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

机译：随机群时间平均大小的大偏差和动态相变，随机群的较大偏差

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摘要

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 Wentzel-Kramers-Brillouin (WKB) 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 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.
机译：我们研究了连续时间马尔可夫跳跃过程所描述的随机群的时间平均大小的大偏差。当稳态中的预期群体尺寸N大时，可以通过使用直接应用于主方程的WebRIZEL-KRamers-Brillouin（WKB）方法来评估时间平均群体大小的大偏差函数（LDF）马尔可夫进程。对于我们识别的一类模型，直接WKB方法预测观察异常小的概率与时间平均群体大小的异常小的值之间的巨大差异。差异来自潜在的经典力学问题的“最优”轨迹的定性变化。直接WKB方法还预测N→∞的极限，LDF的奇异性，其可以被解释为二阶动态相位转变。过渡在有限的n下平滑，但巨大的差距仍然存在。通过覆盖底层确定性模型的吸引的固定点附近的精确主方程的van-kampen系统尺寸扩展捕获平滑效果。我们通过开发WKB方法的不同变体来描述有限N的巨大视差，该方法与Donsker-varadhan大偏差形式主义一起应用，并涉及在1 / N中的前瞻性计算。

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