Classical Physics Paradox Solved Without Relying on Quantum Assumptions

Scientists have long grappled with the Gibbs paradox, a fundamental issue in classical statistical mechanics traditionally addressed by the 1/N!. correction to account for particle indistinguishability. Zheng Zhang from Lanzhou University of Technology and Zheng Zhang from The University of Hong Kong, with colleagues, now offer a novel resolution of this paradox within a purely classical framework, bypassing the need for this conventional correction. Their research, detailed in a recent letter, demonstrates that the paradox dissolves when considering only the equal probability principle and reinterpreting Gibbs entropy as Shannon entropy, a measure of our lack of information rather than system disorder. This informational perspective not only resolves the paradox but also clarifies the relationship between entropy and extractable work during gas mixing, potentially reshaping our understanding of entropy’s role in statistical mechanics.

This work presents a novel resolution relying solely on the equal probability principle inherent in classical ensemble theory, eliminating the need for the 1/N. correction.

The researchers interpret the Gibbs entropy as Shannon entropy, quantifying ignorance rather than disorder, offering a fundamentally informational perspective on the paradox. This approach clarifies the connection between information and extractable work during gas mixing processes, demonstrating that the apparent paradox arises from a misunderstanding of information loss.
The Gibbs paradox emerges when applying classical ensemble theory to ideal gases, resulting in a non-extensive entropy that leads to paradoxical consequences when considering the mixing of identical gases. Traditionally, this has been resolved by incorporating quantum mechanics and the 1/N. correction to restore entropy extensivity.

However, this study demonstrates that maintaining the equal probability principle rigorously avoids the paradox, even in regimes where quantum effects are negligible. The research challenges the conventional belief that non-extensivity of entropy causes the paradox, instead highlighting the importance of strictly adhering to the foundational principles of classical statistics.

This resolution is distinct from previous attempts, as it avoids complicated arguments or subtle principles, offering a straightforward and simple solution. By adopting a purely informational perspective, the researchers reinterpret the Gibbs entropy as a measure of our lack of knowledge about the system’s microscopic state.

This interpretation naturally explains the resolution and reveals a crucial link between information and work in gas mixing. The study finds that entropy changes during mixing depend directly on whether information is lost in the process, aligning with principles of information thermodynamics. Furthermore, this work suggests a paradigm shift in statistical mechanics, prompting a re-evaluation of fundamental concepts like entropy and information.

The researchers believe their informational perspective extends beyond the Gibbs paradox, offering a new framework for understanding statistical systems. By explicitly linking entropy to Shannon entropy, the study provides a deeper understanding of the role of information in determining thermodynamic behaviour and opens avenues for exploring the interplay between information, work, and entropy in various physical processes.

Entropy maximisation and derivation of the microcanonical and grand canonical distributions

A central technique employed within this work involves maximizing entropy to resolve the Gibbs paradox within a classical ensemble framework. The research begins by defining entropy, S = −k ∫ ρ(p, q) ln ρ(p, q) dpdq, and establishing constraints based on available information about the system. Specifically, when energy E, volume V, and particle number N are known, the study maximizes entropy subject to the condition that the integral of E(p,q) multiplied by ρ(p, q) over all phase space equals E.

To determine the stationary point for maximum entropy, a Lagrangian, L = S −λg, was constructed and minimized with respect to ρ. This process yielded the equation ln ρ(p, q) + 1 + λk = 0, leading to the microcanonical distribution ρ(p, q) = C, where C is a constant determined by normalization. Extending this approach, the study considered scenarios with average energy ⟨E⟩ instead of fixed energy, again maximizing entropy but now constrained by both the integral of ρ(p, q) equalling one and the integral of E(p, q)ρ(p, q) equalling ⟨E⟩.

The resulting stationary point led to the canonical distribution ρ(p, q) = e−βE(p,q) Z, where Z is the partition function and β is determined by the constraints. Further generalizing this, the research addressed systems with known average particle number ⟨N⟩, maximizing entropy subject to constraints on the integral of ρ(p, q), the integral of E(p, q)ρ(p, q), and the integral of NρN(p, q).

This ultimately produced the grandcanonical distribution ρN(p, q) = e−βE(p,q)−αN Ξ, with Ξ, β, and α determined by the imposed constraints. To specifically address the Gibbs paradox, the study introduced an additional constraint representing extra information about particle distribution within a divided box.

The box was conceptually split into two equal halves, each containing exactly N particles, imposing a constraint on the phase space accessible to the particles. Applying the maximum entropy principle with this added constraint, the research demonstrated that the Gibbs paradox could be resolved, yielding a distribution consistent with the canonical ensemble and avoiding the conventional 1/N. correction.

Resolving the Gibbs paradox via Shannon entropy and the equal probability principle

Calculations reveal an entropy increase of 2Nk ln 2 in gas mixing processes, a result traditionally addressed by invoking the 1/N. correction to account for particle indistinguishability. This work presents a resolution to the Gibbs paradox within the classical ensemble, relying solely on the equal probability principle and eliminating the need for the 1/N. correction.

The research interprets Gibbs entropy as Shannon entropy, quantifying ignorance rather than disorder, and clarifies the connection between entropy and extractable work during gas mixing. The study considers a system initially divided by a wall into two halves, each containing N particles, with both sides occupying volume V.

When the wall is removed, the system equilibrates to a state with 2N particles in volume 2V, resulting in an entropy increase quantified as 2Nk ln 2. This increase, when applied to gases of different types, is understood as mixing entropy, but appears paradoxical when applied to identical gases. The research demonstrates that the classical ensemble theory is sufficient to predict the correct thermodynamic entropy without requiring the conventional 1/N. correction to the partition function.

Analysis of the gas mixing process involving two different gas types, labelled A and B, reveals a total partition function Z, calculated by integrating over phase space coordinates and momenta. This calculation yields a total entropy before mixing, S(1)t, equivalent to 2Sid(N, V, T), consistent with previous formulations.

After mixing, the total entropy S(2)t remains defined by Sid(2N, 2V, T), again resulting in an entropy change of 2Nk ln 2, interpretable as mixing entropy between different gases. When considering the case of identical gases, the research highlights that particles are distinguishable within classical statistics.

Before removing the wall, the phase space is divided into (2N)./(N.)2 disconnected regions, each representing a possible particle partition. Applying the equal probability principle across these regions leads to a total partition function Z, expressed as (2N)./(N.)2 multiplied by the original partition function Z0. This approach avoids the need for the 1/N. factor and offers a concise explanation within the framework of classical statistical mechanics.

Entropy’s informational basis resolves the classical Gibbs paradox

Scientists have resolved the Gibbs paradox within classical statistical mechanics without relying on the conventional indistinguishability correction factor of 1/N. This resolution stems from applying the equal probability principle and interpreting Gibbs entropy as Shannon entropy, which quantifies a lack of information rather than disorder.

The approach clarifies the relationship between entropy and the potential to extract work during gas mixing processes, offering a new perspective on the fundamental role of entropy in statistical mechanics. This work demonstrates that the original formulation of classical ensemble theory does not inherently produce the Gibbs paradox, suggesting a historical oversight in its interpretation.

The key to this understanding lies in adopting an informational perspective, where entropy’s non-additivity becomes understandable as a consequence of limited knowledge. An equivalent resolution can also be achieved using the maximum entropy principle, further supporting the validity of this informational approach.

The authors acknowledge that defining entropy based on information introduces a degree of subjectivity into thermodynamics, but emphasize that information possesses objective physical consequences, notably in determining the amount of controllable work obtainable from a system. This research signifies a potential paradigm shift, positioning information as a central concept within statistical mechanics and opening possibilities for future applications. Further investigation into the composition law of Shannon entropy, detailed in the appendix, reinforces the theoretical foundation of this informational interpretation.