Anomalous Scaling of Dynamical Large Deviations

Abstract

The typical values and fluctuations of time-integrated observables of nonequilibrium processes driven in steady states are known to be characterized by large deviation functions, generalizing the entropy and free energy to nonequilibrium systems. The definition of these functions involves a scaling limit, similar to the thermodynamic limit, in which the integration time τ appears linearly, unless the process considered has long-range correlations, in which case τ is generally replaced by τ^{ξ} with ξ≠1. Here, we show that such an anomalous power-law scaling in time of large deviations can also arise without long-range correlations in Markovian processes as simple as the Langevin equation. We describe the mechanism underlying this scaling using path integrals and discuss its physical consequences for more general processes.

Anomalous scaling of dynamical large deviations of stationary Gaussian processes

“…The action, calculated from Eq. (22), conforms to the general scaling form (10) and coincides with the action found by NT [9].…”

“…which is nothing but the Euler-Lagrange equation for the action (30), with the constraintx n = a accommodated via a Lagrange multiplier [9]. NT determined the action, corresponding to the (zero-energy) homoclinic solution of Eq.…”

“…[8] for an accessible review. Remarkably, Nickelsen and Touchette (NT) [9] have recently observed that the scaling behavior of ln P(a, T ) can be anomalous. For large T and large a they obtained − ln P(a → ∞, T → ∞) T ξ f (a),…”

“…As in Ref. [9], we will employ the OFM which correctly predicts the large-a asymptotic of − ln P(a, T ) at any fixed V. For non-Markov processes, that we are interested in, the OFM minimization procedure will lead us to a non-local theory, in contrast to the local "classical mechanics" of Ref. [9].…”

Packets of Diffusing Particles Exhibit Universal Exponential Tails

“…Let us first stress out why, mathematically speaking, the numerously observed exponential decay is unexpected from the stand point of a regular random walk and standard Large Deviation approach [20][21][22][23][24][25]. The random walk definition is as follows, at each step a particle can perform a step of size x, while the PDF of x is given by f (x).…”