# A Noether paradigm for thermodynamics

New paper posted on the arXiv:

M. Polettini, G. Bulnes Cuetara, and M. Esposito, Bridging stochastic and macroscopic thermodynamics, arXiv:1602.06555.

The idea is very simple, yet we find it important to discuss this issue explicitly. Old-style, phenomenological thermodynamics (PT) is a nightmare for students, who get lost in a mess of thermodynamic potentials, inexact differentials, vague formulations of universal principles, and phenomenological laws that apply to very specific systems (i.e. ideal gases). However, today Nonequilibrium Statistical Mechanics provides us with a powerful toolbox to build a consistent and comprehensive theory of thermodynamics based on Markov processes (in particular I am fond of Markov processes on networks). Let’s call this theory Stochastic Thermodynamics (ST).

Time is ripe for making ST the way thermodynamics is taught at undergraduate level. However, before we can actually do that legitimately, we should do an effort to re-obtain all of PT from the new formulation. While in principle the ingredients are all in place, much work is yet to do be done. An obvious example of the distance between ST and PT is the fact that the latter is formulated in physical space, and the former in configuration space (our network). In the latter there are several conservation laws of physical quantities such as energy, number of particles, charge, etc. while in the former there is only one conserved quantity: probability. Moreover, in the latter there are a certain number of thermodynamic forces conjugate to thermodynamic fluxes; in the former the fundamental thermodynamic forces are given by Kirchhoff’s Loop Law along all possible cycles in configuration space. The number of such cycles might be huge.

Hence, there must be a mapping from configuration space to physical space, that maps ST to PT in a consistent way, and which is responsible of the emergence of physical conservation laws, and at the same time of cutting down the number of physical forces because there are certain symmetries at work. In this paper we discuss the origin and structure of this linear map. In particular, we find that there is a balance between the number $\mathrm{sym}$ of symmetries of the thermodynamic forces at the stochastic level, and the number $\mathrm{cons}$ of conservation laws at the phenomenological level:

$\mathrm{sym} + \mathrm{cons} = \mathrm{fixed}$

where $\mathrm{fixed}$ is a fixed topological number. I like to think of this relation as revealing of a sort of Noether theorem in thermodynamics: in particular, if by changing the system parameters one macroscopic conservation law is obtained, then this implies that one additional microscopic symmetry must be verified. Of course this is very hand-waving, so more inspection of the actual relationship with the original Noether theorem for conservative systems is under scrutiny. And by the way, check out another formulation for a Noether theorem for stochastic systems:

J. C. Baez, B. Fong, A Noether Theorem for Markov Processes, arXiv:1203.2035

This is quite different from what we have in mind, as the Noether charge here is precisely the total probability.