Non-equilibrium thermodynamics as gauge fixing

So Katagiri
Graduate School of Arts and Sciences, The Open University of Japan, Chiba, Japan
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So Katagiri
Progress of Theoretical and Experimental Physics, Volume 2018, Issue 9, 1 September 2018, 093A02, https://doi.org/10.1093/ptep/pty102

1. Introduction

Onsager’s theory is the most important one in non-equilibrium thermodynamics with linear constitutive equations [1,2], in which constitutive equations for currents are derived from the minimum energy dissipation principle. Later on, this argument was supported by the path-integral representation of the probability distribution [37]. Onsager’s theory holds in the case of linear constitutive equations, but it is not well understood in the non-linear case (for the latest research see Refs. [8,9]). Recently, Sugamoto, along with his collaborators, including the present author, pointed out that the thermodynamic force can be viewed as a gauge field (A. Sugamoto, private communication and Ref. [10]).

In this paper we discuss this statement more definitely by means of gauge fixing, and derive a non-linear constitutive equation by adding the free action of the usual electromagnetism.

The paper contains the following sections. In the next section, we review the electromagnetism in a pure gauge as an useful analogy in later discussions. In Sect. 3, non-equilibrium thermodynamics is introduced using a differential form. In Sect. 4, we examine the gauge properties of the thermodynamic force. We extend the thermodynamic force to a thermodynamical gauge field in Sect. 5. In Sect. 6, we discuss this gauge theory in the path-integral method. The final section is devoted to the discussion.

In addition, the paper contains the following appendices. In Appendix A, the meaning of the time dependence of S(a,t) is discussed. In Appendix B, we examine the restriction from the second law of thermodynamics. In Appendix C, we discuss how non-linear effects work in dimensional analysis. In Appendix D, a simple example is derived in our model.