Aging behaviors of Brownian particle coupled to finite heat bath
Non-equilibrium dynamics and stochastic thermodynamics have emerged as prominent areas of research in recent years,particularly focusing on small systems.These systems often exhibit unique properties due to memory effects and finite-scale considerations,which present challenges within the traditional theoretical framework.The system-plus-heat bath model has been utilized as a phenomenological and intuitive approach for investigating complex multi-body problems.However,there is currently a dearth of research on the dynamics of a particle coupled to a finite-size reservoir.This study aims to fill this gap by investigating the nonergodic and nonequilibrium behaviors of a particle.Specifically,the particle is in contact with one end of a finite coupled os-cillator chain through a harmonic potential.The dynamics of the particle,confined to an external potential,are precisely analyzed using a Hamiltonian microscopic map within the framework of the generalized Langevin equation.It is important to note that an additional harmonic force with a certain frequency only exists in a finite-size heat bath chain and can be disregarded for a sufficiently large number of coupled oscillators.The second moments of the particle's coordinates and velocities are obtained through the inversion of the Laplace transform and the utilization of the fluctuation-dissipation theorem.Furthermore,the long-time asymptotic behaviors of the particle are examined using the final-value theorem.The movement of the particle confined to a nonlinear potential is also simulated through the implementation of the fourth-order stochastic Runge-Kutta algorithm.Notably,it is observed that the steady variance of the particle's coordinate depends on its initial coordinate rather than its initial velocity,dem-onstrating a kind of nonergodic motion.Additionally,the study reveals that nonergodicity and nonequilibrium are distinct concepts,indicating that nonequilibrium does not necessarily imply nonergodicity.The behavioral evolution and establishment of stationary production are contingent upon the particle's initial coordinate rather than its initial velocity.Moreover,the shapes of the stationary probability density function solely depend on the particle's initial coordinate,specifically the first and second moments.Equilibrium is solely attained when the initial distribution of the particle adheres to a Gibbs-Boltzmann distribution.If the particle is not initially thermally equilibrated,it will never reach an equilibrium state due to its inherent nonergodicity.These findings,along with future research in this area,are expected to enhance our understanding of the intricate relationships between nonergodicity and nonequilibrium phenomena.
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