in an [[Physics:Isolated system|isolated system]] internal portions at different temperatures tend to adjust to a single uniform temperature and thus produce equilibrium. A recently developed educational approach avoids ambiguous terms and describes such spreading out of energy as dispersal, which leads to loss of the differentials required for work even though the total energy remains constant in accordance with the [[Physics:First law of thermodynamics|first law of thermodynamics]]<ref>{{cite journal|last1=Lambert|first1=Frank L.|s2cid=97102995|title=Disorder – A Cracked Crutch for Supporting Entropy Discussions|url=http://franklambert.net/entropysite.com/cracked_crutch.html|journal=Journal of Chemical Education|date=February 2002|volume=79|issue=2|pages=187|doi=10.1021/ed079p187|bibcode=2002JChEd..79..187L}}</ref> (compare discussion in next section). Physical chemist Peter Atkins, in his textbook ''Physical Chemistry'', introduces entropy with the statement that "spontaneous changes are always accompanied by a dispersal of energy or matter and often both".<ref name="AtkinsPaula2019">{{cite book |author1=Peter Atkins |author2=Julio de Paula |author3=James Keeler |title=Atkins' Physical Chemistry 11e: Volume 3: Molecular Thermodynamics and Kinetics |url=https://books.google.com/books?id=0UKjDwAAQBAJ&pg=PA89 |year=2019 |publisher=Oxford University Press |isbn=978-0-19-882336-0 |page=89}}</ref>

=== Relating entropy to energy ''usefulness'' ===
It is possible (in a thermal context) to regard lower entropy as a measure of the ''effectiveness'' or ''usefulness'' of a particular quantity of energy.<ref>{{Cite journal|title=Book Review of 'A Science Miscellany'|journal=Khaleej Times|publisher=UAE: Galadari Press|date=23 February 1993|page=xi|author=Sandra Saary |url=http://dlmcn.com/entropy2.html}}</ref>  Energy supplied at a higher temperature (i.e. with low entropy) tends to be more useful than the same amount of energy available at a lower temperature. Mixing a hot parcel of a fluid with a cold one produces a parcel of intermediate temperature, in which the overall increase in entropy represents a "loss" that can never be replaced.

As the entropy of the universe is steadily increasing, its total energy is becoming less useful. Eventually, this is theorized to lead to the [[Astronomy:Heat death of the universe|heat death of the universe]].<ref>{{Cite book |title =Energy and Empire: A Biographical Study of Lord Kelvin | last1= Smith |first1=Crosbie |last2=Wise |first2=M. Norton |publisher=Cambridge University Press |year=1989 |isbn=978-0-521-26173-9 |pages= 500–501 |}}</ref>

=== Entropy and adiabatic accessibility ===
A definition of entropy based entirely on the relation of [[Physics:Adiabatic accessibility|adiabatic accessibility]] between equilibrium states was given by [[Biography:Elliott H. Lieb|E.&nbsp;H.&nbsp;Lieb]] and J.&nbsp;Yngvason in 1999.<ref>{{cite journal |last1=Lieb |first1=Elliott H. |last2=Yngvason |first2=Jakob |title=The physics and mathematics of the second law of thermodynamics |journal=Physics Reports |date=March 1999 |volume=310 |issue=1 |pages=1–96 |doi=10.1016/S0370-1573(98)00082-9 |arxiv=cond-mat/9708200 |bibcode=1999PhR...310....1L |s2cid=119620408}}</ref> This approach has several predecessors, including the pioneering work of [[Biography:Constantin Carathéodory|Constantin Carathéodory]] from 1909<ref>{{cite journal |last1=Carathéodory |first1=C. |title=Untersuchungen über die Grundlagen der Thermodynamik |journal=Mathematische Annalen |date=September 1909 |volume=67 |issue=3 |pages=355–386 |doi=10.1007/BF01450409 |s2cid=118230148 |url=https://zenodo.org/record/1428268 |language=de}}</ref> and the monograph by R.&nbsp;Giles.<ref>{{cite book |author=R. Giles |title=Mathematical Foundations of Thermodynamics: International Series of Monographs on Pure and Applied Mathematics |url=https://books.google.com/books?id=oK03BQAAQBAJ |date=2016 |publisher=Elsevier Science |isbn=978-1-4831-8491-3}}</ref> In the setting of Lieb and Yngvason, one starts by picking, for a unit amount of the substance under consideration, two reference states <math>X_0</math> and <math>X_1</math> such that the latter is adiabatically accessible from the former but not conversely. Defining the entropies of the reference states to be 0 and 1 respectively, the entropy of a state <math>X</math> is defined as the largest number <math>\lambda</math> such that <math>X</math> is adiabatically accessible from a composite state consisting of an amount <math>\lambda</math> in the state <math>X_1</math> and a complementary amount, <math>(1 - \lambda)</math>, in the state <math>X_0</math>. A simple but important result within this setting is that entropy is uniquely determined, apart from a choice of unit and an additive constant for each chemical element, by the following properties: It is monotonic with respect to the relation of adiabatic accessibility, additive on composite systems, and extensive under scaling.

=== Entropy in quantum mechanics ===
{{Main|Physics:Von Neumann entropy}}

In [[Physics:Quantum statistical mechanics|quantum statistical mechanics]], the concept of entropy was developed by [[Biography:John von Neumann|John von Neumann]] and is generally referred to as "[[Physics:Von Neumann entropy|von Neumann entropy]]",

: <math>S = - k_\text{B} \operatorname{Tr}(\rho \ln \rho),</math>

where ''ρ'' is the [[Density matrix|density matrix]], and Tr is the [[Trace (linear algebra)|trace]] operator.

This upholds the [[Physics:Correspondence principle|correspondence principle]], because in the [[Physics:Classical limit|classical limit]], when the phases between the basis states used for the classical probabilities are purely random, this expression is equivalent to the familiar classical definition of entropy,

: <math>S = - k_\text{B} \sum_i p_i \, \ln \, p_i,</math>
i.e. in such a basis the density matrix is diagonal.

Von Neumann established a rigorous mathematical framework for quantum mechanics with his work {{lang|de|Mathematische Grundlagen der Quantenmechanik}}. He provided in this work a theory of measurement, where the usual notion of [[Physics:Wave function collapse|wave function collapse]] is described as an irreversible process (the so-called von Neumann or projective measurement). Using this concept, in conjunction with the [[Density matrix|density matrix]] he extended the classical concept of entropy into the quantum domain.

=== Information theory ===
{{Main|Entropy (information theory)|Entropy in thermodynamics and information theory|Physics:Entropic uncertainty}}

{{quote box
|align=right
|width=30em
|quote=I thought of calling it "information", but the word was overly used, so I decided to call it "uncertainty". [...] Von Neumann told me, "You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage.
|source=Conversation between [[Biography:Claude Shannon|Claude Shannon]] and [[Biography:John von Neumann|John von Neumann]] regarding what name to give to the [[Physics:Attenuation|attenuation]] in phone-line signals<ref>{{cite journal | last1 = Tribus | first1 = M. | last2 = McIrvine | first2 = E. C. | year = 1971 | title = Energy and information | journal = Scientific American | volume = 224 | pages = 178–184 }}</ref>}}

When viewed in terms of [[Information theory|information theory]], the entropy state function is the amount of information in the system that is needed to fully specify the microstate of the system. Entropy is the measure of the amount of missing information before reception.<ref>{{cite book|first=Roger |last=Balian |chapter=Entropy, a Protean concept |editor-last=Dalibard |editor-first=Jean |title=Poincaré Seminar 2003: Bose-Einstein condensation – entropy |year=2004 |publisher=Birkhäuser |location=Basel |isbn=978-3-7643-7116-6 |pages=119–144}}</ref> Often called ''Shannon entropy'', it was originally devised by [[Biography:Claude Shannon|Claude Shannon]] in 1948 to study the size of information of a transmitted message. The definition of information entropy is expressed in terms of a discrete set of probabilities <math>p_i</math> so that
:<math>H(X) = -\sum_{i=1}^n p(x_i) \log p(x_i),</math>
where the base of the logarithm determines the units (for example, the [[Binary logarithm|binary logarithm]] corresponds to [[Bit|bit]]s).

In the case of transmitted messages, these probabilities were the probabilities that a particular message was actually transmitted, and the entropy of the message system was a measure of the average size of information of a message. For the case of equal probabilities (i.e. each message is equally probable), the Shannon entropy (in bits) is just the number of binary questions needed to determine the content of the message.<ref name="Perplexed" />

Most researchers consider information entropy and thermodynamic entropy directly linked to the same concept,<ref>{{Cite book|last=Brillouin|first=Leon|title=Science and Information Theory|year= 1956|publisher=Dover Publications |isbn=978-0-486-43918-1}}</ref><ref name="Georgescu-Roegen 1971">{{Cite book|last=Georgescu-Roegen|first=Nicholas|title=The Entropy Law and the Economic Process|publisher=Harvard University Press|year=1971|isbn=978-0-674-25781-8 |url = https://archive.org/details/entropylawe00nich}}</ref><ref>{{Cite book|last=Chen|first=Jing|title=The Physical Foundation of Economics – an Analytical Thermodynamic Theory|publisher=World Scientific|year=2005|isbn=978-981-256-323-1}}</ref><ref>{{cite journal | last1 = Kalinin | first1 = M.I. | last2 = Kononogov | first2 = S.A. | year = 2005 | title = Boltzmann's constant | journal = Measurement Techniques | volume = 48 | issue = 7| pages = 632–636 | doi=10.1007/s11018-005-0195-9| s2cid = 118726162 }}</ref><ref>{{cite book|last1=Ben-Naim|first1=Arieh|title=Entropy demystified the second law reduced to plain common sense|url=https://archive.org/details/entropydemystifi0000benn|url-access=registration|date= 2008|publisher=World Scientific|location=Singapore|isbn=9789812832269|edition=Expanded}}</ref> while others argue that they are distinct.<ref>{{cite book|first1=Joseph J.|last1=Vallino|first2=Christopher K. |last2=Algar|first3=Nuria Fernández|last3=González|first4=Julie A.|last4=Huber|editor-first1=Roderick C.|editor-last1=Dewar|editor-first2=Charles H. |editor-last2=Lineweaver|editor-first3=Robert K.|editor-last3=Niven|editor-first4=Klaus|editor-last4=Regenauer-Lieb|date= 2013|chapter=Use of Receding Horizon Optimal Control to Solve MaxEP-Based (max entropy production) Biogeochemistry Problems |department=Living Systems as Catalysts|chapter-url=https://books.google.com/books?id=xF65BQAAQBAJ&pg=PA340|title=Beyond the Second Law: Entropy Production & Non-equilibrium Systems|page=340|isbn=978-3642401534|publisher=Springer |access-date=31 August 2019 |quote=Inked page forms pattern w/ information → page entropy < diff page w/ randomized letters. Reduced entropy trivial compared to inked paper entropy. If the paper is burned, it hardly matters in a thermodynamic context if the text contains the meaning of life or only {{sic|jibberish}}.}}</ref> Both expressions are mathematically similar. If <math>W</math> is the number of microstates that can yield a given macrostate, and each microstate has the same ''a priori'' probability, then that probability is <math>p = 1/W</math>. The Shannon entropy (in [[Nat (unit)|nats]]) is
:<math>H = -\sum_{i=1}^W p_i \ln p_i = \ln W,</math>
and if entropy is measured in units of <math>k</math> per nat, then the entropy is given by
:<math>H = k \ln W,</math>
which is the [[Physics:Boltzmann's entropy formula|Boltzmann entropy formula]], where <math>k</math> is the Boltzmann constant, which may be interpreted as the thermodynamic entropy per nat. Some authors argue for dropping the word entropy for the <math>H</math> function of information theory and using Shannon's other term, "uncertainty", instead.<ref>Schneider, Tom, DELILA system (Deoxyribonucleic acid Library Language), (Information Theory Analysis of binding sites), Laboratory of Mathematical Biology, National Cancer Institute, Frederick, MD.</ref>

===Measurement===
The entropy of a substance can be measured, although in an indirect way. The measurement, known as entropymetry,<ref>{{Cite journal|last1=Kim|first1=Hye Jin|last2=Park|first2=Youngkyu|last3=Kwon|first3=Yoonjin|last4=Shin|first4=Jaeho|last5=Kim|first5=Young-Han|last6=Ahn|first6=Hyun-Seok|last7=Yazami|first7=Rachid|last8=Choi|first8=Jang Wook|year=2020|title=Entropymetry for non-destructive structural analysis of LiCoO 2 cathodes|url=http://xlink.rsc.org/?DOI=C9EE02964H|journal=Energy & Environmental Science|language=en|volume=13|issue=1|pages=286–296|doi=10.1039/C9EE02964H|s2cid=212779004|issn=1754-5692}}</ref> is done on a closed system (with particle number N and volume V being constants) and uses the definition of temperature<ref>{{cite book|last1=Schroeder|first1=Daniel V.|title=An introduction to thermal physics|url=https://archive.org/details/introductiontoth00schr_817|url-access=limited|date=2000|publisher=Addison Wesley|location=San Francisco, CA [u.a.]|isbn=978-0-201-38027-9|page=[https://archive.org/details/introductiontoth00schr_817/page/n99 88]|edition=[Nachdr.]}}</ref> in terms of entropy, while limiting energy exchange to heat (<math>dU \rightarrow dQ</math>).

: <math>T := \left(\frac{\partial U}{\partial S}\right)_{V,N} \Rightarrow \cdots \Rightarrow \; dS=dQ/T</math>

The resulting relation describes how entropy changes <math>dS</math> when a small amount of energy <math>dQ</math> is introduced into the system at a certain temperature&nbsp;<math>T</math>.

The process of measurement goes as follows. First, a sample of the substance is cooled as close to absolute zero as possible. At such temperatures, the entropy approaches zero{{snd}}due to the definition of temperature. Then, small amounts of heat are introduced into the sample and the change in temperature is recorded, until the temperature reaches a desired value (usually 25&nbsp;°C). The obtained data allows the user to integrate the equation above, yielding the absolute value of entropy of the substance at the final temperature. This value of entropy is called calorimetric entropy.<ref>{{cite web|title=Measuring Entropy|url=https://www.chem.wisc.edu/deptfiles/genchem/netorial/modules/thermodynamics/entropy/entropy04.htm|website=chem.wisc.edu}}</ref>

==Interdisciplinary applications==
Although the concept of entropy was originally a thermodynamic concept, it has been adapted in other fields of study,<ref name="Pokrovskii 2020"/> including [[Information theory|information theory]], psychodynamics, [[Finance:Thermoeconomics|thermoeconomics]]/[[Social:Ecological economics|ecological economics]], and [[Biology:Evolution|evolution]].<ref name="Brooks" /><ref>{{Cite book|last=Avery|first=John|title=Information Theory and Evolution|publisher=World Scientific |year=2003|isbn=978-981-238-399-0}}</ref><ref>{{Cite book|last=Yockey|first=Hubert, P.|title=Information Theory, Evolution, and the Origin of Life|publisher=Cambridge University Press|year=2005|isbn=978-0-521-80293-2}}</ref><ref>{{cite journal|last1= Chiavazzo |first1=Eliodoro|last2=Fasano|first2=Matteo|last3=Asinari|first3=Pietro|title=Inference of analytical thermodynamic models for biological networks|journal=Physica A: Statistical Mechanics and Its Applications|volume=392|issue=5 |doi= 10.1016/j.physa.2012.11.030|url=https://iris.polito.it/bitstream/11583/2504927/1/BIOaps_final_R01c.pdf |pages=1122–1132|bibcode = 2013PhyA..392.1122C |year=2013|s2cid=12418973 }}</ref><ref>{{Cite book|title = The Unity of Science and Economics: A New Foundation of Economic Theory|last = Chen|first = Jing|publisher = Springer|year = 2015|doi = 10.1007/978-1-4939-3466-9|isbn = 978-1-4939-3464-5|url=https://www.springer.com/us/book/9781493934645}}</ref>

===Philosophy and theoretical physics===
Entropy is the only quantity in the physical sciences that seems to imply a particular direction of progress, sometimes called an [[Physics:Arrow of time|arrow of time]]. As time progresses, the second law of thermodynamics states that the entropy of an [[Physics:Isolated system|isolated system]] never decreases in large systems over significant periods of time. Hence, from this perspective, entropy measurement is thought of as a clock in these conditions.{{Citation needed|date=February 2023}}

=== Biology ===
Chiavazzo ''et&nbsp;al.'' proposed that where cave spiders choose to lay their eggs can be explained through entropy minimization.<ref>{{cite journal|last1=Chiavazzo|first1=Eliodoro|last2=Isaia|first2=Marco|last3=Mammola|first3=Stefano|last4=Lepore|first4=Emiliano|last5=Ventola|first5=Luigi|last6=Asinari|first6=Pietro|last7=Pugno|first7=Nicola Maria|year=2015|title=Cave spiders choose optimal environmental factors with respect to the generated entropy when laying their cocoon|journal=Scientific Reports|volume=5|pages=7611|bibcode=2015NatSR...5E7611C|doi=10.1038/srep07611|pmc=5154591|pmid=25556697}}</ref>

Entropy has been proven useful in the analysis of base pair sequences in DNA. Many entropy-based measures have been shown to distinguish between different structural regions of the genome, differentiate between coding and non-coding regions of DNA, and can also be applied for the recreation of evolutionary trees by determining the evolutionary distance between different species.<ref>{{Cite journal|last1=Thanos|first1=Dimitrios|last2=Li|first2=Wentian|last3=Provata|first3=Astero|date=1 March 2018|title=Entropic fluctuations in DNA sequences|journal=Physica A: Statistical Mechanics and Its Applications|volume=493|pages=444–457|doi=10.1016/j.physa.2017.11.119|bibcode=2018PhyA..493..444T|issn=0378-4371}}</ref>

=== Cosmology ===
Assuming that a finite universe is an isolated system, the second law of thermodynamics states that its total entropy is continually increasing. It has been speculated, since the 19th century, that the universe is fated to a [[Astronomy:Heat death of the universe|heat death]] in which all the energy ends up as a homogeneous distribution of thermal energy so that no more work can be extracted from any source.

If the universe can be considered to have generally increasing entropy, then – as [[Biography:Roger Penrose|Roger Penrose]] has pointed out – [[Physics:Gravity|gravity]] plays an important role in the increase because gravity causes dispersed matter to accumulate into stars, which collapse eventually into [[Astronomy:Black hole|black hole]]s. The entropy of a black hole is proportional to the surface area of the black hole's [[Astronomy:Event horizon|event horizon]].<ref>{{Cite book|last=von Baeyer|first=Christian, H.|title=Information–the New Language of Science|publisher=Harvard University Press|year=2003|isbn=978-0-674-01387-2|url=https://archive.org/details/informationnewla00vonb}}</ref><ref>{{Cite journal|author=Srednicki M|title=Entropy and area|journal=Phys. Rev. Lett. |volume=71|issue=5|pages=666–669|date=August 1993|pmid=10055336|doi=10.1103/PhysRevLett.71.666 |bibcode=1993PhRvL..71..666S |arxiv=hep-th/9303048|s2cid=9329564}}</ref><ref>{{Cite journal|author=Callaway DJE|title=Surface tension, hydrophobicity, and black holes: The entropic connection|journal=Phys. Rev. E|volume=53|issue=4|pages=3738–3744|date=April 1996|pmid=9964684|doi= 10.1103/PhysRevE.53.3738 |arxiv=cond-mat/9601111|bibcode=1996PhRvE..53.3738C|s2cid=7115890|author-link=David J E Callaway}}</ref> [[Biography:Jacob Bekenstein|Jacob Bekenstein]] and [[Biography:Stephen Hawking|Stephen Hawking]] have shown that black holes have the maximum possible entropy of any object of equal size. This makes them likely end points of all entropy-increasing processes, if they are totally effective matter and energy traps.<ref>{{cite book|first1=T.K.|last1=Sarkar|first2=M.|last2=Salazar-Palma|first3=Eric L.|last3=Mokole|date=2008|chapter=A Look at the Concept of Channel Capacity from a Maxwellian Viewpoint|chapter-url=https://books.google.com/books?id=chfvTMRsv38C&pg=PA162 |title= Physics of Multiantenna Systems & Broadband Processing |page=162|isbn=978-0470190401|publisher=Wiley|access-date= 31 August 2019}}</ref> However, the escape of energy from black holes might be possible due to quantum activity (see [[Physics:Hawking radiation|Hawking radiation]]).

The role of entropy in cosmology remains a controversial subject since the time of [[Biography:Ludwig Boltzmann|Ludwig Boltzmann]]. Recent work has cast some doubt on the heat death hypothesis and the applicability of any simple thermodynamic model to the universe in general. Although entropy does increase in the model of an expanding universe, the maximum possible entropy rises much more rapidly, moving the universe further from the heat death with time, not closer.<ref>{{Cite book|last=Layzer|first=David|title=Cosmogenesis : The Growth of Order in the Universe|publisher=Oxford University Press|year=1990}}</ref><ref>{{Cite book|last=Chaisson|first=Eric J.|title=Cosmic Evolution: The Rise of Complexity in Nature|publisher=Harvard University Press|year=2001|isbn=978-0-674-00342-2|url=https://archive.org/details/cosmicevolutionr00chai}}</ref><ref>{{Cite book|editor-last1=Lineweaver|editor-first1=Charles H.|editor-last2=Davies|editor-first2=Paul C. W.|editor-last3=Ruse|editor-first3=Michael|title=Complexity and the Arrow of Time|publisher=Cambridge University Press|year=2013|isbn=978-1-107-02725-1}}</ref> This results in an "entropy gap" pushing the system further away from the posited heat death equilibrium.<ref>{{Cite book|last=Stenger|first=Victor J.|title=God: The Failed Hypothesis|publisher=Prometheus Books|year=2007|isbn=978-1-59102-481-1}}</ref> Other complicating factors, such as the energy density of the vacuum and macroscopic [[Physics:Quantum mechanics|quantum]] effects, are difficult to reconcile with thermodynamical models, making any predictions of large-scale thermodynamics extremely difficult.<ref>{{Cite book|author=Benjamin Gal-Or|title=Cosmology, Physics and Philosophy|publisher=Springer Verlag|year=1987|isbn=978-0-387-96526-0}}</ref>

Current theories suggest the entropy gap to have been originally opened up by [[Astronomy:Inflation (cosmology)|the early rapid exponential expansion]] of the universe.<ref name="Albrecht">{{cite encyclopedia |year=2004 |title=Cosmic Inflation and the Arrow of Time |encyclopedia=Science and Ultimate Reality: From Quantum to Cosmos |publisher=Cambridge University Press |location=Cambridge, UK |url=https://arxiv.org/ftp/astro-ph/papers/0210/0210527.pdf |access-date=28 June 2017 |last=Albrecht |first=Andreas |editor-last=Barrow |editor-first=John D. |arxiv=astro-ph/0210527 |bibcode=2002astro.ph.10527A |postscript=none |editor-last2=Davies |editor-first2=Paul C.W. |editor-link2=Paul Davies |editor-last3=Harper |editor-first3=Charles L.}}. In honor of John Wheeler's 90th birthday.</ref>

=== Economics ===

Romanian American economist Nicholas Georgescu-Roegen, a progenitor in economics and a [[Philosophy:Paradigm shift#Kuhnian paradigm shifts|paradigm founder]] of [[Social:Ecological economics|ecological economics]], made extensive use of the entropy concept in his magnum opus on ''The Entropy Law and the Economic Process''.<ref name="Georgescu-Roegen 1971" /> Due to Georgescu-Roegen's work, the [[Physics:Laws of thermodynamics|laws of thermodynamics]] form an [[Social:Ecological economics#Methodology|integral part of the ecological economics school]].<ref>{{cite journal|last1=Cleveland |first1=Cutler J. |last2=Ruth |first2=Matthias |year=1997 |title=When, where, and by how much do biophysical limits constrain the economic process? A survey of Nicholas Georgescu-Roegen's contribution to ecological economics |journal=Ecological Economics |volume=22 |issue=3 |location=Amsterdam |publisher=[[Company:Elsevier|Elsevier]] |doi=10.1016/s0921-8009(97)00079-7 |pages=203–223|doi-access=free }}</ref>{{rp|204f}}<ref>{{cite book |last1=Daly |first1=Herman E. |last2=Farley |first2=Joshua |author-link1=Herman Daly |date= 2011 |title=Ecological Economics. Principles and Applications. |edition=2nd |url=http://library.uniteddiversity.coop/Measuring_Progress_and_Eco_Footprinting/Ecological_Economics-Principles_and_Applications.pdf |format=PDF contains full book |location=Washington |publisher=Island Press |isbn=978-1-59726-681-9 }}</ref>{{rp|29–35}} Although his work was blemished somewhat by mistakes, a full chapter on the economics of Georgescu-Roegen has approvingly been included in one elementary physics textbook on the historical development of thermodynamics.<ref>{{cite book |last=Schmitz |first=John E.J. |date=2007 |title=The Second Law of Life: Energy, Technology, and the Future of Earth As We Know It.|url=https://secondlawoflife.wordpress.com/contents |format=Link to the author's science blog, based on his textbook |location=Norwich |publisher=William Andrew Publishing |isbn=978-0-8155-1537-1 }}</ref>{{rp|95–112}}

In economics, Georgescu-Roegen's work has generated the term [[Philosophy:Pessimism#Entropy pessimism|'entropy pessimism']].<ref>{{cite journal |last=Ayres |first=Robert U. |year=2007 |title=On the practical limits to substitution |url=http://pure.iiasa.ac.at/id/eprint/7800/1/IR-05-036.pdf |journal=Ecological Economics |volume=61 |location=Amsterdam |publisher=[[Company:Elsevier|Elsevier]] |doi=10.1016/j.ecolecon.2006.02.011 |pages=115–128|s2cid=154728333 }}</ref>{{rp|116}} Since the 1990s, leading ecological economist and [[Social:Steady-state economy#Herman Daly's concept of a steady-state economy|steady-state theorist]] Herman Daly – a student of Georgescu-Roegen – has been the economics profession's most influential proponent of the entropy pessimism position.<ref>{{cite journal |last=Kerschner |first=Christian |year=2010 |title=Economic de-growth vs. steady-state economy |url=http://degrowth.org/wp-content/uploads/2012/11/Kerschner-2010.pdf |journal=Journal of Cleaner Production |volume=18 |issue=6 |location=Amsterdam |publisher=[[Company:Elsevier|Elsevier]] |doi=10.1016/j.jclepro.2009.10.019 |pages=544–551}}</ref>{{rp|545f}}<ref>
{{cite journal |url=http://www.greattransition.org/publication/economics-for-a-full-world |title=Economics for a Full World |author=Daly, Herman E. |year=2015 |journal=Scientific American |volume=293 |issue=3 |pages=100–7 |doi=10.1038/scientificamerican0905-100 |pmid=16121860 |s2cid=13441670 |access-date=23 November 2016 }}</ref>

== See also ==
{{colbegin}}
* Boltzmann entropy
* [[Physics:Brownian ratchet|Brownian ratchet]]
* [[Physics:Configuration entropy|Configuration entropy]]
* [[Physics:Conformational entropy|Conformational entropy]]
* [[Physics:Entropic explosion|Entropic explosion]]
* [[Physics:Entropic force|Entropic force]]
* [[Finance:Entropic value at risk|Entropic value at risk]]
* [[Physics:Entropy and life|Entropy and life]]
* [[Entropy unit]]
* [[Free entropy]]
* Harmonic entropy
* [[Info-metrics]]
* [[Negentropy]] (negative entropy)
* [[Physics:Phase space|Phase space]]
* [[Principle of maximum entropy]]
* [[Physics:Residual entropy|Residual entropy]]
* [[Thermodynamic potential]]
{{colend}}

== Notes ==
{{Reflist|group=note}}

== References ==
{{Reflist|35em}}
* {{cite web |last1=David |first1=Kover |title=Entropia – fyzikálna veličina vesmíru a nášho života |url=https://www.stejfree.sk/entropia-najmenej-chapany-vzorec-vesmiru-a-nasho-zivota/ |website=stejfree.sk |date=August 14, 2018 |access-date=13 April 2022 |url-status=dead |archive-url=https://web.archive.org/web/20220527213522/https://www.stejfree.sk/entropia-najmenej-chapany-vzorec-vesmiru-a-nasho-zivota/ |archive-date= May 27, 2022 }}

== Further reading ==
* {{cite book |last=Adam |first=Gerhard |author2=Otto Hittmair |title=Wärmetheorie |publisher=Vieweg, Braunschweig |year=1992 |isbn=978-3-528-33311-9|}}
* {{cite book |last=Atkins |first=Peter |author2=Julio De Paula |title=Physical Chemistry |edition=8th |publisher=Oxford University Press |year=2006 |isbn=978-0-19-870072-2}}
* {{cite book |author=Baierlein, Ralph |title=Thermal Physics |publisher=Cambridge University Press |year=2003 |isbn=978-0-521-65838-6 |url-access=registration |url=https://archive.org/details/thermalphysics00ralp }}
* {{cite book |last=[[Biography:Arieh Ben-Naim|Ben-Naim]] |first=Arieh |year=2007 |title=Entropy Demystified |publisher=World Scientific |isbn=978-981-270-055-1}}
* {{cite book |last=Callen |first=Herbert, B |title=Thermodynamics and an Introduction to Thermostatistics |edition=2nd |publisher=John Wiley and Sons |year=2001 |isbn=978-0-471-86256-7}}
* {{cite book |author=Chang, Raymond |title=Chemistry |url=https://archive.org/details/chemistry00chan_0 |url-access=registration |edition=6th |location=New York |publisher=McGraw Hill |year=1998 |isbn=978-0-07-115221-1}}
* {{cite book |last1=Cutnell |first1=John, D. |last2=Johnson |first2=Kenneth, J. |title=Physics |url=https://archive.org/details/physi1998cutn |url-access=registration |edition=4th |publisher=John Wiley and Sons, Inc. |year=1998 |isbn=978-0-471-19113-1}}
* {{cite book |last=Dugdale |first=J. S. |year=1996 |title=Entropy and its Physical Meaning |edition=2nd |publisher=Taylor and Francis (UK); CRC (US) |isbn=978-0-7484-0569-5}}
* {{cite book |last=Fermi |first=Enrico |year=1937 |title=Thermodynamics |publisher=Prentice Hall |isbn=978-0-486-60361-2}}
* {{cite book |title=The Refrigerator and the Universe |url=https://archive.org/details/refrigeratoruniv0000gold |url-access=registration |publisher=Harvard University Press |year=1993 |isbn=978-0-674-75325-9 |author1=Goldstein, Martin |author2=Inge, F }}
* {{cite book |last=Gyftopoulos |first=E.P. |author2=G.P. Beretta |year=2010 |title=Thermodynamics. Foundations and Applications |publisher=Dover |isbn=978-0-486-43932-7}}
* {{cite book |last=Haddad |first=Wassim M. |author2=Chellaboina, VijaySekhar |author3=Nersesov, Sergey G. |title=Thermodynamics – A Dynamical Systems Approach |publisher=Princeton University Press |year=2005 |isbn=978-0-691-12327-1}}
* {{cite book | last = Johnson | first = Eric | title = ''Anxiety and the Equation: Understanding Boltzmann's Entropy'' | publisher = The MIT Press | year = 2018 | isbn = 978-0-262-03861-4}}
* {{cite book |last=Kroemer |first=Herbert |author2=Charles Kittel |year=1980 |title=Thermal Physics |edition=2nd |publisher=W. H. Freeman Company |isbn=978-0-7167-1088-2}}
* Lambert, Frank L.; [http://franklambert.net/entropysite.com/]
* {{cite book |last=Müller-Kirsten |first=Harald J. W. |year=2013 |title=Basics of Statistical Physics |edition=2nd |isbn=978-981-4449-53-3 |publisher=World Scientific |location=Singapore}}
* {{cite book |last=Penrose |first=Roger |year=2005 |title=The Road to Reality: A Complete Guide to the Laws of the Universe |isbn=978-0-679-45443-4 |publisher=A. A. Knopf |location=New York |url=https://archive.org/details/roadtorealitycom00penr_0 }}
* {{cite book |last=Reif |first=F. |year=1965 |title=Fundamentals of statistical and thermal physics |publisher=McGraw-Hill |isbn=978-0-07-051800-1 |url=https://archive.org/details/fundamentalsofst00fred }}
* {{cite book |author=Schroeder, Daniel V. |title=Introduction to Thermal Physics |publisher=New York: Addison Wesley Longman |year=2000 |isbn=978-0-201-38027-9}}
* {{cite book |last=Serway |first=Raymond, A. |title=Physics for Scientists and Engineers |publisher=Saunders Golden Subburst Series |year=1992 |isbn=978-0-03-096026-0 |url=https://archive.org/details/physicsforscient00serw }}
* Sharp, Kim (2019). ''Entropy and the Tao of Counting: A Brief Introduction to Statistical Mechanics and the Second Law of Thermodynamics'' (SpringerBriefs in Physics). Springer Nature. {{ISBN|978-3030354596}}.
* Spirax-Sarco Limited, [http://www.spiraxsarco.com/resources/steam-engineering-tutorials/steam-engineering-principles-and-heat-transfer/entropy-a-basic-understanding.asp Entropy – A Basic Understanding] A primer on entropy tables for steam engineering
* {{cite book |author1=vonBaeyer |author2=Hans Christian |title=Maxwell's Demon: Why Warmth Disperses and Time Passes |publisher=Random House |year=1998 |isbn=978-0-679-43342-2|title-link=Maxwell's demon }}

== External links ==
* [http://www.scholarpedia.org/article/Entropy "Entropy"] at ''[[Scholarpedia]]''
* [http://ocw.mit.edu/courses/chemistry/5-60-thermodynamics-kinetics-spring-2008/video-lectures/lecture-9-entropy-and-the-clausius-inequality/ Entropy and the Clausius inequality] MIT OCW lecture, part of 5.60 Thermodynamics & Kinetics, Spring 2008
* [https://www.youtube.com/watch?v=ER8d_ElMJu0 Entropy and the Second Law of Thermodynamics] – an A-level physics lecture with 'derivation' of entropy based on Carnot cycle
* Khan Academy: entropy lectures, part of [https://www.youtube.com/playlist?list=PL1A79AF620ABA411C Chemistry playlist]
** [https://www.youtube.com/watch?v=xJf6pHqLzs0 Entropy Intuition]
** [https://www.youtube.com/watch?v=dFFzAP2OZ3E More on Entropy]
** [https://www.youtube.com/watch?v=sPz5RrFus1Q Proof: S (or Entropy) is a valid state variable]
** [https://www.youtube.com/watch?v=WLKEVfLFau4 Reconciling Thermodynamic and State Definitions of Entropy]
** [https://www.youtube.com/watch?v=PFcGiMLwjeY Thermodynamic Entropy Definition Clarification]
* {{cite web|last=Moriarty|first=Philip|title=S Entropy|url=http://www.sixtysymbols.com/videos/entropy.htm|work=Sixty Symbols|publisher=[[Biography:Brady Haran|Brady Haran]] for the [[Organization:University of Nottingham|University of Nottingham]]|author2=Merrifield, Michael|year=2009}}
* [https://www.youtube.com/watch?v=glrwlXRhNsg The Discovery of Entropy] by Adam Shulman. Hour-long video, January 2013.
* [http://oyc.yale.edu/physics/phys-200/lecture-24 The Second Law of Thermodynamics and Entropy] – Yale OYC lecture, part of Fundamentals of Physics I (PHYS 200)
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[[Category:Entropy| ]]
[[Category:Physical quantities]]
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{{Sourceattribution|Entropy}}