Abstract
Computing the stochastic entropy production associated with the evolution of a stochastic dynamical system is a well-established problem. In a small number of cases such as the Ornstein–Uhlenbeck process, of which we give a complete exposition, the distribution of entropy production can be obtained analytically. For a general potential it is much harder. A recent development in solving the Fokker–Planck equation, in which the solution is written as a product of positive functions, addresses any system governed by the condition of detailed balance, thereby permitting nonlinear potentials. Using examples in one and higher dimension, we demonstrate how such a framework is very convenient for the computation of stochastic entropy production in diffusion processes.
Highlights
Computing the stochastic entropy production associated with the evolution of a stochastic dynamical system is a well-established problem
The notion of the production of entropy as physical systems evolve dates from the time of Boltzmann and Gibbs, and underpins basic ideas of thermodynamics and statistical mechanics, including the celebrated second law of thermodynamics describing the irreversibility of events on the macroscale [1]
This paper is, the rst application in physics of the new method for approximately solving Fokker–Planck equations, and we apply it to the problem of stochastic entropy production in a range of different diffusive systems, all corresponding to the spreading of probability density from an earlier point source, but with different force elds and different stationary states
Summary
The notion of the production of entropy as physical systems evolve dates from the time of Boltzmann and Gibbs, and underpins basic ideas of thermodynamics and statistical mechanics, including the celebrated second law of thermodynamics describing the irreversibility of events on the macroscale [1]. In a recent paper [26], a new method of dealing with Fokker–Planck PDEs was developed, in which the general thesis is that one does better to write the solution as a product of terms, all of which are positive This naturally models the logarithm of the phase space density, so it obviates the dif culties described above, and is potentially very suitable for dealing with problems that pertain to entropy production. This paper is, the rst application in physics of the new method for approximately solving Fokker–Planck equations, and we apply it to the problem of stochastic entropy production in a range of different diffusive systems, all corresponding to the spreading of probability density from an earlier point source, but with different force elds and different stationary states (see gure 3) Our nal section (section 5) gives our conclusions and suggests opportunities for further research
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