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The law of action and reaction for the effective force in a nonequilibrium colloidal system

We study a nonequilibrium Langevin many-body system containing two 'test' particles and many 'background' particles. The test particles are spatially confined by a harmonic potential, and the background particles are driven by an external driving force. Employing numerical simulations of the model, we formulate an effective description of the two test particles in a nonequilibrium steady state. In particular, we investigate several different definitions of the effective force acting between the test particles. We find that the law of action and reaction does not hold for the total mechanical force exerted by the background particles, but that it does hold for the thermodynamic force defined operationally on the basis of an idea used to extend the first law of thermodynamics to nonequilibrium steady states.


Introduction
Colloids are microscopic particles suspended in liquid. Generally, they are on the order of micrometers in size, and thus they are much larger than molecules, but still small enough to exhibit Brownian motion. Colloids are convenient to study experimentally because unlike molecules, they can be easily observed with ordinary microscopes. In recent years, as the technology used in their manipulation has developed, experimental studies of colloids have advanced greatly [1,2,3,4,5,6].
In colloidal many-body systems, melting, freezing, glass transitions and nonequilibrium statistical mechanics have been studied.
Among studies of nonequilibrium steady states (NESSs), Dzubeilla et al. investigated a model that describes two fixed colloidal particles (test particles) in an environment containing many driven particles (background particles). They reported that in a NESS, the effective interaction forces they defined between the two fixed particles violate the law of action and reaction [7]. This result implies that an effective potential seems not to be constructed in NESS.
The effective interaction force defined in Ref. [7] represents the two-body effect of the total force exerted by the background particles, which is extracted by subtracting from it the total force in the case that there is only colloidal particle. Although their definition seems plausible because of its simplicity, we expect that there is room to define a different type of effective force in NESS. With this motivation, we attempt to provide another idea for an effective interaction force between the two fixed particles.
In this paper, we reconsider effective forces in NESSs by studying a system similar to that used in Ref [7]. First, we review the concept of effective forces in equilibrium from three points of view, statistical mechanics, mechanics, and thermodynamics. Next, we investigate effective forces for nonequilibrium systems by considering the results of numerical experiments. Particularly, we demonstrate that in the NESS we consider, the law of action and reaction holds for the effective force defined according to a conjectured thermodynamic relation applied to NESSs.

Model
The model we study describes a system in which two 'test' particles are trapped side by side in the center of the system by a harmonic potential and N 'background' particles are driven by an external force (see Fig. 1). The test particles are unaffected by the driving force, and the background particles are unaffected by the trapping potential. Hereafter, we refer to the test particles as particle 1 and particle 2.
For the sake of simplicity, we consider the idealized situation of a two-dimensional system of length L x and L y in the x and the y directions, with periodic boundary conditions in both the directions. Then, letting R 1 and R 2 be the positions of particle 1 and particle 2, we consider particle 2 fixed for simplicity, with particle 1 trapped by the harmonic potential where R c corresponds to the center of the potential, and k is a spring constant chosen to be sufficiently large that particle 1 can be regarded as almost fixed. It is important to note that, although it may seem simpler to fix particle 1, the dynamical degrees of freedom of particle 1 must be taken into account when we consider the distribution function of the effective description in which the degrees of freedom of the background particles are integrated. [If particle 1 is also fixed, no variable remains in the distribution function of the effective description (see the next section)]. As a further simplification, it is assumed that there is no direct interaction between particles 1 and 2 and that there is no direct interaction among the background particles. Then, we stipulate that the interaction between particle 1 or particle 2 and a background particle is represented by a short-ranged potential function V int ( R i − r k ), where r k is the position of the k-th background particle (k = 1, · · · , N ), of the form (2) With the system as described above, the motion of particle 1 is described by the Langevin equation where γ is a friction constant, and ξ(t) = (ξ x (t), ξ y (t)) represents Gaussian white noise that satisfies Here, the Boltzmann constant is set to unity, and T (= 1/β) is the temperature of the environment. The motion of the k-th background particle is described by where f = (f, 0) is an external driving force, and ξ k (t) = (ξ k,x (t), ξ k,y (t)) is Gaussian white noise that satisfies Below, all the quantities are converted into dimensionless forms by normalizing σ, γ and T to unity. We set the parameters used for our numerical experiments as L x = 11, L y = 6, N = 200, k = 50, V 0 = 7, 0 ≤ f ≤ 0.3, R c = (3, 3) and R 2 = (4.5, 3). Also, all the numerical results in this paper is obtained by a finite difference method of the Langevin equations with a time step 2.5 × 10 −4 .

Review of the effective description
Before analyzing nonequilibrium cases, we review the effective description of particles 1 and 2 in equilibrium, in which the stationary distribution function of the model is the canonical distribution where Z is a normalization constant. In the equilibrium case, employing the wellestablished framework of statistical mechanics, we can derive the effective Hamiltonian of particles 1 and 2 by integrating (7) over the degrees of freedom of the background particles.
For the canonical distribution p c ( R 1 , { r k }; R 2 ) given by (7) and the interaction given by (2), there exists a function R 1 − R 2 , which we write V eff ( R 1 − R 2 ), satisfying the equation It is natural to refer to the function V eff ( R 1 − R 2 ) as the effective interaction potential. The fact that this effective interaction potential depends on only the relative displacement R 1 − R 2 is due to the the spatial symmetry of the system. Then, as the left-hand side of (8) depends on only R 1 and R 2 , it is interpreted as the steady state distribution for the effective description: Clearly the effective potential V eff ( R 1 − R 2 ) obtained in the above manner is properly defined, as this P eff ( R 1 ; R 2 ) is in canonical form.
With the above argument establishing the proper definition of V eff ( R 1 − R 2 ), the statistical mechanical force between particles 1 and 2 is defined naturally by we immediately find the relation which represents the law of action and reaction for statistical mechanical forces in equilibrium. The statistical mechanical force defined above is closely related to the mechanical force exerted by all the background particles, Indeed, using (8) and (10), we can prove the relation where s f =0 represents the statistical average in the steady state with f = 0 (see Subsection 3.1 for the proof of (13)). From (11) and (13), we obtain the law of action and reaction for F mec The statistical mechanical force in equilibrium can also be understood within a thermodynamic framework by considering the case in which an operation is performed on the system. We now explain this understanding. As an example, we consider the simple operation of shifting the position of particle 2 as R 2 → R 2 + ∆ R 2 instantaneously at time t = 0. (See Fig. 2 (a) depicting the case f = 0.) After the application of this operation, the system is left until a new equilibrium state is realized. Then, the work done by the external agent carrying out this operation, W op1 , satisfies the first law of thermodynamics: Here op1 f represents the statistical average with respect to this operation. The quantities Q 1 and Q b are the energies transfered to the heat bath from the degrees of freedom of particle 1 and the background particles. We refer to each of these transferred quantities of energy as heat. According to Ref. [8], the quantities represent the heat transferred per the unit time, and therefore Q 1 and Q b can be defined as The symbol • in (16) and (17) indicates that the product here is the stochastic Stieltjes integral in the Stratonovich sense [8]. Now, using the work, W op1 , we define the thermodynamic force by Then, integrating the Langevin equations (3) and (5), and using the expressioṅ to represent the operation, we can prove the relation (see Subsection 3.2 for the proof of (22)).
In the same manner that we defined the thermodynamic force F thermo 2 , we can also define F thermo 1 by exchanging the roles of particles 1 and 2. (See Fig. 2 depicting the case f = 0.) That is, particle 1 is fixed, while particle 2 is trapped by the harmonic potential. Because this exchange is equivalent to the spatial reflection with respect to R 2 /2, we find (23) Then, from (14), (22) and (23), we derive From the above results, we conclude that in the equilibrium case, because , we can regard any one of these quantities as the effective interaction force between particles 1 and 2. (13) and (22) Proof of (13): The differentiation of (8) with respect to R 2 yields

Proofs of
Integrating this with respect to R 1 , we derive Using the definitions (10) and (12), we obtain (13).
Proof of (22): From (3) and (5) with f = 0, we obtain the energy balance equations: where k = 1, · · · , N . The addition of (28) and (29) with k = 1, · · · , N leads to where we have used (21). Here, we perform the time integration and consider its the statistical average. Noting that the third term of the right hand side of (30) becomes Comparing this with the definition of F thermo 2 given in (15), we obtain

Nonequilibrium case
In this section, which consists of three subsections, we study the nonequilibrium case. We first note that the canonical distribution cannot be employed in the derivation of an effective description for a NESS, in contrast to the equilibrium case. In the case of NESSs, the relations among F stat are not yet understood. In this section, we investigate these three forces as defined above in the nonequilibrium, f = 0 case. We hope that this will allow us to obtain the proper effective description of particles 1 and 2 in a NESS. We first present the numerically computed results determining F mec i and F thermo i as defined in (12)  f =0 , which is consistent with (14). However, it is seen that for f = 0, these quantities are not equal, and indeed the discrepancy between them increases monotonically as a function of f . Hence, the law of action f in the nonequilibrium case, unlike in the equilibrium case. This asymmetry is interpreted as follows. Particle 2 is partially shielded from the effect of the background particles by particle 1 in the case f > 0 (see Fig. 1).

Thermodynamic definition
Let us now reconsider the first law of thermodynamics represented by (15), which provides the definition of F thermo i (i = 1, 2). In contrast to equilibrium cases, (15) cannot be employed directly for NESSs, because J b , defined by (17), takes a nonzero value even when there is no operation applied to the system, and hence Q b becomes infinite. (Recall that the background particles are driven by the external driving force, f . ) In order to extend thermodynamic considerations to NESSs, we introduce the idea of the net heat generated by an operation, following the framework of steady state thermodynamics proposed by Oono and Paniconi [9]. In this framework, the heat necessary to maintain a steady state (the so-called house keeping heat) is regarded as being independent of the change undergone by the system caused by the operation, and the excess heat is defined by subtracting the house-keeping heat from the total heat dissipated to the heat bath. They conjecture that thermodynamics can be extended to NESSs by using this excess heat.
Here, as one example, in Fig. 4, By replacing Q b in (15)   the first law of thermodynamics: Using this W op1 , we define the thermodynamic force in the present NESS by (20).
In Figs. 5 and 6, we plot ∆V t op1 f f as functions of ∆R 2 in the case f = 0.3. It is seen that these quantities depend linearly on ∆R 2 in the range plotted. This implies that ∆R 2 = 0.25 is sufficiently small to be used in (20) to numerically compute the force. Using this value, we obtain Also, we find that Q 1 op1 f is sufficiently small to be ignored (within the numerical accuracy of our simulations). Using these values, we estimate Next, by exchanging the roles of particles 1 and 2 (see Fig. 2 (b)), we can define F thermo 1 in the same way:

Statistical mechanical definition
In this subsection, we investigate the interaction force between particles 1 and 2 in a NESS from a statistical mechanical point of view, employing the steady state distribution function, p s ( R 1 , { r k }; R 2 ). It is known that p s ( R 1 , { r k }; R 2 ) is given by with where A denotes the statistical average of A over noise histories under the condition that the initial condition ( R 1 (0), { r k (0)}) is fixed as the argument ( R 1 , { r k }) of the distribution function p s ( R 1 , { r k }; R 2 ). The expression (38) indicates that the deviation of p s ( R 1 , { r k }; R 2 ) from the canonical distribution, p c ( R 1 , { r k }; R 2 ), is represented by the entropy production βΣ τ . This form of the steady state distribution is similar to that obtained by Zubarev and McLennan [10,11]. (Regarding the derivation for stochastic systems, see Subsection 4.2 of Ref. [12].) Now, referring to the equilibrium case, we define an effective potential V eff ( With this effective potential, we define the statistical mechanical force as Then, differentiating (40) with respect to R 2 , assuming that the relation ∂ Note that (42) is valid to linear order in f because the both hand sides of (42) become We expect that (42) might be approximately valid even for large f , due to many-body effects. In the argument below, we assume (42). On the other hand, using the energy balance equation obtained from the Langevin equations (3) and (5), we can rewrite (34) as F thermo Then, recalling that op1 f is the statistical average with respect to the instantaneous f as a function of ∆R 2 , and satisfies the relation We further assume that Then, (49) and (50) imply that there exists a functionṼ eff ( R 1 − R 2 ) such that It is a future problem to develop a theoretical argument for the suggestion (49).

Discussion
In this paper, we have studied three types of force, F stat f , and hence the law of action and reaction holds for forces of the former kind but not the latter kind. We also demonstrated that F thermo i defined with respect to a form of the first law of thermodynamics hypothesized to hold for our NESS is equal to F stat i s f derived from the expression for the steady state distribution under the assumption represented by (42).
The law of action and reaction holds for any force given by the derivative of a potential function. Thus, the fact that this law holds with respect to the thermodynamic force defined by (34) implies that in the NESS we considered, there may exist a potential function associated with this force. It is important to clarify the physical conditions under which the thermodynamic force obtained using the method employed in this paper indeed is a potential force.
In equilibrium, a potential function is related to the thermodynamic free energy. For this reason, it is natural to conjecture that even in a NESS, the potential force measured experimentally is related to to thermodynamics. Such a conjecture led us to introduce the thermodynamic force based on the idea of Oono and Paniconi [9]. Here we remark that the framework proposed by Oono and Paniconi is embodied in the nonequilibrium Langevin model [13]. In that work, Hatano and Sasa derived an identity which yields an extended second law relating the Shannon entropy with the excess heat they defined. Recently, the validity of this identity was confirmed experimentally for a bead system [14]. In the analysis employed in the present paper, we conjecture thatQ b defined in (33) has a certain connection with the excess heat defined in Ref. [13]. However, we do not yet understand this relation, though apparently the quantityQ b employed here takes a simpler form than that proposed by Hatano and Sasa.
As another important subject related to the present work, we now consider the problem of force decomposition. As seen in Subsection 4.1 (see Fig. 3), the law of action and reaction does not hold for the total mechanical force acting on the test particle, F mec i s f . If the potential force in a NESS can be defined properly on the basis of thermodynamic considerations, it might be possible to decompose F mec i s f into its potential and non-potential parts. Furthermore, because F mec i fluctuates in time, we would like to decompose F mec i into a fluctuating dissipative force, a fluctuating interaction potential force and the remaining part.
With regard to the decomposition of fluctuating forces, recently, a significant result has been obtained for a model of a Brownian particle that is driven by an external driving force and subject to a spatially periodic potential. As this particle is driven in one direction, it experiences repeated 'collisions' with the periodic potential barriers. Studying a finite time average of the force exerted by the periodic potential under nonequilibrium conditions, we have found that a simple orthogonality condition provides a proper force decomposition of the time averaged force into a dissipative force and a non-dissipative part [15]. This result has led us to construct a mathematical technique for re-expressing a Langevin equation in a form in which the response function appears explicitly [16]. Using this technique, an interesting equality that connects energy dissipation with the violation of a fluctuation-response relation has been proved [17]. Considering these developments, the problem of decomposing F mec i s f can be regarded as a natural extension of the study presented in Ref. [15], involving the elimination of the dynamical degrees of freedom.
To summarize the most important point of this work, we have found that the law of action and reaction with respect to thermodynamic forces has a deep connection with fundamental aspects of nonequilibrium statistical mechanics. We believe that the numerical findings reported in this paper will stimulate further theoretical and experimental studies.