The Einstein relation generalized to non-equilibrium
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r The Einstein relation generalized to non-equilibrium
V. Blickle, ∗ T. Speck, ∗ C. Lutz, U. Seifert, and C. Bechinger
2. Physikalisches Institut, Universit¨at Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany II. Institut f¨ur Theoretische Physik, Universit¨at Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
The Einstein relation connecting the diffusion constant and the mobility is violated beyond thelinear response regime. For a colloidal particle driven along a periodic potential imposed by lasertraps, we test the recent theoretical generalization of the Einstein relation to the non-equilibriumregime which involves an integral over measurable velocity correlation functions.
PACS numbers: 05.40.-a,82.70.Dd
A comprehensive theory of systems driven out of equi-librium is still lacking quite in contrast to the univer-sal description of equilibrium systems by the Gibbs-Boltzmann distribution. Linear response theory providesexact relations valid, however, only for small deviationsfrom equilibrium [1]. The arguably most famous linearresponse relation is the Einstein relation D = k B T µ, (1)involving the diffusion constant D , the mobility µ , andthe thermal energy k B T [2]. In his original derivationfor a suspension in a force field, Einstein balances thediffusive current with a linear drift. The Einstein re-lation embodies a deep connection between fluctuationscausing diffusion and dissipation responsible for frictionexpressed by a finite mobility.In the present Letter, we report on the extension ofthe classical Einstein relation beyond the linear responseregime using a driven colloidal particle as a paradigmaticsystem. Our previous theoretical work [3] and its presentexperimental test thus introduce a third type of exact re-lation valid for and relevant to small driven systems cou-pled to a heat bath of constant temperature T . The pre-viously discovered exact relations comprise, first, the fluc-tuation theorem [4, 5] which quantifies the steady stateprobability of observing trajectories of negative entropyproduction. Second, the Jarzynski relation [6] expressesthe free energy difference between different equilibriumstates by a nonlinear average of the work spent in driv-ing such a transition [7]. Both the fluctuation theoremand the Jarzynski relation as well as their theoreticalextensions [8, 9, 10] have been tested in various experi-mental systems such as micro-mechanically manipulatedbiomolecules [11, 12], colloids in time-dependent lasertraps [13, 14, 15], Rayleigh-Benard convection [16], me-chanical oscillators [17], and optically driven single two-level systems [18]. Such exact relations (and the study oftheir limitations) are fundamentally important since theyprovide the first elements of a future more comprehensivetheory of non-equilibrium systems.For a non-equilibrium extension of the Einstein rela-tion (1), consider the overdamped motion x ( t ) of a par-ticle moving along a periodic one-dimensional potential V ( x ) governed by the Langevin equation˙ x ( t ) = 1 γ F ( x ( t )) + ξ ( t ) (2)with F = − ∂V /∂x + f and f a non-conservative force.The friction coefficient γ determines the correlations h ξ ( t ) ξ ( t ′ ) i = 2( k B T /γ ) δ ( t − t ′ ) of the white noise ξ .Therefore Eq. (2) describes a colloidal bead driven tonon-equilibrium under the assumption that the fluctuat-ing forces arising from the heat bath are not affected bythe driving.For the crucial quantities D and µ , it is convenient toadapt definitions which can be used both in equilibriumand beyond linear response, i.e., in a non-equilibriumsteady state characterized by f = const. = 0. The diffu-sion coefficient is given by D = lim t →∞ [ h x ( t ) i − h x ( t ) i ] / (2 t ) , (3)where h· · ·i denotes the ensemble average. Both theoret-ical work [19] and a recent experiment [20] have shownthat the force-dependent diffusion constant can be sub-stantially larger than its equilibrium value. The mobility µ = ∂ h ˙ x i ∂f (4)quantifies the response of the mean velocity h ˙ x i to a smallchange of the external force f . If the response is takenat f = 0, which corresponds to equilibrium, one has thelinear response relation (1). How does the Einstein rela-tion change for f = 0, i.e., what is the relation betweena force-dependent diffusion constant D ( f ) and a force-dependent mobility µ ( f )? Is there a simple relation atall? We have recently shown that under non-equilibriumconditions the Einstein relation (1) has to be replacedby [3] D = k B T µ + Z ∞ d τ I ( τ ) , (5)where the second term on the right hand side is givenby an integral over a known “violation function” I ( τ )involving measurable velocity correlations to be discussedin detail below. Such a relation is complementary to FIG. 1: (a) Experimental setup. (b) Typical trajectory ofthe angular particle position for a mean particle revolutiontime ≃ . introducing an effective temperature which replaces T inEq. (1) in an attempt to keep its simple form [21, 22].It has the advantage that knowledge of I ( τ ) offers us abetter understanding of the crucial characteristics of thenon-equilibrium steady state that causes the breakdownof the Einstein relation (1).In our experiment we subject a single colloidal sil-ica bead with 1 . µ m diameter to a non-equilibriumsteady state by forcing it along a toroidal trap ( R =1 . µ m) created by tightly focussed rotating opticaltweezers [23, 24] (see Fig. 1). This is achieved by fo-cusing the beam of a Nd:YAG laser ( λ = 532 nm) with amicroscope objective (100x, NA=1.3) into a sample cellcontaining a highly diluted aqueous suspension of silicaparticles with 1 . µ m diameter. A pair of galvanomet-ric driven mirrors with telescope optics deflects the beamalong a circular path and thus confines the silica bead toan effectively one-dimensional motion. Depending on thevelocity of the rotating trap three different regimes can bedistinguished [23]. (i) For small velocities friction forcesare much smaller than the trapping force, the trappedparticle is able to follow the trap. (ii) With increasingvelocity the trap is not strong enough to compensate theviscous force of the fluid, the particle escapes from thelaser trap. However, every time the laser passes the parti-cle it is still dragged a small distance along the circle andmoves with a constant mean velocity around the torus.(iii) As the focus speed increases (quasi)-equilibrium con-ditions are established and the particle is able to diffuse FIG. 2: (a) Reconstructed potential V ( x ). (b) Tilted poten-tial. The colloidal particle is subjected to a constant drivingforce f ≃ .
06 pN and the periodic potential V ( x ). freely along the torus. With the trap rotation frequencyset to 567 Hz the experiments are performed in the in-termediate regime (ii) where the particle is observed tocirculate with a constant mean velocity. Since the dis-placement of the particle by a single kick depends on thelaser intensity and is approximately 10 nm, under ourexperimental conditions the spatial (50 nm) and tem-poral (80 ms) resolution of digital video microscopy isnot sufficient to resolve single ”kicking” events. There-fore the particle can be considered to be subjected toa constant force f along the angular direction x . Ad-ditionally the scanning motion is synchronized with anelectro-optical modulator (EOM) which allows the peri-odic variation of the laser intensity along the toroid. Inthe experiment the tweezers intensity P is weakly mod-ulated (∆ P/P ≤ V ( x ) act-ing on the particle when moving along the torus. As theresult, the particle moves in a tilted periodic potential.Both the potential V ( x ) and the driving force f are notknown from the input values to the EOM but must bereconstructed as described in detail below.The central quantitity of Eq. (5) is the violation func-tion I ( τ ) which can be written as [3] I ( τ ) = h [ ˙ x ( t + τ ) − h ˙ x i ][ v s ( x ( t )) − h ˙ x i ] i . (6)It correlates the actual velocity ˙ x ( t ) with the local meanvelocity v s ( x ) subtracting from both the global mean ve-locity h ˙ x i = 2 πRj s that is given by the net particle flux j s through the torus. In one dimension for a steady state,the current must be the same everywhere and hence j s is a constant. The offset t is arbitrary because of time-translational invariance in a steady state and in the fol-lowing we set t = 0. The local mean velocity v s ( x ) isthe average of the stochastic velocity ˙ x over the subset oftrajectories passing through x . An equivalent expressionis j s = v s ( x ) p s ( x ) connecting the current with the proba-bility density p s ( x ). The local mean velocity can thus beregarded as a measure of the local violation of detailedbalance. Since in equilibrium detailed balance holds andtherefore v s ( x ) = h ˙ x i = 0, the violation (6) vanishes andEq. (5) reduces to Eq. (1).For an experimental test of the non-equilibrium Ein- FIG. 3: a) Experimentally measured violation function I ( τ ) (solid line). b) Comparison of the velocities involved in theviolation function I ( τ ). For an ideal cosine potential, we sketch the probability distribution p s ( x ) (solid gray line), the localmean velocity v s ( x ) together with the drift velocity and their mean h ˙ x i versus the angular particle position. The drift velocityis the deterministic part F/γ of the actual velocity ˙ x . The sign change in I ( τ ) at (2), (3), and (4) can be understood as follows.In a steady state, a single particle trajectory will start with highest probability in the shaded region and, for an illustration,we choose its maximum as starting point (1) determining the value v s ( x ( t )) in Eq. (6). Neglecting thermal fluctuations, theparticle would follow the dashed line and during a small time step τ the product F ( x ( t + τ )) v s ( x ( t )) is positive. If the particlepasses (2), the product would become negative. The sign changes again if the particle passes (3) and then (4) and so on due tothe periodic nature of the potential. Thermal noise and averaging over all trajectories does not change this behavior responsiblefor the oscillations of I ( τ ). stein relation (5), we measure trajectories of a single col-loidal particle for different driving forces f by adjustingthe intensity transmitted through the EOM. From a lin-ear fit to the data we first determine the mean global ve-locity h ˙ x i . Next, we extract the mean local velocity v s ( x )from the histogram p s ( x ) with the coordinate x confinedto 0 ≤ x ≤ πR . Since measurements are performed witha sampling rate of 80 ms, we cannot directly access thevelocity ˙ x ( t ) experimentally. To calculate the violationintegral I ( τ ), we decompose ˙ x ( t ) into a randomly fluctu-ating Brownian part and a drift term, see Eq. (2). Wethen transform I ( τ ) as I ( τ ) = h [ v s ( x ( τ )) − k B Tγ ∂ Φ ∂x ( x ( τ ))] v s ( x (0)) i − h ˙ x i + h ξ ( τ ) v s ( x (0)) i . (7)The generalized potential Φ( x ) is determined via themeasured stationary probability distribution, p s ( x ) =exp[ − Φ( x )] [3]. For τ >
0, the last term vanishes becausethen ξ ( t + τ ) and x ( t ) are uncorrelated. Thus the function I ( τ ) depends on two measurable quantities, the current j s and the stationary probability distribution p s ( x ).The potential V and the driving force f are determinedby integrating the force F = − ∂V∂x + f = γv s − k B T ∂ Φ ∂x (8)along the torus. We obtain f = γ πR Z πR d x v s ( x ) (9) and V ( x ) = k B T Φ( x ) + Z x d x ′ [ f − γv s ( x ′ )] (10)up to an irrelevant constant. In Eq. (9), terms involving V and Φ are zero due to the periodicity of our system.Both, the potential V ( x ) and the tilted potential V ( x ) − f x are shown in Fig. 2. The mobility µ = ∆ h ˙ x i / ∆ f isdetermined from the change of the global mean velocity∆ h ˙ x i upon a small variation of the force ∆ f .With the experimentally determined quantities, wemeasure the violation function I ( τ ) shown as solid line inFig. 3a for f = 0 .
06 pN. It clearly displays the two timescales present in the system. First, the driving leads toan oscillatory behavior with a period equal to the meanrevolution time ≃ . ≃ . I ( τ ) in more detail it is helpful to compare the dif-ferent velocities involved in the violation function I ( τ )which are sketched in Fig. 3b.After numerical integration of the experimentally de-termined I ( τ ) we finally calculate the diffusion coefficientaccording to Eq. (5). To quantify the relative impor-tance of the violation integral we plot the two terms ofthe right hand side of Eq. (5) separately for five differentvalues of the driving force in Fig. 4. Their sum is in goodagreement with the independently measured diffusion co-efficient directly obtained from the particles trajectoryusing Eq. (3). As the maximal error for the indepen- FIG. 4: Experimental test of Eq. (5) for different drivingforces f . The open bars show the measured diffusion co-efficients D . The stacked bars are mobility (gray bar) andintegrated violation (hatched bar), respectively. dent measurements we estimated from our data ±
3% forthe diffusion coefficient D , up to ±
10% for the violationintegral, and ±
7% for the mobility µ .We emphasize that under our experimental parametersthe violation term dominates the diffusion coefficient (upto 80%) and must not be ignored. In Fig. 4 one observesa non-monotonic dependence of the violation integral onthe driving force. This is due to the fact that the maximaof µ ( f ) and D ( f ) do not occur at the same driving forcebut are slightly offset [19]. This implies for the violationfunction a maximum followed by a minimum as a func-tion of f . For very small driving forces, the bead is closeto equilibrium and its motion can be described using lin-ear response theory. As a result, the violation integralis negligible. Experimentally, this regime is difficult toaccess since D and µ become exponentially small andcannot be measured at reasonable time scales for smallforces and potentials as deep as 40 k B T (cf. Fig. 2a).For much larger forces, the relative magnitude of the vi-olation term becomes smaller as well. In this limit, theimposed potential becomes irrelevant and the spatial de-pendence of the local mean velocity, which is the sourceof the violation term, vanishes. The fact that in ourregime the violation term is of the same order of magni-tude as the mobility proves that we are indeed probingthe regime beyond linear response. Still, the descriptionof the colloidal motion by a Markovian (memory-less)Brownian motion with drift as implicit in our analysisremains obviously a faithful representation since the the-oretical results are derived from such a framework. The Einstein relation generalized to non-equilibrium aspresented and tested here for the driven motion along asingle coordinate could be considered as a paradigm. Ex-tending such an approach to interacting particles and re-solving frequency dependent versions of Eq. (6) [3] whilecertainly experimentally challenging will provide furtherinsight into crucial elements of a future systematic theoryof non-equilibrium systems. ∗ These authors contributed equally to this work.[1] R. Kubo, M. Toda, and N. Hashitsume,
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