PRL 112, 180605 (2014)

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Nonequilibrium Statistical Mechanics of the Heat Bath for Two Brownian Particles 1

Caterina De Bacco,1 Fulvio Baldovin,2 Enzo Orlandini,2 and Ken Sekimoto3,4*

Laboratoire de Physique Théorique et Modéles Statistiques, CNRS et Université Paris-Sud 11, UMR8626, Bâtiment 100, 91405 Orsay Cedex, France 2 Dipartimento di Fisica e Astronomia Galileo Galilei and Sezione INFN, Università di Padova, Via Marzolo 8, I-35100 Padova, Italy 3 Matières et Systèmes Complexes, CNRS-UMR7057, Université Paris-Diderot, 75205 Paris, France 4 Gulliver, 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 bathmediated static forces between the particles. Based on the general sum rule of the linear response theory, we derive an explicit relation linking these forces to the friction kernel describing the particles’ dynamics. The relation is analytically confirmed in the case of two solvable models and could be experimentally challenged. Our results point out that the inclusion of the environment as a part of the whole system is important 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 fluctuating environment can mediate static forces between the objects constituting its borders. Through a unique combination of the generalized Langevin equation and the linear response theory, we uncover a link between such interactions and the correlated Brownian motions with memory, both of which reflect the spatiotemporal nonlocality of the heat bath. The more fine details of Brownian motion are experimentally revealed, the more deviations from the idealized Wiener process are found (see, for example, Ref. [3]). When two Brownian particles are trapped close to each other in a heat bath (see Fig. 1), the random forces on those objects are no more independent noises but should be correlated. Based on the projection methods [4–6], we expect the generalized Langevin equations to apply [7–10]:

where J and J 0 are either 1 or 2 independently. This model [Eq. (1)] is a pivotal benchmark model for the correlated Brownian motion, although the actual Brownian motions could be more complicated (see, for example, Refs. [3,12]). But, “the physical meaning of the random force autocorrelation function is in this case far from clear…” even now, and “A proper derivation of the effective potential could be of great help in clarifying this last point” [10]. In addition to the bare potential U 0 ðX1 ; X2 Þ independent of the heat bath, the potential U, which is in fact the free energy as a function of XJ , may contain a bath-mediated interaction potential Ub ðX1 ; X2 Þ so that UðX1 ; X2 Þ ¼ U 0 ðX1 ; X2 Þ þ U b ðX1 ; X2 Þ: In this Letter, we propose the relation

2 Z t d2 XJ ðtÞ ∂U X dX 0 ðτÞ MJ dτ ¼− − K J;J0 ðt − τÞ J 2 ∂XJ J0 ¼1 0 dτ dt

þ ϵJ ðtÞ;

K 1;2 ð0Þ ¼ − (1)

where XJ (J ¼ 1 and 2) are the positions of the Brownian particles with the mass being M J , and K J;J0 ðsÞ and ϵJ ðtÞ are, respectively, the friction kernel and the random force. UðX 1 ; X2 Þ is the static interaction potential between the Brownian particles. If the environment of the Brownian particles at the initial time t ¼ 0 is in canonical equilibrium at temperature T, the noise and the frictional kernel should satisfy the fluctuation-dissipation (FD) relation of the second kind with the Onsager symmetries [7,11]: hϵJ ðtÞϵJ0 ðt0 Þi ¼ kB TK J;J0 ðt − t0 Þ;


K J;J0 ðsÞ ¼ K J0 ;J ðsÞ ¼ K J;J0 ð−sÞ;




∂ ∂ U ðX ; X Þ; ∂X1 ∂X 2 b 1 2


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



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


© 2014 American Physical Society

PRL 112, 180605 (2014)

XJ ¼ hXJ ieq . This relation implies that the bath-mediated static interaction is always correlated with the frictional one. Our approach is to regard the heat bath as the weakly nonequilibrium system which is both perturbed and sensed by the mesoscopic Brownian particles. From this point of view, Eq. (5) is deduced from the so-called “general sumrule theorem” [13] of the linear response theory of nonequilibrium statistical mechanics [14]. While the FD relation of the second kind [Eq. (2)] is well known as an outcome of this theory, the other aspects have not been fully explored. Below, we give a general argument supporting Eq. (5) and then give two analytically solvable examples for which the claim holds exactly. General argument.—While the spatial dimensionality is not restrictive in the following argument, we will use the notations as if the space were one dimensional. Suppose we observe the force F1;2 on the J ¼ 1 particle as we move the J ¼ 2 particle from hX2 ieq at t ¼ −∞ to X2 ðtÞ at t. Because of the small perturbation X2 ðtÞ − hX2 ieq , the average force at that time hF1;2 it is deviated from its equilibrium value hF1;2 ieq . The linear response theory relates these two through the response function Φ1;2 ðsÞ as Z hF1;2 it − hF1;2 ieq: ¼



Φ1;2 ðt − τÞ½X2 ðτÞ − hX2 ieq dτ: (6)

(Within the linear theory, the force is always measured at X1 ¼ hX 1 ieq .) The complex admittance χ 1;2 ðωÞ ¼ χ 01;2 ðωÞ þ iχ 001;2 ðωÞ is defined as the Fourier-Laplace transformation of Φ1;2 ðsÞ: Z χ 1;2 ðωÞ ¼



eiωs−εs Φ1;2 ðsÞds;

Φ1;2 ðtÞ ¼ −




χ 001;2 ðωÞ dω ¼ χ 01;2 ð0Þ; π ω

dK 1;2 ðtÞ ; dt


or, in other words, K 1;2 is the relaxation function corresponding to Φ1;2 . With this linkage between the Langevin description and the linear response theory, the static reversible response χ 01;2 ð0Þ of the force hF1;2 i − hF1;2 ieq to the static displacement X2 − hX2 ieq can be identified with the rhs of Eq. (5). As for the lhs of Eq. (8), we can show by Eqs. (9) and (7) that it is equal to K 1;2 ð0Þ. The argument presented here is to be tested both analytically or numerically and experimentally. At least for the two models presented below, the claim [Eq. (5)] is analytically confirmed. Solvable model I: Hamiltonian system.—As the first example that confirms the relation (5), we take up a Hamiltonian model inspired by the classic model of Zwanzig [8]; see Fig. 2(a). Instead of a single Brownian particle [8], we put the two Brownian particles with masses MJ (J ¼ 1; 2) which interact with the “bath” consisting of light mass “gas” particles. While Fig. 2(a) gives the general idea, the solvable model is limited to the one-dimensional space. Each gas particle, e.g., the ith one, has a mass mi (≪ M J ) and is linked to at least one of the Brownian particles J ¼ 1 or 2 through Hookean springs of the spring constant mi ω2i;J ð> 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 consists of three parts H ¼ HB þ Hb þ H bB , with (a)


where ε is a positive infinitesimal number (i.e., þ0). If χ 1;2 ð∞Þ ¼ 0 (see the section titled Discussion), the causality of Φ1;2 ðtÞ, or the analyticity of χ 1;2 ðωÞ in the upper half complex plane of ω, imposes the general sum rule [13] P

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where P on the left-hand side (lhs) is denoted as taking the principal value of the integral across ω ¼ 0. The significance 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 heat bath to the fluctuating Brownian particles, which underlies Eq. (1), is essentially the same as the response to externally specified perturbations described by Eq. (6). Thus, the comparison of Eq. (6) with Eq. (1) gives

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


PRL 112, 180605 (2014)

P21 P2 þ 2 þ U 0 ðX1 ; X2 Þ; 2M 1 2M 2

HB ¼ Hb ¼ HbB ¼


X p2 i ; 2m i i 2 X mi X i


ω2i;J ðqi − XJ − li;J Þ2 ;



where the pairs ðXJ ; PJ ¼ M J dXJ =dtÞ and ðxi ; pi ¼ mi dxi =dtÞ denote, respectively, the positions and momenta of the heavy (J) and light (i) particles. The Brownian particles obey the following dynamics: MJ

d2 XJ ∂U0 X ¼− þ mi ω2i;J ðqi − XJ − li;J Þ: 2 ∂XJ dt i

2 X d2 qi ¼ −m ω2i;J ðqi − XJ ðtÞ − li;J Þ; i dt2 J¼1

X mi ω2i;J ω2i;J0 ω~ 2i


~ i sÞ cosðω



and the noise term ϵJ ðtÞ is   X ~ i tÞ dq~ i ð0Þ sinðω 2 ϵJ ðtÞ ≡ ; mi ωi;J q~ i ð0Þ cosðω~ i tÞ þ ~i dt ω i (15) with ω~ 2i ≡ ω2i;1 þ ω2i;2 and q~ i ðtÞ ≡ qi ðtÞ −

2 X ω2i;J J¼1

ω~ 2i

½li;J þ XJ ðtÞ:

Ub ðX1 − X2 Þ ¼


To our knowledge, this is the first concrete model that demonstrates Eq. (1). Only those gas particles linked to both Brownian particles satisfy ω2i;1 ω2i;2 > 0 and contribute to K 1;2 ðsÞ. While the generalized Langevin form [Eq. (1)] holds for an individual realization without any ensemble average, the statistics of ϵJ ðtÞ must be specified. We assume

kb ðX − X2 þ Lb Þ2 ; 2 1


where kb ¼

X mi ω2i;1 ω2i;2 ~ 2i ω


can be solved in supposing that the histories of XJ ðsÞ (J ¼ 1 and 2) for 0 ≤ s ≤ t are given. In order to assure the compatibility with the initial canonical equilibrium of the heat bath, we assume the vanishing initial velocity for the Brownian particles dXJ =dtjt¼0 ¼ 0. Substituting each qi in Eq. (12) by its formal solution thus obtained, the dynamics of XJ ðtÞ is rigorously reduced to Eq. (1), where the friction kernels K J;J0 ðsÞ are K J;J0 ðsÞ ¼

that at t ¼ 0, the bath variables q~ i ð0Þ and p~ i ð0Þ [¼ pi ð0Þ because we defined dXJ =dtjt¼0 ¼ 0] belong to the canonical ensemble of a temperature T with the weight ∝ exp½−ðH b þ HbB Þ=kB T. Then, the noises ϵJ ðtÞ satisfy the FD relation of the second kind [Eq. (2)] and the Onsager symmetries [Eq. (3)]. In this solvable model, the heat-bath-mediated static potential Ub which supplements U0 to make U ¼ U 0 þ Ub is found to be


Given the initial values of ðqi ; pi Þ at t ¼ 0, the Hamilton equation for ðqi ðtÞ; pi ðtÞÞ, which reads mi

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Lb ¼


1 X mi ω2i;1 ω2i;2 ðli;1 − li;2 Þ : kb i ω~ 2i


Note that U b depends on X 1 and X 2 only through X1 − X2 ; that is, it possesses the translational symmetry (see below). While this form appears in the course of deriving Eq. (1), its origin can be simply understood from the following identity: HbB ¼

X mi ω~ 2 i



q~ 2i þ Ub ðX1 − X2 Þ:


Finally, our claim [Eq. (5)] is confirmed by Eq. (14) for K 1;2 ð0Þ and by Eqs. (17) and (18) for the U00b ðXÞ ¼ kb . In the standard language of the linear response theory, the “displacement” A conjugate to the external parameter X2 ðtÞ − hX2 ieq is A ¼ P 2 m ω i i;2 ðqi − X 2 − li;2 Þ and the flux as the response is i P B ¼ i mi ω2i;1 ðqP i − X 1 − li;1 Þ [14]. Direct calculation gives χ 1;2 ðωÞ ¼ i ðmi ω2i;1 ω2i;2 Þ=½ω~ 2i − ðω þ iεÞ2 . A remark is in order about the translational symmetry of Ub ðXÞ. In the original Zwanzig model [8], the factor corresponding to qi − XJ − li;J in Eq. (11) was qi − ci XJ with an arbitrary constant ci and the natural length li;J set to be 0 arbitrarily. In order that the momentum in the heat bath is locally conserved around two Brownian particles, we needed to set ci ¼ 1 and explicitly introduce the natural length li;J , especially for those gas particles which are coupled to the both Brownian particles, i.e., with ω2i;1 ω2i;2 > 0. We note that the so-called dissipative particle dynamics modeling [16–18] also respects the local momentum conservation. Solvable model II: Langevin system.—The second example that confirms the relation (5) is constructed by modifying the first one; see Fig. 2(b). There, we replace the Hamiltonian evolution of each light mass particle [Eq. (13)] by the overdamped stochastic evolution governed by the Langevin equation:


PRL 112, 180605 (2014) 0 ¼ −γ i


2 X dqi þ ξi ðtÞ − mi ω2i;J ðqi − XJ ðtÞ − li;J Þ; dt J¼1

(20) where γ i is the friction constant with which the ith gas particle is coupled to an “outer-heat" bath of the temperature T. ξi ðtÞ is the Gaussian white random force from the outer-heat bath obeying hξi ðtÞi ¼ 0, and hξi ðtÞξi0 ðt0 Þi ¼ 2γ i kB Tδðt − t0 Þδi;i0 . This outer-heat bath may represent those degrees of freedom of the whole-heat bath which are not directly coupled to the Brownian particles, while the variables ðqi ; pi Þ represent that freedom of our primary interest as the “system.” (A similar idea has already been proposed in different contexts; see Secs. 6.3 and 7.1 of Ref. [19] and also Refs. [20–22].) Integrating Eq. (20) for qi ðtÞ and substituting the result into the rhs of Eq. (12), we again obtain Eqs. (1) and (2) with the same bath-mediated static potential as before, i.e., Ub defined by Eqs. (17) and (18). (In this overdamped model, mi ω2i;J simply represents the spring constant between the ith light mass and the Jth Brownian particle.) The friction kernel and the noise term of the present model are, however, different: instead of Eqs. (14) and (15), they read, respectively, K J;J0 ðsÞ ¼

X mi ω2i;J ω2i;J0 ω~ 2i


Z X 2 mi ωi;J ϵJ ðtÞ ¼ i


e−ðjsj=τi Þ ;

e−ðs=τi Þ ξi ðt − sÞds; γi



~ 2i Þ. Because the forms of K 1;2 ð0Þ as where τi ¼ γ i =ðmi ω well as U b ðXÞ are unchanged from the first model, our claim [Eq. (5)] is again confirmed. Discussion: Implication of Eq. (5).—In applications, we should note that the only those couplings whose causality is explicitly retained contribute to the general sum-rule theorem of the linear response theory (see, for example, Secs. 3.1.2 and 3.5.1 of Ref. [14], where χ ∞ μν corresponds to such coupling to be excluded). That is, if the force-velocity relation K J;J0 contains the contributions whose delay can be neglected, those contributions do not participate in Eq. (5). For example, the Stokesian fluid models supplemented by the thermal random forces satisfying the FD relation [23–25] contain such an instantaneous part of K J;J0 , called the friction tensor, without accompanying the Casimir-like potential force. We also remark that the sum rule relies only on the causality of the response function. The relation of the type of Eq. (5) might, therefore, be generalizable to some cases of nonequilibrium bath [26]. The above solvable models, although artificial, represent certain nonlocal aspects of the more realistic heat baths. The cross frictional kernel K 1;2 ðsÞ and the bath-mediated potential Ub ðXÞ are generated by those microscopic

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degrees of freedom which couple to both the Brownian particles. This picture is reminiscent of the quantum system interacting with electromagnetic fields (see, for example, Ref. [27]). In Eq. (5), both the cross-memory term (lhs) and the bathmediated interaction term (rhs) should generally depend on the distance between the Brownian particles, while that was not the case for the above solvable models. If these terms are a smooth function of the particles’ relative position X1 − X2 , we may reconstruct Ub from the data of K 1;2 ð0Þ with different values of hX1 ieq − hX2 ieq , up to integration constants. For example, the Casimir force may obey an inverse power law of the distance between the Brownian particles if the radii of Brownian particles as well as the separation between them are appropriate. Equation (5) implies that, in such a case, K 1;2 will also show an inverse power law as function of the particles’ separation. From an operational point of view, the relation (5) implies that we cannot control the friction kernels or friction coefficients without changing the bath-mediated interaction between the Brownian particles. As a demonstration, if all the ωi;J of the light particles are changed by a multiplicative factor λ, i.e., ωi;J ↦λωi;J , then both K J;J0 ðsÞ and U b ðXÞ should be changed to λ2 K J;J0 ðλsÞ and λ2 U b ðXÞ, respectively. Especially about the work W of operations, Eq. (5) implies that the work W K to change the off-diagonal friction kernel K 1;2 cannot be isolated from the work W U to change the bath-mediated interaction potential Ub . In the above solvable models, the total work W ¼ W K þ W U to change the parameters fωi;J g can be given as the Stieltjes integrals along the time evolution of the whole degrees of freedom: 2 Z XX ∂HbB W¼ dωi;J ðtÞ; ∂ωi;J i J¼1 Γ


R where Γ indicates to integrate along the process where all the dynamical variables pi0 ; q~ i0 and X J in the integrals evolves according to the system’s dynamics under time dependent parameters fωi;J g. The operational inseparability of the work into W K and W U justifies the fact that, on the level of the stochastic energetics [19], we could not access the work to change the friction coefficients. On the microscopic level, however, the above models allow us to identify W K : First, W U is given by the above framework [19]: WU ¼

2 Z XX ∂U b dωi;J ðtÞ ∂ω i;J i J¼1 Γ


because U 0 does not depend on ωi;J . Combining Eq. (24) with Eq. (23) as well as the identity (19), the kinetic part of the work W K is found to be


PRL 112, 180605 (2014) WK ¼

X  ~ 2i0 2 mi0 ω ∂ q~ i0 dωi;J ðtÞ; 2 Γ ∂ωi;J i0

2 Z XX i



where q~ i are defined in Eq. (16). The result again shows that, unless we have access to the microscopic fluctuations in the heat bath, W K is not measurable. In conclusion, we propose, with supporting examples, that a bath-mediated effective potential between the Brownian particles Ub should accompany the off-diagonal frictional memory kernel K 1;2 ðsÞ with a particular relation (5) due to the general sum rule of the linear response theory. This relation should be tested experimentally and/or numerically on the one hand, and the generalization to other models [3,12] should be explored on the other hand. For example, in the reaction dynamics of protein molecules or of colloidal particles, nonlocal fluctuations of the solvent may play important roles both kinetically and statically. The consciousness of the environment as a part of the whole system is important not only in the ecology but also at the micron- or nanoscale physics. This work is supported by the Marie Curie Training Network NETADIS (FP7, Grant No. 290038) for C. D. B. K. S. acknowledges Antoine Fruleux for fruitful discussions. C. D. B. and K. S. thank ICTP (Trieste, Italy) for providing them with the opportunity to start the collaboration.

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Nonequilibrium statistical mechanics of the heat bath for two Brownian particles.

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 p...
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