Generalized Fluctuation-Dissipation relations holding in non-equilibrium dynamics
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Generalized Fluctuation-Dissipation relations holding innon-equilibrium dynamics
Lorenzo Caprini Scuola di Scienze e Tecnologie, Università di Camerino - via Madonna delle Carceri, 62032, Camerino,ItalyE-mail: [email protected]
Gennuary 2019
Abstract.
We derive generalized Fluctuation-Dissipation Relations (FDR) holding for a generalstochastic dynamics that includes as subcases both equilibrium models for passive colloids and non-equilibrium models used to describe active particles. The relations reported here differ from previousformulations of the FDR because of their simplicity: they require only the microscopic knowledgeof the dynamics instead of the whole expression of the steady-state probability distribution functionthat, except for linear interactions, is unknown for systems displaying non-vanishing currents. Fromthe response function, we can extrapolate generalized versions of the Mesoscopic Virial equation andthe equipartition theorem, which still holds far from equilibrium. Our results are tested in the caseof equilibrium colloids described by underdamped or overdamped Langevin equations and for modelsdescribing the non-equilibrium behavior of active particles. Both the Active Brownian Particle and theActive Ornstein-Uhlenbeck particle models are compared in the case of a single particle confined in anexternal potential. eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics
1. Introduction
The Fluctuation-Dissipation relations (FDR) represent a fundamental topic in statistical physics with along history. It dates back to the pioneering work of Einstein about the relation between mobility anddiffusivity. Einstein’s picture was unified by Kubo [1] that through its linear response theory was ableto predict the transport coefficients and received an outstanding contribution by the pivotal Onsager’swork on reciprocal relations [2] holding near the equilibrium. In these cases, the equilibrium feature ofthe dynamics and the consequent validity of the detailed balance lead to simple results that played acrucial role in many areas of physics.The relation between the response function due to a small perturbation and suitable correlationsevaluated in the unperturbed system still represents a fundamental topic to explore non-equilibriumphysics leading to the challenging issue of obtaining generalized versions of the FDR holdingindependently of the detailed balance [3]. In the last forty years, several formulations of generalized FDRhave been derived using different approaches. Vulpiani et al. [4] and Agarwal [5] obtained independentlygeneralized FDR for chaotic deterministic and stochastic systems, respectively. Similar formulationsconnect the response functions to well-known observables in the framework of stochastic thermodynamics,such as the entropy production [6, 7]. Moreover, both the relations remain, somehow, implicit sinceexplicitly depend on the steady-state probability distribution function, which is typically unknownfor non-equilibrium dynamics. Successively, path-integral approaches starting from the probabilityassociated with a stochastic trajectory have been employed to derive a new kind of relations, stillholding far from the equilibrium. First examples have been obtained for systems composed by discretespin variables [8] and for Langevin dynamics [9–13], both in the overdamped and underdamped regimes.Using this approach, Maes et al. focused on the different roles of entropic and frenetic contributions(see, here, for a recent review [14]) that distinguish for their parity under time-reversal transformation.The path-integral technique leads also to another formulation of the FDR connecting the response to acorrelation that involves the noise. The method is known as
Malliavin weight sampling [15] (see also theNovikov theorem [16]) and has been mostly employed in the context of glassy systems to calculate thesusceptibility and the effective temperature [17–19]. While the Malliavin weight method is particularlyefficient and works also for many-body systems, often lacks transparency in the physical meaning of thecorrelations that are involved in the numerical calculation.Both the approaches have been recently applied in the context of an emergent class of non-equilibrium dynamics, introduced to describe many biological and physical systems in the framework ofActive Matter [20–24]. These systems usually store energy from the environment, for instance throughmechanical agents or chemical reactions, to produce directed motion, and represent a good platformto test any version of the generalized FDR [25, 26]. Specifically, the approach of Ref. [3] has beenapplied to active matter systems in the limit of small activity (in particular, small persistence time) [27].Extending this approach far from equilibrium has the same level of complexity of solving the non-equilibrium active dynamics. In the same spirit, near-equilibrium FDR have been derived using path-integral techniques [28] leading to a near-equilibrium expression for the susceptibility. More generalresults holding also far from equilibrium, both for small and large activities, have been obtained afterthe generalization of the Malliavin weight sampling procedure to active particle dynamics [29]. Forinstance, this technique has been employed to numerically calculate i) the effective temperature of activesystems [30–34], with a recent attention to phase-separation [35], and ii) the transport coefficients, such eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics
2. The Response function due to a small perturbation
To define the response function and successively derive our version of the FDR, we introduce a generalstochastic dynamics which describes the evolution of a set of N variables, namely x = ( x , x , ..., x N ) .The set of stochastic differential equations, of which we refer to as the unperturbed dynamics , is thefollowing: ˙ x i = F i ( x , t ) + σ ij · ξ j , (1)where we have adopted the Einstein summation convention. The term F i ( x , t ) contains all thedeterministic contributions ruling the dynamics of x i and in the following will be denoted as a force .This is a general function that could depend on the whole set of the state variables and could evencontain an explicit dependence on the time, t . This choice of the force allows us to describe bothequilibrium systems characterized by Boltzmann distributions and non-equilibrium systems with non-vanishing steady-state currents induced by F i ( x , t ) . The general dynamics (1) includes a broad rangeof non-equilibrium models that have been largely employed to describe systems of biological and/ortechnological interest, for instance in the context of active matter. These examples will be explicitlydiscussed in the final part of Sec. 4. The term ξ j is a white noise with zero average and unit variance,such that h ξ j ( t ) · ξ i ( s ) i = 2 δ ji δ ( t − s ) , where δ ij is the Kronecker function and δ ( t − s ) is the Dirac- δ function. Finally, the term σ ij is a generalmatrix that determines the amplitudes of each noise term (and that could also be non-symmetric). Itssquare gives rise to the diffusion matrix, D ij = σ ik σ kj . The dynamics (1) has a very general form andcould include also deterministic variables if the matrix σ is singular, as in the usual case of a particledescribed in terms of position and velocity. From now, we choose σ as a general matrix with constantelements, restricting our analysis to the case of additive noise and, thus, excluding any multiplicativedynamics. eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics h i ( t ) smaller than the other force contributionsso that the perturbed variables, x hi , which will be denoted by the superscript h , evolves as: ˙ x hi = F i ( x h , t ) + σ ij ξ j + h i ( t ) . (2)As a result, the set of perturbed variables x h deviate from the unperturbed set of variables x by δx j (0) .We choose h j ( t ) = δx j (0) δ ( t ) , where δ ( t ) is the Dirac function and δx j (0) = x hj − x j is the deviation ofthe perturbed variable from the unperturbed one. The response function, due to this small perturbation,of a general observable A ( x , t ) , that depends on the whole set of variables and explicitly on the time, isdefined as: R A,x j ( t ) = h A ( x ( t )) i h − h A ( x ( t )) i δx j (0) ≡ δ h A ( x h ) i δh j (0) (cid:12)(cid:12)(cid:12) h n =0 , (3)where δ/δh j is the functional derivative with respect to h j calculated at h j = 0 . The response functioncan be calculated numerically through Eq. (3) that requires the knowledge of the perturbed dynamics.
3. Generalized Fluctuation-Dissipation Relations
As mentioned in the introduction, the idea of expressing the response due to a small perturbation interms of unperturbed correlations has a long history. Here, we report a version of the generalized FDRholding independently of the presence of non-vanishing currents and, thus, valid also for non-equilibriumdynamics. This version of the FDR does not depend explicitly on the probability distribution functionbut requires the explicit knowledge of the microscopic dynamical details, i.e. the knowledge of σ ij and F i ( x ) . Using a path-integral formalism and assuming the stationarity of the time-properties, theresponse function, R A,x j ( t ) , associated to the dynamics (1) and (2), can be expressed as: R A,x j ( t ) = − D − jm (cid:20) h A ( t ) F m (0) i + ddt h A ( t ) x m (0) i (cid:21) , (4)where the dependence on x in A ( t ) = A ( x ( t )) and F ( x ( t )) = F ( t ) has been omitted for notationalconvenience and the average on the right-hand side, h·i , is calculated through the unperturbed dynamics.Further details about the derivation of Eq. (4) are reported in Appendix A. In addition, we remark thatthese exact relations are not simply expressed by the temporal correlation between A and anotherobservable at variance with equilibrium. Our FDR explicitly contains the time-derivative of theobservable A and the state variable. Fixing A ( x ) = x k allows us to consider the expression for theresponse matrix of element R x k ,x j ( t ) . In this specific case, the relation (4) further simplifies leading tothe following steady-state expression that is derived in Appendix B: R x k ,x j ( t ) = − D − jm [ h x k ( t ) F m (0) i + hF k ( t ) x m (0) i ] . (5)The relations (4) and (5) are general for every equilibrium and non-equilibrium dynamics of the form (1).We remark that the response matrix elements assumes a form particularly simple: they are expressedas temporal correlations between the state variables and the forces that rule the dynamics, combinedby the elements of the diffusion matrix. In the diagonal case, D ij = δ ij D j , in particular, Eq. (5) turnsto be: R x k ,x j ( t ) = − D j [ h x k ( t ) F j (0) i + hF k ( t ) x j (0) i ] . (6) eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics In this section, we show that we can extract a relation between suitable steady-state correlationsfrom the response matrix and our version of generalized FDR. Indeed, the response matrix at theperturbation time, R x k x j (0) , is not arbitrary because of its definition (3). In particular, R x k x j (0) = δ jk since its diagonal elements are unitary, while the cross elements vanish because of the causality condition.Therefore, evaluating the FDR, Eq. (5), at the perturbation time, leads to the following tensorial relation: δ jk = − D − jm [ h x k F m i + hF k x m i ] . (7)Equation (7) establishes a set of exact relations between special equal-time averages that are functionsof the state variables of the dynamics through x k and F m ( x ) (that is expressed as F m for notationalconvenience). These relations hold for both equilibrium and non-equilibrium systems and their physicalinterpretation will be clarified in the explicit examples reported in Sec. 4. In particular, we anticipatethat Eqs. (7) represent a generalization of the Equipartition theorem and the Mesoscopic Virial equation.Despite these equations can be obtained via other methods in many interesting cases, we stress thatthey are also contained in our version of the FDR, from which their derivation is straightforward.In the case of diagonal diffusion, such that D ij = D j δ ij , Eq. (7) assumes a simpler form. Theserelations provide general constraints for the matrix, M , of elements M jk = h x j F k i , involving the steady-state correlation between the state variable x j and the deterministic force that determines the evolutionof x k . The diagonal elements of M satisfy: D j = −h x k F j i δ jk , (8)that can be interpreted as a generalized version of the equipartition theorem as illustrated in Sec. 4.Instead, the off-diagonal elements of M are constrained by the following relation: h x k F j i = −h x j F k i , (9)with k = j . Therefore, the matrix M is anti-symmetric. As we can see in Sec. 4, the relation (9)represents a generalized version of the Mesoscopic Virial equation. These relations have been derived forthe specific case of a particle, following the Langevin dynamics, by Falasco et al. [41] using a differentapproach while, here, are extended to a more general dynamics and connected to our version of thegeneralized FDR. eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics
4. Examples
To test our general results, we start by considering the equilibrium dynamics describing the motion of apassive colloidal particle in a solvent. In this case, the generalized FDR need to be consistent with thewell-known FDR holding at equilibrium. Specifically, assuming that the colloid is in equilibrium withthe solvent at temperature, T , and neglecting hydrodynamics interactions, the dynamics for the particleposition, x , and the particle velocity, v , reads: ˙ x = v (10) m ˙ v = − γ v + F + p γT ξ , (11)where m is the mass of the colloid, γ the drag coefficient and T the solvent temperature that satisfiesthe Einstein relation with the diffusion coefficient, γD t = T . The term F accounts for external forcesdue to a potential, such that F = −∇ U ( x ) , while the term − γ v ≡ F s is the Stokes force proportionalto the velocity. This term balances the injection of energy due to the collisions of the solvent particlesthat are modeled through a white noise. The diffusion matrix is diagonal, such that σ ij = δ ij √ T γ/m ,since there are no temperature gradient. In this case, the response of an observable A ( v , x ) due to theadditional perturbative force h j reads: R A,v j ( t ) = h A ( x ( t )) i h − h A ( x ( t )) i δv hj (0) = m δ h A ( x h ) i δh j (cid:12)(cid:12)(cid:12) h j =0 , (12)where we remind that the average h·i h is realized through the perturbed measure and the Latin indicesare used to denote the Cartesian components of the vectors, here and in the next examples. Thus, inthis case, the set of variables is composed of the d Cartesian components of position and velocity where d is the dimension of the system.Applying the general formula (5) with the dynamics (10), i.e. replacing F j = − γv j /m − ∇ x j U/m ,leads to the following result for the response matrix: R v k ,v j ( t ) = mT h v k ( t ) v j (0) i + m T γ (cid:2) h v k ( t ) ∇ x j U (0) i + h∇ x k U ( t ) v j (0) i (cid:3) . (13)Since, by definition, the system is in equilibrium, the detailed balance holds and we can furthermanipulate Eq. (13) by using the time-reversibility of the steady-state correlation such that h v k ( t ) ∇ x j U (0) i = −h∇ x j U ( t ) v k (0) i . In addition, we can use the symmetry among different Cartesiancomponents, such that h∇ x k U ( t ) v j (0) i = h∇ x j U ( t ) v k (0) i , that is valid for central potentials. Usingthese properties, the square brackets in Eq. (13) vanish and we obtain the well-known equilibriumresult, R v k ,v j ( t ) = mT h v k ( t ) v j (0) i . Finally, choosing A ( x , v ) = x k one can calculate the cross terms ofthe response matrix (coupling position and velocity) starting from Eq. (4): R x k ,v j ( t ) = m T h x k ( t ) v j (0) i + m T γ h x k ( t ) ∇ x j U (0) i − m T γ h v k ( t ) v j (0) i . (14)We observe that in equilibrium systems the relations (14) vanish term by term except for j = k whereonly the second and the third terms survive. Using the time-reversibility, the equation of motion andtricks similar to those employed to manipulate Eq. (13), also in this case, we can recover the well-knownresult holding in equilibrium, that is R x k ,v j ( t ) = mT h x k ( t ) v j (0) i . eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics The dynamics of an equilibrium colloidal particle is often described byan overdamped stochastic differential equation for the position, x , because the inertial forces play anegligible role. In this case, the evolution of each colloid is described by the following equation: γ ˙ x = −∇ U + p γT ξ . (15)In the overdamped case, one can calculate the response function of an observable A ( x ) perturbingdirectly the particle position that is a noisy variable, i.e. the dynamics (15). Therefore, the responsefunction, R A,x j ( t ) , is defined as: R A,x j ( t ) = h A ( x ( t )) i h − h A ( x ( t )) i δx j (0) = γ δ h A ( x h ) i δh j (cid:12)(cid:12)(cid:12) h n =0 . (16)Now, the set of variables involved in the FDR contains only the d Cartesian components of the position.After identifying F k = −∇ x k U/γ and σ jk = δ jk p T /γ , we can apply formula (5) so that the responsefunction reads: R x k ,x j ( t ) = 12 T (cid:2) h x k ( t ) ∇ x j U (0) i + h∇ x k U ( t ) x j (0) i (cid:3) . (17)The well-known FDR can be recovered again by using the time-reversibility so that h∇ x k U ( t ) x j (0) i = h x j ( t ) ∇ x k U (0) i . The absence of currents also implies that the system is invariant for changes of Cartesiancomponents so that h x j ( t ) ∇ x k U (0) i = h x k ( t ) ∇ x j U (0) i . In this way, Eq. (17) reduces to the well-knownequilibrium result, R x k ,x j ( t ) = T h x k ( t ) ∇ x j U (0) i . In the case of a passive underdampedcolloid, following the dynamics (10), the relation (7) turns to be: m h v k v j i + mγ h v j ∇ x k U i = T δ kj . (18)The diagonal elements of this relation for j = k can be further manipulated since h v k ∇ x k U i =1 /t f R t f d/dtU ( x ( t )) dt = [ U ( x ( t f )) − U ( x (0))] /t f , that is the potential energy difference from the initialand the final state. Since this term gives a negligible contribution in the steady-state ( t f → ∞ ), therelation (18) trivially holds and states that h v k i = T , in agreement with the equilibrium distribution ∝ exp ( − U/T − m P k v k / T ) . The equation for the off-diagonal terms implies that h v k v j i = − γ h v j ∇ x k U i .where each correlation is zero. The cross-correlation of the response matrix, coupling position andvelocity, i.e. Eq. (14) at the perturbation time, leads to the following relation: h x j v k i + h x j ∇ x k U i − m h v j v k i = 0 . (19)if j = k , the first term vanishes because is a boundary term, such that h x k v k i = 1 /t f R t f d/dt x ( t ) / dt =[ x ( t f ) − x (0)] / (2 t f ) , that is irrelevant for large times ( t f → ∞ ). Moreover, the second term of Eq. (19)can be identified as the virial pressure and is related to the kinetic energy by this formula. If U contains also an interacting potential with other colloidal particles, this equation is nothing but theVirial mesoscopic equation, that has been derived in Ref. [41] using a different method.In the case of passive overdamped colloids following the dynamics (15), we can apply the relation (7),obtaining: δ jk T = h x j ∇ x k U i . (20) eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics m/γ issmall, just by considering the different contributions in powers of m/γ . Again, this equation holds sincethe equilibrium distribution is ∝ exp ( − U/T ) and states that the Virial pressure is determined by thesolvent temperature. Active particles are usually described by stochastic equations that resemble those of passive colloidsmoving in viscous solvents except for the addition of a time-dependent stochastic force called self-propulsion or simply active force. Usually, the active force is chosen to reproduce the typical time-persistence of the active trajectory at a coarse-grained level that neglects its mechanical or chemicalorigin (which depends on the system under consideration). This force, except for a few special cases,breaks the detailed balance [42, 43] condition producing a non-vanishing entropy production [44–49].Therefore, active dynamics are good platforms to evaluate generalized FDR in far-equilibrium systems.The most popular and simple models to reproduce the self-propulsions through a stochastic processare the Active Brownian Particles (ABP) dynamics [50–58] and the Active Orstein Uhlembeck particles(AOUP) one [59–65]. Both have been used to reproduce the non-equilibrium phenomenology of self-propelled particles. In the ABP case, the self-propulsion force, f a has a constant modulus and reads: f a = γv n , being v the swim velocity induced by the self-propulsion and γ the viscous solvent. The term n i = (cos θ i , sin θ i ) is a unit vector representing the particle orientation since θ is the orientationalangle that evolves via a Brownian motion: ˙ θ = p D r ξ , where D r is the rotational diffusion coefficient and ξ is a white noise with zero average and unit variance.According to the AOUP scheme, the self-propulsion of each particle is described by a vectorial Ornstein-Uhlenbeck process: τ ˙ f a = − f a + γv √ τ η , (21)where τ is the persistence time of the process, η is a vector of white noises with zero average andunit variance, and the other parameters have been already introduced. Here, the term v γ is thevariance of the self-propulsion whose square root also represents the average value of its modulus, which,thus, provides the same average swim velocity of the ABP. Despite the different shapes of ABP andAOUP models, they share important time-dependent properties that are considered responsible for theircommon phenomenology. Even if many experimental systems of active matter have microscopic sizes [21]and usually move in environments with large viscosity (in such a way that inertial forces are negligible),recently, the effects of inertia [66] have been highlighted in many experimental active systems, such asvibro-robots [67], Hexbug crawlers and camphor surfers [68] and vibration-driven granular particles [69–71] (in the granular case, the response function has been also calculated experimentally [72]). To includethe active force in these physical systems, the active Langevin model has been introduced [66, 67, 73–76]so that the equation of motion of the active particle is described by its position, x , and velocity, v : ˙ x = v (22) m ˙ v = − γ v + F + f a + p T γ w , (23) eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics w is a white noise vector with zero average and unit variance, f a is the active force discussedabove, and the other terms have been already introduced below Eq. (10).Both for ABP and AOUP active forces, we can obtain a generalized FDR for the elements of theresponse matrix, defined by Eq. (12), by applying the formula (5) with F j = − γv j /m − ∇ x j U/m + f aj /m and σ ij = δ ij √ T γ/m : R v k ,v j ( t ) = mT h v k ( t ) v j (0) i + m T γ (cid:0)(cid:10) v k ( t ) ∇ x j U (0) (cid:11) + h∇ x k U ( t ) v j (0) i (cid:1) − m T γ (cid:0)(cid:10) v k ( t )f aj (0) (cid:11) + h f ak ( t ) v j (0) i (cid:1) . (24)The first term in the right-hand-side of Eq. (24) represents the response in the equilibrium regimes, i.e.for v → that corresponds to the well-known results for passive Brownian particles reported in Eq. (13).The second and third terms are non-equilibrium contributions of the response that exactly balance atequilibrium where the detailed balance holds since these terms are odd under time-reversal symmetry.The fourth and fifth terms of the second line, instead, are truly non-equilibrium contributions thatexplicitly contain the self-propulsion force. We remark that we do not need to specify the parity undertime-reversal transformation of the active force since this information is not required for the calculationof the response. In a similar way, we can calculate the cross elements of the response matrix, choosing A ( x , v ) = x k in Eq. (24), obtaining: R x k ,v j ( t ) = m T h x k ( t ) v j (0) i + m T γ h x k ( t ) ∇ x j U (0) i− m T γ h x k ( t )f aj (0) i− m T γ h v k ( t ) v j (0) i . (25)We remark that both Eqs. (24) and (25) hold far from the equilibrium without restriction in theparameters of the self-propulsion, at variance with other approaches where the active force is consideredas a small perturbation [77, 78].We check our theoretical results by studying the elements of the response matrix, R v j ,v j ( t ) and R v j ,x j ( t ) , confining the system through a linear and a quartic potential in two dimensions. The time iscalculated in unit of t ∗ = m/γ . In both cases, the cross elements of the response function ( R v k ,v j ( t ) and R x k ,v j ( t ) with k = j ) are zero for symmetric arguments: indeed, each correlation appearing in the FDRshould be invariant under the transformation x j → − x j , v j → − v j and f aj → − f aj at fixed j . Sinceall the terms appearing in the cross elements of Eq. (27) are odd under this transformation, the onlypossibility is that R v k ,v j ( t ) = R x k ,v j ( t ) = 0 if k = j .The response in the harmonic passive case, with U ( x ) = k | x | / (where k is the potential constant),can be analytically solved because the velocity correlation appearing in the FDR can be calculated asa function of t and depends on the inertial time t ∗ = m/γ and on the frequency ω = k/m − ( t ∗ ) − / ,as known in the literature. As shown in Fig. 1 (a), the profile of R v j ,v j ( t ) and R x j v j ( t ) both for theAOUP and the ABP dynamics remains the same as a result of the linearity of the force. This occurseven if, in both cases, the functional form of the FDR changes because of the non-vanishing timecorrelation between x k and f ak . Fig. 1 (b) reports a similar study when passive or active dynamics areconfined by the quartic potential, U ( x ) = k | x | / . In this case, there are no analytical solutions for h v j ( t ) v j (0) i (and for the other correlations) neither in the passive nor in the active cases, because of thenon-linearity of the dynamics. Therefore, the validity of the FDR is checked numerically by comparingthe elements of R calculated by their definition (12) and by the FDR, and shows a good agreement.Besides, the functional forms of R v j ,v j ( t ) and R x j ,v j ( t ) in the active cases (both ABP and AOUP) display eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics Figure 1.
Elements of the response matrix, R v j ,v j ( t/t ∗ ) and R x j ,v j ( t/t ∗ ) , obtained numerically usingthe definition (12) (colored lines) and using the relation (24) and (25) (dashed black lines). Inpanel (a), R v j v j ( t/t ∗ ) and R x j v j ( t/t ∗ ) are studied for the harmonic confinement, U ( x ) = k | x | / , forpassive, AOUP and ABP particles. The shapes of R v j ,v j ( t/t ∗ ) and R x j ,v j ( t/t ∗ ) are reported for asingle set of parameters (also in the active cases) since the only dependence on them is contained in t ∗ = m/γ and ω = k/m − ( t ∗ ) − / . In addition, we have simply plotted the passive profile, R ( t ) =exp ( − t/t ∗ ) [cos ( ωt ) − sin ( ωt ) / (2 t ∗ ω )] , for the passive system. Panels (b) and (c) report R v j ,v j ( t/t ∗ ) and R x j ,v j ( t/t ∗ ) , respectively, for a system confined through a quartic potential, U ( x ) = k | x | / . Here, theadditional dotted black lines are eye-guides The other parameters are k = 3 , γ = 1 , T = 1 , τ = 1 and v = 1 . more pronounced oscillations that also occur for smaller times with respect to the passive profile of theresponse. Additionally, the difference between AOUP and ABP dynamics appears only in the limit v ≫ T (and increases with the growth of v /T , while in the opposite limit (not shown) the AOUP andABP responses become equal to each other before converging to the passive profiles when the activeforce is negligible. We also study the active dynamics directlyin the overdamped regime. Since the inertial forces are usually negligible in many experimental activesystems, the overdamped limit has been largely employed in most of the numerical studies about activematter and, thus, deserves particular attention. The resulting dynamics is a stochastic differentialequation for the particle position x : γ ˙ x = F + f a + p T γ w . (26)Once the velocities have been eliminated, the positions evolve through a stochastic dynamics and we cancalculate the response function, defined by Eq. (16). Taking σ ij = δ ij √ D t and F j = −∇ x j U/γ + f aj /γ , weapply the general Eq.(5) to calculate R x k x j obtaining the following FDR for the elements of the responsematrix: R x k ,x j ( t ) = 12 T (cid:0)(cid:10) x k ( t ) ∇ x j U (0) (cid:11) + h∇ x k U ( t ) x j (0) i (cid:1) − T (cid:0)(cid:10) x k ( t )f aj (0) (cid:11) + h f ak ( t ) x j (0) i (cid:1) . (27)The first and the second terms are the equilibrium-like contributions of the response that coincides onlyif the detailed balance holds and that are otherwise different. Instead, the second and third terms are the eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics Figure 2.
Response function, R xx ( t/t ∗ ) , obtained numerically using the definition (16) (colored lines)and using the relation (27) (dashed black lines). In panel (a), R xx ( t/t ∗ ) is studied for the harmonicconfinement, U ( x ) = k | x | / , in the cases of passive, AOUP and ABP particles. In this case, we reportthe expression for a single set of parameters (also in the active cases) since the only dependence on themis contained in t ∗ = t/γ . In addition, we have simply plotted the exponential profile, exp ( − t/t ∗ ) , for thepassive system. In panel (b), we show R xx ( t/t ∗ ) confined through a quartic potential, U ( x ) = k | x | / .We show the active response for different values of the ratio v /T comparing ABP and AOUP models.The passive case, obtained for v = 0 , is temperature independent since the only dependence on theparameters is contained in t ∗ = √ γ/ √ T k and, thus, is simply shown for T = 1 . Finally, the inset showsthe comparison between Eq. (27) for T /v = 10 − and Eq. (C.1) for T = 0 , showing the good agreementbetween the two FDR. Here, the additional dashed black lines are eye-guides to evidence the differenttime regimes. The other parameters are k = 3 , γ = 1 , τ = 1 and v = 1 . non-equilibrium contributions involving the time-correlation of active force and position that disappearsin the equilibrium limit, v → .To check the results also in the overdamped case, we numerically study the response functionconsidering the same confining potentials studied in the underdamped case: i) quadratic potential, U ( x ) = k | x | / and ii) quartic potential U ( x ) = k | x | / where k is the potential constant. In bothcases, the cross elements of the response function ( R x i ,x j with i = j ) are zero for the same symmetricarguments already explained for the underdamped dynamics. The time, t , is evaluated in unit of thetypical time, t ∗ , that rules the response decay of the passive overdamped system, given by t ∗ = γ/k forthe harmonic potential and t ∗ = √ γ/ √ T k for the quartic potential. With this time rescaling, R x i ,x j ( t ) does not depend on the model parameters, in the passive case.In both cases, the response function evaluated numerically from the perturbed dynamics (seedefinition (12)) is compared with the FDR, Eq. (27), showing a good agreement for different valuesof v /T both for the AOUP and ABP models. This confirms the validity of our exact relations also innon-equilibrium dynamics. Fig. 2 (a) illustrates the response function in the harmonic case, where thedecay is exponential, R x j x j ( t ) = e − t/t ∗ , as analytically predicted in Ref. [79] for the athermal AOUP. Inthe harmonic case, we observe that there are no differences between AOUP, ABP, and passive systems.As a consequence, the shape of the active force is irrelevant despite the non-Gaussian form of the activeforce in the ABP model. In Fig. 2 (b), the response function in the quartic potential case shows aricher behavior. The rescaled R x j x j ( t/t ∗ ) has an exponential profile that does not depend on the choice eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics k/γ and T . This profile coincides with the active one, in the equilibrium limit v ≪ T (shownfor v /T = 10 − ), where AOUP and ABP cannot be distinguished simply because the active force isnegligible. Increasing the ratio v /T , the active response starts decreasing faster even if there are noclear differences between AOUP and ABP models, that appear only for further values of v /T . Ingeneral, the decay of R x j x j ( t ) is faster for the ABP model than the AOUP one, and the differencebetween the two models increases when v /T grows. In this regime, the decay is characterized by twodistinct time-regimes, as explicitly shown in the inset of Fig. 2 (b). As also discussed in [40] for T = 0 ,these two regimes can be easily explained because an active particle (in the large persistence regime,considered here, for v ≫ T ) confined in a quartic potential accumulates on a circular crown far fromthe potential minimum showing pronounced non-gaussianity in the distribution [80]. We observe thatthe formulation (27) of the FDR reported in this work does not coincide with the recent one, obtainedfor an AOUP particle with zero solvent-temperature. Indeed, the FDR for T = 0 , reported in [40]involves the second derivative of the potential that is not contained in Eq. (27). Moreover, Eq. (27) isnot well-defined at T = 0 even if can be numerically evaluated for T arbitrarily small. In the inset ofFig. 2 (b), the expression (27) and the formulation of Ref. [40] (that for completeness is reported inAppendix C) reveals a good agreement between the two formulations of the AOUP response functionwhen T ≪ v revealing the convergence of the two generalized FDR in this limit. In the case of an active particle in theunderdamped regime following the dynamics (22), the relation (18) (for a passive colloid) turns to be: m h v k i + mγ h v j ∇ x k U i δ jk = T + mγ h v j f ak i δ jk , (28)where we have reported the relation for j = k for simplicity. In practice, the interpretation of the termsinvolved in this equation does not change with respect to Eq. (18), except for the presence of a newterm, i.e. h v j f ak i , appearing in the generalized version of the equipartition theorem. This can be easilyinterpreted as the work done by the active force that is responsible for the increase of the particle kineticenergy. A similar scenario occurs by generalizing Eq. (19) to the active dynamics. In particular, taking j = k , we obtain: h x j ∇ x k U i δ jk = h x j f ak i δ jk + m h v k i . (29)Now, the generalized Virial equation contains a new term that depends on the active force via itscorrelation with the particle position, which is proportional to minus the swim pressure (See [81–83]).The term on the left-hand side of Eq. (29) is proportional to the Virial pressure (as in the case of passivecolloids). We remark that, in the active case, the Virial pressure is not simply determined by the kineticenergy but is affected by the swim pressure.In a similar way, we can apply Eq. (7) to the overdamped active dynamics, Eq. (26), obtaining aset of relations, that we report for j = k , for simplicity: h x j ∇ x k U i δ jk = δ jk T + 1 γ h x j f ak i δ jk . (30)Eq. (30) is the equation of state (mesoscopic virial equation) for the active dynamics, that can also beobtained by Eqs. (28) and (29) in the limit m/γ ≪ . Here, the virial pressure is modified by the swimpressure as in Eq. (29) and the kinetic energy has been replaced by the solvent temperature. eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics
5. Comparison with other versions of the FDR
Under very general hypothesis, the response function due to a small perturbation for the generaldynamics (1) can be expressed in terms of suitable temporal correlations that involves the log-derivativeof the steady-state probability distribution, P s ( x ) , [3, 26]. This result has been independently derivedby Agarwal [5] in the context of stochastic processes and Vulpiani et al. [4] for chaotic deterministicdynamics, and reads: R A,x j ( t ) = − (cid:28) A ( t ) ddx j log P s ( x ) (cid:12)(cid:12)(cid:12) s =0 (cid:29) . (31)This relation allows us to express the response function in terms of a temporal correlation that has a verysimple form. The application of Eq. (31) does not require the dynamical knowledge of the deterministicor stochastic contributions appearing in Eq. (1) since the knowledge of the steady-state distribution isenough to express the generalized FDR. However, P s ( x ) is known for a few cases: i) linear dynamicswith additive noise ii) equilibrium dynamics characterized by zero currents. Indeed, in general, when thedetailed balance does not hold the distribution is unknown and the use of Eq. (31) requires the numericalcalculation of ddx j log P s ( x ) . Therefore, this relation remains, somehow, an implicit relation. We remarkthat, through this approach, one can calculate the response function directly from the experimentaldata in the absence of perturbation but, also in this case, still requires to recognize the leading variablesappearing in the dynamics. On the contrary, our exact relations (4) and (5) for a general observableand for A = x k , respectively, reveal also that the response function cannot be easily expressed in thesame form of Eq. (31), i.e. R A,x j ( t ) = h A ( t ) C (0) i , except when the detailed balance holds. Indeed, Eqs. (4) and (5) contain additional terms that cannoteasily be recast onto this form. Finding the functional form of C is a problem with the same difficultyof finding the functional form of the steady-state probability distribution of a non-equilibrium system.Several later formulations of the FDR based on a path-integral approach focused on the importanceof the time-reversal symmetry in the different contributions of the response function. For instance,in [14], the response function has been decomposed in terms of an entropic and a frenetic contributions.However, this decomposition goes beyond the aim of this study and, in general, cannot be achievedunless one knows the parity under the time-reversal transformation of each variable appearing in thedynamics. This parity is often unknown, as occurs for the active force appearing in the dynamics [63]and could depend on the physical system under consideration. Our formulation of the FDR does notneed this information and is expressed in a simple and compact form.
6. Conclusions
In this paper, we have derived a new version of the generalized Fluctuation-Dissipation relations (FDR)that holds both for equilibrium and non-equilibrium dynamics. The advantage of our relations is thatthey are expressed in a very compact and simple form in terms of time correlations between the observedvariable and the force ruling the dynamics of the perturbed variable. For this reason, our FDR onlyrequires the knowledge of the deterministic forces and the diffusion matrix appearing in the dynamicsand does not need the numerical calculation of the steady-state probability distribution, at variance with eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics
Acknowledgement
The author warmly thanks A. Sarracino, A. Puglisi and U. Marini Bettolo Marconi for interestingdiscussions and acknowledges support from the MIUR PRIN 2017 Project No. 201798CZLJ
Appendix A. Derivation of Eq. (4)
To derive Eq. (4), we employ a path-integral approach to estimate the probability of the trajectoryassociated to the unperturbed dynamics (1). In the following, we use the compact notation, x = { x } T t ,to denote the time-history of the single trajectory between the initial time, t , and the final time, T . Theexplict introduction of a source of noise in the dynamics, produces a probability, P [ x | x ] , of observinga path x given the initial state x . In the following, we consider Gaussian noises, η , which are entirelyspecified by mean values and correlations and that satisfies η i = σ ij ξ j . Under these assumptions, theprobability of observing the noise path, η , reads: P [ η | η t ] ∝ exp (cid:20) − Z T ds D − ij η j ( s ) η i ( s ) (cid:21) , (A.1)where we dropped an irrelevant normalization factor and used the Einstein convention for repeatedindices.Observing that the functional derivative with respect to the perturbation, h j , is equivalent to thefunctional derivative with respect to the noise, η j , we obtain an expression for the response functionstarting from its definition (3): R A,x j ( t − s ) = δ h A ( x h ( t )) i δh j ( s ) (cid:12)(cid:12)(cid:12) h n =0 = Z t D [ η ] P [ η | η t ] δδη j ( s ) A ( x ) = (A.2) = − Z t D [ η ] A ( x ) δδη j ( s ) P [ η | η t ] , (A.3)where in the last equality we have just performed an integration by parts. Using the expression (A.1)and performing the derivative, we get: R A,x j ( t − s ) = 12 h A ( t ) D − kj η k ( s ) i . (A.4) eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics η in terms of the state variables x through a changeof variables, so that, formally, we have η = η [ x , ˙ x ] . By replacing this relation into (A.4), we obtain: R A,x j ( t − s ) = 12 (cid:10) A ( t ) D − kj [ ˙ x k ( s ) − F k ( s )] (cid:11) . (A.5)Because of the stationarity of the correlations, the following relation holds: (cid:10) A ( t ) D − kj ˙ x k ( s ) (cid:11) = dds (cid:10) A ( t ) D − kj x k ( s ) (cid:11) = − ddt (cid:10) A ( t ) D − kj x k ( s ) (cid:11) , so that we can express Eq. (A.5) as R A,x j ( t − s ) = − D − kj (cid:20) h A ( t ) F k ( s ) i + ddt h A ( t ) x k ( s ) i (cid:21) , (A.6)that corresponds to Eq. (4) after choosing s = 0 . Appendix A.1. Assuming the detailed balance
Assuming the equilibrium condition or the detailed balance means the possibility of flipping the time inthe temporal correlation appearing in Eq. (A.6). In particular, using this property, we get: h A ( t ) x k ( s ) i = ± h x k ( t ) A ( s ) i , where the plus or minus sign is needed if the product between x k A is even or odd under time-reversaltransformation, respectively. In this way, Eq. (A.6) reads R A,x j ( t − s ) = − D − kj (cid:20) h A ( t ) F k ( s ) i ∓ ddt h x k ( t ) A ( s ) i (cid:21) = − D − kj [ h A ( t ) F k ( s ) i ∓ hF k ( t ) A ( s ) i ] = − D − kj h A ( t ) F k ( s ) i , (A.7)where, in the last equality, we have used again the reversibility condition and that F k needs to havethe same parity of x k in equilibrium dynamics. This equilibrium result is in agreement with the otherversion of the FDR [3, 26], given by Eq. (31). Indeed, if the detailed balance holds the distributionassociated with the dynamics (1) is simply: P s ∝ exp (cid:18)Z dx i D − ki F k (cid:19) . Appendix B. Derivation of Eq. (5)
To derive Eq. (5), we start fom the formula (4). Choosing A ( x ( t )) = x k , we obtain: R x k ,x j ( t ) = − D − mj (cid:20) h x k ( t ) F m (0) i + ddt h x k ( t ) x m (0) i (cid:21) . (B.1)Replacing d/dt x k with the equation of motion (1), we get: R x k ,x j ( t ) = − D − mj [ h x k ( t ) F m (0) i + hF k ( t ) x m (0) i ] , (B.2) eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics h η k ( t ) x m (0) i = 0 . This trick can be also used in the more general case A = A ( x ( t )) : ddt h A ( x ( t )) x m (0) i = (cid:10) ˙ x j ( t ) ∇ x j A ( x ( t )) x m (0) (cid:11) = (cid:10) F j ( t ) ∇ x j A ( x ( t )) x m (0) (cid:11) + (cid:10) η j ( t ) ∇ x j A ( x ( t )) x m (0) (cid:11) . (B.3)Moreover, in this case, the correlation involving the noise does not vanish because the causality conditioncannot be applied. At variance with the specific case reported in Eq. (5), this last general relation isnot simply expressed in terms of state variables and, thus, has the same level of complexity as Eq. (4). Appendix B.1. Assuming the detailed balance
Even if we have shown the result for a general observable A , it is instructive to use the time-reversibilityto further manipulate Eq. (5). In particular, if the product F j ( t ) x k (0) is even under time-reversaltransformation, the second term in Eq.(5) becomes hF j ( t ) x k (0) i = h x k ( t ) F j (0) i , while, if the product F j ( t ) x k (0) is odd, the following relation holds: hF j ( t ) x k (0) i = −h x k ( t ) F j (0) i . Thus, the response matrix can be expressed as: R x j ,x n ( t ) = − D − nk [ h x j ( t ) F k (0) i ± h x k ( t ) F j (0) i ] . (B.4)Further manipulation of this expression can be obtained accounting for the symmetry of the system.For instance, if the equilibrium is guaranteed by a force due to an external potential that depends onlyon the distance qP j x j , the system is invariant for the inversion of each component and F j needs to anodd function of x j . Thus, the non-vanishing elements of the response matrix are those with j = n and F j and h x j ( t ) F k (0) i = h x k ( t ) F j (0) i . In this way, Eq. (B.4) leads to the well-known equilibrium result. Appendix C. FDR for zero solvent temperature
In this Appendix, we report the FDR obtained in the case of overdamped active particles evolving withthe AOUP model with vanishing solvent temperature, i.e. the dynamics (22) with T = 0 and active forceevolving via Eq. (21). In this case, the formulation (27) of the FDR does not hold since the equationof motion is not of the form (2). Indeed, the perturbation, h , affects the dynamics of a state variable x j with a deterministic equation of motion, since the noise appears only in the evolution of the activeforce.For completeness, we report the FDR expression derived in [40], holding for the athermal AOUP,that has been employed in the inset of Fig. 2 (b), for the quartic potential case: D a γ R x j x i = h x j ( t ) ∇ x i U (0) i + h∇ x i U ( t ) x j (0) i (C.1) + τ h v j ( t ) ∇ x j ∇ x k U (0) v k (0) i + τ h v k ( t ) ∇ x i ∇ x k U ( t ) v j (0) i , where the particle velocity, defined as v j = ˙ x j , satisfies the following relation: γv j = f aj − ∇ x j U eneralized Fluctuation-Dissipation relations holding in non-equilibrium dynamics References [1] Kubo R 1957
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