Applicability of Jarzynski’s Equality in a Two-Dimensional Lattice Gas System: Numerical Simulation and Thermodynamic Analysis

Abstract

This thesis investigates the Jarzynski Equality in microscopic systems driven far from equilibrium states, based on key idea in stochastic thermodynamics. The study explores how fluctuations influence energy transfer, change in free energy, and irreversibility during finite-time transformations. To do this, we introduced a two dimensional piston lattice gas model, where interacting particles in thermal contact are confined by immobile walls and manipulated via a movable piston. The movement of the piston drives the system out of equilibrium, and simulations combining stochastic particle dynamics with deterministic piston behaviour under Markovian assumptions, establish a comprehensive framework for probing both equilibrium and nonequilibrium phenomena. Simulations were carried out across a wide range of system sizes (N = 5, 10, 20 & 30) and piston velocities (v = 0.05 to 1). For each scenario, ensemble averages of energy, work, and heat were computed, and the Jarzynski Equality was applied to obtain equilibrium free-energy differences from the nonequilibrium measurements. At slow piston speeds, the system remains close to equilibrium, and the Jarzynski Equality provides a description of the reversible free energy changes. At high speeds, higher the average work exceeds the free energy difference, indicating strong dissipation and irreversibility. Analysis of work distributions shows systematic broadening and asymmetry with increasing piston velocity and system size. These distributions reveal that, though average trajectories yield works values well above the free energy difference, rare low-work events are required to satisfy the exponential average stipulated by Jarzynski's relation. The limited occurrence of these trajectories under higher drive rates is responsible for the deviation of observed results from the equality of strongly nonequilibrium regimes. It is demonstrated that Jarzynski's Equality remains applicable for finite systems with appropriate sampling of the full statistical ensemble, including the complete set of rare fluctuations. The study also shows how driving speed, system size, and relaxation dynamics control entropy production and the crossover of dynamics from reversible to irreversible behaviour. In summary, this work connects microscopic stochastic dynamics with the macroscopic laws of thermodynamics. The piston lattice gas model developed here a computational framework to study nonequilibrium thermodynamics, fluctuation theorems, and the statistical basis of the second law. The findings confirm of recovering equilibrium properties from ensemble averaged microscopic behaviours even under nonequilibrium conditions and highlight the broad significance of Jarzynski's Equality in modern statistical physics.