Understanding entropy production via a thermal zero-player game

TL;DR

Problem: understanding entropy production bounds in driven nonequilibrium many-body systems. Approach: introduce ICEg, a self-driven lattice game coupled to a thermal bath, and define a discrete entropy surrogate $S(t)$ and relative EP $S_{prod}$; analyze with Metropolis and Glauber dynamics and perform coupled finite-size scaling with $EP(\beta,N,M)=N^c M^d f(u)$, $u=\beta N^a M^b$, where $f(u)$ is chosen as $f(u)=Au+B$. Findings: observe a transition to an entropic regime and a universal bound on the EP rate that is independent of $\beta$ and lattice size; data collapse supports universality; Glauber dynamics shows stronger temperature dependence than Metropolis. Significance: ICEg provides a physically plausible test bed for stochastic thermodynamics and game-theoretic dynamics, enabling numeric exploration of entropy production bounds and potential generalization to broader self-driven stochastic systems.

Abstract

Understanding the natural bounds of entropy production for driven nonequi- librium dynamics in many-body systems reveals how the fundamentals of thermodynamics manifest in these regimes across a wide variety of systems. In this direction, we propose and study the dynamics of a thermal zero-player entropy game, the Ising-Conway Entropy Game (ICEg), a self-driven system exhibiting characteristics of lattice gases, Ising models, and discrete games. We show that there is a universal bound on the entropy production rate, independent of temperature and lattice size. The thermalized game is shown to be physically interesting and a plausible testbed for studying the fundamentals of stochastic thermodynamics.

Figures (5)

• Figure 1: An example evolution of the game is visualized for $N=50$, $M=10$ at $\beta=1.0$. Each row represents a time window along the x-axis, with the states of the sites arranged along the y-axis. The next row continues the evolution from the previous row. (a) Glauber Dynamics (b) Metropolis Dynamics
• Figure 2: (a) Evolution of the entropy measure for $N=50$, $M=10$ with Glauber dynamics and different temperatures, including standard errors. (b) Evolution of the entropy measure for $N=50$, $M=10$ with Metropolis dynamics and different temperatures, including standard errors.
• Figure 3: (a) Evolution of entropy measure for $N=40$, $M=10$ with Glauber dynamics and different temperatures with standard errors. (b) Evolution of entropy measure for $N=40$, $M=10$ with Metropolis dynamics and different temperatures with standard errors.
• Figure 4: Entropy production with respect to lowest temperature over range of temperatures and different settings with standard errors. (a) EP for Metropolis dynamics (b) EP for Glauber dynamics (c) Cumulative Distribution Function (CDF) for Metropolis and Glauber EPS, showing statistically significant difference.
• Figure 5: Entropy production finite size scaling analysis: (a) Metropolis dynamics (b) Glauber Dynamics