Large deviations in boundary-driven systems: Numerical evaluation and effective large-scale behavior

Abstract

We study rare events in systems of diffusive fields driven out of equilibrium by the boundaries. We present a numerical technique and use it to calculate the probabilities of rare events in one and two dimensions. Using this technique, we show that the probability density of a slowly varying configuration can be captured with a small number of long wave-length modes. For a configuration which varies rapidly in space this description can be complemented by a local equilibrium assumption.PACS numbers: 05.70.Ln In many cases the typical size of fluctuations in a physical system with N degrees of freedom is of order 1/ √ N . Larger fluctuations are rare, and their probability scales as, where φ is an intensive function of the state ρ. The function φ [ρ] is known as the large-deviation function (LDF) and is of fundamental interest in statistical mechanics. In equilibrium systems, φ is equal to the free-energy density. Away from equilibrium, a simple expression for φ is in general not known, and it may be affected by details of the system's dynamics. Besides its fundamental interest for non-equilibrium physics, the function φ is important in various applications, e.g. for calculating escape rates from metastable states, with applications ranging from chemistry and population dynamics to cosmology [1][2][3][4][5][6]9].In a non-equilibrium steady-state, to compute the probability of a rare-event, one must calculate the dynamics leading up to that event [10]. This is in general a difficult task, even more so for spatially extended systems, where only a handful of analytical solutions exist [11]. If a general understanding is to emerge, additional methods beyond exact solutions need to be considered. Indeed, recent years have seen a considerable effort to develop numerical techniques to calculate the LDF in a variety of systems [12][13][14][15][16].In this Letter, we study the LDF in bulk-conserving diffusive systems, which are driven out of equilibrium by the boundaries. These describe a broad range of transport phenomena, including electronic systems, ionic conductors, and heat conduction [17,18]. We show that the LDF in such systems can be efficiently evaluated numerically for a general interacting system in one and two dimensions, giving us access to previously unavailable information. This is done by searching for the most probable history ρ (x, t) of the conserved density function ρ leading to a rare state ρ f (x). Importantly, using the numerical technique we show that for many non-trivial cases, the LDF of a slowly varying configuration ρ f (x) can be calculated by considering only histories ρ (x, t) which are slowly varying in space, i.e. which are given by the sum of only a few long wave-length modes. This implies that the long wave-length structure of the LDF can be understood using an effective finite-dimensional theory, instead of the full infinite dimensional one. In addition, we find that a local equilibrium assumption can capture much of the short wave length structure. This could suggest a simple fram...

Large Deviations in Single-File Diffusion

“…For arbitrary σ(ρ) and D(ρ), the governing equations (7a)-(7b) are intractable, so one has to resort to numerical methods [35,42]. For small values of λ, however, a perturbative expansion of p(x, t) and q(x, t) with respect to λ can be performed [35].…”

Tagged Particle in Single-File Diffusion

“…The MFT provides a significant step towards constructing a general theoretical framework for non-equilibrium systems [44,53]. Over the past decade the MFT has been successfully applied to numerous systems [46][47][48][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70]. A perfect agreement between the MFT and microscopic calculations has been observed whenever results from both approaches were available.…”

Cusp Singularities in Boundary-Driven Diffusive Systems

“…This is now possible using the numerical technique described in [13], and used in the present paper.…”

“…4 appear to be quite smooth in x; this is sensible, as the field is constantly diffusing, making enduring, sharp spatial gradients improbable. This was studied in [13]. The smoothness motivates us to introduce toy models with a finite number of degrees of freedom, which capture many of the essential features of the field models described above.…”

“…In spite of this, the properties of φ [ρ f ] remain poorly understood. Much of what is known is based on a handful of exact solutions for specific models [10][11][12], and numerical techniques [13]. In particular, it was recently realized that φ [ρ f ] can be non-differentiable [21].…”