Universal Geometric Theory of Mesoscopic Stochastic Pumps and Reversible Ratchets

Abstract

We construct a unifying theory of geometric effects in mesoscopic stochastic kinetics. We demonstrate that the adiabatic pump and the reversible ratchet effects, as well as similar new phenomena in other domains, such as in epidemiology, all follow from very similar geometric phase contributions to the effective action in the stochastic path integral representation of the moment generating function. The theory provides the universal technique for identification, prediction, and calculation of pumplike phenomena in an arbitrary mesoscopic stochastic framework.

Geometrical expression of excess entropy production

“…We note that several formulae that are similar to Eq. (16) have been obtained for different setups [20][21][22][24][25][26].…”

“…In the context of the full counting statistics (and also stochastic ratchets), it has been reported [19][20][21][22][23][24][25][26] that several phenomena in classical stochastic processes are analogous to the Berry's geometrical phase in quantum mechanics [27,28]. In this analogy, the above-mentioned vector potential corresponds to the gauge field that induces the Berry phase.…”

Artificial Brownian motors: Controlling transport on the nanoscale

“…These systems do not require a periodic arrangement of components, nor is thermal Brownian motion an issue for their operation. In particular, adiabatic turnstiles and pumps for charge and other degrees of freedom have attracted considerable interest both experimentally ͑Kouwenhoven et Pothier et al, 1992;Switkes et al, 1999;Hohberger et al, 2001͒ andtheoretically ͑Thouless, 1983;Spivak et al, 1995;Brouwer, 1998;Zhou et al, 1999;Shutenko et al, 2000;Vavilov et al, 2001;Brandes and Vorrath, 2002;Aono, 2003;Moskalets and Büttiker, 2004;Sinitsyn and Nemenman, 2007͒. Realizations of artificial pumps on the nanoscale often involve coupled quantum dots or superlattices. Most notably, in such peristaltic devices the number of transferred charges, or, more generally, the number of transporting units per cycle, is directly linked to the cycle period.…”

Adiabatic coarse-graining and simulations of stochastic biochemical networks

“…An entire such chain will contribute a single effective Hamiltonian term, H μν ({N}, {χ}, {χ C }), to the full CGF of the slow fluxes, where {N}, {χ}, and {χ C } are the slow species abundances and the conjugate counting variables. If necessary, the geometric correction to the CGF, S μν geom ({N}, {χ}, {χ C }), can be written out as well (15). Overall,…”

“…3, 7, and 10-13, none possesses all 3 of the above features, leaving room for substantial improvements. The method we propose here reaches the goal by building upon the stochastic path integral (SPI) technique from mesoscopic physics (14,15). To make the SPI applicable to biology, significant modifications are needed.…”