Einstein Relation Generalized to Nonequilibrium

Abstract

The Einstein relation connecting the diffusion constant and the mobility is violated beyond the linear response regime. For a colloidal particle driven along a periodic potential imposed by laser traps, we test the recent theoretical generalization of the Einstein relation to the nonequilibrium regime which involves an integral over measurable velocity correlation functions. DOI: 10.1103/PhysRevLett.98.210601 PACS numbers: 05.40.ÿa, 82.70.Dd A comprehensive theory of systems driven out of equilibrium is still lacking quite in contrast to the universal description of equilibrium systems by the GibbsBoltzmann distribution. Linear response theory provides exact relations valid, however, only for small deviations from equilibrium [1]. The arguably most famous linear response relation is the Einstein relationinvolving the diffusion constant D, the mobility , and the thermal energy k B T [2]. In his original derivation for a suspension in a force field, Einstein balances the diffusive current with a linear drift. The Einstein relation embodies a deep connection between fluctuations causing diffusion and dissipation responsible for friction expressed by a finite mobility.In the present Letter, we report on the extension of the classical Einstein relation beyond the linear response regime using a driven colloidal particle as a paradigmatic system. Our previous theoretical work [3] and its present experimental test thus introduce a third type of exact relation valid for and relevant to small driven systems coupled to a heat bath of constant temperature T. The previously discovered exact relations comprise, first, the fluctuation theorem [4,5] which quantifies the steady state probability of observing trajectories of negative entropy production. Second, the Jarzynski relation [6] expresses the free energy difference between different equilibrium states by a nonlinear average of the work spent in driving such a transition [7]. Both the fluctuation theorem and the Jarzynski relation as well as their theoretical extensions [8][9][10] have been tested in various experimental systems such as micromechanically manipulated biomolecules [11,12], colloids in time-dependent laser traps [13][14][15], Rayleigh-Benard convection [16], mechanical oscillators [17], and optically driven single two-level systems [18]. Such exact relations (and the study of their limitations) are fundamentally important since they provide the first elements of a future more comprehensive theory of nonequilibrium systems.For a nonequilibrium extension of the Einstein relation (1), consider the overdamped motion xt of a particle moving along a periodic one-dimensional potential Vx governed by the Langevin equationwith F ÿ@V=@x f and f a nonconservative force. The friction coefficient determines the correlations htt 0 i 2k B T=t ÿ t 0 of the white noise . Therefore, Eq. (2) describes a colloidal bead driven to nonequilibrium under the assumption that the fluctuating forces arising from the heat bath are not affected by the driving.For the crucial quantiti...

Generalized Einstein or Green-Kubo Relations for Active Biomolecular Transport

“…In such a state, the Stokes-Einstein relation no longer holds true. The difference between diffusion constant and mobility, however, can be expressed by an integral over an experimentally measurable correlation function [2].…”

Nonequilibrium version of the Einstein relation

“…For completeness we note that extensions of the fluctuation-dissipation phenomenology far from equilibrium have been considered in [30][31][32][33], but from a different perspective. In [34] it has been pointed out that a violation of the ESR is expected in a Markovian network, however this has not been explicitly demonstrated for a model of interest, neither related to the Sinai anomaly.…”

Brownian Motion of Boomerang Colloidal Particles

“…Our experimental observations show that the mean displacements are biased towards the center of hydrodynamic stress (CoH), and that the mean-square displacements exhibit a crossover from short time faster to long time slower diffusion with the short-time diffusion coefficients dependent on the points used for tracking. A model based on Langevin theory elucidates that these behaviors are ascribed to a superposition of two diffusive modes: the ellipsoidal motion of the CoH and the rotational motion of the tracking point with respect to the CoH.Brownian motion as a general phenomenon of the diffusion processes has inspired extensive research [1][2][3][4][5][6][7][8][9][10][11][12] due to both its interesting physics and practical applications such as in microrheology [13][14][15][16], selfpropelled microswimmers [17] and particle and molecular separation [18][19][20]. Inspired by the diverse geometric shapes of biological macromolecules, Brenner and others have extended the hydrodynamic theory of Brownian motion to particles with irregular shapes [21][22][23][24][25][26].…”