Path integrals and perturbation theory for stochastic processes

Dickman, Ronald; Vidigal, Ronaldo

doi:10.1590/s0103-97332003000100005

Cited by 31 publications

(48 citation statements)

References 25 publications

(29 reference statements)

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“…Thus it is of interest to study further examples of this class, and to apply new methods of analysis to such models. Absorbing-state transitions have been studied via mean-field theory, series expansion [2], renormalization group [9], perturbation theory [12] and numerical simulation. Recently, an analysis based on the exact (numerical) determination of the quasistationary (QS) probability distribution was proposed and applied to models in the DP class [13].…”

Section: Introductionmentioning

confidence: 99%

Conserved directed percolation: exact quasistationary distribution of small systems and Monte Carlo simulations

Filho¹,

Dickman²

2011

J. Stat. Mech.

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We study symmetric sleepy random walkers, a model exhibiting an absorbing-state phase transition in the conserved directed percolation (CDP) universality class. Unlike most examples of this class studied previously, this model possesses a continuously variable control parameter, facilitating analysis of critical properties. We study the model using two complementary approaches: analysis of the numerically exact quasistationary (QS) probability distribution on rings of up to 22 sites, and Monte Carlo simulation of systems of up to 32000 sites. The resulting estimates for critical exponents β, β/ν ⊥ , and z, and the moment ratio m 211 = ρ 2 / ρ 2 (ρ is the activity density), based on finite-size scaling at the critical point, are in agreement with previous results for the CDP universality class. We find, however, that the approach to the QS regime is characterized by a different value of the dynamic exponent z than found in the QS regime.

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Section: Introductionmentioning

confidence: 99%

Conserved directed percolation: exact quasistationary distribution of small systems and Monte Carlo simulations

Filho¹,

Dickman²

2011

J. Stat. Mech.

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“…(108) of Ref. [9]), so that the probability generating function at time zero is Φ 0 (z) = e p(z−1) . Here, the kernel U (r) t is given by the functional integral,…”

Section: Perturbation Theory For the Malthus-verhulst Processmentioning

confidence: 99%

“…where we have introduced w ≡ λ−1 (equal to −w as defined in [9]). We recognize the exponential of ζ plus the first term in the integrand as F [ψ, φ] z=1 ; the remaining terms then represent −S I , the new effective interaction, which will be treated perturbatively.…”

Section: Perturbation Theory For the Malthus-verhulst Processmentioning

confidence: 99%

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Functional-integral based perturbation theory for the Malthus-Verhulst process

Moloney¹,

Dickman²

2006

Braz. J. Phys.

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We apply a functional-integral formalism for Markovian birth and death processes to determine asymptotic corrections to mean-field theory in the Malthus-Verhulst process (MVP). Expanding about the stationary meanfield solution, we identify an expansion parameter that is small in the limit of large mean population, and derive a diagrammatic expansion in powers of this parameter. The series is evaluated to fifth order using computational enumeration of diagrams. Although the MVP has no stationary state, we obtain good agreement with the associated quasi-stationary values for the moments of the population size, provided the mean population size is not small. We compare our results with those of van Kampen's Ω-expansion, and apply our method to the MVP with input, for which a stationary state does exist. We also devise a modified Fokker-Planck approach for this case.

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“…The analogy of the master equation to quantum descriptions has been introduced by Doi, and several authors have developed the formalism [6,7,8]. The field theoretic approach has revealed the anomalous kinetics in reaction-diffusion systems incorporating the renormalization group method [9,10], and the analytical scheme has been applied to various phenomena [11,12,13,14,15]. In addition, the variational method based on the second quantization description developed by Sasai and Wolynes [2] is hopeful in order to investigate the fluctuation and discrete effects in the small size systems.…”

Section: Introductionmentioning

confidence: 99%

Approximation scheme for master equations: Variational approach to multivariate case

Ohkubo

2008

The Journal of Chemical Physics

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We study an approximation scheme based on a second quantization method for a chemical master equation. Small systems, such as cells, could not be studied by the traditional rate equation approach because fluctuation effects are very large in such small systems. Although a Fokker-Planck equation obtained by the system size expansion includes the fluctuation effects, it needs large computational costs for complicated chemical reaction systems. In addition, discrete characteristics of the original master equation are neglected in the system size expansion scheme. It has been shown that the usage of the second quantization description and a variational method achieves tremendous reduction in the dimensionality of the master equation approximately, without loss of the discrete characteristics. We here propose a new scheme for the choice of variational functions, which is applicable to multivariate cases. It is revealed that the new scheme gives better numerical results than old ones and the computational cost increases only slightly.

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Path integrals and perturbation theory for stochastic processes

Cited by 31 publications

References 25 publications

Conserved directed percolation: exact quasistationary distribution of small systems and Monte Carlo simulations

Conserved directed percolation: exact quasistationary distribution of small systems and Monte Carlo simulations

Functional-integral based perturbation theory for the Malthus-Verhulst process

Approximation scheme for master equations: Variational approach to multivariate case

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