A giant disparity and a dynamical phase transition in large deviations of the time-averaged size of stochastic populations

Abstract

We study large deviations of the time-averaged size of stochastic populations described by a continuous-time Markov jump process. When the expected population size $N$ in the steady state is large, the large deviation function (LDF) of the time-averaged population size can be evaluated by using a WKB (after Wentzel, Kramers and Brillouin) method, applied directly to the master equation for the Markov process. For a class of models that we identify, the direct WKB method predicts a giant disparity between the probabilities of observing an unusually small and an unusually large values of the time-averaged population size. The disparity results from a qualitative change in the "optimal" trajectory of the underlying classical mechanics problem. The direct WKB method also predicts, in the limit of $N\to \infty$, a singularity of the LDF, which can be interpreted as a second-order dynamical phase transition. The transition is smoothed at finite $N$, but the giant disparity remains. The smoothing effect is captured by the van-Kampen system size expansion of the exact master equation near the attracting fixed point of the underlying deterministic model. We describe the giant disparity at finite $N$ by developing a different variant of WKB method, which is applied in conjunction with the Donsker-Varadhan large-deviation formalism and involves subleading-order calculations in $1/N$.

arXiv comment10 pages, 7 figures

References in corpus (13)

• The large deviation approach to statistical mechanics
• Nonequilibrium Markov processes conditioned on large deviations
• Nonequilibrium microcanonical and canonical ensembles and their equivalence
• Extinction of metastable stochastic populations
• Introduction to dynamical large deviations of Markov processes
• Non equilibrium current fluctuations in stochastic lattice gases
• WKB theory of large deviations in stochastic populations
• Cumulants and large deviations of the current through non-equilibrium steady states
• Switching rates of multi-step reactions
• Anomalous scaling of dynamical large deviations
• Effective driven dynamics for one-dimensional conditioned Langevin processes in the weak-noise limit
• Large-deviation theory for a Brownian particle on a ring: a WKB approach
• Large deviations of a long-time average in the Ehrenfest Urn Model

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