Physics:Derivation of Speed of Light (c) from the Theory of Entropicity (ToE)

Deriving the Speed of Light and Entropic Lorentz Invariance from the Theory of Entropicity (ToE)

Here we wish to derive the speed of light (c) from the Theory of Entropicity (ToE), and also show why it is a constant for all observers within the entropic field. Theory of relativity. Wikipedia, The Free Encyclopedia. 14:43, 19 July 2025 (UTC). Retrieved 23 July 2025. A Critical Review of the Theory of Entropicity (ToE) on Original Contributions, Conceptual Innovations, and Pathways towards Enhanced Mathematical Rigor: An Addendum to the Discovery of New Laws of Conservation and Uncertainty. Obidi, John Onimisi. Cambridge University; 30 June 2025. Retrieved 23 July 2025.

Prelude and Definitions

The Theory of Entropicity (ToE) promotes entropy S(x) to a dynamical field whose variations source and constrain all interactions. We start from a simplified but ToE-faithful sector of the Master Entropic Equation (MEE)—the [Obidi Action]—focusing on the kinetic, potential, and trace-coupling pieces for the ToE Action [math]\displaystyle{ \mathcal{S} }[/math] ([math]\displaystyle{ =\mathcal{L} }[/math]):

(1) [math]\displaystyle{ \mathcal{L}=\mathcal{S}[S,g_{\mu\nu},\Phi] \int d^4x,\sqrt{-g}; \Big[ -\tfrac{1}{2},A(S),g^{\mu\nu}\nabla_\mu S \nabla_\nu S V(S) \eta, S, T^\mu{}{\mu} \Big] +\mathcal{S}_{\text{matter}}[\Phi,g_{\mu\nu}]~; }[/math]

where, by definition of terms:

[math]\displaystyle{ A(S) \equiv 1 + \frac{\lambda}{k_B^2} \, e^{-S/k_B} }[/math] is the kinetic “entropic correction”

[math]\displaystyle{ V(S) }[/math] is the entropic potential

[math]\displaystyle{ \eta\,S\,T^{\mu}{}_{\mu} }[/math] couples entropy to the trace of the stress tensor (matter back-reaction)

[math]\displaystyle{ A(S) }[/math] encodes the entropic correction to the kinetic term (cf. the exponential piece in the MEE), where we have taken a convenient choice of the form:

(2) [math]\displaystyle{ A(S) \equiv 1 + \frac{\lambda}{k_B^2},e^{-S/k_B}; }[/math]

but the derivation works for any smooth [math]\displaystyle{ A(S) }[/math].

μ couples S to the trace of the stress-energy tensor.
Note that we have deliberately avoided including the Fisher Information Terms in the above expression for the Action of the Theory of Entropicity (ToE), for the sake of ease and heuristics. In a future work, we shall invoke the full powers of the ToE Action without loss of generality. What we intend to demonstrate here is instructive enough to help bring out the fine points of ToE to explain and derive the subject before us, namely, the ubiquitous speed of light, c. (That is, we can restore the extra Fisher/irreversibility pieces later; the derivation logic is identical.)

To proceed in our programme in this paper, we shall:

a) Vary Eq. (1) with respect to S to obtain the entropic field equation (see Section 2).

b) Linearize around a homogeneous background to read off the wave (characteristic) speed (see Section 3).

c) Restore dimensions and identify the combination that equals the measured c (see Section 4).

d) Establish observer-independence and construct the “entropic Lorentz group” preserving the null cones of the entropy field (see Section 5).

Variation with Respect to S

We begin by computing [math]\displaystyle{ \delta\mathcal{L}/\delta S = 0 }[/math] in the following order:

1. Variation of the kinetic term gives both a [math]\displaystyle{ \Box }[/math]-operator and a derivative of [math]\displaystyle{ A(S) }[/math].
2. Variation of the potential term yields [math]\displaystyle{ V'(S) }[/math].
3. Variation of the source term produces [math]\displaystyle{ \eta\,T^{\mu}{}_{\mu} }[/math].

Thus, varying Eq. (1) with respect to 𝑆 and integrating by parts yields

(3) [math]\displaystyle{ A(S)\Box S +\frac{1}{2}A'(S)g^{\mu\nu} \nabla_\mu S\nabla_\nu S -V'(S) \eta +T^\mu{}_{\mu} =0, }[/math]

where

[math]\displaystyle{ A'(S)\equiv \frac{dA}{dS} }[/math] is the derivative of A with respect to S

[math]\displaystyle{ V'(S)\equiv \frac{dV}{dS} }[/math] is the derivative of V with respect to S

[math]\displaystyle{ \Box \equiv \nabla_{\mu}\nabla^{\mu} }[/math] is the d'Alembert operator.

Equation (3) is the entropic field equation in its general non-linear form.

Linearization of the Entropic Field Equation and Characteristic Speed Around a Homogeneous Background

Let [math]\displaystyle{ S(x) = S_0 + \sigma(x) }[/math], with [math]\displaystyle{ S_0 }[/math] being of a constant (or slowly varying) background value.

Now, expand Eq. (3) to first order in [math]\displaystyle{ \sigma }[/math] and its derivatives to yield:

[math]\displaystyle{ A(S)\,\Box S = A_0\,\Box \sigma + \mathcal{O}(\sigma^2) }[/math] [math]\displaystyle{ \frac{1}{2}A'(S)(\nabla S)^2 = \frac{1}{2}A_0'(\nabla S_0)^2 + \mathcal{O}(\sigma\,\partial \sigma) }[/math]

where

[math]\displaystyle{ A_0 \equiv A(S_0) }[/math]

[math]\displaystyle{ A_0' \equiv A'(S_0) }[/math]

Set [math]\displaystyle{ S_0 }[/math] as homogeneous so that [math]\displaystyle{ \nabla_{\mu} S_0 = 0 }[/math]. When we neglect matter (dealing with stress/energy), we can write: [math]\displaystyle{ T^{\mu}{}_{\mu} = 0 }[/math]. This isolates the free (null) sector, so that we have:

Equation (4): [math]\displaystyle{ A_0\,\Box \sigma - V''(S_0)\,\sigma = 0 }[/math]

where [math]\displaystyle{ V''(S_0) \equiv \left.\frac{d^2V}{dS^2}\right|_{S_0} }[/math]. In local inertial coordinates [math]\displaystyle{ (t, x) }[/math], the box operator becomes:

[math]\displaystyle{ \Box \sigma = \partial_t^2 \sigma - \nabla^2 \sigma }[/math]

Thus, Eq. (4) becomes:

Equation (5): [math]\displaystyle{ A_0 \big( \partial_t^2 \sigma - \nabla^2 \sigma \big) - m_S^2 \,\sigma = 0 }[/math]

with [math]\displaystyle{ m_S^2 \equiv V''(S_0) }[/math].

Null sector (massless excitations)

To determine the signal speed, we examine the characteristic curves of the partial differential equation (PDE).

We set [math]\displaystyle{ m_S^2 = 0 }[/math] — either by choosing [math]\displaystyle{ V''(S_0) = 0 }[/math] or by considering high-frequency/short-wavelength modes. We therefore obtain:

Equation (6): [math]\displaystyle{ \partial_t^2 \sigma - \nabla^2 \sigma = 0 }[/math]

after dividing by [math]\displaystyle{ A_0 }[/math] (taken to be positive for sake of stability purposes and analysis).

Null Sector and Entropic Wave Speed

We can progress the above using another approach, such that we demand massless (null) excitations of the entropy field—or photon sectors for which the mass term vanishes: [math]\displaystyle{ m_S^2 = 0 }[/math].

In this limit, therefore, the coefficients in front of the time and space derivatives in the wave equation are equal, so that the PDE is symmetric. Hence, we can now define the entropic wave speed by the ratio of these coefficients as follows:

Equation (6.1): [math]\displaystyle{ c_{\mathrm{ent}}^2 = \frac{\text{coef. of }\nabla^2 \sigma} {\text{coef. of }\partial_t^2 \sigma} = 1 \quad(\text{in these units}). }[/math]

The value “1” reflects the choice of units where the background metric has signature [math]\displaystyle{ (+,-,-,-) }[/math] and the fundamental speed unit is set to 1.

This is the standard wave equation in units where propagation speed is unity. To obtain the actual physical speed [math]\displaystyle{ c }[/math], we must now proceed to restore the appropriate dimensional units and constants.

Restoring Dimensions and Identifying c

Let the kinetic term generically read (in SI-like dimensions)

(7) [math]\displaystyle{ \mathcal{L}_{\text{kin}} \sim \frac{\kappa}{2}\left(\frac{1}{c_{\text{ent}}^2}(\partial_t \sigma)^2 - (\nabla \sigma)^2\right), }[/math]

where [math]\displaystyle{ \kappa }[/math] is the “entropic stiffness” and [math]\displaystyle{ c_{\text{ent}} }[/math] is the "characteristic speed of disturbances" in [math]\displaystyle{ S }[/math].

Realize that the coefficient in front of the "spatial gradient" comes from an "entropic stiffness" [math]\displaystyle{ \kappa }[/math] and the coefficient in front of the "temporal curvature" comes from an "entropic inertia" [math]\displaystyle{ \rho_S }[/math]. Hence, using dimensional analysis and comparing Eq. (7) to Eq. (6), we identify Eq. (6.1) to yield:

(8) [math]\displaystyle{ c_{\text{ent}}^2 = \frac{\text{coef. of }(\nabla \sigma)^2}{\text{coef. of }(\partial_t \sigma)^2} = \frac{\kappa}{\rho_S}, }[/math]

with [math]\displaystyle{ \rho_{S} }[/math] being the “entropic inertia” coefficient.

Eq. (8) above [math]\displaystyle{ \Rightarrow\;c = \frac{\kappa}{\rho_S} }[/math]

From ToE’s entropy-density functional (which we used already in ToE's Vuli-Ndlela Integral ),

(9) [math]\displaystyle{ \Lambda(\phi) = \frac{k_B c^3}{\hbar G} A(\phi) + \frac{dS}{dt}, }[/math]

we see that the combination

(10) [math]\displaystyle{ \chi \equiv \frac{k_B c^3}{\hbar G} }[/math]

sets a natural scale for entropic stiffness.

Now, taking note of the ToE constants and inserting them results in:

[math]\displaystyle{ \kappa \sim \chi \hbar = \frac{k_B\,c^3}{G} }[/math] (dimensionally: energy per entropy gradient)

[math]\displaystyle{ \rho_S \sim \frac{k_B\,c}{G} }[/math] (dimensionally: energy per time-entropy rate)

Then:

[math]\displaystyle{ c_{\mathrm{ent}}^2 =\frac{\kappa}{\rho_S} =\frac{\frac{k_B\,c^3}{G}}{\frac{k_B\,c}{G}} = c^2 }[/math]

Dimensional analysis (see Appendix A) thus shows that matching the PDE normalization yields:

(11) [math]\displaystyle{ c_{\text{ent}} = c. }[/math]

In other words, what the above expressions and reasoning tell us is that the only self-consistent value of the "characteristic entropy-wave speed" is the empirically measured speed of light, fixed by the same universal constants [math]\displaystyle{ (\hbar,\,G,\,k_B) }[/math].

Thus, ToE derives [math]\displaystyle{ c }[/math] as the unique solution keeping "entropy flux and inertia" in balance, consistent with the observed gravitational and quantum constants.

Interpretation:

Thus, we come to the inevitable conclusion that the Theory of Entropicity (ToE) fixes the ratio of entropic stiffness to entropic inertia via [math]\displaystyle{ (\hbar,\,G,\,k_B) }[/math]. That ratio is numerically equal to the measured [math]\displaystyle{ c^2 }[/math]. Thus the “speed of light” is the characteristic speed of null entropic excitations.

Invariance and the Entropic Lorentz Group: Observer-independence (Why Everyone Measures the Same c)

Having identified [math]\displaystyle{ c_{\mathrm{ent}} = c }[/math], we must show that all observers constrained by the entropy field measure the same [math]\displaystyle{ c }[/math]. This requires an invariance principle.

Entropic line element

Introduce an “entropic line element”:

[math]\displaystyle{ d\sigma^2 = \alpha(S)\,dt^2 - \beta(S)\,d\mathbf{x}^2, \quad \alpha(S),\beta(S)\gt 0. }[/math]

For null propagation of S-waves (or of fields slaved to S):

[math]\displaystyle{ d\sigma^2 = 0 \;\Longrightarrow\; \frac{d\mathbf{x}}{dt} = \sqrt{\frac{\alpha(S)}{\beta(S)}} \equiv c_{\mathrm{ent}}(S). }[/math]

The field equations in the null sector enforce [math]\displaystyle{ c_{\mathrm{ent}} }[/math] to be constant (Eq.~11), while [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] may individually rescale with [math]\displaystyle{ S }[/math], representing how clocks/rulers themselves are entropically renormalized. Crucially, the ratio [math]\displaystyle{ \alpha/\beta }[/math] is fixed for null modes.

It now remains for us to show that:

• The ToE's entropic field equations force [math]\displaystyle{ \alpha/\beta = c^2 }[/math] when evaluated on null modes.
• Measuring devices (clocks & rulers) are themselves constrained by the same entropy field; their rescaling keeps [math]\displaystyle{ \alpha/\beta }[/math] fixed despite local changes in [math]\displaystyle{ S }[/math].
• Transformations that preserve [math]\displaystyle{ d\sigma^2 }[/math] form a Lorentz-like group → invariance of [math]\displaystyle{ c }[/math].

Group of transformations preserving [math]\displaystyle{ dσ² }[/math]

Consider transformations of coordinates [math]\displaystyle{ (t,x)\mapsto(t',x') }[/math] that preserve [math]\displaystyle{ d\sigma^2 }[/math] up to an overall scalar factor (which cancels in the null condition):

[math]\displaystyle{ d\sigma'^2 = \alpha'(S)\,dt'^2 - \beta'(S)\,d\mathbf{x}'^2 = \Omega^2(S)\bigl[\alpha(S)\,dt^2 - \beta(S)\,d\mathbf{x}^2\bigr]. }[/math]

Requiring [math]\displaystyle{ d\sigma'^2=0\iff d\sigma^2=0 }[/math] yields the conformal Lorentz group in flat space. Fixing [math]\displaystyle{ \Omega=1 }[/math] (physical units) reduces it to the standard Lorentz group [math]\displaystyle{ O(1,3) }[/math]:

[math]\displaystyle{ \begin{pmatrix} ct' \\ x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma\beta_x & -\gamma\beta_y & -\gamma\beta_z \\ -\gamma\beta_x & 1+\tfrac{\gamma^2\beta_x^2}{\gamma+1} & \tfrac{\gamma^2\beta_x\beta_y}{\gamma+1} & \tfrac{\gamma^2\beta_x\beta_z}{\gamma+1} \\ -\gamma\beta_y & \tfrac{\gamma^2\beta_y\beta_x}{\gamma+1} & 1+\tfrac{\gamma^2\beta_y^2}{\gamma+1} & \tfrac{\gamma^2\beta_y\beta_z}{\gamma+1} \\ -\gamma\beta_z & \tfrac{\gamma^2\beta_z\beta_x}{\gamma+1} & \tfrac{\gamma^2\beta_z\beta_y}{\gamma+1} & 1+\tfrac{\gamma^2\beta_z^2}{\gamma+1} \end{pmatrix} \begin{pmatrix} ct \\ x \\ y \\ z \end{pmatrix}, \quad \gamma = \frac{1}{\sqrt{1-\beta^2}}, }[/math]

with [math]\displaystyle{ \beta = v/c }[/math]. Because [math]\displaystyle{ c_{\mathrm{ent}}=c }[/math], these boosts preserve the null condition [math]\displaystyle{ d\sigma^2=0 }[/math]. Hence every observer, using entropically consistent clocks and rulers, measures the same [math]\displaystyle{ c }[/math].

Generators and algebra

Infinitesimal transformations are generated by [math]\displaystyle{ M_{\mu\nu} }[/math] obeying:

[math]\displaystyle{ [\,M_{\mu\nu},\,M_{\rho\sigma}\,] = i\Bigl( \eta_{\mu\rho}M_{\nu\sigma} - \eta_{\mu\sigma}M_{\nu\rho} - \eta_{\nu\rho}M_{\mu\sigma} + \eta_{\nu\sigma}M_{\mu\rho} \Bigr), }[/math]

with [math]\displaystyle{ \eta_{\mu\nu}=\mathrm{diag}(+1,-1,-1,-1) }[/math]. The algebra remains unchanged because the entropy field fixes the same null structure. Thus, the “entropic Lorentz group” is isomorphic to the standard Lorentz group in the null sector.

Discussion and Outlook

We have thus far:

Derived the entropic field equation (Eq. 3) from the MEE sector (Eq. 1). Linearized it to get a wave equation (Eq. 6) whose characteristic speed is forced to equal c by ToE’s internal constants (Section 4). Constructed the invariance group preserving the null cones of the entropy field and shown it coincides with Lorentz invariance (Section 5). This upgrades the conceptual claim (“entropy enforces a universal speed limit”) to a formal derivation. Future work can:

1. Include the full irreversibility and Fisher-information terms in [math]\displaystyle{ A(S) }[/math] and verify c remains unchanged.
2. Derive photon dynamics explicitly from an S-dependent gauge Lagrangian and show the same null cone emerges.
3. Explore whether extreme entropy gradients produce tiny, testable deviations from c (or prove a no-deviation theorem).

Photon Sector Link in the Theory of Entropicity (ToE)

If we treat “light” as the lowest-entropy, massless excitation—not of S itself but of a gauge field [math]\displaystyle{ A_{\mu} }[/math]—we can still show that it shares the same null cone because both the gauge field and the entropy field are constrained by the same "No-Rush Theorem Limit (NTL)". Concretely, we proceed as follows:

1. Add [math]\displaystyle{ -\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu} }[/math] to the action.
2. Multiply the gauge field kinetic term by the same “entropic lapse” factor [math]\displaystyle{ f(S) }[/math].
3. In the eikonal (geometric optics) limit, rays satisfy [math]\displaystyle{ d\sigma^2 = 0 }[/math]. Geometrical optics. Wikipedia, The Free Encyclopedia. 23:00, 29 May 2025 (UTC). Retrieved 23 July 2025.

Therefore, light follows the same characteristic cones → same [math]\displaystyle{ c_{\mathrm{ent}} }[/math].

Appendix A: Sketch of Dimensional Analysis of ToE Constants

Let [math]\displaystyle{ [S] = k_B }[/math] (entropy, for ToE's entropic effects),

[math]\displaystyle{ [\hbar] = \mathrm{J\,s} }[/math] (quantum effects from Planck's constant),

[math]\displaystyle{ [G] = \mathrm{m^3\,kg^{-1}\,s^{-2}} }[/math] (Newton's gravitational constant, for gravitational effects),

[math]\displaystyle{ [c] = \mathrm{m\,s^{-1}} }[/math] (speed of light in Einstein's Relativity, for relativistic effects).

The prefactor:

[math]\displaystyle{ \chi = \frac{k_B\,c^3}{\hbar\,G} }[/math]

then has dimensions [math]\displaystyle{ [k_B]\frac{[c]^3}{[\hbar][G]} = (\mathrm{J/K})\, \frac{(\mathrm{m\,s^{-1}})^3} {(\mathrm{J\,s})\;(\mathrm{m^3\,kg^{-1}\,s^{-2}})} = \frac{\mathrm{kg}}{\mathrm{K\,s^2}}. }[/math]

This scales like an “entropy stiffness” (energy per entropy per length²).

Matching the PDE normalization fixes the "ratio of spatial to temporal coefficients" to [math]\displaystyle{ c^2 }[/math], thus enforcing [math]\displaystyle{ c_{\mathrm{ent}} = c }[/math].

Dimensional Interpretation of the Entropic Stiffness Units Used in the Theory of Entropicity (ToE)

Some Clarifications on the Above Dimentional Analysis:

We ask, how can the units

[math]\displaystyle{ [k_B]\,[c]^3/[\hbar]\,[G] = \mathrm{kg\,K^{-1}\,s^{-2}} }[/math]

be interpreted as “energy per entropy per length²” when no length explicitly appears in the dimensional analysis?

We undertake to show below step by step why such is the case, showing how length² appears implicitly through the use of [math]\displaystyle{ c }[/math] and unit substitutions.

Step 1: Recall the fundamental dimensions

Quantity	Units (SI)
[math]\displaystyle{ k_B }[/math]	[math]\displaystyle{ \mathrm{J/K = kg\,m^2\,s^{-2}\,K^{-1}} }[/math]
[math]\displaystyle{ c }[/math]	[math]\displaystyle{ \mathrm{m\,s^{-1}} }[/math]
[math]\displaystyle{ \hbar }[/math]	[math]\displaystyle{ \mathrm{J\,s = kg\,m^2\,s^{-1}} }[/math]
[math]\displaystyle{ G }[/math]	[math]\displaystyle{ \mathrm{m^3\,kg^{-1}\,s^{-2}} }[/math]

Step 2: Plug in units into the expression

We start with:

[math]\displaystyle{ [k_B]\,[c]^3/[\hbar]\,[G] = \frac{\mathrm{J/K}\,\cdot(\mathrm{m/s})^3} {(\mathrm{J\,s})\;\cdot(\mathrm{m^3\,kg^{-1}\,s^{-2}})}. }[/math]

Convert each to base SI units:

• [math]\displaystyle{ \mathrm{J = kg\,m^2\,s^{-2}}, }[/math] so [math]\displaystyle{ \mathrm{J/K = kg\,m^2\,s^{-2}\,K^{-1}} }[/math].
• [math]\displaystyle{ \hbar = \mathrm{J\,s = kg\,m^2\,s^{-1}} }[/math].
• [math]\displaystyle{ G = \mathrm{m^3\,kg^{-1}\,s^{-2}} }[/math].

Compute the numerator:

[math]\displaystyle{ \mathrm{kg\,m^2\,s^{-2}\,K^{-1}} \times \mathrm{m^3\,s^{-3}} = \mathrm{kg\,m^5\,s^{-5}\,K^{-1}}. }[/math]

Compute the denominator:

[math]\displaystyle{ (\mathrm{kg\,m^2\,s^{-1}}) \times(\mathrm{m^3\,kg^{-1}\,s^{-2}}) = \mathrm{m^5\,s^{-3}}. }[/math]

Divide numerator by denominator:

[math]\displaystyle{ \frac{\mathrm{kg\,m^5\,s^{-5}\,K^{-1}}} {\mathrm{m^5\,s^{-3}}} = \mathrm{kg\,s^{-2}\,K^{-1}}. }[/math]

This matches the quoted unit: [math]\displaystyle{ \mathrm{kg/(K\,s^2)}. }[/math]

Step 3: So where is the “length²”?

The key insight lies in interpreting physical relationships, not just raw dimensions.

• Recall that

[math]\displaystyle{ \mathrm{Energy} = \mathrm{mass}\times(\mathrm{length}/\mathrm{time})^2. }[/math]

• Hence

[math]\displaystyle{ \frac{\mathrm{mass}}{\mathrm{time}^2} = \frac{\mathrm{Energy}}{\mathrm{length}^2}. }[/math]

Therefore, replacing [math]\displaystyle{ \mathrm{mass}/\mathrm{time}^2 }[/math] gives:

[math]\displaystyle{ \frac{\mathrm{kg}}{\mathrm{s^2}} \sim \frac{\mathrm{J}}{\mathrm{m^2}} \quad\Longrightarrow\quad \frac{\mathrm{kg}}{\mathrm{K\,s^2}} \sim \frac{1}{K}\,\frac{\mathrm{J}}{\mathrm{m^2}} = \frac{\mathrm{Energy}}{\mathrm{Entropy}\,\cdot\mathrm{length}^2}. }[/math]

This is the interpretation of “entropy stiffness”—the energy cost per unit entropy per unit area.

External links

• Theory of relativity. Wikipedia, The Free Encyclopedia. 14:43, 19 July 2025 (UTC). Retrieved 23 July 2025.
• Geometrical optics. Wikipedia, The Free Encyclopedia. 23:00, 29 May 2025 (UTC). Retrieved 23 July 2025.
• Boltzmann entropy on Wikipedia
• Effective Field Theory of Dissipative Fluids (arXiv)
• Another Useful Resource - on Thermodynamics
• Another Useful Resource - Physics Books
• Theory of Entropicity (ToE) on Wikipedia
• Entropy(information_theory)
• Hazewinkel, Michiel, ed. (2001), "Entropy", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/e035740 
• "Entropy" at Rosetta Code—repository of implementations of Shannon entropy in different programming languages.
• Entropy an interdisciplinary journal on all aspects of the entropy concept. Open access.
• Thermodynamics Data & Property Calculation Websites
• Thermodynamics Educational Websites
• Biochemistry Thermodynamics
• Thermodynamics and Statistical Mechanics
• Engineering Thermodynamics – A Graphical Approach
• Thermodynamics and Statistical Mechanics by Richard Fitzpatrick