Physics:Explaining the Mpemba Effect in the Theory of Entropicity (ToE)

The Mpemba Effect and Its Entropic Explanation in the Theory of Entropicity (ToE)

Introduction

The Mpemba Effect is a counterintuitive thermodynamic phenomenon in which, under certain conditions, hot water freezes faster than cold water. Despite appearing to contradict classical thermodynamics, this effect has been observed experimentally and continues to attract scientific interest. In this article, we explore the historical origins of the effect and then present a comprehensive explanation using the framework of the Theory of Entropicity (ToE), first formulated by John Onimisi Obidi.

Historical Background

The Mpemba Effect is named after Erasto Bartholomeo Mpemba, a Tanzanian secondary school student who in 1963 observed that hot milk froze faster than cold milk when making ice cream. He reported this observation to visiting physicist Dr. Denis Osborne, who initially doubted the claim. However, Osborne later conducted experiments and confirmed the phenomenon. The two jointly published a paper in 1969 titled "Cool?" in the journal Physics Education.

Although anecdotal reports of the effect date back to Aristotle, Roger Bacon, and René Descartes, it was Mpemba’s insight and collaboration with Osborne that gave it scientific visibility and inspired rigorous investigation.

This is the full incident, for historical correctness and completeness:

In 1963, Erasto Bartholomeo Mpemba, then a student at Magamba Secondary School in Tanganyika (now Tanzania), was taking a cooking course in which participants prepared ice cream by mixing a powdered premix into water and freezing it. Pressed for time, Mpemba skipped the usual cooling step and placed his warm milk mixture directly into the freezer. To his surprise, it froze faster than those of his classmates.

Mpemba’s physics teacher declared the result impossible, and his classmates ridiculed the claim. A few years later, during a guest lecture at Mpemba’s school, the visiting British physicist Denis Osborne (1932–2014) from the University of Dar es Salaam heard Mpemba pose the question that had long puzzled him: “If two equal volumes of water—one at 35 °C and the other at 100 °C—are frozen under identical conditions, why does the hotter sample sometimes freeze first?”

Osborne was also caught off guard, but later invited Mpemba to the university in Dar es Salaam to test his observations. In these tests, the pair were able to find some evidence for Mpemba's claim.

Their collaborative experiments yielded the first scientific evidence for what would become known as the Mpemba effect. Subsequent studies have since confirmed that, under certain conditions, warmer water can indeed freeze faster than cooler water, contrary to prior intuition.

Classical Explanations and Challenges

Classically, one would expect the colder water to freeze faster, since it is closer in temperature to 0°C. However, experiments show that:

Under certain conditions, the initially hotter water freezes faster.

The effect depends on initial conditions such as volume, container shape, environmental exposure, dissolved gases, and convection.

Various hypotheses have been proposed:

Enhanced evaporation

Reduced dissolved gas concentration

Supercooling suppression

Convection current acceleration

Frost layer formation dynamics

These explanations, while useful, do not provide a unified thermodynamic principle. The Theory of Entropicity (ToE) fills this gap.

Theory of Entropicity (ToE) Perspective

The Theory of Entropicity (ToE) posits that entropy is not merely a scalar quantity but a dynamical field that underlies all physical interactions. Within this framework, heat flow, phase transitions, and even information processing are governed by gradients and flows of this entropic field. ToE incorporates:

Irreversibility and time asymmetry

Entropic inertia and stiffness

Finite-time interaction constraints (No-Rush Theorem)

Entropic non-Markovianity

Applying these ideas to the Mpemba Effect leads to a coherent and elegant explanation.

Entropic Field and Action

The entropic field obeys an action principle derived from the Master Entropic Equation (MEE). A simplified form of the action relevant to thermal processes is:

[math]\displaystyle{ \mathcal{S}[S] = \int d^4x\,\sqrt{-g}\left[ -\frac{1}{2} A(S)\, g^{\mu\nu} \partial_\mu S\, \partial_\nu S - V(S) + \eta S T^\mu{}_\mu \right], }[/math]

where:

[math]\displaystyle{ A(S) }[/math] governs the kinetic behavior ("entropic stiffness").

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

[math]\displaystyle{ \eta }[/math] is the trace of the matter stress-energy tensor.

Variation of the terms yields the entropic field equation:

[math]\displaystyle{ A(S) \Box S + \frac{1}{2} A'(S) (\partial S)^2 - V'(S) + \eta T^\mu{}_{\mu} = 0. }[/math]

In thermal systems like water, the entropy field gradients drive thermal and phase transition dynamics.

Mechanisms of the Mpemba Effect in the Theory of Entropicity (ToE)

==== (i) Entropic Gradient Drives Faster Relaxation ==== Hot water has a higher entropy content and therefore a steeper entropic gradient with respect to its environment. This gradient drives a faster entropy flux into the surroundings. The entropy field responds dynamically:

[math]\displaystyle{ \frac{\partial S}{\partial t} \propto -\nabla \cdot (A(S) \nabla S), }[/math]which leads to a more rapid reconfiguration of the system into lower-entropy states (like ice).

(ii) Entropic Inertia and Irreversibility

From ToE, every entropic system possesses a kind of entropic inertia—a resistance to changes in entropy. However, for systems with high entropy (hot water), this inertia couples strongly to irreversible flow:

[math]\displaystyle{ \mathcal{S}_{\text{irr}}[S] = \int dt\, \frac{\gamma}{2} \left(\frac{dS}{dt}\right)^2, }[/math]where is an irreversibility coefficient. The initial conditions of the hot system prepare it for a faster entropic descent, surpassing the slower evolution of colder water.

(iii) Entropic Memory and Non-Markovianity

ToE posits that systems retain a form of memory—encoded in the entropy field. The evolution of water is not Markovian; it depends on the entropic path taken.

[math]\displaystyle{ \frac{d\rho(t)}{dt} = \int_0^t K(t-s)\,\rho(s)\,ds, }[/math]with the entropic memory kernel. Hot water has more entropic momentum and shorter delay times (as per the No-Rush Theorem), allowing it to traverse phase space more quickly.

(iv) Reduced Entropic Constraints

Hot water holds fewer dissolved gases, reducing the internal configurational entropy. This results in a lower entropic barrier to freezing:

[math]\displaystyle{ \Delta S_{\text{freeze}}^{\text{hot}} \lt \Delta S_{\text{freeze}}^{\text{cold}}. }[/math]With fewer microstates to reorder, the entropic collapse into the solid phase proceeds faster.

(v) Obidi’s No-Rush Theorem

According to this No-Rush Theorem, no physical interaction occurs instantaneously; all changes require finite entropic delay:

[math]\displaystyle{ \Delta t_{\text{min}} \propto \frac{1}{\nabla S}. }[/math]Since hot water has a higher , its is shorter—allowing interactions to occur more rapidly, facilitating faster freezing.

Implications and Future Research in the Theory of Entropicity (ToE) on the Mpemba Effect

The ToE explanation of the Mpemba effect transcends conventional thermal models. It introduces:

A unified field-theoretic explanation for nonequilibrium thermodynamics.

Predictive power for other counterintuitive thermal processes.

A bridge between physical entropy and computational processes (e.g., in AI).

Future work may involve:

Quantifying and for specific water samples.

Modeling freezing front dynamics under varying profiles.

Investigating entropic hysteresis loops and their measurable signatures.

See Also

Theory of Entropicity

Non-Markovian Dynamics

Irreversibility in Thermodynamics

No-Rush Theorem

References

E. B. Mpemba and D. G. Osborne, "Cool?" Physics Education 4, 172–175 (1969).

J. O. Obidi, "Master Entropic Equation and the Theory of Entropicity," (2025).

V. V. Vlasov, "Memory kernels in statistical mechanics," J. Stat. Phys. 98, 343–358 (2000).

L. D. Landau and E. M. Lifshitz, "Statistical Physics" (Course of Theoretical Physics, Vol. 5).

External Links

Wikipedia: Mpemba Effect

Mpemba articles at Journal of Thermodynamics