Finite Size Effects and Metastability in Zero-Range Condensation

A Markov process fluctuating away from its typical behavior can be represented in the long-time limit by another Markov process, called the effective or driven process, having the same stationary states as the original process conditioned on the fluctuation observed. We construct here this driven process for diffusions spending an atypical fraction of their evolution in some region of state space, corresponding mathematically to stochastic differential equations conditioned on occupation measures. As an illustration, we consider the Langevin equation conditioned on staying for a fraction of time in different intervals of the real line, including the positive half-line which leads to a generalization of the Brownian meander problem. Other applications related to quasi-stationary distributions, metastable states, noisy chemical reactions, queues, and random walks are discussed. C 2016 AIP Publishing LLC.

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“…The dynamics leading to this condensation and metastable phases related to it have been studied using occupation conditioning in [54][55][56].…”

Section: Discussionmentioning

confidence: 99%

Diffusions conditioned on occupation measures

Angeletti

Touchette

2016

Journal of Mathematical Physics

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“…As noted in the introduction, in the fluid phase all random variables contribute with small values to the sums M L and V L and the marginal distribution takes the form (17). In this Section we study the fluid phase in more detail using two alternative approaches: the saddle-point method and large deviation theory.…”

Section: Fluid Phase µmentioning

confidence: 99%

Condensation transition in joint large deviations of linear statistics

Szavits-Nossan¹,

Evans²,

Majumdar³

2014

J. Phys. A: Math. Theor.

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Abstract. Real space condensation is known to occur in stochastic models of mass transport in the regime in which the globally conserved mass density is greater than a critical value. It has been shown within models with factorised stationary states that the condensation can be understood in terms of sums of independent and identically distributed random variables: these exhibit condensation when they are conditioned to a large deviation of their sum. It is well understood that the condensation, whereby one of the random variables contributes a finite fraction to the sum, occurs only if the underlying probability distribution (modulo exponential) is heavy-tailed, i.e. decaying slower than exponential. Here we study a similar phenomenon in which condensation is exhibited for non-heavy-tailed distributions, provided random variables are additionally conditioned on a large deviation of certain linear statistics. We provide a detailed theoretical analysis explaining the phenomenon, which is supported by Monte Carlo simulations (for the case where the additional constraint is the sample variance) and demonstrated in several physical systems. Our results suggest that the condensation is a generic phenomenon that pertains to both typical and rare events.

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“…As we discuss in Appendix A, non-monotonicity in homogeneous systems with finite critical density can be related, on a heuristic level, to convexity properties of the canonical entropy. For condensing systems with zero-range dynamics, it has been shown that this is related to the presence of metastable states, resulting in the non-monotone behaviour of the canonical stationary current/diffusivity [25]. This corresponds to a first order correction of a hydrodynamic limit leading to an ill-posed equation with negative diffusivity in the case of reversible dynamics.…”

Section: Introductionmentioning

confidence: 99%

Monotonicity and condensation in homogeneous stochastic particle systems

Rafferty¹,

Chleboun²,

Großkinsky³

2018

Ann. Inst. H. Poincaré Probab. Statist.

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We study stochastic particle systems that conserve the particle density and exhibit a condensation transition due to particle interactions. We restrict our analysis to spatially homogeneous systems on finite lattices with stationary product measures, which includes previously studied zero-range or misanthrope processes. All known examples of such condensing processes are non-monotone, i.e. the dynamics do not preserve a partial ordering of the state space and the canonical measures (with a fixed number of particles) are not monotonically ordered. For our main result we prove that condensing homogeneous particle systems with finite critical density are necessarily non-monotone. On finite lattices condensation can occur even when the critical density is infinite, in this case we give an example of a condensing process that numerical evidence suggests is monotone, and give a partial proof of its monotonicity.

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Finite Size Effects and Metastability in Zero-Range Condensation

Cited by 29 publications

References 45 publications

Diffusions conditioned on occupation measures

Diffusions conditioned on occupation measures

Condensation transition in joint large deviations of linear statistics

Monotonicity and condensation in homogeneous stochastic particle systems

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