Direct Evaluation of Large-Deviation Functions

Abstract

We introduce a numerical procedure to evaluate directly the probabilities of large deviations of physical quantities, such as current or density, that are local in time. The large-deviation functions are given in terms of the typical properties of a modified dynamics, and since they no longer involve rare events, can be evaluated efficiently and over a wider ranges of values. We illustrate the method with the current fluctuations of the Totally Asymmetric Exclusion Process and with the work distribution of a driven Lorentz gas.

Renormalized time scale for anticipating and lagging synchronization

“…Moreover, the Euler-Lagrange equation (2.22) implies that 27) that is, we also obtain the Euler-Lagrange equation corresponding to the original Lagrangian L. The definition of p j in eq. (2.26) guarantees that p j = ∂L/∂j, as in the case of the usual variational problem with a Lagrangian including only first-order derivatives.…”

“…3, where the dissipation distribution has been measured directly but we are unable to gather enough statistics in the tails of the pdf to obtain clear-cut results in the non-gaussian regime. A recent series of works have addressed this issue, developing an efficient method to measure directly LDFs in many particle systems [27][28][29]. The method is based on a modification of the dynamics so that the rare events responsible of the large deviation are no longer rare [27], and it has been developed for discrete- [27] and continuous-time Markov dynamics [28].…”

“…In particular, if U C C is the transition rate from configuration C to C of the associated stochastic process, the modified dynamics is defined asŨ C C (s) = U C C exp(s d C C ), where d C C is the energy dissipated in the elementary transition C → C . It can be then shown [11,[27][28][29] that the natural logarithm of the largest eigenvalue of matrixŨ (s) gives µ(s), which in turn can be Legendre-transformed back to obtain a Monte Carlo estimate of G(d). The method of refs.…”

“…The method of refs. [27][28][29] thus provides a way to measure µ(s) by evolving a large number M of copies or clones of the system using the modified dynamicsŨ (s). This method is exact in the limit M → ∞, but in practice we are able to simulate a large but finite population of clones, typically M ∈ [10 3 , 10 4 ].…”

“…The chances are that our results remain valid for more complex dissipative media described at the mesoscopic level by a similar evolution equation. Here we report analytical and simulation results for the statistics of the dissipated energy in this general class of models using both standard simulations and an advanced Monte Carlo method [27]. The latter allows the sampling of the tails of the distribution, and implies simulating a large number of clones of the system.…”