AOUP in the presence of Brownian noise: a perturbative approach
AAOUP in the presence of Brownian noise: aperturbative approach
David Martin
Laboratoire Matière et Systèmes Complexes, UMR 7057 CNRS/P7, Université deParis, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, FranceE-mail: [email protected]
Abstract.
By working in the small-persistence-time limit, we determine the steady-state distribution of an Active Ornstein Uhlenbeck Particle (AOUP) experiencing, inaddition to self-propulsion, a Gaussian white noise modelling a bath at temperature T . This allows us to derive quantitative formulas for the spatial probability density ofa confined particle and for the current induced by an asymmetric periodic potential.These formulas disentangle the respective roles of the passive and active noises on thesteady state of AOUPs, showing that both the correction to the Gibbs-Boltzmann dis-tribution and the ratchet current are of order 1 /T . Thus, signatures of nonequilibriumvanish in the limit of large translational diffusion. We probe the range of validity of ouranalytical derivations by numerical simulations. Finally, we explain how the methodpresented here to tackle perturbatively an Ornstein Uhlenbeck (OU) noise could befurther generalized beyond the Brownian case. Keywords : Statistical Physics, Stochastic dynamics, Active Matter, non-equilibirumprocesses
Submitted to:
J. Stat. Mech.
1. Introduction
One of the challenges of Active Matter is to understand and predict the emergingproperties of assemblies composed of individual agents able to produce mechanical workby locally dissipating energy [1]. By their very nature, such systems break detailedbalance and thus lie within the realm of out-of-equilibrium physics. At a theoreticallevel, the modelling of the individual units of active systems reveal their nonequilibriumnature by involving non-Gaussian and colored noises differing from the familiar Wienerprocess of equilibrium thermal physics. On the one hand, this approach has beenfruitful and numerical simulations have shown that intriguing phenomena, prohibitedin equilibrium physics, arise for such noises : accumulation near walls [2, 3], emergenceof currents in asymmetric periodic potentials [4–7], collective motion [8, 9], motilityinduced phase separation [10, 11]... On the other hand, from a theoretical standpoint, a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p OUP in the presence of Brownian noise: a perturbative approach perse [12–18]. In this paper, we focus on an Active Ornstein Uhlenbeck Particle evolving inone space dimension, subjected to an external potential φ ( x ) and further experiencingan additional thermal noise. Its position x ( t ) and self-propulsion v ( t ) evolve accordingto the following system of Langevin equations :˙ x = − ∂ x φ + √ T η + v (1)˙ v = − vτ + √ Dτ η . (2)In the above dynamics (1)-(2), η and η are two uncorrelated Gaussian white noisesof unit variance, T is the amplitude of the thermal noise while D and τ control theamplitude and the persistence of the self-propulsion respectively. When T = 0, (1)-(2)correspond to the workhorse AOUP model which has been used to model transportproperties of active colloids [19] as well as collective cell dynamics [20, 21]. On thetheoretical side, there has been fundamental interest in its steady-state distribution,which has been characterized both in the limit of small τ [22–26] and in the limit ofhigh τ [25–27]. However, these theoretical approaches ignore the physically relevantpresence of an underlying thermal noise and the steady state distribution of (1)-(2)remains elusive for a generic combination of T and D . Indeed, such a combinationof different noise sources arises in experiments and has already been used to describea passive tracer immersed in a bath of active colloids [28]. In this paper, we aim atfilling this gap by computing perturbatively the stationary probability density of anAOUP experiencing an additional thermal noise in the small persistence time limit.Note that for τ = 0, the self-propulsion v falls back onto a Wiener process of amplitude D . In this particular case, the dynamics (1)-(2) is an equilibrium one with temperature T + D . Thus, intuitively, one could hope to find analytical formulas that smoothlydeparts from thermal equilibrium when τ is small. We develop here such a perturbativeexpansion and our main result is an analytical prediction of the steady-state distribution P s ( x, v ) as a series in τ / . Building on it, we make quantitative predictions abouttwo emerging quantities: the marginal in space of the probability density and thecurrent in an asymmetric periodic ratchet [7, 19, 29]. We find both the correction tothe Gibbs-Boltzmann distribution and the ratchet current to decay as 1 /T , entailingthat equilibrium physics, beyond the specific case τ = 0, is also recovered in the limit T (cid:29) D .
2. Systematic construction of the probability density function
We first present the derivation of the steady-state distribution P s ( x, v ) as a series inpowers of τ / . We start from the Fokker-Planck operator L corresponding to (1)-(2), OUP in the presence of Brownian noise: a perturbative approach L = ∂ x ( ∂ x Φ) − v∂ x + ∂ v (cid:18) vτ (cid:19) + Dτ ∂ vv + T ∂ xx . (3)We now rescale v as ˜ v = √ τ v . Expressed in terms of the rescaled variable, P s ( x, ˜ v )evolves according to˜ LP s ( x, ˜ v ) = 0 (4)with the operator ˜ L being defined as :˜ L = ∂ x ( ∂ x Φ) − ˜ v √ τ ∂ x + ∂ ˜ v (cid:18) ˜ vτ (cid:19) + Dτ ∂ ˜ v ˜ v + T ∂ xx . (5)In the remainder of this work, the tilde notation for v and L will be omitted for notationalsimplicity. We first note that the Fokker-Planck operator (5) can be written as : L = 1 τ L + 1 √ τ L + L . (6)Where L , L and L are given by L = D ∂ ∂ v + ∂∂v v L = − ∂∂x v L = ∂∂x ∂φ∂x + T ∂ xx . (7) L is the Fokker-Planck generator of the Ornstein-Uhlenbeck process, and its n th eigenfunction P n is related to the n th physicists’ Hermite polynomial H n ( v ) =( − n e v ∂ nv e − v : P n ( v ) = e − v D H n (cid:16) v √ D (cid:17) √ n n !2 πD . (8)The family { P n } are eigenfunctions of the operator L satisfying L P n = − nP n (9)and they are further orthogonal to the family { H n } as δ k,n = Z + ∞−∞ H k (cid:16) v √ D (cid:17) √ k k ! P n ( v ) dv . (10)We use the P n ’s to search for the solution of the stationnary distribution P s under theform of: P s ( x, v ) = X n P n ( v ) A n ( x ) . (11)Using the orthogonality property (10), the A n ’s can be obtained as A n ( x ) = Z P s ( x, v ) H n (cid:16) v √ D (cid:17) √ n n ! dv . (12)Injecting (11) into (4) and using (9), we find that A n is a solution of X n P n ( v ) ∂ x ( ∂ x φA n ) + X n P n ( v ) T ∂ xx A n − X n nP n ( v ) τ A n − X n vP n ( v ) √ τ ∂ x A n = 0 . (13) OUP in the presence of Brownian noise: a perturbative approach H n +1 ( v ) = 2 vH n ( v ) − nH n − ( v ),we decompose vP n into a sum of P n +1 and P n − vP n = q ( n + 1) DP n +1 + √ nDP n − . (14)We are now in position to project equation (13) onto H k and use the orthogonalityrelation (10). This leads us to the following recursion relation for the A n ’s0 = − nA n − √ τ q ( n + 1) D∂ x A n +1 − √ τ √ nD ∂ x A n − + τ ∂ x ( ∂ x φA n ) + τ T ∂ xx A n . (15)We now look for the A n ’s as series in powers of τ / . We shall assume that A k containsonly integer powers of τ and that its first nonzero contribution is of order τ k . Likewise,we shall assume that A k +1 contains only half-integer powers of τ and that its firstnonzero contribution is of order τ k +1 / . We thus propose the scaling ansatz A = A ( x ) + τ A ( x ) + τ A ( x ) + ... (16) A = τ / A ( x ) + τ / A ( x ) + τ / A ( x ) + ... (17) A = τ A ( x ) + τ A ( x ) + τ A ( x ) + ... (18) ... Let us now show that the A ji can be computed recursively. Looking at (15) for n = 0,we get ∂ x A = r τD [ ∂ x ( ∂ x φA ) + T ∂ xx A ] . (19)Equating coefficients of order τ k/ on both sides of (19) and integrating the x variableleads to: A k = 1 √ D h ∂ x φA k − + T ∂ x A k − i + b k . (20)Further equating coefficients of order τ k/ in (15), we obtain: A kn = − q ( n + 1) Dn ∂ x A k − n +1 − s Dn ∂ x A k − n − + ∂ x (cid:16) ∂ x φA k − n (cid:17) n + Tn ∂ xx A k − n . (21)Taking k = n in (21) and using that A jn = 0 for j ≤ n yields the expression of A nn as afunction of A : A nn = − s Dn ∂ x A n − n − = ( − n D n/ √ n ! ∂ nx A . (22)Using expression (20) for k = 1 and expression (22) for n = 1, we obtain a closedequation on A : ∂ x φ A + ( T + D ) ∂ x A = − b √ D . (23)Since A corresponds to the equilibrium stationary measure when τ = 0 we must have Z + ∞−∞ P s ( x, v ) | τ =0 dv = A = c e − φT + D (24)with c fixed by normalization: c = (cid:18)Z + ∞−∞ e − φT + D dx (cid:19) − . (25) OUP in the presence of Brownian noise: a perturbative approach b is self-consistently fixed to zero such that (24) is a solution of (23). Wenow set out to compute the next order correction A . Applying (21) for n = 1 and k = 3 gives: A = −√ D∂ x A − √ D∂ x A + ∂ x (cid:16) ∂ x φA (cid:17) + T ∂ xx A . (26)In (26), we can use (22) to express A and A as a function of A and (20) to express A as a function of A . We thus obtain a differential equation for A : ∂ x φA T + D + ∂ x A = − D ∂ x A T + D + D∂ x ( ∂ x φ∂ x A ) T + D + T D∂ x A T + D − b √ DT + D . (27)Using (24), we can integrate (27) and determine the expression of A A = c e − φT + D D − ( ∂ x φ ) T + D ) + ∂ xx φT + D ! + c e − φT + D − b √ DT + D e − φT + D Z x e φT + D dx (28)where c is defined in (25). Equation (28) involves two integration constants: c and b . While c is found by normalization, requiring R + ∞−∞ A ( x ) dx = 0, b is fixed byboundary conditions on A as we shall see in the next sections. We remark that (28)shares a common feature with the distribution of other active models [30, 31]: it isnon-local. Indeed, a perturbation of the potential δφ ( x ) localized around position x will affect the steady-state at position x located far away from x . This strongly differsfrom Boltzmann distribution and leads to intriguing phenomena, for example in bathsof bacteria [32]. The recursion can be iterated up to an arbitrary order in τ to findboth the A k ’s and the A ki ’s for i >
0. In addition to the previous constants c i and b i +1 for i < k , which were determined for lower orders, A k generically depends on twonew integration constants : c k and b k +1 . The former, c k , is found by requiring thenormalization of A k while the latter b k +1 is fixed by boundary conditions for A k . Forexample, the differential equation on A is found by applying (21) for ( n = 2 , k = 4)and ( n = 1 , k = 5). Its solution not only depends on c and b , which were previouslydetermined upon computing A , but also on two new integration constants : c and b . The constant c is found by requiring normalization R + ∞−∞ A = 0 and b is fixed byenforcing the correct boundary conditions for A . While the explicit expressions of the A k rapidly become cumbersome, their systematic derivation can be implemented witha software such as Mathematica [33]. For illustration purposes, we report the completeexpression of P s ( x, v ), with its integration constants, up to the order τ in (A.1).
3. Confining potential: explicit computation and numerics
The marginal in space of P s ( x, v ) can be used to quantify how the steady-statedistribution departs from the Boltzmann weight as τ increases : P s ( x ) = Z + ∞−∞ P s ( x, v ) dv = A = X k A k τ k . (29)Here we consider the special case of a confining potential φ , and we require that, for all k ≥
1, lim x →±∞ A k ( x ) = 0 (30) OUP in the presence of Brownian noise: a perturbative approach Z + ∞−∞ A k ( x ) = 0 . (31)For a simple harmonic confinement, we note that the complete steady-state distribution P s ( x, v ) remains gaussian and we report its expression in Appendix B. In the remainderof this paper, we will focus on the more general case of anharmonic potentials. Weremark that equation (30) imposes b k +1 = 0 for all k ≥ c k for all k ≥
1. The function A k is then uniquely determined. For example, using (28) and thedefinition of c (25), A reads A = c e − φT + D D − ( ∂ x φ ) T + D ) + ∂ xx φT + D ! − c D T + D ) e − φT + D Z + ∞−∞ ∂ xx φ e φT + D dx . (32)When T = 0 in (32), we recover the steady-state of an Active Ornstein Uhlenbeck(AOUP) particle [22]. The cumbersome expression of the full marginal in space P s ( x )up to order τ is reported in appendix A.2. Our ansatz (16) rests on the hypothesisthat P s ( x ) is an analytic function in √ τ which need not necessarily hold for an arbitrarypotential. To check this hypothesis, we have to verify whether the series admits a finiteradius of convergence. We do this for a potential φ ( x ) = x /
4, at fixed D and T and fortwo different values of τ . For τ = 0 .
01, we show in Figure 1 that the truncation of (29)to order τ is well-behaved and quantitatively agrees with the stationary distributionobtained numerically. However, for τ = 0 .
2, Figure 1 shows the successive orders of thetruncation to be typical of a divergent series: adding one order in τ increases the seriesby a larger amount than the sum of the previous terms, thus leading to wild oscillations.While such a result seems disappointing, it does not mean that the full series fails incapturing the steady state. Mathematically speaking, it only entails that the finitetruncation yields a poor approximation of the full series and that more work shouldbe carried out to extract physical behaviors. To regularize our diverging truncatedsequence, we resort to a Padé-Borel summation method. We first introduce the Boreltransform B N associated to (29): B N ( τ ) = N X k =0 A k k ! τ k . (33)The finite- N truncation of the series (29) is exactly recovered from its N th -Boreltransform B N by applying a Laplace inversion : N X k =0 A k τ k = Z ∞ B N ( ωτ ) e − ω dω . (34)The Laplace inversion of expression (33) for B N indeed leads back to the divergentfinite truncation that we wanted to regularize. To avoid such a fate, one has to finda nonpolynomial approximation of B N ( τ ) whose Taylor expansion coincides with theknown terms in (33). In the Padé-Borel method, it is achieved by approximating B N with a rational fraction F N = Q N /R N , where Q N and R N are polynomials in τ of order N/ B N ( τ ) = Q N ( τ ) /R N ( τ ) + o ( τ N ). The Borel resummation at order N of (29), B rN , is defined by replacing B N in (34) by its Padé approximant F N : B rN = Z ∞ Q N ( ωτ ) R N ( ωτ ) e − ω dt . (35) OUP in the presence of Brownian noise: a perturbative approach − − . . . . . xP s ( x ) order 3order 4order 5order 6numerics (a) − − . . . . . xP s ( x ) order 3order 4order 5order 6numerics (b) − − . . . . . xP s ( x ) order 6order 7order 8numerics (c) − − . . . . . xP s ( x ) order 8numerics (d) Figure 1.
Steady-state distribution of (1)-(2) in a confining potential φ ( x ) = x / Top:
For τ = 0 .
01, the finite truncation of (29) converges and agrees with the numerics (a) . Its corresponding Borel resummation B r also coincides with simulation data (b) . Bottom:
For τ = 0 .
2, the finite truncation of (29) is rapidly diverging away (c) .However, the Borel resummation B r accurately follows the data (d) . Parameters : D = T = 1, dt = 10 − , time = 10 . Finally, the series (29) is formally obtained from the limit of B rN when N → ∞ . In thisarticle, we estimate (29) while keeping N finite and we will not evaluate B rN beyond N = 8. Interestingly, for τ = 0 .
2, while the truncated sequence of (29) is divergent, itsBorel resummation B r agrees quantitatively with numerical estimates of the steady-statedistribution. In Figure 2, we plot the Borel resummations B r and the correspondingnumerics for different values of T . When T (cid:28) D , the dynamics (1)-(2) is stronglyout-of-equilibrium and the probability density differs significantly from the Boltzmannweight with the presence of two humps. When T (cid:29) D , self-propulsion is washed outby thermal noise, the dynamics draws closer to equilibrium and the two humps of thedistribution are smoothened out. Note that the Borel resummation B r accurately fitsthe numerics without any free parameter.
4. Emerging current: analytical formula and numerics
An interesting marker of a non-equilibrium dynamics is the ratchet mechanism by whichasymmetric periodic potentials may lead to steady currents. We consider here such a
OUP in the presence of Brownian noise: a perturbative approach − − . . . . . xP s ( x ) T = 0 . T = 0 . T = 1 T = 1 . T = 3 . Figure 2.
Steady-state distribution of (1)-(2) in a confining potential φ ( x ) = x / T . Plain curves correspond to Borel resummations B r whilesymbols are obtained from numerical simulations of (1)-(2). In dashed lines, we plotthe Gibbs-Boltzmann distributions for the two limiting cases T = 0 . T = 3 . B r always fitsthe data accurately without any free parameters. Parameters : D = 1, dt = 10 − ,time = 10 . potential φ of period L and we use our perturbative expansion to compute the steady-state current J , defined as J = h ˙ x i (36) J = Z L Z ∞−∞ − ∂ x φ + v √ τ ! P s ( x, v ) dxdv (37) J = − X k ≥ Z L ∂ x φA k dx + √ D √ τ X k ≥ Z L A k +11 dx (38) J = X k ≥ T Z L ∂ x A k dx + L √ D X k> b k +1 τ k . (39)To go from (38) to (39), we used the expression of A k +11 in (20). We require the marginalin space P s ( x ) to be periodic, which entails A k to be periodic for all k ≥ J then simplifies into : J = L √ D X k> b k +1 τ k . (40)While the { b k } all vanished in the previous section as a result of confinement (30),they do not for a periodic potential. Indeed, the value of b k is fixed upon requiring theperiodicity of A k − . Thus, different boundary conditions lead to different distributions,highlighting once again the nonlocal nature of the steady state. We report the expressionof the marginal in space P s ( x ) for a periodic potential up to order τ in (A.3)-(A.4).Using it, we find that Lb τ is the first non-vanishing contribution to the current : J = DLτ T + D ) R L φ (1)2 φ (3) dx R L e φT + D dx R L e − φT + D dx + o ( τ ) . (41) OUP in the presence of Brownian noise: a perturbative approach . . . . . . . τ [10 − ] | J | [ − ] T = 0 . T = 1 . T = 1 . /τ | J | / τ T = 0 . T = 1 . T = 1 . Figure 3.
Current J induced by a ratchet potential φ ( x ) = sin( πx/
2) + sin( πx )for different values of T . Plain curves correspond to prediction (41) while dots arenumerical simulations with error bars given by the standard deviation. Left: J as afunction of τ . Right:
J/τ as a function of 1 /τ . Parameters : D = 1, dt = 15 . − ,time = 5 . . The above formula reduces to the recently computed expression of J for an AOUPparticle when T = 0 [24]. It is interesting to note that, as T → ∞ , J vanishes as J ∝ /T . Physically, when the amplitude T of the thermal noise is much bigger than theamplitude D of the self-propulsion, the particle does not "feel" the nonequilibrium natureof its dynamics and J dies out. In Figure 3, we compare our quantitative prediction(41) with the results of numerical simulations for a potential φ ( x ) = sin( πx/
2) + sin( πx )and different values of T . We find quantitative agreement at small τ for τ < . J in (40) could also be regularized using Borel resummationto extend the quantitative range of agreement between theory and simulations to highervalues of τ , but we leave such a regularization for futur works.
5. Conclusion
We developed theoretical insights for an AOUP submitted to an additional Browniannoise (1)-(2). First, we devised a recurrence scheme allowing us to compute its stationaryditribution at an arbitrary order in τ . We then used this result to show that activity-induced phenomena, such as spatial accumulation near walls or the emergence of current,vanish in the limit T (cid:29) D as an inverse power law 1 /T . The regime of high translationaldiffusion thus provides a second route, besides the regime of low τ , to recover equilibriumphysics. Alternatively, it is possible to derive the marginal in space P s ( x ) up to order τ by using a Markovian approximation for the evolution operator: such a method hasbeen developed in parallel to this article [34]. We further remark that our perturbativeapproach can be generalized to more complex dynamics than the Brownian case. Letus consider a stochastic dynamics S whose corresponding Fokker-Planck operator is L S . OUP in the presence of Brownian noise: a perturbative approach D to S amounts to add L + L to L S . The startingpoint for our perturbative expansion, A , is defined by the equation ( L S + D∂ xx ) A = 0.As long as A defined this way is known analytically, the recurrence can be carriedout and the effect of the OU noise can be taken into account perturbatively in τ . Forexample, using our method, one could try to assess the effect of a colored noise on anunderdamped Langevin dynamics. Acknowledgments
We thanks Julien Tailleur, Frédéric van Wijland, Cesare Nardini and Thibaut Arnoulxde Pirey for insightful discussions, continuous support as well as enlightening feedback.
Appendix A. Full steady-state distribution
In this appendix, we report the steady-state probability density P s ( x, v ) up to order τ . e φT + D P s ( x, v ) = c + √ τ P ( v ) c √ Dφ (1) ( x ) T + D + τ P ( v ) (cid:20) − c Dφ (1)2 T + D ) + c + Dc φ (2) T + D − √ Db T + D Z x e φ ( z ) T + D dz (cid:21) + τ (cid:20) P ( v ) c D √ (cid:18) φ (1)3 ( T + D ) − φ (1) φ (2) ( T + D ) + φ (3) T + D (cid:19) + P ( v ) (cid:18) b De φT + D T + D − φ (1) b D ( T + D ) Z x e φ ( z ) T + D dz + √ Dc φ (1) T + D − c D φ (1)3 T + D ) + c √ D ( D − T ) φ (1) φ (2) ( T + D ) + c √ DT φ (3) T + D (cid:19)(cid:21) + τ P ( v ) (cid:20) c Dφ (2) T + D − c Dφ (1)2 T + D ) − D b ( T + D ) Z x e φ ( z ) T + D φ (2) ( z ) dz + c D T + D ) φ (1)4 − c D ( D − T ) φ (1)2 φ (2) T + D ) + b D ( T + D ) Z x (cid:18)Z s e φ ( z ) T + D dz (cid:19) (cid:16) φ (1) ( s ) φ (2) ( s ) − ( T + D ) φ (3) ( s ) (cid:17) ds + c + Dc T + D ) Z x φ (1)2 ( z ) φ (3) ( z ) dz − Dc ( D + 2 T )( T + D ) φ (3) φ (1) − √ Db T + D Z x e φ ( z ) T + D dz + Dc ( D − T ) φ (2)2 T + D ) + Dc ( D + 2 T ) φ (4) T + D ) (cid:21) (A.1)In (A.1), c is defined by (25) while c , c , b and b are integration constants whoseexpressions must be adapted to the boundary conditions. For a confining potential, P s ( x, v ) must vanish for x → ±∞ , and thus b = b = 0 yielding the following spatialdistribution : e φT + D P s ( x ) = c + τ (cid:20) − c Dφ (1)2 T + D ) + c + Dc φ (2) T + D (cid:21) + τ (cid:20) c Dφ (2) T + D − c Dφ (1)2 T + D ) + c + c D T + D ) φ (1)4 − c D ( D − T ) φ (1)2 φ (2) T + D ) + Dc T + D ) Z x φ (1)2 ( z ) φ (3) ( z ) dz − Dc ( D + 2 T )( T + D ) φ (3) φ (1) + Dc ( D − T ) φ (2)2 T + D ) + Dc ( D + 2 T ) φ (4) T + D ) (cid:21) (A.2) OUP in the presence of Brownian noise: a perturbative approach c and c are finally found by normalization, requiring R + ∞−∞ P s ( x ) dx = 1.For a periodic potential of period L , the spatial distribution must respect P s ( x + L ) = P s ( x ). This condition implies b = 0, but b = 0 and P s reads : e φT + D P s ( x ) = c + τ (cid:20) − c Dφ (1)2 T + D ) + c + Dc φ (2) T + D (cid:21) + τ (cid:20) c Dφ (2) T + D − c Dφ (1)2 T + D ) + c + c D T + D ) φ (1)4 − c D ( D − T ) φ (1)2 φ (2) T + D ) + Dc T + D ) Z x φ (1)2 ( z ) φ (3) ( z ) dz − Dc ( D + 2 T )( T + D ) φ (3) φ (1) + Dc ( D − T ) φ (2)2 T + D ) + Dc ( D + 2 T ) φ (4) T + D ) − √ Db T + D Z x e φ ( z ) T + D dz (cid:21) (A.3) b = D T + D ) R L φ (1)2 φ (3) dx R L e φT + D dx R L e − φT + D dx (A.4)Once again, c and c are then found by normalization. Note that in expression (A.1), v corresponds to the rescaled variable ˜ v . To get the exact steady-state distributionassociated to (1)-(2), one has thus to make the change of variable v → √ τ v . Appendix B. Harmonic potential
We report hereafter the steady-state distribution for the special case of a harmonicpotential φ ( x ) = κx / P s ( x, v ) = √ ab − c π e − ax − bv + cvx (B.1)With the constants a , b and c defined as : a = κ (1 + κτ ) D + T (1 + κτ ) ) b = D (1 + κτ ) + T (1 + κτ ) D ( D + T (1 + κτ ) ) c = κ √ τ (1 + κτ ) D + T (1 + κτ ) (B.2)Note that in expression (B.1), v corresponds to the rescaled variable ˜ v . To get theexact steady-state distribution associated to (1)-(2), one has thus to make the changeof variable v → √ τ v . Appendix C. Numerical methods
To simulate dynamics (1), we used a discretized Heun scheme while dynamics (2) wasintegrated exactly using Gillespie’s method [35]. The obtained algorithm iterates asfollows :1 µ = exp( − dt /τ ) ;2 σ x = q D (1 − µ ) /τ ;3 Y = q Dτ (dt /τ − − µ ) + 0 . − µ )) − τ D (1 − µ ) / (1 − µ ) ;4 Y = √ τ D (1 − µ ) / √ − µ ;5 T = √ T dt ; OUP in the presence of Brownian noise: a perturbative approach
126 Y = x = 0 ;7 v = q D/τ ∗ n o r m a l _ d i s t r i b u t i o n ( 0 , 1 ) ;89 while ( t < t o t a l t i m e ) {10 η = n o r m a l _ d i s t r i b u t i o n ( 0 , 1 ) ;11 η = n o r m a l _ d i s t r i b u t i o n ( 0 , 1 ) ;12 η = n o r m a l _ d i s t r i b u t i o n ( 0 , 1 ) ;13 Y = τ ∗ v ∗ (1 − µ ) + Y ∗ η + Y ∗ η ;14 v = v ∗ µ + σ x ∗ η ;15 x = x − dt ∗ ∂ x φ ( x ) + Y + T ∗ η ;16 x += Y + T ∗ η − ∗ dt ∗ ( ∂ x φ ( x ) + ∂ x φ ( x ) ) ;17 t += d t ; }At step (17), x(t) is stored in the variable x. The steady-state marginal in space of thedistribution P s ( x ) was then obtained by recording the particle’s position recurrentlyinto an histogram. The current J was computed using the distance travelled by theparticle divided by the duration of the simulation : the error bar on J thus correspondsto the standard deviation. Such a definition for the current was heuristically found toconverge faster than computing J = h− ∂ x φ + v/ √ τ i with recurrent recordings. References [1] Marchetti M C, Joanny J F, Ramaswamy S, Liverpool T B, Prost J, Rao M and Simha R A 2013
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