Effective thermodynamics of two interacting underdamped Brownian particles

Starting from the stochastic thermodynamics description of two coupled underdamped Brownian particles, we showcase and compare three different coarse-graining schemes leading to an effective thermodynamic description for the first of the two particles: marginalization over one particle, bipartite structure with information flows and the Hamiltonian of mean force formalism. In the limit of time-scale separation where the second particle with a fast relaxation time scale locally equilibrates with respect to the coordinates of the first slowly relaxing particle, the effective thermodynamics resulting from the first and third approach are shown to capture the full thermodynamics and to coincide with each other. In the bipartite approach, the slow part does not, in general, allow for an exact thermodynamic description as the entropic exchange between the particles is ignored. Physically, the second particle effectively becomes part of the heat reservoir. In the limit where the second particle becomes heavy and thus deterministic, the effective thermodynamics of the first two coarse-graining methods coincides with the full one. The Hamiltonian of mean force formalism however is shown to be incompatible with that limit. Physically, the second particle becomes a work source. These theoretical results are illustrated using an exactly solvable harmonic model.Comment: 18 pages, 4 figure

Stochastic Thermodynamics for Underdamped Brownian Particles: Equivalent Measures, Reversed Stochastic Processes and Feynman-Kac Techniques

Underdamped stochastic thermodynamics provides a handy tool to study a large class of stochastic processes operating out of equilibrium. Colloidal particles in a laser trap, molecular motors and feedback processes are some of the prominent examples. In the present work we give a mathematical framework for the study of the thermodynamic properties of these phenomena. We focus on Markovian stochastic processes in continuous time and space, and show how the techniques of equivalent measures combined with stochastic solutions of partial differential equations, obtained through Feynman-Kac formula, can be used to derive exact relations between forward and backward diffusion processes. We prove a theorem which allows us to derive the time evolution of an arbitrary path quantity in a simple and systematic way. We further consider a fairly general underdamped stochastic model, and study its nonequilibrium thermodynamic properties at both single trajectory and average levels. For this model, we establish several integral and detailed fluctuation theorems for thermodynamic quantities such as work and entropy production, amongst others. Some of these theorems directly parallel those already obtained in the context of overdamped and master equations, while others are novel. We also discuss some special cases of our model which are directly related to physical systems such as active Brownian particles, feedback processes and isoenergetic stochastic processes. The formalism we develop, and the general model considered here constitute a unified and extended framework for the study of the thermodynamics of underdamped processes, encompassing several physical systems and applications