Non-equilibrium statistical mechanics of the heat bath for two Brownian particles
Caterina De Bacco, Fulvio Baldovin, Enzo Orlandini, Ken Sekimoto
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Non-equilibrium statistical mechanics of the heat bath for two Brownian particles
Caterina De Bacco, Fulvio Baldovin, Enzo Orlandini, and Ken Sekimoto Laboratoire de Physique Th`eorique et Mod`eles Statistiques,CNRS et Universit´e Paris-Sud 11, UMR8626, Bˆat.100, 91405 Orsay Cedex, France Dipartimento di Fisica e Astronomia G. Galilei and Sezione INFN,Universit`a di Padova, Via Marzolo 8, I-35100 Padova, Italy Mati`eres et Syst`emes Complexes, CNRS-UMR7057, Universit´e Paris-Diderot, 75205 Paris, France Gulliver, CNRS-UMR7083, ESPCI, 75231 Paris, France
We propose a new look at the heat bath for two Brownian particles, in which the heat bath as a ‘system’ is bothperturbed and sensed by the Brownian particles. Non-local thermal fluctuation give rise to bath-mediated staticforces between the particles. Based on the general sum-rule of the linear response theory, we derive an explicitrelation linking these forces to the friction kernel describing the particles’ dynamics. The relation is analyticallyconfirmed in the case of two solvable models and could be experimentally challenged. Our results point out thatthe inclusion of the environment as a part of the whole system is important for micron- or nano-scale physics.
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 en-vironment can mediate static forces between the objects con-stituting its borders. Through a unique combination of thegeneralized Langevin 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 non-locality of the heat bath.The more fine details of Brownian motion are experimen-tally revealed, the more deviations from the idealized Wienerprocess are found (see, for example, [3]). When two Brow-nian particles are trapped close to each other in a heat bath(see Fig.1), the random forces on those objects are no moreindependent noises but should be correlated. Based on theprojection methods [4–6] we expect the generalized Langevinequations to apply [7–10]: M J d X J ( t ) dt = − ∂ U ∂ X J − X J ′ = Z t K J , J ′ ( t − τ ) dX J ′ ( τ ) d τ d τ + ǫ J ( t ) , (1)where X J ( J = M J , and K J , J ′ ( s ) and ǫ J ( t ) are, re-spectively, the friction kernel and the random force. U ( X , X )is the static interaction potential between the Brownian parti-cles. If the environment of the Brownian particles at the initialtime t = T , thenoise and the frictional kernel should satisfy the fluctuation-dissipation (FD) relation of the second kind with the Onsagersymmetries [7, 11]: h ǫ J ( t ) ǫ J ′ ( t ′ ) i = k B T K J , J ′ ( t − t ′ ) , (2) K J , J ′ ( s ) = K J ′ , J ( s ) = K J , J ′ ( − s ) , (3)where J and J ′ are either 1 or 2 independently. This model(1) is a pivotal benchmark model for the correlated Brownianmotion, although the actual Brownian motions could be morecomplicated (see, for example, [3, 12]). But “the physical J=1 J=2
FIG. 1. Two Brownian particles (filled disks, J = J = meaning of the random force autocorrelation function is in thiscase far from clear...” even now and “A proper derivation ofthe effective potential could be of great help in clarifying thislast point” [10]. In addition to the bare potential U ( X , X )independent of the heat bath, the potential U , which is in factthe free energy as function of X J , may contain a bath-mediatedinteraction potential U b ( X , X ) so that U ( X , X ) = U ( X , X ) + U b ( X , X ) . (4)In this Letter we propose the relation K , (0) = − ∂∂ X ∂∂ X U b ( X , X ) , (5)where the both sides of this relation should be evaluated at theequilibrium positions of the Brownian particles, X J = h X J i eq . This relation implies that the bath-mediated static interactionis always correlated with the frictional one. Our approach isto regard the heat bath as the weakly non-equilibrium systemwhich is both perturbed and sensed by the mesoscopic Brow-nian particles. From this point of view (5) is deduced from socalled ‘general sum-rule theorem’ [13] of the linear responsetheory of non-equilibrium statistical mechanics [14]. Whilethe FD relation of the second kind (2) is well known as anoutcome of this theory, the other aspects have not been fullyexplored. Below we give a general argument supporting (5),and then give two analytically solvable examples for whichthe claim holds exactly.
General argument —
While the spatial dimensionality isnot restrictive in the following argument, we will use the no-tations as if the space were one-dimensional. Suppose we ob-serve the force F , on the J = J = h X i eq at t = −∞ to X ( t ) at t . Due to thesmall perturbation X ( t ) −h X i eq , the average force at that time, h F , i t , is deviated from its its equilibrium value, h F , i eq . Thelinear response theory relates these two through the responsefunction, Φ , ( s ) as (cid:10) F , (cid:11) t − (cid:10) F , (cid:11) eq . = Z t −∞ Φ , ( t − τ ) ( X ( τ ) − h X i eq ) d τ. (6)(Within the linear theory the force is always measured at X = h X i eq .) The complex admittance χ , ( ω ) = χ ′ , ( ω ) + i χ ′′ , ( ω )is defined as the Fourier-Laplace transformation of Φ , ( s ) : χ , ( ω ) = Z + ∞ e i ω s − ε s Φ , ( s ) ds , (7)where ε is a positive infinitesimal number (i.e., + χ , ( ∞ ) =
0, which is the case in the present , the causalityof Φ , ( t ) , or the analyticity of χ , ( ω ) in the upper half com-plex plane of ω , impose the general sum rule [13], P Z + ∞−∞ χ ′′ , ( ω ) ω d ωπ = χ ′ , (0) , (8)where P on the left hand side (l.h.s.) denotes to take the prin-cipal value of the integral across ω =
0. The significance of(8) is that it relates the dissipative quantity (l.h.s.) and the re-versible static response (right hand side (r.h.s.)) of the system.Now we suppose, along the thought of Onsager’s mean re-gression hypothesis [15], that the response of the heat bath tothe fluctuating Brownian particles, which underlies (1), is es-sentially the same as the response to externally specified per-turbations described by (6). Thus the comparison of (6) with(1) gives Φ , ( t ) = − dK , ( t ) dt , (9)or, in other words, K , is the relaxation function correspond-ing to Φ , . With this linkage between the Langevin descrip-tion and the linear response theory, the static reversible re-sponse χ ′ , (0) of the force h F , i − h F , i eq to the static dis-placement X − h X i eq can be identified with the r.h.s. of (5).As for the l.h.s. of (8), we can show by (9) and (7) that it isequal to K , (0) . The argument presented here is to be testedboth analytically/numerically and experimentally. At least forthe two models presented below the claim (5) is analyticallyconfirmed.
Solvable model I: Hamiltonian system—
As the first ex-ample that confirms the relation (5) we take up a Hamilto-nian model inspired by the classic model of Zwanzig [8], see Fig. 2(a). Instead of a single Brownian particle [8] we put thetwo Brownian particles with masses M J ( J = ,
2) which in-teract with the ‘bath’ consisting of light mass ‘gas’ particles.While Fig. 2(a) gives the general idea, the solvable model islimited to the one-dimensional space. Each gas particle, e.g. i -th one, has a mass m i ( ≪ M J ) and is linked to at least one ofthe Brownian particles, J = m i ω i , J ( >
0) and the natural length, ℓ i , J .In Fig. 2(a) these links are represented by the dashed lines.The Hamiltonian of this purely mechanical model consists ofthree parts, H = H B + H b + H bB , with H B = P M + P M + U ( X , X ) , (10) H b = X i p i m i , H bB = X i m i X J = ω i , J ( q i − X J − ℓ i , J ) , (11)where the pairs ( X J , P J = M J dX J / dt ) and ( x i , p i = m i dx i / dt )denote, respectively, the positions and momenta of the heavy( J ) and light ( i ) particles. The Brownian particles obey thefollowing dynamics : M J d X J dt = − ∂ U ∂ X J + X i m i ω i , J ( q i − X J − ℓ i , J ) . (12) (a)(b) FIG. 2. (a)
Hamiltonian model of two Brownian particles which isanalytically solvable for one dimensional space with harmonic cou-pling. Each light mass particle (thick dot) is linked to at least one ofthe Brownian particles (filled disks) with Hookean springs (dashedlines). (b)
Langevin model of two Brownian particles. Unlike theHamiltonian model, each light mass particle receives the randomforce and frictional force from the background (shaded zone) andits inertia is ignored.
Given the initial values of ( q i , p i ) at t =
0, the Hamiltonequations for ( q i ( t ) , p i ( t )) , which reads m i d q i dt = − m i X J = ω i , J ( q i − X J ( t ) − ℓ i , J ) , (13)can be solved in supposing that the histories of X J ( s ) ( J = ≤ s ≤ t are given. In order to assure the compat-ibility with the initial canonical equilibrium of the heat bath,we assume the vanishing initial velocity for the Brownian par-ticles, dX J / dt | t = = . Substituting each q i in (12) by its for-mal solution thus obtained, the dynamics of X J ( t ) is rigorouslyreduced to (1), where the friction kernels K J , J ′ ( s ) are K J , J ′ ( s ) = X i m i ω i , J ω i , J ′ ˜ ω i cos( ˜ ω i s ) , (14)and the noise term ǫ J ( t ) is ǫ J ( t ) ≡ X i m i ω i , J ( ˜ q i (0) cos( ˜ ω i t ) + d ˜ q i (0) dt sin( ˜ ω i t )˜ ω i ) , (15)with ˜ ω i ≡ ω i , + ω i , and˜ q i ( t ) ≡ q i ( t ) − X J = ω i , J ˜ ω i [ ℓ i , J + X J ( t )] . (16)To our knowledge this is the first concrete model that demon-strates (1). Only those gas particles linked to the both Brow-nian particles satisfy ω i , ω i , > K , ( s ).While the generalized Langevin form (1) holds for an indi-vidual realization without any ensemble average, the statis-tics of ǫ J ( t ) must be specified. We assume that at t = q i (0) and ˜ p i (0) ( = p i (0) because we defined dX J / dt | t = =
0) belong to the canonical ensemble of a tem-perature T with the weight ∝ exp( − ( H b + H bB ) / k B T ) . Thenthe noises ǫ J ( t ) satisfy the FD relation of the second kind (2).and the Onsager symmetries (3).In this solvable model, the heat bath-mediated static poten-tial U b which supplements U to make U = U + U b is foundto be U b ( X − X ) = k b X − X − L b ) , (17)where k b = X i m i ω i , ω i , ˜ ω i , L b = k b X i m i ω i , ω i , ( ℓ i , − ℓ i , )˜ ω i . (18)Note that U b depends on X and X only through X − X , thatis, it possesses the translational symmetry (see later). Whilethis form appears in the course of deriving (1), its origin canbe simply understood from the following identity: H bB = X i m i ˜ ω i q i + U b ( X − X ) . (19) Finally, our claim (5) is confirmed by (14) for K , (0) and by(17) and (18) for the U ′′ b ( X ) = k b . In the standard language ofthe linear response theory, the ‘displacement’ A conjugate tothe external parameter X ( t ) −h X i eq is A = P i m i ω i , ( q i − X − ℓ i , ) and the flux as the response is B = P i m i ω i , ( q i − X − ℓ i , )[14]. Direct calculation gives χ , ( ω ) = P i ( m i ω i , ω i , ) / [ ˜ ω i − ( ω + i ε ) ] . A remark is in order about the translational symmetry of U b ( X ). In the original Zwanzig model [8], the factor corre-sponding to q i − X J − ℓ i , J in (11) was q i − c i X J with an ar-bitrary constant c i and the natural length ℓ i , J set to be zeroarbitrarily. In order that the momentum in the heat bath is lo-cally conserved around two Brownian particles, we needed toset c i = ℓ i , J , es-pecially for those gas particles which are coupled to the bothBrownian particles, i.e. with ω i , ω i , > . We note that theso-called dissipative particle dynamics modeling [16–18] alsorespects the local momentum conservation.
Solvable model II: Langevin system.—
The second examplethat confirms the relation (5) is constructed by modifying thefirst one, see Fig. 2(b). There, we replace the Hamiltonianevolution of each light mass particle (13) by the over-dampedstochastic evolution governed by the Langevin equation;0 = − γ i dq i dt + ξ i ( t ) − m i X J = ω i , J ( q i − ℓ i , J − X J ( t )) , (20)where γ i is the friction constant with which the i -th gas par-ticle is coupled to a ‘outer’-heat bath of the temperature T .ξ i ( t ) is the Gaussian white random force from the outer-heatbath obeying h ξ i ( t ) i = , and h ξ i ( t ) ξ i ′ ( t ′ ) i = γ i k B T δ ( t − t ′ ) δ i , i ′ . This outer-heat bath may represent those degrees of freedomof the whole heat bath which are not directly coupled to theBrownian particles, while the variables ( q i , p i ) represent thosefreedom of our primary interest as the ‘system’. (Similar ideahas already been proposed in different contexts, see [19] §6.3and §7.1, and also [20–22].) Integrating (20) for q i ( t ) and sub-stituting the result into the r.h.s. of (12), we again obtain (1)and (2) with the same bath-mediated static potential as before,i.e., U b defined by (17) and (18). (In this over-damped model, m i ω i , J simply represents the spring constant between the i -thlight mass and the J -th Brownian particle.) The friction kerneland the noise term of the present model are, however, differ-ent: instead of (14) and (15), they read, respectively, K J , J ′ ( s ) = X i m i ω i , J ω i , J ′ ˜ ω i e − | s | τ i , (21) ǫ J ( t ) = X i m i ω i , J Z ∞ e − s τ i γ i ξ i ( t − s ) ds , (22)where τ i = γ i / ( m i ˜ ω i ) . Because the form of K , (0) as well as U b ( X ) are unchanged from the first model, our claim (5) isagain confirmed. Discussion : Implication of (5) —