Minimum Action Principle for Entropy Production Rate of Far-From-Equilibrium Systems

Abstract

The Boltzmann distribution connects the energetics of an equilibrium system with its statistical properties, and it is desirable to have a similar principle for non-equilibrium systems. Here, we derive a variational principle for the entropy production rate (EPR) of far-from-equilibrium discrete state systems, relating it to the action for the transition probability measure of discrete state processes. This principle leads to a tighter, non-quadratic formulation of the dissipation function, speed limits, the thermodynamic-kinetic uncertainty relation, the large deviation rate functional, and the fluctuation relation, all within a unified framework of the thermodynamic length. Additionally, the optimal control of non-conservative transition affinities using the underlying geodesic structure is explored, and the corresponding slow-driving and finite-time optimal driving exact protocols are analytically computed. We demonstrate that discontinuous endpoint jumps in optimal protocols are a generic, model-independent physical mechanism that reduces entropy production during finite-time driving of far-from-equilibrium systems.

Figures (1)

• Figure 1: (Leftmost panel) An illustration and example for a three-state (red-orange-blue) unicyclic graph. The magenta curved arrow denotes the control of the non-conservative part $F_{or}$ of the transition affinity $A_{or}$. (a) Lagrangian $\mathcal{L}[j, \chi]$ (\ref{['eq:doi_peliti_transition_lagrangian']}) for fixed $J_\gamma=3.5, T_\gamma=4$ (cyan) and $J_\gamma=2.5, T_\gamma=4$ (orange). The corresponding most likelihood transition affinity $\chi^*$ as a vertical dotted lines. (b) Comparison between exact $I = 2x\tanh^{-1}(x)$, dynamical $I_{D}= 2x\sinh^{-1}(x)$ and Gaussian $I_{G} = 2x^2$ rate functional, where, $x=J_\gamma/T_\gamma$ is the current precision. (c) $\mathcal{G}(F_\alpha): F_\alpha \to t$. Comparison between $\mathcal{G}^{cEQ}(F_\alpha)$, $\mathcal{G}^{fEQ}(F_\alpha)$ and $\mathcal{G}^{lin}(F_\alpha)$. For the fixed $v_{qs}$, the $F_\alpha^f - F_\alpha^i = 1$ is considered for the close-to-equilibrium $F_\alpha^i=1$ and far-from-equilibrium $F_\alpha^i = 3.5$ and the corresponding $\Delta \tau_c$ and $\Delta \tau_f$ are plotted. (d) The finite-time optimal protocol $\mathcal{G}_{\tau}$ is plotted for the different values of $\tau$ with the same initial and final value condition (shown by the dotted blue lines).