Condensation in the Inclusion Process and Related Models

Großkinsky, Stefan; Redig, Frank; Vafayi, Kiamars

doi:10.1007/s10955-011-0151-9

Cited by 62 publications

(110 citation statements)

References 27 publications

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“…Moreover, master equations and the SSA have helped to understand the formation of traffic jams on highways [186,187], the walks of molecular motors along cytoskeletal filaments [113][114][115][188][189][190], and the condensation of bosons in driven-dissipative quantum systems [178,191,192]. The master equation that was found to describe the coarse-grained dynamics of these bosons coincides with the master equation of the (asymmetric) inclusion process [193][194][195][196]. Transport processes are commonly modelled in terms of the (totally) asymmetric simple exclusion process (ASEP or TASEP) [197][198][199][200].…”

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confidence: 99%

Master equations and the theory of stochastic path integrals

2017

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Abstract. This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of masters equations most often rely on lownoise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a "generating functional", which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a "forward" and a "backward" path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon expanding the forward and the backward path integrals around stationary paths, we then discuss and extend a recent method for the computation of rare event probabilities. Besides, we also derive path integral representations for processes with continuous state spaces whose forward and backward master equations admit Kramers-Moyal expansions. A truncation of the backward expansion at the level of a diffusion approximation recovers a classic path integral representation of the (backward) Fokker-Planck equation. One can rewrite this path integral in terms of an Onsager-Machlup function and, for purely diffusive Brownian motion, it simplifies to the path integral of Wiener. To make this review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory and do not assume any knowledge of the latter. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit. arXiv:1609.02849v2 [cond-mat...

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confidence: 99%

Master equations and the theory of stochastic path integrals

2017

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“…(2) and (3), we utilized the results in Eqs. (31)(32) to explore how occupancy correlations between sites i and j decay with the distance ∆ = j − i between them (i < j). For large ∆, this decay was seen to be roughly exponential; but Eqs.…”

Section: Discussionmentioning

confidence: 99%

Occupancy correlations in the asymmetric simple inclusion process

Bonomo

Reuveni

2019

Phys. Rev. E

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The asymmetric simple inclusion process (ASIP) -a lattice-gas model for unidirectional transport with irreversible aggregation -has been proposed as an inclusion counterpart of the asymmetric simple exclusion process and as a batch service counterpart of the tandem Jackson network. To date, analytical tractability of the model has been limited: while the average particle density in the model is easy to compute, very little is known about its joint occupancy distribution. To partially bridge this gap, we study occupancy correlations in the ASIP. We take an analytical approach to this problem and derive an exact formula for the covariance matrix of the steady-state occupancy vector. We verify the validity of this formula numerically in small ASIP systems, where Monte-Carlo simulations can provide reliable estimates for correlations in reasonable time, and further use it to draw a comprehensive picture of spatial occupancy correlations in ASIP systems of arbitrary size.

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“…Even in much simpler model systems such as exclusion processes or zero-range processes in one dimension, unexpected phenomena continue to surprise physicists and mathematicians, including condensation [1,2,3,4,5,6,7] and unusual fluctuation phenomena [8,9,10,11,12].…”

Section: Introductionmentioning

confidence: 99%

Emergence of particle clusters in a one-dimensional model: connection to condensation processes

Burman¹,

Carpenter²,

Jack³

2017

J. Phys. A: Math. Theor.

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Abstract. We discuss a simple model of particles hopping in one dimension with attractive interactions. Taking a hydrodynamic limit in which the interaction strength increases with the system size, we observe the formation of multiple clusters of particles, with large gaps between them. These clusters are correlated in space, and the system has a self-similar (fractal) structure. These results are related to condensation phenomena in mass transport models and to a recent mathematical analysis of the hydrodynamic limit in a related model.

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Condensation in the Inclusion Process and Related Models

Cited by 62 publications

References 27 publications

Master equations and the theory of stochastic path integrals

Master equations and the theory of stochastic path integrals

Occupancy correlations in the asymmetric simple inclusion process

Emergence of particle clusters in a one-dimensional model: connection to condensation processes

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