Erratum: Large deviations of a long-time average in the Ehrenfest urn model (2018 <i>J. Stat. Mech</i>. 053202)

Meerson, Baruch; Zilber, Pini

doi:10.1088/1742-5468/aae84d

Cited by 4 publications

(14 citation statements)

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“…As NT found, for n = 1 and 2 and in the regime T τ , the optimal path x(t) stays, for most of the time, very close to the unique stable fixed point on the phase plane of the effective classical mechanics. (An identical behavior was previously predicted, by a different version of the OFM [14], for the continuous-time Ehrenfest urn model [15] and its extensions.) As a result, the classical action S T f (a) is proportional to T , immediately leading to Eq.…”

Section: Introductionsupporting

confidence: 81%

Anomalous scaling of dynamical large deviations of stationary Gaussian processes

Meerson

2019

Phys. Rev. E

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Employing the optimal fluctuation method (OFM), we study the large deviation function of longtime averages (1/T ) T /2 −T /2 x n (t)dt, n = 1, 2, . . . , of centered stationary Gaussian processes. These processes are correlated and, in general, non-Markovian. We show that the anomalous scaling with time of the large-deviation function, recently observed for n > 2 for the particular case of the Ornstein-Uhlenbeck process, holds for a whole class of stationary Gaussian processes.

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Section: Introductionsupporting

confidence: 81%

Anomalous scaling of dynamical large deviations of stationary Gaussian processes

Meerson

2019

Phys. Rev. E

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“…A similar property, and the ensuing large deviation function (LDF) that encodes it, have received much recent attention from physicists in the context of continuous diffusion processes, described by Langevin equations [12][13][14][15]. Recently this characterization has been * pizilber@gmail.com † naftali.smith@mail.huji.ac.il ‡ meerson@mail.huji.ac.il extended [16] to one of the best known jump processes in physics: the Ehrenfest Urn Model [17]. The basic version of the Ehrenfest Urn Model involves two urns with N identical balls.…”

Section: Introductionmentioning

confidence: 99%

“…The balls hop randomly, one ball at a time, from one urn to the other. For non-interacting balls the LDF of the long-time average of the number of balls in a given urn can be calculated exactly [16] by using the Donsker-Varadhan (DV) large-deviation formalism [18], which we briefly review below. If the balls interact, the DV formalism can still be used numerically [16], but it does not allow for analytical solution.…”

Section: Introductionmentioning

confidence: 99%

“…For non-interacting balls the LDF of the long-time average of the number of balls in a given urn can be calculated exactly [16] by using the Donsker-Varadhan (DV) large-deviation formalism [18], which we briefly review below. If the balls interact, the DV formalism can still be used numerically [16], but it does not allow for analytical solution. However, when the total number of balls N is very large, one can circumvent the DV method and employ instead a WKB approximation, directly applied to the master equation [19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

“…It is natural to call the function f (a) the intensive rate function. For many stochastic populations the intensive rate function f (a) is an analytic function of a, and the Ehrenfest Urn Model provides one such example [16]. As we will see shortly, another such example is provided by the immigration-death process ∅ A, a simple and well studied reversible jump process.…”

Section: Introductionmentioning

confidence: 99%

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

2019

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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]

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Large deviations conditioned on large deviations I: Markov chain and Langevin equation

Derrida

Sadhu

2019

J Stat Phys

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We present a systematic analysis of stochastic processes conditioned on an empirical measure Q T defined in a time interval [0, T ] for large T . We build our analysis starting from a discrete time Markov chain. Results for a continuous time Markov process and Langevin dynamics are derived as limiting cases. We show how conditioning on a value of Q T modifies the dynamics. For a Langevin dynamics with weak noise, we introduce conditioned large deviations functions and calculate them using either a WKB method or a variational formulation. This allows us, in particular, to calculate the typical trajectory and the fluctuations around this optimal trajectory when conditioned on a certain value of Q T .

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Erratum: Large deviations of a long-time average in the Ehrenfest urn model (2018 J. Stat. Mech. 053202)

Cited by 4 publications

References 0 publications

Anomalous scaling of dynamical large deviations of stationary Gaussian processes

Anomalous scaling of dynamical large deviations of stationary Gaussian processes

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

Large deviations conditioned on large deviations I: Markov chain and Langevin equation

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