Nonequilibrium Statistical Mechanics of the Heat Bath for Two Brownian Particles

Caterina De Bacco,1 Fulvio Baldovin,2 Enzo Orlandini,2 and Ken Sekimoto3,4*1Laboratoire de Physique Théorique et Modéles Statistiques, CNRS et Université Paris-Sud 11,

UMR8626, Bâtiment 100, 91405 Orsay Cedex, France2Dipartimento di Fisica e Astronomia Galileo Galilei and Sezione INFN, Università di Padova, Via Marzolo 8, I-35100 Padova, Italy

3Matières et Systèmes Complexes, CNRS-UMR7057, Université Paris-Diderot, 75205 Paris, France4Gulliver, CNRS-UMR7083, ESPCI, 75231 Paris, France(Received 14 November 2013; published 9 May 2014)

We propose a new look at the heat bath for two Brownian particles, in which the heat bath as a “system”is both perturbed and sensed by the Brownian particles. Nonlocal thermal fluctuations give rise to bath-mediated static forces between the particles. Based on the general sum rule of the linear response theory, wederive an explicit relation linking these forces to the friction kernel describing the particles’ dynamics. Therelation is analytically confirmed in the case of two solvable models and could be experimentallychallenged. Our results point out that the inclusion of the environment as a part of the whole system isimportant for micron- or nanoscale physics.

DOI: 10.1103/PhysRevLett.112.180605 PACS numbers: 05.40.Jc, 05.20.Dd, 05.40.Ca, 45.20.df

Introduction.—Known as the thermal Casimir interactions[1] or the Asakura-Oosawa interactions [2], a fluctuatingenvironment can mediate static forces between the objectsconstituting its borders. Through a unique combination of thegeneralizedLangevin equation and the linear response theory,we uncover a link between such interactions and the corre-lated Brownian motions with memory, both of which reflectthe spatiotemporal nonlocality of the heat bath.The more fine details of Brownian motion are exper-

imentally revealed, the more deviations from the idealizedWiener process are found (see, for example, Ref. [3]).When two Brownian particles are trapped close to eachother in a heat bath (see Fig. 1), the random forces on thoseobjects are no more independent noises but should becorrelated. Based on the projection methods [4–6], weexpect the generalized Langevin equations to apply [7–10]:

MJd2XJðtÞdt2

¼ −∂U∂XJ

−X2J0¼1

Zt

0

KJ;J0 ðt − τÞ dXJ0 ðτÞdτ

dτ

þ ϵJðtÞ; (1)

where XJ (J ¼ 1 and 2) are the positions of the Brownianparticles with the mass being MJ, and KJ;J0 ðsÞ and ϵJðtÞare, respectively, the friction kernel and the random force.UðX1; X2Þ is the static interaction potential between theBrownian particles. If the environment of the Brownianparticles at the initial time t ¼ 0 is in canonical equilibriumat temperature T, the noise and the frictional kernel shouldsatisfy the fluctuation-dissipation (FD) relation of thesecond kind with the Onsager symmetries [7,11]:

hϵJðtÞϵJ0 ðt0Þi ¼ kBTKJ;J0 ðt − t0Þ; (2)

KJ;J0 ðsÞ ¼ KJ0;JðsÞ ¼ KJ;J0 ð−sÞ; (3)

where J and J0 are either 1 or 2 independently. This model[Eq. (1)] is a pivotal benchmark model for the correlatedBrownian motion, although the actual Brownian motionscould be more complicated (see, for example, Refs. [3,12]).But, “the physical meaning of the random force autocor-relation function is in this case far from clear…” even now,and “A proper derivation of the effective potential could beof great help in clarifying this last point” [10]. In addition tothe bare potential U0ðX1; X2Þ independent of the heat bath,the potential U, which is in fact the free energy as afunction of XJ, may contain a bath-mediated interactionpotential UbðX1; X2Þ so that

UðX1; X2Þ ¼ U0ðX1; X2Þ þUbðX1; X2Þ: (4)

In this Letter, we propose the relation

K1;2ð0Þ ¼ −∂

∂X1

∂∂X2

UbðX1; X2Þ; (5)

where both sides of this relation should be evaluated atthe equilibrium positions of the Brownian particles

J=1 J=2

FIG. 1 (color online). Two Brownian particles (filled disks,J ¼ 1 and J ¼ 2) are trapped by an external potential, such asthrough optical traps (vertical cones), and interact through boththe direct and the heat-bath-mediated interactions.

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XJ ¼ hXJieq. This relation implies that the bath-mediatedstatic interaction is always correlated with the frictionalone. Our approach is to regard the heat bath as the weaklynonequilibrium system which is both perturbed and sensedby the mesoscopic Brownian particles. From this point ofview, Eq. (5) is deduced from the so-called “general sum-rule theorem” [13] of the linear response theory of non-equilibrium statistical mechanics [14]. While the FDrelation of the second kind [Eq. (2)] is well known asan outcome of this theory, the other aspects have not beenfully explored. Below, we give a general argument support-ing Eq. (5) and then give two analytically solvableexamples for which the claim holds exactly.General argument.—While the spatial dimensionality is

not restrictive in the following argument, we will use thenotations as if the space were one dimensional. Suppose weobserve the force F1;2 on the J ¼ 1 particle as we move theJ ¼ 2 particle from hX2ieq at t ¼ −∞ to X2ðtÞ at t. Becauseof the small perturbation X2ðtÞ − hX2ieq, the average forceat that time hF1;2it is deviated from its equilibrium valuehF1;2ieq. The linear response theory relates these twothrough the response function Φ1;2ðsÞ as

hF1;2it − hF1;2ieq: ¼Z

t

−∞Φ1;2ðt − τÞ½X2ðτÞ − hX2ieq�dτ:

(6)

(Within the linear theory, the force is always measuredat X1 ¼ hX1ieq.) The complex admittance χ1;2ðωÞ ¼χ01;2ðωÞ þ iχ001;2ðωÞ is defined as the Fourier-Laplace trans-formation of Φ1;2ðsÞ:

χ1;2ðωÞ ¼Z þ∞

0

eiωs−εsΦ1;2ðsÞds; (7)

where ε is a positive infinitesimal number (i.e., þ0). Ifχ1;2ð∞Þ ¼ 0 (see the section titled Discussion), the cau-sality of Φ1;2ðtÞ, or the analyticity of χ1;2ðωÞ in the upperhalf complex plane of ω, imposes the general sum rule [13]

PZ þ∞

−∞

χ001;2ðωÞω

dωπ

¼ χ01;2ð0Þ; (8)

where P on the left-hand side (lhs) is denoted as taking theprincipal value of the integral across ω ¼ 0. The signifi-cance of Eq. (8) is that it relates the dissipative quantity(lhs) and the reversible static response [right-hand side(rhs)] of the system.Now, we suppose, along the thought of Onsager’s mean

regression hypothesis [15], that the response of the heatbath to the fluctuating Brownian particles, which underliesEq. (1), is essentially the same as the response to externallyspecified perturbations described by Eq. (6). Thus, thecomparison of Eq. (6) with Eq. (1) gives

Φ1;2ðtÞ ¼ −dK1;2ðtÞ

dt; (9)

or, in other words, K1;2 is the relaxation function corre-sponding to Φ1;2. With this linkage between the Langevindescription and the linear response theory, the static revers-ible response χ01;2ð0Þ of the force hF1;2i − hF1;2ieq to thestatic displacement X2 − hX2ieq can be identified with therhs of Eq. (5). As for the lhs of Eq. (8), we can show byEqs. (9) and (7) that it is equal to K1;2ð0Þ. The argumentpresented here is to be tested both analytically or numericallyand experimentally. At least for the two models presentedbelow, the claim [Eq. (5)] is analytically confirmed.Solvable model I: Hamiltonian system.—As the first

example that confirms the relation (5), we take up aHamiltonian model inspired by the classic model ofZwanzig [8]; see Fig. 2(a). Instead of a single Brownianparticle [8], we put the two Brownian particles with massesMJ (J ¼ 1; 2) which interact with the “bath” consisting oflight mass “gas” particles. While Fig. 2(a) gives the generalidea, the solvable model is limited to the one-dimensionalspace. Each gas particle, e.g., the ith one, has a mass mi(≪ MJ) and is linked to at least one of the Brownianparticles J ¼ 1 or 2 through Hookean springs of thespring constant miω

2i;Jð> 0Þ and the natural length li;J.

In Fig. 2(a), these links are represented by the dashed lines.The Hamiltonian of this purely mechanical model consistsof three parts H ¼ HB þHb þHbB, with

(a)

(b)

FIG. 2 (color online). (a) Hamiltonian model of two Brownianparticles which is analytically solvable for one-dimensional spacewith harmonic coupling. Each light mass particle (thick dot) islinked to at least one of the Brownian particles (filled disks) withHookean springs (dashed lines). (b) Langevin model of twoBrownian particles. Unlike the Hamiltonian model, each lightmass particle receives the random force and frictional force fromthe background (shaded zone) and its inertia is ignored.

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HB ¼ P21

2M1

þ P22

2M2

þ U0ðX1; X2Þ; (10)

Hb ¼Xi

p2i

2mi;

HbB ¼Xi

mi

2

X2J¼1

ω2i;Jðqi − XJ − li;JÞ2; (11)

where the pairs ðXJ; PJ ¼ MJdXJ=dtÞ and ðxi; pi ¼midxi=dtÞ denote, respectively, the positions and momentaof the heavy (J) and light (i) particles. The Brownianparticles obey the following dynamics:

MJd2XJ

dt2¼ −

∂U0

∂XJþXi

miω2i;Jðqi − XJ − li;JÞ: (12)

Given the initial values of ðqi; piÞ at t ¼ 0, the Hamiltonequation for ðqiðtÞ; piðtÞÞ, which reads

mid2qidt2

¼ −mi

X2J¼1

ω2i;Jðqi − XJðtÞ − li;JÞ; (13)

can be solved in supposing that the histories of XJðsÞ(J ¼ 1 and 2) for 0 ≤ s ≤ t are given. In order to assure thecompatibility with the initial canonical equilibrium of theheat bath, we assume the vanishing initial velocity for theBrownian particles dXJ=dtjt¼0 ¼ 0. Substituting each qi inEq. (12) by its formal solution thus obtained, the dynamicsof XJðtÞ is rigorously reduced to Eq. (1), where the frictionkernels KJ;J0 ðsÞ are

KJ;J0 ðsÞ ¼Xi

miω2i;Jω

2i;J0

~ω2i

cosð ~ωisÞ (14)

and the noise term ϵJðtÞ is

ϵJðtÞ≡Xi

miω2i;J

�~qið0Þ cosð ~ωitÞ þ

d ~qið0Þdt

sinð ~ωitÞ~ωi

�;

(15)

with ~ω2i ≡ ω2

i;1 þ ω2i;2 and

~qiðtÞ≡ qiðtÞ −X2J¼1

ω2i;J

~ω2i½li;J þ XJðtÞ�: (16)

To our knowledge, this is the first concrete model thatdemonstrates Eq. (1). Only those gas particles linked toboth Brownian particles satisfy ω2

i;1ω2i;2 > 0 and contribute

to K1;2ðsÞ. While the generalized Langevin form [Eq. (1)]holds for an individual realization without any ensembleaverage, the statistics of ϵJðtÞmust be specified. We assume

that at t ¼ 0, the bath variables ~qið0Þ and ~pið0Þ [¼ pið0Þbecause we defined dXJ=dtjt¼0 ¼ 0] belong to the canoni-cal ensemble of a temperature T with the weight∝ exp½−ðHb þHbBÞ=kBT�. Then, the noises ϵJðtÞ satisfythe FD relation of the second kind [Eq. (2)] and the Onsagersymmetries [Eq. (3)].In this solvable model, the heat-bath-mediated static

potential Ub which supplements U0 to makeU ¼ U0 þ Ubis found to be

UbðX1 − X2Þ ¼kb2ðX1 − X2 þ LbÞ2; (17)

where

kb ¼Xi

miω2i;1ω

2i;2

~ω2i

;

Lb ¼1

kb

Xi

miω2i;1ω

2i;2ðli;1 − li;2Þ~ω2i

: (18)

Note that Ub depends on X1 and X2 only through X1 − X2;that is, it possesses the translational symmetry (see below).While this form appears in the course of deriving Eq. (1), itsorigin can be simply understood from the followingidentity:

HbB ¼Xi

mi ~ω2i

2~q2i þ UbðX1 − X2Þ: (19)

Finally, our claim [Eq. (5)] is confirmed byEq. (14) for K1;2ð0Þ and by Eqs. (17) and (18) for theU00

bðXÞ ¼ kb. In the standard language of the linearresponse theory, the “displacement” A conjugate tothe external parameter X2ðtÞ − hX2ieq is A ¼P

imiω2i;2ðqi − X2 − li;2Þ and the flux as the response is

B ¼ Pimiω

2i;1ðqi − X1 − li;1Þ [14]. Direct calculation

gives χ1;2ðωÞ ¼P

iðmiω2i;1ω

2i;2Þ=½ ~ω2

i − ðωþ iεÞ2�.A remark is in order about the translational symmetry of

UbðXÞ. In the original Zwanzig model [8], the factorcorresponding to qi−XJ −li;J in Eq. (11) was qi − ciXJwith an arbitrary constant ci and the natural length li;J setto be 0 arbitrarily. In order that the momentum in the heatbath is locally conserved around two Brownian particles,we needed to set ci ¼ 1 and explicitly introduce the naturallength li;J, especially for those gas particles which arecoupled to the both Brownian particles, i.e., withω2i;1ω

2i;2 > 0. We note that the so-called dissipative particle

dynamics modeling [16–18] also respects the local momen-tum conservation.Solvable model II: Langevin system.—The second exam-

ple that confirms the relation (5) is constructed by modi-fying the first one; see Fig. 2(b). There, we replace theHamiltonian evolution of each light mass particle [Eq. (13)]by the overdamped stochastic evolution governed by theLangevin equation:

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0 ¼ −γidqidt

þ ξiðtÞ −mi

X2J¼1

ω2i;Jðqi − XJðtÞ − li;JÞ;

(20)

where γi is the friction constant with which the ithgas particle is coupled to an “outer-heat" bath of thetemperature T. ξiðtÞ is the Gaussian white random forcefrom the outer-heat bath obeying hξiðtÞi ¼ 0, andhξiðtÞξi0 ðt0Þi ¼ 2γikBTδðt − t0Þδi;i0 . This outer-heat bathmay represent those degrees of freedom of the whole-heatbath which are not directly coupled to the Brownianparticles, while the variables ðqi; piÞ represent that freedomof our primary interest as the “system.” (A similar idea hasalready been proposed in different contexts; see Secs. 6.3and 7.1 of Ref. [19] and also Refs. [20–22].) IntegratingEq. (20) for qiðtÞ and substituting the result into the rhs ofEq. (12), we again obtain Eqs. (1) and (2) with the samebath-mediated static potential as before, i.e., Ub defined byEqs. (17) and (18). (In this overdamped model, miω

2i;J

simply represents the spring constant between the ithlight mass and the Jth Brownian particle.) The frictionkernel and the noise term of the present model are,however, different: instead of Eqs. (14) and (15), theyread, respectively,

KJ;J0 ðsÞ ¼Xi

miω2i;Jω

2i;J0

~ω2i

e−ðjsj=τiÞ; (21)

ϵJðtÞ ¼Xi

miω2i;J

Z∞

0

e−ðs=τiÞ

γiξiðt − sÞds; (22)

where τi ¼ γi=ðmi ~ω2i Þ. Because the forms of K1;2ð0Þ as

well as UbðXÞ are unchanged from the first model, ourclaim [Eq. (5)] is again confirmed.Discussion: Implication of Eq. (5).—In applications, we

should note that the only those couplings whose causality isexplicitly retained contribute to the general sum-ruletheorem of the linear response theory (see, for example,Secs. 3.1.2 and 3.5.1 of Ref. [14], where χ∞μν corresponds tosuch coupling to be excluded). That is, if the force-velocityrelationKJ;J0 contains the contributions whose delay can beneglected, those contributions do not participate in Eq. (5).For example, the Stokesian fluid models supplementedby the thermal random forces satisfying the FD relation[23–25] contain such an instantaneous part of KJ;J0 , calledthe friction tensor, without accompanying the Casimir-likepotential force. We also remark that the sum rule relies onlyon the causality of the response function. The relation of thetype of Eq. (5) might, therefore, be generalizable to somecases of nonequilibrium bath [26].The above solvable models, although artificial, represent

certain nonlocal aspects of the more realistic heat baths.The cross frictional kernel K1;2ðsÞ and the bath-mediatedpotential UbðXÞ are generated by those microscopic

degrees of freedom which couple to both the Brownianparticles. This picture is reminiscent of the quantum systeminteracting with electromagnetic fields (see, for exam-ple, Ref. [27]).In Eq. (5), both the cross-memory term (lhs) and the bath-

mediated interaction term (rhs) should generally depend onthe distance between the Brownian particles, while that wasnot the case for the above solvable models. If these terms area smooth function of the particles’ relative position X1 − X2,we may reconstruct Ub from the data of K1;2ð0Þ withdifferent values of hX1ieq − hX2ieq, up to integration con-stants. For example, the Casimir force may obey an inversepower law of the distance between the Brownian particles ifthe radii of Brownian particles as well as the separationbetween them are appropriate. Equation (5) implies that, insuch a case, K1;2 will also show an inverse power law asfunction of the particles’ separation.From an operational point of view, the relation (5)

implies that we cannot control the friction kernels orfriction coefficients without changing the bath-mediatedinteraction between the Brownian particles. As a demon-stration, if all the ωi;J of the light particles are changed by amultiplicative factor λ, i.e., ωi;J↦λωi;J, then both KJ;J0 ðsÞand UbðXÞ should be changed to λ2KJ;J0 ðλsÞ and λ2UbðXÞ,respectively.Especially about the work W of operations, Eq. (5)

implies that the work WK to change the off-diagonalfriction kernel K1;2 cannot be isolated from the workWU to change the bath-mediated interaction potentialUb. In the above solvable models, the total work W ¼WK þWU to change the parameters fωi;Jg can be given asthe Stieltjes integrals along the time evolution of the wholedegrees of freedom:

W ¼Xi

X2J¼1

ZΓ

∂HbB

∂ωi;Jdωi;JðtÞ; (23)

whereRΓ indicates to integrate along the process where all

the dynamical variables pi0 ; ~qi0 and XJ in the integralsevolves according to the system’s dynamics under timedependent parameters fωi;Jg. The operational inseparabilityof the work into WK and WU justifies the fact that, on thelevel of the stochastic energetics [19], we could not accessthe work to change the friction coefficients. On the micro-scopic level, however, the above models allow us to identifyWK: First, WU is given by the above framework [19]:

WU ¼Xi

X2J¼1

ZΓ

∂Ub

∂ωi;Jdωi;JðtÞ (24)

because U0 does not depend on ωi;J. Combining Eq. (24)with Eq. (23) as well as the identity (19), the kinetic part ofthe work WK is found to be

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WK ¼Xi

X2J¼1

ZΓ

∂∂ωi;J

�Xi0

mi0 ~ω2i0

2~q2i0

�dωi;JðtÞ; (25)

where ~qi are defined in Eq. (16). The result again shows that,unless we have access to the microscopic fluctuations in theheat bath, WK is not measurable.In conclusion, we propose, with supporting examples,

that a bath-mediated effective potential between theBrownian particles Ub should accompany the off-diagonalfrictional memory kernel K1;2ðsÞ with a particular relation(5) due to the general sum rule of the linear response theory.This relation should be tested experimentally and/ornumerically on the one hand, and the generalization toother models [3,12] should be explored on the other hand.For example, in the reaction dynamics of protein moleculesor of colloidal particles, nonlocal fluctuations of the solventmay play important roles both kinetically and statically.The consciousness of the environment as a part of thewhole system is important not only in the ecology but alsoat the micron- or nanoscale physics.

This work is supported by the Marie Curie TrainingNetwork NETADIS (FP7, Grant No. 290038) for C. D. B.K. S. acknowledges Antoine Fruleux for fruitful discus-sions. C. D. B. and K. S. thank ICTP (Trieste, Italy) forproviding them with the opportunity to start thecollaboration.

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