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# 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

Onsagers 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]. Onsagers 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 $S\left(a,t\right)$ 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.

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