Stochastic continuity equation and related processes

Stochastic continuity equation and related processes Gabriele Bassi c Armando Bazzani a Helmut Mais b Giorgio Turchetti a a Dept. of Physics Univ. of Bologna, INFN Sezione di Bologna, ITALY b DESY, Hamburg, GERMANY c Dept. of Mathematics and Statistics, Univ. of New Mexico, U.S.A Abstract We consider the processes defined by a Langevin equation and the associated continuity equation. The average of the density function, solution of the continuity equation, satisfies the Fokker-Planck equation. For a volume preserving vector field the same equation is satisfied by the average of the integer powers of the density, which are the moments of the related probability density. For a generic vector field the Fokker-Planck equation for the moments is slightly modified. We first illustrate the problem in the simple case of a free particle subject to a white noise, since the averages can be computed by an elementary procedure using the factorization property of the correlation functions of the noise. The probabilistic meaning of the moments is discussed and the comparison between the analytical results and the numerical simulation is shown. The case of a generic Langevin equation is treated by computing the averages via a Dyson expansion after observing that, for a volume preserving vector field, any power of the density function satisfies the same continuity equation with the appropriate initial conditions. As a consequence the results obtained for the free particle are easily extended. An alternative approach is based on the characteristics of the continuity equation; the probability density of this process in an extended phase space still satisfies a Fokker-Planck equation and its moments coincide with the previous definition. Key words: Stochastic processes, continuity equation, moments. 1 Corresponding author: G. Turchetti Phone ( 39-519195, Email turchetti@bo.infn.it. Preprint submitted to Elsevier Science 16 March 1

Introduction Stochastic processes are commonly used to build phenomenological models of physical and biological systems, in which a small subsystem is isolated taking into account its interaction with the fluctuating environment. The Langevin equation, continuous version of a random walk, is an ordinary first order differential equation with a deterministic vector field plus a white noise component. The probability density of the process satisfies the Fokker-Planck equation; its fundamental solution is the weighted sum over all paths connecting the initial and final points in the extended phase space. We consider here the continuity equation associated to the Langevin equation. This is a first order linear partial differential equation, which defines a stochastic process whose properties are analyzed by considering the average of its integer powers. These are the moments of the probability density associated to the stochastic process, and determine it completely. What we propose is an extension of the stochastic Liouville equation previously considered [1; ; 3], and discussed in the context of beam dynamics [4]. It is important to remark that there are physical processes, such as the Brownian motion, in which each particle experiences a different realization of the noise (describing random collisions with the medium). In this case the physical process is described by the Langevin equation and, even though the stochastic continuity equation for the density ρ is mathematically well defined, only the average ρ, satisfying the Fokker-Planck equation, has a physical meaning as the probability density of the process. The evaluation of ρ at a given location is useful from the numerical viewpoint since it allows to compute the average ρ there with a high accuracy. However there are physical processes in which all the particles experience the same noise: consider for instance a bunch of particles crossing a magnetic element in an accelerator, where the current has small random fluctuations. In this case the density ρ, satisfying the continuity equation, describes the physical process and ρ has to be interpreted as the average over many distinct runs, each one providing a different realization of the stochastic process, for the same initial bunch. The variance of ρ and more generally all its cumulants provide the required information on the physical process. The investigation of the stochastic continuity equation (SCE) we propose extends the previous works [1; ; 4] on the stochastic Liouville equation and shows that the moments ρ n satisfy a simple generalization of the Fokker-Planck equation satisfied by ρ. We first analyze the simple case of a free particle with white noise and Gaussian initial distribution. The average of the density and any of its integer powers ρ n satisfies the same Fokker-Planck equation The direct proof is based on a series expansion and the average of even powers of a Wiener process, which have a simple form. The extension from the free particle case to any vector field

with an additive or multiplicative white noise is based on a Dyson expansion in the interaction representation, a method first suggested in [1] and used in [5] to obtain a generalized Fokker-Planck equation in the case of correlated noise. When the vector field is volume preserving, a non trivial case in dimension d, the average of any integer power of the fluctuating density ρ n satisfies the Fokker-Planck equation, with the appropriate initial condition ρ n in. To end up we consider an extended phase space (x, ρ) and we investigat the evolution of a probability density P (x, ρ, t) associated to the random process fluctuating variables x, ρ. The probability density P (ρ, x, t) has the property that ts moments of order n with respect to ρ coincide with the averages of the n-th power of the fluctuating density ρ(x, t; ξ). The function P satisfies the Fokker-Planck equation associated to the generalized Langevin equation in the extended (x, ρ) phase space. Concerning the mathematical properties of the fluctuating density we notice that at any point x of the phase space the average ρ is the transition probability from the initial point to the point x of the process defined by Langevin equation. However the mean square deviation σ = ρ ρ and higher cumulants of the fluctuating density are not zero. This is easy to check in the free particle case where σ is a bimodal distribution vanishing for t. A fluctuating density ρ, which evolves according to the SCE, differs from the standard probability density associated to Langevin equation. The difference is evident since in the first case for any realization of the stochastic process there is a rigid transport of all the points in the support of the initial distribution, whereas the Langevin equation propagates any initial point with an independent realization of the stochastic process. In some problems related to accelerators and plasma physics one can be interested in evaluating the probability that some particles reach a specific region in phase space. A Monte-Carlo aproach to the Langevin equation usually requires many realizations of the noise to keep the statistical error small, since the whole distribution function is computed. A direct integration of the Fokker-Planck equation using a fixed grid in phase space (Liouville approach) can introduce interpolating errors, that destroy some structures of the distribution due for instance of the Hamiltonian character of the deterministic dynamics. Once the initial distribution is known analytically the Monte-Carlo approach allows to compute the average of the fluctuating density, solution of the SCE, in the region of interest, without any grid interpolation. The density at the given point is equal to the initial density at the back propagated point for the volume preserving flows and the error due to the Monte-Carlo averaging process is inversely proportional to the number of realizations. The plan of the paper is the following: in section 1 we consider the SCE for the density function ρ(x, t; ξ) and its integer powers, and the equation satisfied by their averages; in section we analyze in detail the case of a free 3

particle subject to a white noise; in section 3 we compare the Montecarlo solution of the Fokker-Planck equation with the average ρ(x, t; ξ) of the SCE, analyzing the related errors; in section 4 we discuss the space correlations and the equations they satisfy; in section 5 we introduce the probability density P (ρ, x, t), the equations it satisfies and its moments; in appendix A we recall the basic properties of the Wiener and Uhlenbeck processes; in appendix B we compute the space correlations for the free particle and damped particle with a white noise; in appendix C we compute the average ρ for a damped particle with a white or correlated noise. 4

1 Stochastic continuity equation The Langevin equation describes the evolution of a point subject to a vector field Φ = a + bξ(t) in a phase space R d where a is the deterministic component and ξ(t) is a white noise, formal time derivative of a Wiener process w(t). dx dt = Φ(x, t; ξ) a(x, t) + b(x, t) ξ(t) (1) A regularization of the stochastic process, achieved by replacing the white noise ξ(t) with a Uhlenbeck process, see Appendix A, would allow to make our formal derivation rigorous by letting the correlation time vanish as a final step. To the stochastic differential equation we associate the stochastic continuity equation, ρ t + (Φρ) = () x which is well defined for any realization of a regular stochastic process. Given an initial distribution ρ in (x) and letting x = S t,t (x, ξ) be the evolution semi-group, the solution can be written as ρ(x, t; ξ) = ρ in (x ) µ(x, t; ξ) (3) where µ is the Jacobian of the flow from x at time t to x at time t. In the volume preserving case Φ/x = the solution simplifies, since ρ is constant along any trajectory, and reads ρ(x, t; ξ) = ρ in (x ) (4) This is the case of the free particle Φ = ξ(t), we shall analyze in detail, where ρ(x, t; ξ) = ρ in (x w(t) + w(t )) (5) In the general case we write the continuity equation () in the following operator form ρ t = ( A(t) + ξ(t) B(t) ) ρ A = a(x, t) x B and its solution as a Dyson series according to ρ(x, t; ξ) = T exp t ( ( t = x b(x, t) (6) A(s) + ξ(s) B(s) ) ds ρ in (x) (7) where T is the time ordering operator defined by T (O(t 1 )O(t )) = O(t )O(t 1 ) if t t 1 and T (O(t 1 )O(t )) = O(t 1 )O(t ) if t 1 > t. In [5] it was proved 5

that density averaged with respect to the white noise ξ satisfies the following equation ( ) t ρ = A + B ρ (8) which is just the Fokker-Planck equation corresponding to the Stratonovich interpretation of the Langevin equation (1). In order to characterize the stochastic process ρ(x, t; ξ) the average is clearly not sufficient and to this end we consider the integer powers of the density and their averages ρ n (x, t; ξ). 1.1 The moments In the case of a volume preserving vector field Φ/x = the integer powers of the density ρ n (x, t; ξ) for n > 1 satisfy the same equation as for n = 1, the initial conditions being ρ n t + Φ ρn x = (9) ρ n (x, t, ξ) = ρ n in (x) (1) as a consequence the solution can be written as a Dyson series ρ n (x, t; ξ) = T exp t ( ( t and their averages still satisfy the equation A(s) + ξ(s) B(s) ) ds ρ n in (x) (11) ( ) t ρn = A + B ρ n (1) with initial conditions ρ n (x, t ; ξ) = ρ n in (x) (13) Notice that in (1) A = a and B = b for a volume preserving flow. In x x the case of a generic flow equation (9) is replaced by ρ n t + Φ ρn x + nρn Φ x = (14) and the equation for the averages is still given by (11) where the operators A and B are given by A n = (n 1) a x x a B n = (n 1) b x x b (15) 6

The free particle example We consider the simplest Langevin equation defined by dx dt = ξ(t) x(t) = x + w(t) w(t ) (16) and a Gaussian initial distribution ρ in (x) = 1 πt exp ) ( x In this case we know that the density ρ(x, t; ξ) is given by (5) namely t (17) ( ρ(x, t; ξ) = 1 x w(t) + w(t ) ) exp (18) πt t Letting the averages of the n th power be labeled according to ρ (n) (x, t) = ρ n (x, t; ξ) (19) by a straightforward, even though tedious, calculation we show that ρ (1) (x, t) = 1 ) exp ( x πt t () By the same procedure we prove that ρ (n) (x, t) satisfy the Fokker-Planck equation ρ (n) = 1 ρ (n) (1) t x with initial conditions ) ρ (n) (x, t ) = ρ n 1 in (x) = ( (πt ) exp nx () n/ t.1 Direct computation of ρ (n) (x, t) We start by expanding the exponential ρ (1) = 1 πt n= ( 1) n (x w(t) w(t )) n = (t ) n n! = 1 πt n= ( 1) n (t ) n n x (n k) (t t ) k (k 1)!! k= n! ( ) n k 7

= 1 πt m= x m k= ( 1) k+m (t ) (t t k (k 1)!! ) k+m (m + k)! ( ) m + k k = 1 π m= x m ( 1) m (t ) m+1/ m! ( ) t t k 1 (k 1)!! (m + k)! m! k (m + k)! (k)!(m)! = k= t = 1 π m= x m ( 1) m (t ) m+1/ m! ( ) t t k 1 k= t (m + k 1)!! k k! m m! (m)! = = 1 π m= x m ( 1) m (t ) m+1/ m! ( ) t t k 1 (m + k 1)!! k (m 1)!! k! k= t = = 1 π m= x m ( 1) m (t ) m+1/ m! ( 1 t t t ) m 1 = 1 πt exp ( ) x t (3) In summing the series we have used the following identity n= n a n k= b n k c k = b m a n c n m (4) m= n=m where we have set m = n k and interchanged the summation order. From k = n m it follows m n <. Changing again the summation index in the second sum from n to k = n m it follows now that k < and consequently we have n= n a n k= b n k c k = b m a k+m c k (5) m= k= We have used the following formula for the Taylor expansion of an algebraic function (1 x) m 1 = x k 1 1 (m + k 1)!! (6) k k! (m 1)!! k= The series converges within y < 1 namely our previous series converges for t t t < 1 < t < t (7) Obviously for t > t the result still holds with an analytic continuation performed after summing the series. The computation of ρ () follows a similar 8

.4.4 <ρ> σ -5 x 5-5 x 5 Fig. 1. Left panel: plot of the mean ρ (1) (x, t) of the density ρ(x, t; ξ) solution of the stochastic Liouville equation for a free particle with white noise for an initial Gaussian density ρ in (x) = e x / /(π) 1/. The three curves correspond to t = 1.1 (green), 1.1 (red), (blue) respectively. Right panel: plot of the variance σ(x, t) for the same values of t and the same initial condition as the right panel. procedure and the result is given by ( ) ρ () 1 (x, t) = π exp x t t t t t (8) It is not hard to verify that it satisfies the Fokker-Planck equation (1) with initial condition (), where n =. In a similar way we can prove that ρ (n) (x, t) is given by ( ) ρ (n) t (x, t) = (πt ) n(t n t n ) exp x (t t n ) (9) where t n = t n 1 n 3 Numerical results and error analysis It is immediate to check that the mean square deviation of the fluctuating density ρ(x, t; ξ) is not zero σ (x, t) = ρ () (x, t) ( ρ (1) (x, t) ) (3) A simple computation shows that the first two even moments with respect to x (the odd ones vanish) of σ (x, t) are given by 1 σ dx σ (x, t) = 1 π ( 1 t 1 t ) (31) 9

and x σ dx x σ (x, t) = 1 4 π ( t t t t ) (3) Both vanish for t = t, whereas for t the former 1 σ reaches a constant limit, the latter x σ diverges linearly. In figure 1 we show the behavior of ρ (1) (x, t) and the variance σ(x, t) for a Gaussian initial distribution (19) with t = 1 using the analytical results (), (8), (3). It is interesting to observe that the variance is a bimodal distribution with a minimum at the origin. 3.1 Error on ρ In Figure we plot the initial density corresponding to the Gaussian initial distribution ρ(x, t ) given by () for t = 1/, Monte-Carlo simulated with N = 1 4 points, and its evolution ρ(x, t ) at time t =, obtained by propagating each initial point with with a distinct Wiener trajectory. Even though N is reasonably large, the discrepancy, with respect to the exact result due to statistical error, is visible at the initial time and at the next one. In the next frame of figure five realizations of the process ρ(x, t; ξ) are shown, namely the analytical initial distribution translated according to five realization of the Wiener process. The error in the computation of averages is inversely proportional to the number N of realizations. In figure 3 the average density ρ(x, t), obtained by back propagating ρ(x, t) with N = 1 4 distinct Wiener trajectories and taking the average, is shown and compared with the exact result. In this case the error cannot be appreciated within the graphical resolution since it is smaller than 1 4 ; this is true also for the mean square deviation σ (x, t) which is compared with the exact resuklt in the right frame. The order N 1 accuracy for the average density ρ at a given point, obtained from N realization of the fluctuating density ρ is a general property. The numerical implementation is straightforward if the vector field is volume preserving and consists in the use equation (4). This corresponds to evaluate ρ(x, t), we suppose to be analytically known, as the initial density ρ in at the back propagated initial point x determined by a suitable integrator with the desired level of accuracy (in the generic case the computation of the jacobian function µ(x, t) is also required). Conversely the solution of the Langevin equation is obtained by approximating the initial density with N particles and letting each of them evolve in time with an independent realization of the noise. Let us consider for example the one dimensional case and suppose that the 1

.4.35.3 numer. t=.5 analit. t=.5 numer. t= analit. t=.4.35.3 initial density mean density < ρ (x) >.5. ρ (x).5..15.15.1.1.5.5-6 -4-4 6 x -3 - -1 1 3 x Fig.. Left: 1-dimensional diffusion for different realizations of the noise (Brownian motion): comparison between the exact and numerical solution at time t =.5, with N = 1 4. Right: 1-dimensional diffusion for same realizations of the noise (stochastic Liouville equation): initial distribution at t =.5 (red), mean distribution at t = 1 (blue), distributions obtained for one realization of the noise (black). support of the particles density is an interval of length L in the time interval t t T. In any space subinterval of center x and length x the average number of particles is N = N ρ(x, t) x with an uncertainty given by the statistical fluctuation ( N) 1/. Supposing that ρ is nearly constant e have ρ L 1. The space resolution x/l is equal to the statistical resolution ( N) 1/, if the number of particles is N (L/ x) 3. Using the stochastic Liouville equation in order in order to evaluate ρ(x, t) in the single point x with an accuracy ɛ we need to compute N 1/ɛ trajectories, along which we back-propagate the density to its initial value, which is supposed to be known analytically. In this case the accuracy does not decrease at the points where the density is very low, as it happens in the standard Langevin scheme since the statistical fluctuation N/N is proportional to ρ 1/. For example to obtain ρ on a grid of 1 3 points with 1 3 accuracy we need 1 9 Wiener trajectories with the standard Monte-Carlo approach to solve the Fokker-Planck equation, just 1 6 using the stochastic continuity equation. 4 Space correlations The fluctuating densities are space correlated. The two and more generally the n points space correlation function for the stochastic continuity equations (SCE) satisfy Fokker-Planck like equations which are obtained in a way very similar to the one followed for the average of the density and its integer powers. 11

Indeed starting from equation (6) for the SCE we write t ρ(x, t) = (A x(t)+ξ(t)b x (t))ρ(x, t) A x (t) = a(x, t) x B x(t) = b(x, t) x (33) it is not hard to show that the product of the density at two distinct points x, y satisfies the following equation t ρ(x, t)ρ(y, t) = ( A(t) + ξ(t)b(t) ) ρ(x, t)ρ(y, t) A = A x + A y B = B x + B y (34) We should notice that Aρ(x)ρ(y) in the limit y x becomes x ρ (x); taking the limit separately on the operator and the function gives a wrong result. The solution can be written in the same way as (7) where the the operators A, B are defined according to (34) and the initial condition is replaced by ρ in (x)ρ in (y). Taking the average and denoting for brevity ρ (x, y, t) = ρ(x, t)ρ(y, t) the two point space correlation function, the equation it satisfies is t ρ (x, y, t) = (A + B ) ρ (x, y, t) (35) with initial condition ρ (x, y, ) = ρ in (x) ρ in (y). More generally denoting ρ n (x 1,..., x n, t) = ρ 1 (x, t) ρ n (x n, t) the n points correlation function, the equation it satisfies is similar to (35) and reads t ρ n(x 1,..., x n, t) = (A + B ) ρ n(x 1,..., x n, t) A = n i=1 x i a(x i, t) B = with initial conditions ρ n (x 1,..., x n, ) = ρ in (x 1 ) ρ in (x n ). n i=1 x i b(x i, t) (36) 5 The probability distribution The process associated to the fluctuating density ρ(x, t; ξ) can be determined from a probabilistic view point by associating a probability density P (x, ρ, t). At each time t indeed our field is specified by a space coordinate x and the value ρ it takes there. We may also introduce a conditional probability P (x, ρ, t x, ρ, t ) equal to the weighted sum of all the paths joining in an extended phase space the initial point (x, ρ, t ) with the final point (x, ρ, t). The phase space (x, ρ) R d+1 has one extra dimension with respect to the ordinary phase space x R d where the Langevin equation is defined. The initial 1

< ρ (x) >.4.35.3.5..15 initial distr analitycal numerical variance of ρ(x).5..15.1 analitycal numerical.1.5.5-6 -4-4 6 X -6-4 - 4 6 X Fig. 3. Left: 1-dimensional diffusion: comparison between the exact and numerical mean density at time t =. In red the initial distribution at t =.5 with N = 1 4. Right: the same for the variance of the density. probability distribution is P (x, ρ) = P (x, ρ, t ), not to be confused with the initial density ρ in (x). If the initial density is ρ in (x) with probability 1 then P (x, ρ) = δ(ρ ρ in (x)) (37) The normalization condition reads for any t t dρ dx ρ P (x, ρ, t) = 1 (38) The relation between the initial probability and its value at time t is P (x, ρ, t) = dρ dx P (x, ρ, t x, ρ, t ) P (x, ρ ) (39) where the transition probability satisfies the standard normalization condition and the group property dρ 1 dρ dx P (x, ρ, t x, ρ, t ) = 1 dx 1 P (x, ρ, t x 1, ρ 1, t 1 ) P (x 1, ρ 1, t 1 x, ρ, t ) = P (x, ρ, t x, ρ, t ) (4) To determine the transition probability we have to consider the characteristics of the SCE in the extended phase space (x, ρ). We assume for simplicity that b/x = so that the divergence of the vector field φ is a/x =. In this 13

case the evolution equations read dx dt = a(x, t) + ξ(t) b(x, t) dρ dt = ρ a x (41) We consider the continuity equation in the extended phase space for the fluctuating density Π(x, ρ, t; ξ) which satisfies the equation Π t = (A + A ρ + ξ(t)b) Π (4) where A = a(x, t) x A ρ = a x ρ ρ Denoting by P the average of Π with respect to ξ, B = b(x, t) x (43) ( P t = A + A ρ + 1 ) B n P (44) In the volume preserving case A ρ = and consequently the equation satisfied by P and its moments ρ n = ρ n P dρ with respect to ρ is the Fokker- Planck equation (1). As a consequence we may identify P with the probability density P in extended phase space. In the volume non preserving case it is not difficult to see that the equation satisfied by the moments ρ n is t ρn + a x ρn = (A n + 1 ) B ρ n (45) where A n = A (n 1) a/x is the operator defined by (15) and B n = B having assumed that b/x =. This result easily follows by observing that ρ n B ρ P dρ = n a/x n ρ n after an integration by parts. The relation with ρ n is simple only if a/x is constant. In the one dimensional case letting α be this scalar constant it turns out that ρ n = e α(t t ) ρ n so that the probability density in extended phase space is given by P = e α(t t ) P. 5.1 The free particle with damping In order to clarify the above discussion we analyze in detail the case of the free particle with damping in which the equation (41) reduces to and the equation for P is ẋ = α x + ξ(t) ρ = αρ (46) t P = α x (x P ) α ρ (ρp ) + 1 P (47) x 14

The equation is solved by separation of variables and we write its fundamental solutions as P (x, ρ, t x, ρ, t ) = G(x, t x, t ) δ(ρ ρ e α(t t ) ) (48) where G(x, t x, t ) is the fundamental solution of the Fokker-Planck equation for the Uhlenbeck process G(x, t x, t ) = ( 1 πσ (t t ) exp (x x e α(t t) ) ) σ (t t ) σ (t) = 1 e α t α (49) As a consequence recalling that P = e α(t t ) P the probability density in the extended phase space at time t is given by P (x, ρ, t) = e α(t t ) dρ dx G(x, t x, t )δ(ρ ρ e α(t t ) ) P (x, ρ ) (5) For an initial probability distribution with support on a unique distribution ρ in (x) we have P (x, ρ) = δ(ρ ρ in (x)) In figure 4 we plot P (x, ρ, t) as function of ρ having chosen x = 1 and α = 1 and ρ in (x)) equal to the gaussian distribution (17). The moments of this probability distribution are easily computed ρ n = e α(t t ) ρ n dρ dρ dx G(x, t x, t ) δ(ρ ρ e α(t t ) ) P (x, ρ ) = = e α(t t ) dρ e nα(t t ) ρ n dx G(x, t x, t ) δ(ρ ρ in (x )) = = e (n 1)α(t t ) dx G(x, t x, t ) ρ n in (x ) (51) In appendix C we carry out the explicit calculation of the second moment ρ using (51) also in the case where the white noise ξ(t) is replaced by a correlated noise. 6 Conclusions The SCE associated to a Langevin equation is a more general process in which a given initial density function is propagated by any realization of the process. The probability density associated to the Langevin equation is the average of 15

3 5 P(x=1,ρ,t) 15 t=.51 1 t=.8 t=.51 5 t=.6.1..3.4.5.6.7 ρ Fig. 4. Plot of the probability density P (x, ρ, t) for the free particle with damping (α = 1) and white noise as a function of ρ at x = 1 and t =.51, t =.51, t =.6, t =.8 for an initial probability density at t =.5 concentrated on a Gaussian. The circles correspond to the numerical result while the lines to the analytical result. the density function ρ computed over all the realizations. Naturally the average of the density function does not exhaust all the information, which is captured instead by the average of all the integer powers of the density. This density evolves following the Fokker-Planck equation associated to the Langevin equation for a volume preserving vector field. The averages are equivalent to the moments of a probability density in the phase space whose dimension is increased by one, by including the ρ coordinate. This probability also satisfies a Fokker-Planck equation associated to a new Langevin equation in the extended phase space, describing the characteristics of the continuity equation. The meaning of the process corresponding to the SCE has been illustrated in the case of the free particle where all the computations are easily carried out. From the numerical viewpoint the calculation of the probability density, given initially in analytic form, is fast and accurate at a given space point, since one can average the value there of the fluctuating density. The standard numerical procedure requires in any case the propagation of all the points of the initial probability density, which is usually done with a Monte-Carlo procedure computing for each point a stochastic trajectory, which is an autonomous realization. We did not analyze the case of a correlated noise, since 16

the extension is rather straightforward, at least conceptually. One can treat the correlated noise as an additional variable, by including the white noise driven differential equation to the previous set of differential equations. It is possible to avoid the increase of the phase space dimensionality, however the probability densities do not longer satisfy an equation in closed form and the approximation schemes required are applicable only to noise of small amplitude and with rapidly decaying correlations. Acknowledgments We wish to thank Prof. Ellison for useful discussions. A. Bazzani and G. Turchetti wish to thank DESY for hospitality during the preparation of this work, which was completed within the framework of the COFIN project 3 Order and chaos in nonlinear extended systems: coherent structures, weak stochasticity and anomalous transport. G. Bassi wishes to thank DESY for a doctoral fellowship and the U.S. Dept. of Energy for for a partial support by contract DE-FG3-99ER4114. 17

7 Appendix A: Wiener and Uhlenbeck processes The time evolution of a stochastic process is determined by a deterministic field and its fluctuations (noise). The white noise ξ(t), formal time derivative of a Wiener process w(t), describes a fluctuating field, whose correlations have an instantaneous decay ξ(t) = ξ(t)ξ(t ) = δ(t t ) (A1) The presence of memory corresponds to a non instantaneous decay of correlations and the decay rate characterizes the fluctuating field. The fluctuating part χ(t) of the process x(t) defined by dx dt = αx(t) + ξ(t) (A) is asymptotic to the Ornstein-Ulhenbeck process for large t and reduces to the Wiener process w(t) for α =. As a consequence letting x(t ) = x be the initial condition we have x(t) = e α(t t ) x + χ(t) χ(t) = and for α = we have χ(t) = w(t) w(t ) t t t e α(t s) ξ(s)ds (A3) t ξ(s)ds (A4) By using the following property of the white noise correlations ξ(s 1 )ξ(s ) ξ(s n 1 )ξ(s n ) = i 1...i n δ(s i1 s i ) δ(s in 1 s in ) (A5) where the sum runs over all the (n 1)!! pairings of the indices, it is not difficult to show that the following relations hold for t t χ n (t) = ( ) 1 e α(t t ) n (n 1)!! (w(t) w(t )) n = (t t ) n (n 1)!! α The most general Langevin equation can be written as (A6) dx dt = Φ(x, t; ξ) a(x, t) + b(x, t) ξ(t) (A7) where the stochastic field may be regularized by replacing ξ(t) with αχ(t), a regular process whose limit is the white noise when α. 18

8 Appendix B: Space correlations for a free and damped particle In order to illustratrate the computation of space correlations, we consider the free particle case in which equation (35) explicitly reads t ρ (x, y, t) = 1 ( x + ) ρ (x, y, t) (B1) y In order to find the solution corresponding to ρ in (x) = δ(x) we perform a Fourier decomposition ρ (x, ty, t) = (π) R e 1 (kx+ky) t ik xx ik yy dk x dk y (B) It is evident that this is a solution of equation (B1). Rotating the coordinates with u = (k x + k y )/ and v = (k x k y )/ we obtain ρ (x, y, t) = (π) R e u t iu x+y du R x y i e v dv = x+y e ( ) 1 t δ(x y) (B3) πt If the initial distribution a Gaussian ρ (x) = (πt ) 1/ exp ( ) x t we may calculate the space correlation as ρ () = ρ in (x w(t) + w(t ))ρ in (y w(t) + w(t )) rather than solving (B1) ρ () (x, y, t) = ρ in (x w)ρ in (y w)p(w, t t )dw (B5) where p(w, t t ) = 1 π(t t ) e w (t t ) (B6) It follows that = ρ(x, t)ρ(y, t) = dw 1 (x w) πt ) e πt ) e t 1 1 π t(x ( +y ) (t t )xy t e (t t ) 1 ) = t t t π(t t ) (y w) 1 t π(t t ) e e ( x+y ) 1 t t 1 w (t t ) e (x y) 4t 4πt (B7) Remark that in the limit t the new result (B5) agrees with the previous one (B3). Moreover fot y = x we recover ρ () (x, t) ρ (x, t) given by (8). 19

8.1 Damped particle with white noise If the initial distribution is a Gaussian ρ in (x) = (πσ )1/ exp ( ) x σ we calculate the space correlation for ẋ = αx + ξ(t) which is given by (B5) where 1 w p(w, t t ) = πσ(t t ) e σ(t t ) ; σ(t) = 1 α (1 e α t ) It follows that ρ(x, t)ρ(y, t) = = πσ e α(t t ) e α(t t ) πσ (t t ) π σ σ (t t ) + e α(t t ) σ For x = y we get e α(t t ) exp dwe 1 σ (e α(t t ) [(x w) +(y w) ]) e ( (B8) w σ (t t ) (x y) σ (t t )e α(t t ) + (x + y )σ σ (σ (t t ) + e α(t t ) σ ) ( (B9) ρ (x, t) = exp π σ σ (t t ) + e α(t t ) σ σ (t t ) + e α(t t ) σ (B1) in agreement with the result (C4) obtained in appendix C. x ) )

9 Appendix C: The second moment ρ for a damped particle In order to give a simple explicit example we consider a point subjected to a damping force and random uncorrelated collisions. The corresponding process for the point velocity, we denote by x, is defined by equation (A). The average density ρ and the average of its square ρ satisfy the equations defined by (1) where A, B are replaced by A n, B n defined by (15) for n = 1,. These equations explicitly read t ρ = α x (x ρ ) + 1 x ρ (C1) and t ρ = α x (x ρ ) + α ρ + 1 x ρ (C) which differs from the previous one by the divergence of the deterministic field. In this case there is no Langevin equation associated to the process defined by the fluctuating field ρ and in order to compute the mean ρ we evaluate the second moment with respect to ρ of the probability density P (x, ρ, t) according to (48) Choosing an initial Gaussain density with variance σ ) 1 ρ in (x) = exp ( x πσ σ (C3) and inserting into (46) we obtain ρ = πσ e α(t t ) πσ (t t ) dx exp ( (x x e α(t t ) ) σ (t t ) ) exp ) ( x σ = e α(t t ) = exp π σ σ (t t ) + e α(t t ) σ σ (t t ) + e α(t t ) σ (C4) If we choose σ = σ (t ) then the denominators in the exponent and the argument of the second square root becomes σ (t t ) + σ (t) since the following identity holds σ (t t )+e α(t t) σ (t ) = σ (t). In the limit α the result (3) for the second moment of the free particle is recovered. For the moment ρ n the result is obtained in a similar way. ( x ) If the white noise is replaced by a Uhlenbeck noise according to dx dt = αx + y(t) dy dt = βy(t) + βξ(t) (C5) 1

choosing y() = so that y(t) = we have a new process, which tends to the previous one in the limit β. The process x(t) is Gaussian and and writing the solution of (C5) as x(t) = x e αt + t e α(t τ) y(τ) dτ y(t) = β t dτe β(t τ) ξ(τ) (C6) the variance is easily computed after writing x x e αt according to = t K(t, s)ξ(s)ds σ (t) = t K (t, s)ds = [ β (1 e αt ) β3 (1 e (α+β)t ) α(β α) (β α )(β αβ) + β3 (1 e βt ] ) (β αβ) (C7) The evolution for ρ is still the same dρ/dt = αρ and the probability density P is still expressed by P = e α(t t ) P where P is given by (48) and in (49) σ (t) is given by (C7). As a consequence the second moment ρ is still given by (C4) where the variance is defined by (C7).

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