Geometrical optics of constrained Brownian motion: three short stories. (18th September 2019)

• Record Type: Journal Article
• Authors: Meerson, Baruch; Smith, Naftali R
• Is Part Of: Journal of physics. Volume 52:Number 41(2019)
• Publication Date: 2019-09-18
• Subjects: Brownian motion -- large deviations -- dynamical phase transitions -- optimal fluctuation method; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/ab3f0f
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: The optimal fluctuation method—essentially geometrical optics—gives a deep insight into large deviations of Brownian motion. Here we illustrate this point by telling three short stories about Brownian motions, 'pushed' into a large-deviation regime by constraints. In story 1 we compute the short-time large deviation function (LDF) of the winding angle of a Brownian particle wandering around a reflecting disk in the plane. Story 2 addresses a stretched Brownian motion above absorbing obstacles in the plane. We compute the short-time LDF of the position of the surviving Brownian particle at an intermediate point. Story 3 deals with survival of a Brownian particle in 1 + 1 dimension against absorption by a wall which advances according to a power law, where . We also calculate the LDF of the particle position at an earlier time, conditional on the survival by a later time. In all three stories we uncover singularities of the LDFs which have a simple geometric origin and can be interpreted as dynamical phase transitions. We also use the small-deviation limit of the geometrical optics to reconstruct the distribution of typical fluctuations. We argue that, in stories 2 and 3, this is the Ferrari–Spohn distribution.