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UFN, 2019, Volume 189, Number 5, Pages 529–543 (Mi ufn6165)  

METHODOLOGICAL NOTES

Green's function method in the theory of Brownian motors

V. M. Rozenbauma, I. V. Shapochkinab, L. I. Trakhtenbergcd

a Chuiko Institute of Surface Chemistry, National Academy of Sciences of Ukraine, Kiev
b Belarusian State University, Minsk
c Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow
d Lomonosov Moscow State University

Abstract: We present the main results of the theory of Brownian motors obtained using the authors' approach, in which a Brownian particle moving in a slightly fluctuating potential profile is considered. By using the Green's function method, the perturbation theory in small fluctuations of potential energy is constructed. This approach allows obtaining an analytic expression for the mean particle velocity that is valid for two main types of Brownian motors (flashing and rocking ratchets) and any time dependence (stochastic or deterministic) of the fluctuations. The advantage of the proposed approach lies in the compactness of the description and, at the same time, in the variety of motor systems analyzed with its help: the overwhelming majority of known analytic results in the theory of Brownian motors follow from this expression. The mathematical derivations and analysis of those results are the main subject of these methodological notes.

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation -18-118012390045-2
Russian Foundation for Basic Research 18-57-00003
18-29-02012
Belarusian Republican Foundation for Fundamental Research 18-022
Grant of President of Belarus
This study was carried out in the framework of State Assignment 0082-2018-0003 (registration number AAAA-A18-118012390045-2) and was supported by the Russian Foundation for Basic Research (projects 18-57-00003 and 18-29-02012-mk) and the Foundation for Fundamental Research of Belarus (project F18R-022). I V Sh expresses her gratitude for the support of her work through a grant from the President of the Republic of Belarus in the area of education.

Author to whom correspondence should be addressed

DOI: https://doi.org/10.3367/UFNr.2018.04.038347

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English version:
Physics–Uspekhi, 2019, 62:5, 496–509

Bibliographic databases:

PACS: 05.40.-a, 05.60.Cd
Received: November 7, 2017
Revised: April 19, 2018
Accepted: April 23, 2018

Citation: V. M. Rozenbaum, I. V. Shapochkina, L. I. Trakhtenberg, “Green's function method in the theory of Brownian motors”, UFN, 189:5 (2019), 529–543; Phys. Usp., 62:5 (2019), 496–509

Citation in format AMSBIB
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