Nonequilibrium fluctuations for linear diffusion dynamics

Abstract

We present the theoretical study on non-equilibrium (NEQ) fluctuations for diffusion dynamics in high dimensions driven by a linear drift force. We consider a general situation in which NEQ is caused by two conditions: (i) drift force not derivable from a potential function and (ii) diffusion matrix not proportional to the unit matrix, implying non-identical and correlated multi-dimensional noise. The former is a well-known NEQ source and the latter can be realized in the presence of multiple heat reservoirs or multiple noise sources. We develop a statistical mechanical theory based on generalized thermodynamic quantities such as energy, work, and heat. The NEQ fluctuation theorems are reproduced successfully. We also find the time-dependent probability distribution function exactly as well as the NEQ work production distribution P (W) in terms of solutions of nonlinear differential equations. In addition, we compute low-order cumulants of the NEQ work production explicitly. In two dimensions, we carry out numerical simulations to check out our analytic results and also to get P (W). We find an interesting dynamic phase transition in the exponential tail shape of P (W), associated with a singularity found in solutions of the nonlinear differential equation. Finally, we discuss possible realizations in experiments.

Stochastic echo phenomena in nonequilibrium systems

“…A conservative force is given by a gradient of a potential function V (x) as f c (x) = −∇ x V (x), while a nonconservative force does not have a potential function. A nonconservative force drives a system out of equilibrium [6][7][8]. Along the line of Eq.…”

“…Suppose that the system follows a path x(t) under a protocol λ(t). The total entropy production along the path is related to the irreversibility as [6,8,10] …”

“…When a thermodynamic system is exerted by a timedependent force or a nonconservative force, it is driven into a nonequilibrium state [6][7][8]. We will represent a nonequilibrium driving with a protocol λ(t), which may depend on time or not.…”

“…So, a path x(t) under a protocol λ(t) and its time-reversed path x R (t) under a time-reversed protocol λ R (t) ≡ λ(τ − t) are not equally probable. In fact, the difference in the statistical weights between them is directly related to the entropy production, which is the basis for the various nonequilibrium fluctuation theorems in stochastic thermodynamics [6][7][8][9][10][11][12][13][14].…”

Modified saddle-point integral near a singularity for the large deviation function

“…A typical consequence is the existence of nonzero current in state space. It has been noticed that nonzero current accompanies an incessant production of work, hence heat and entropy [1]- [9], each of which satisfies the fluctuation theorem (FT) given at specific initial distributions [10]- [15]. Such timeintegrated quantities exhibit rare but huge fluctuations which are prominent in small systems.…”