2021/02/28 by Emil Mallmin, E Mallmin, Johan du Buisson +3
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Boundary (topology) #Bounded function #Diffusion #Large deviations theory #Limit (mathematics) #Markov chain #Markov process #Observable #Random walk #Rate function #Stochastic processes and statistical mechanics #cond-mat.stat-mech #math.PR #stochastic dynamics and bifurcation
paper · pdf · doi:10.1088/1751-8121/ac039a
published as J. Phys. A: Math. Theor. 54 295001 (2021) · 24 pages, 2 figures
openalex created_date 2021/02/15 · openalex publication_date 2021/05/20 · arxiv created 2021/06/19 · arxiv updated 2021/06/22 · openalex updated_date 2026/08/05
Abstract We study the large deviations of current-type observables defined for Markov diffusion processes evolving in smooth bounded regions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:math> with reflections at the boundaries. We derive for these the correct boundary conditions that must be imposed on the spectral problem associated with the scaled cumulant generating function, which gives, by Legendre transform, the rate function characterizing the likelihood of current fluctuations. Two methods for obtaining the boundary conditions are presented, based on the diffusive limit of random walks and on the Feynman–Kac equation underlying the evolution of generating functions. Our results generalize recent works on density-type observables, and are illustrated for an N -particle single-file diffusion on a ring, which can be mapped to a reflected N -dimensional diffusion.