Thermodynamic derivation of reciprocal relations

TL;DR

This paper challenges the traditional statistical basis of reciprocal relations in thermodynamics, proposing a nonstatistical perspective that derives these relations from fundamental mathematical theorems applicable to all systems and states.

Contribution

It introduces a nonstatistical thermodynamic framework and derives reciprocal relations from mathematical principles rather than statistical fluctuations or microscopic reversibility.

Findings

Reciprocal relations follow from the symmetry of second derivatives of analytic functions.

Thermodynamics applies universally, including microscopic and non-equilibrium systems.

Entropy is a well-defined, intrinsic property in this framework.

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.