2015/01/31 by Kay Jörg Wiese · 1 citation
Mathematics · Physics and Astronomy · #Classical mechanics #Computer science #Diffusion process #Field (mathematics) #Integral equation #Mathematical analysis #Mathematics #Mean field theory #Motion (physics) #Path (computing) #Path integral formulation #Physics #Quantum mechanics #Spectroscopy and Quantum Chemical Studies #Statistical Mechanics and Entropy #Statistical physics #Stochastic differential equation #Stochastic process #Theoretical and Computational Physics #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.93.042117
published as Phys. Rev. E 93, 042117 (2016) · 29 pages, 33 figures. This is a pedagogic introduction to stochastic processes, their modeling, and effective field theory. Version 2: writing improved + a new appendix
arxiv created 2016/03/25 · openalex publication_date 2016/04/14 · arxiv updated 2016/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.