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 → ∞, a singularity of the LDF, which can be interpreted as a secondorder 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:1901.09384v2 [cond-mat.stat-mech]

Large deviation of long time average for a stochastic process: an alternative method

“…This result has been obtained recently [8] by using a WKB method. If needed, we can check the above result by solving exactly equation (13) using the methods of characteristics : setting φ = exp(N u), the solution, for the initial condition φ(z, t = 0) = 1 is :…”

“…this expression has been obtained by other methods in [8]. We stress that this expression is only correct for single step processes with first order polynomial rates.…”

“…Many interesting stochastic processes are quadratic in n (see for example [8,14] where dynamical phase transition are observed ). The dPGFk method for these cases leads to parabolic PDEs for φ() .…”

“…need to solve the PDE but can restrict the analysis to some particular points z * where the long time limit can be obtained through an ordinary differential equation. Many interesting stochastic processes are quadratic in n (see for example [8,14] where dynamical phase transitions are observed). The dPGFk method for these cases leads to parabolic PDEs for φ().…”

Large deviations and dynamical phase transitions in stochastic chemical networks

“…we take h = 0, for the sake of simplicity. An example of a dynamical phase transition involving a density bias can be found in [43].…”