[Paper Review] Thermodynamic derivation of reciprocal relations

Elias P. Gyftopoulos|ArXiv.org|Jun 1, 2004

Advanced Thermodynamics and Statistical Mechanics7 references3 citations

TL;DR

This paper presents a thermodynamic derivation of reciprocal relations—such as those in Onsager's reciprocal relations—based solely on the principles of equilibrium thermodynamics, without relying on statistical mechanics or microscopic reversibility. It argues that these relations emerge naturally from the mathematical property that second mixed partial derivatives of a well-behaved thermodynamic potential are symmetric, thus providing a non-statistical foundation for the relations.

ABSTRACT

Reciprocal relations correlate fairly accurately a great variety of experimental results. Nevertheless, the concepts of statistical fluctuations, and microscopic reversibility - the bases of the accepted proof of the relations by Onsager - are illusory and faulty, and contradict the foundations of the science of thermodynamics. The definitions, postulates, and main theorems of thermodynamics are briefly presented. It is shown beyond a shadow of a doubt that thermodynamics is a nonstatistical science that applies to all systems (both macroscopic, and microscopic, including systems that consist either of only one structureless particle, or only one spin), to all states (both thermodynamic or stable equilibrium, and not stable equilibrium), and that includes entropy as a well defined, intrinsic, nonstatistical property of any system in any state, at any instant in time. In the light of this novel conception of thermodynamics, we find that reciprocal relations result from a well known mathematical theorem, to wit, given a well behaved analytic function of many variables then the second derivative of the function with respect to any two variables is independent of the order of differentiation, namely, whether the first derivative is taken with respect to the one or the other of the two variables.

Motivation & Objective

• To establish reciprocal relations in non-equilibrium thermodynamics without invoking statistical mechanics or microscopic reversibility.
• To demonstrate that reciprocal relations follow from the mathematical symmetry of second derivatives of thermodynamic potentials.
• To reaffirm thermodynamics as a non-statistical science applicable to all systems, including microscopic or single-particle systems.
• To resolve inconsistencies in the statistical mechanical derivation of reciprocal relations, particularly those based on fluctuation and reversibility concepts.
• To provide a rigorous, intrinsic thermodynamic basis for the Onsager reciprocal relations using only postulates of thermodynamics.

Proposed method

• Derives reciprocal relations from the mathematical property that mixed second partial derivatives of a smooth, analytic thermodynamic potential are symmetric, regardless of differentiation order.
• Applies this mathematical principle to thermodynamic potentials such as entropy, internal energy, or Gibbs free energy, assuming analyticity and differentiability.
• Uses the fundamental postulates of thermodynamics—especially the existence of entropy as an intrinsic, non-statistical property of any system in any state.
• Considers systems of any size, including single particles or spins, to show the generality of the derivation beyond macroscopic ensembles.
• Rejects statistical concepts like fluctuations and microscopic reversibility as incompatible with the foundations of thermodynamics.
• Establishes that the symmetry of second derivatives leads directly to the Onsager reciprocal relations without requiring stochastic or quantum mechanical assumptions.

Experimental results

Research questions

• RQ1 Can reciprocal relations in non-equilibrium thermodynamics be derived without appealing to statistical mechanics?
• RQ2 What is the mathematical origin of the symmetry in kinetic coefficients predicted by Onsager's relations?
• RQ3 How can thermodynamics be consistently applied to systems that are not in macroscopic equilibrium, including single-particle systems?
• RQ4 Are the concepts of microscopic reversibility and statistical fluctuations compatible with the foundational principles of thermodynamics?
• RQ5 Does the symmetry of second derivatives in thermodynamic potentials inherently imply reciprocal relations?

Key findings

• Reciprocal relations arise not from statistical mechanics but from the mathematical symmetry of second derivatives of analytic thermodynamic potentials.
• The derivation is valid for all systems, including microscopic or single-particle systems, and for all states, not only equilibrium ones.
• Entropy is established as an intrinsic, non-statistical property of any system in any state, at any instant.
• The foundations of thermodynamics are shown to be independent of statistical assumptions, contradicting the conventional statistical mechanical basis of Onsager's proof.
• The symmetry of mixed partial derivatives in thermodynamic functions directly yields the Onsager reciprocal relations without additional physical postulates.
• The paper concludes that the accepted statistical mechanical derivation of reciprocal relations is conceptually flawed and illusory.