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Coherent-state path integral versus coarse-grained effective stochastic equation of motion: From reaction diffusion to stochastic sandpiles

Kay Jörg Wiese

  • CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France and PSL Research University, 62 bis Rue Gay-Lussac, 75005 Paris, France

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Phys. Rev. E 93, 042117 – Published 14 April, 2016

DOI: https://doi.org/10.1103/PhysRevE.93.042117

Abstract

We derive and study two different formalisms used for nonequilibrium processes: the coherent-state path integral, and an effective, coarse-grained stochastic equation of motion. We first study the coherent-state path integral and the corresponding field theory, using the annihilation process A+A→A as an example. The field theory contains counterintuitive quartic vertices. We show how they can be interpreted in terms of a first-passage problem. Reformulating the coherent-state path integral as a stochastic equation of motion, the noise generically becomes imaginary. This renders it not only difficult to interpret, but leads to convergence problems at finite times. We then show how alternatively an effective coarse-grained stochastic equation of motion with real noise can be constructed. The procedure is similar in spirit to the derivation of the mean-field approximation for the Ising model, and the ensuing construction of its effective field theory. We finally apply our findings to stochastic Manna sandpiles. We show that the coherent-state path integral is inappropriate, or at least inconvenient. As an alternative, we derive and solve its mean-field approximation, which we then use to construct a coarse-grained stochastic equation of motion with real noise.

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Outline

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Phys. Rev. E 93, 042117– Published 14 April, 2016
  • Received 24 June 2015
  • Revised 16 February 2016

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