Generalized Langevin equations : Anomalous diffusion and probability distributions

We study the motion of a particle governed by a generalized Langevin equation. We show that, when no fluctuation-dissipation relation holds, the long-time behavior of the particle may be from stationary to superdiffusive, along with subdiffusive and diffusive. When the random force is Gaussian, we derive the exact equations for the joint and marginal probability density functions for the position and velocity of the particle and find their solutions. @S1063-651X~96!10706-6#


I. INTRODUCTION
This paper deals with systems that present anomalous diffusion.It has been shown that in many systems the mean square displacement does not grow linearly as in normal diffusion but in the anomalous form given by ͗X 2 ͑ t ͒͘ϳt ͑ t→ϱ ͒, where 1 ͓1,2͔.There is current interest in understanding the physical mechanism leading to anomalous diffusion and it is thus found that chaotic systems can present superdiffusion with Ͼ1 ͓3͔ while subdiffusion with Ͻ1 is encountered in constrained systems such as fractals ͓1,2͔.We have recently shown that the consideration of inertial effects also results in superdiffusion ͓4-6͔.Moreover, we have also observed that Gaussian 1/f driving noise leads to a subdiffusive behavior for the undamped free Brownian particle ͓7,8͔.
On the other hand, generalized Langevin equations have been used recently to describe the dynamics of particles in percolation clusters that present anomalous diffusion ͓9͔.This equation is a nonlocal equation that, in the absence of a deterministic field, can be written in the form where ␤(tϪtЈ) is the dissipative memory kernel and F(t) is a random force that we assume to be zero centered and stationary, i.e., ͗F͑t͒F͑tЈ͒͘ϭC͉͑tϪtЈ͉͒ϭC͑͒,

͑1.2͒
where C() is the correlation function ͓10,11͔.When the system described by Eq. ͑1.1͒ is in the equilibrium state, then the functions ␤(t) and C(t) are related to each other by means of the fluctuation-dissipation theorem ͓11͔: where k B is the Boltzmann constant and T is the absolute temperature of the environment ͑heat bath͒.In this case the random force is sometimes referred to as ''internal noise.''Note that, from a physical point of view, one of the effects of the noise being internal is that the relaxation time of the system is essentially the same as the correlation time of the noise.However, in nonequilibrium systems the driving noise and the dissipation may have different origin and no fluctuation-dissipation relation holds.In such a case F(t) will be referred to as ''external noise.'' In this paper, we will show how the long-time behavior of the mean square displacement for systems described by generalized Langevin equations ͑GLE's͒ depends on the properties of the correlation function and of the memory kernel.We consider nonequilibrium systems although the equilibrium situation is easily recovered by imposing fluctuationdissipation relation.When both the correlation function and the dissipative kernel are long-time tail functions, a variety of limiting behaviors is obtained.The exponents of the longtime tails determine whether the system becomes stationary, subdiffusive, diffusive, superdiffusive, or even logarithmic, i.e., ͗X 2 (t)͘ϳlnt.
When the noise is Gaussian, and due to the linearity of Eq. ͑1.1͒, it is also possible to derive exact expressions and Fokker-Planck equations for the joint probability density, p(x,v,t), of the displacement and the velocity and for the marginal densities p(x,t) and p (v,t).
The paper is organized as follows.In Sec.II, we set the general analysis of the process governed by Eq. ͑1.2͒.In Sec.III we study the long-time behavior of the variances and discuss the anomalous diffusion.In Sec.IV we analyze two relevant examples.In Sec.V we obtain the probability density functions of the process when the driving noise is Gaussian.Conclusions are drawn in Sec.VI and technical aspects are in the Appendices.

II. GENERAL ANALYSIS
In what follows we will assume that the driving noise F(t) is stationary, zero centered, and has a correlation function given by C(t).We write the formal solution to Eq. ͑1.1͒ in the form where where x 0 ϭX(0) and v 0 ϭX ˙(0) are the initial position and the initial velocity of the particle.The relaxation function H(t) is the inverse form of the Laplace transform where is the Laplace transform of the dissipative memory kernel.
We will only consider random systems whose velocity relaxes to a stationary state with zero average velocity.Thus, h(t) goes to zero when t→ϱ.For systems driven by internal noise the relaxation function h(t) is related with the longtime behavior of the autocorrelation function of the velocity.This relation is proved in Appendix A and reads From Eqs. ͑2.1͒ and ͑2.5͒ and the symmetry property of the correlation function, i.e., C(tϪtЈ)ϭC(tЈϪt), we obtain the explicit expression of the variances of process ͑1.1͒ and These are general equations valid for either internal and external noise.However, when the driving noise is internal Eqs.͑2.8͒-͑2.10͒reduce to the following more convenient form ͓12͔: and

III. ASYMPTOTIC BEHAVIOR AND ANOMALOUS DIFFUSION
We will now discuss with some generality the asymptotic behavior of the mean square displacement in terms of the asymptotic expansions of the correlation function C(t) and of the memory kernel ␤(t).
We will not study all possible limiting behaviors of the correlation function.Instead we will only consider two simple but very general models, which are described below.In both cases we will assume that the correlation function is a locally integrable function in (0,M ) for any positive M .
͑a͒ Finite noise intensity.In this case the noise intensity defined by ͓13͔ is finite and nonvanishing.Hence, C()ϭo( Ϫ1 ) when →ϱ. Once we know the leading behavior of C() as →ϱ, the behavior of its Laplace transform C ˆ(s) as s→0 can be obtained using Abelian theorems, which give the small s behavior of C ˆ(s) if one knows the asymptotic behavior of C() as →ϱ ͓14͔.In the case of a finite noise intensity one can easily see that ͓15͔ where K is given by Eq. ͑3.1͒.͑b͒ Long-time tail noise.Now the correlation function decays as a power law of the form where 0Ͻ␣Ͻ1.The Laplace transform of the correlation function behaves as ͓15͔ As we have mentioned in Sec.II we will only consider dissipative kernels ␤(t) that lead to stationary states for the velocity process.Moreover, we restrict ourselves to those kernels ␤(t) leading to relaxation functions, h(t), that finally become monotonic, that is, there exists some t 0 у0 such that dh(t)/dt has definite sign for all tуt 0 .Again, ␤(t) is supposed to be locally integrable.Equation ͑2.6͒ suggests considering the following cases: ͑a͒ Finite relaxation time.That is, the inverse of the relaxation time is finite and positive.In this case ͓see Eq. ͑3.2͔͒ ␤ ˆ͑s ͒ϳEϩo͑ 1 ͒ ͑ s→0 ͒. ͑3.6͒ ͑b͒ Slow relaxation.We now assume that the friction kernel decays as where 0Ͻ␥Ͻ1.Therefore, its Laplace transform behaves as ͓see Eq. ͑3.4͔͒ The behavior of h ˆ(s) is readily obtained from Eq. ͑2.6͒ and from the expansions of ␤ ˆ(s) when s→0 ͓cf.Eqs.͑3.6͒ and ͑3.8͔͒.We now want to know the long-time behavior of h(t) and H(t).This is accomplished by the use of Tauberian theorems, which can be considered converse Abelian theorems and they consist of a collection of results that roughly give the asymptotic behavior of a function f (t) as t→ϱ if one knows the small s behavior of its Laplace transform ͓14͔.For the two kinds of memory friction introduced above we have ͓16͔ the following: ͑a͒ Finite relaxation time.In this case, and ͑b͒ Slow relaxation.Now we have and In Appendix B we verify these relations for a particular example.
The substitution of expansions ͑3.10͒ and ͑3.12͒ into Eqs.͑2.8͒ and ͑2.9͒ allows us to obtain the long-time behavior of the variances of process ͑1.1͒.We can easily see that the variance of the velocity becomes a constant for all cases discussed above.Moreover, the average velocity converge to zero because h(t) goes to zero as t→ϱ.Hence, the velocity process reaches a stationary state with zero average value.When the driving noise is internal then the stationary state is the equilibrium state.In this case the variance of the velocity can be obtained in a more straightforward way.This is the case, for instance, of a free Brownian particle of unit mass in a medium at temperature T where lim t→ϱ vv 2 ͑t͒ϭk B T.

͑3.13͒
The behavior of the mean square displacement, xx 2 (t), shows a much richer diversity.The normal diffusion behavior arises when both the intensity of the noise K, Eq. ͑3.1͒, and the inverse relaxation time E, Eq. ͑3.5͒, are finite.In such a case one obtains from Eq. ͑2.8͒ that Let us now derive this expression.We first note that Eq. ͑2.8͒ can be written as H͑tzЈ͒C͑tzϪtzЈ͒dzЈ.

͑3.15͒
From Eq. ͑3.10͒, we see that, for friction kernels of type ͑a͒, H(t) converges to 1/E when t→ϱ.Therefore, Reversing the order of integration we get When t→ϱ we finally have and for driving noises with finite intensity K we obtain Eq. ͑3.14͒.
When the correlation function presents a long-time tail and E is finite, the X(t) process is superdiffusive because and 0Ͻ␣Ͻ1.The derivation of this expression is outlined in Appendix C. The existence of the tail indicates that the random force varies slowly and persists driving the system in the same direction.As a consequence, the system becomes superdiffusive.
Nevertheless, when the relaxation is also slow ͓i.e., ␤ of type ͑b͔͒ then the number of possible limiting behaviors increases notably.Thus when the driving noise has a finite intensity K then the long-time behavior of X(t) depends solely on the exponent ␥ of the relaxation tail.The possible limiting behaviors of the variance, as a function ␥, are ͑cf.Appendix C͒ (t→ϱ).It is remarkable that for ␥ϭ1/2 the mean square displacement grows as slowly as the logarithmic behavior.
When both the correlation function and the memory kernel show long-time tails the process may become asymptotically stationary to superdiffusive, including subdiffusive, diffusive, and logarithmic.The fact that no fluctuation-dissipation relation holds means that the exponents ␣ and ␥ are independent.Thus the long-time behavior of the variance xx 2 (t) will depend on both exponents.Following a similar reasoning we have ͑see Appendix C͒ (t→ϱ).In the third case where 2␥Ͼ␣ we distinguish three different regimes: ͑i͒ subdiffusion, if 2␥Ϫ␣Ͻ1; ͑ii͒ normal diffusion, if 2␥Ϫ␣ϭ1; and ͑iii͒ superdiffusion, if 2␥Ϫ␣Ͼ1.We show in Fig. 1 the long-time behavior of the mean square displacement as a function of the exponents ␣ and ␥.
We note that for internal noise, the fluctuation-dissipation relation implies that the tail exponents of C(t) and ␤(t) are equal, that is, ␣ϭ␥.In this case, the long-time behavior of This result can also be obtained in a different way ͓9͔.We first observe that the properties of the mean square displacement are determined by the velocity process.In effect, the variance of the displacement can be written in the form We define Taking into account Eq. ͑3.12͒ we see that the long-time behavior of I 2 (t) is The asymptotic behavior of I 1 (t) is obtained as follows.We first write I 1 (t) in the form The asymptotic expression of the first integral on the righthand side of this equation is obtained by taking the limit when s→0 of h ˆ(s) ͓17͔.From Eq. ͑3.11͒, we see that Using Eq. ͑3.12͒ we see that as t→ϱ the second integral in Eq. ͑3.29͒ behaves as

͑3.32͒
Taking into account that the quantities A and B are related by Aϭk B TB we finally obtain Eq. ͑3.22͒.
In the next section we work out exactly two cases where the long-time behavior becomes diffusive.

IV. EXAMPLES
We discuss in this section two examples that can be solved analytically.The expression for the mean square displacement at all times makes it possible to identify different regimes in the behavior of these systems.Unfortunately, we have not been able to solve exactly any case with a long-time tail.In a previous study, Muralidhar et al. reduced the velocity correlation function for a power law tail kernel up to quadrature ͓18͔.This particular case refers to internal noise and the quadrature has to be approximated in order to obtain the standard anomalous diffusive behavior.
We now assume that the stationary driving noise F(t) is exponentially correlated, where D is the noise intensity and c is the correlation time.This is a relevant example of driving noise with finite intensity.

A. Internal noise
Let us first assume that F(t) is internal noise.In this case the frictional kernel is given by ␤(t)ϭC(t)/k B T ͑note that this example represents a Brownian particle͒.The function H(t) is given by ͓cf.Eq. ͑2.3͔͒ where and

͑4.4͒
The substitution of Eq. ͑4.2͒ into Eqs.͑2.11͒-͑2.13͒results, after some algebra, in the following explicit expressions for the variances of the particle

͑4.7͒
We see that when t→ϱ, This result is in agreement with Eqs.͑3.13͒ and ͑3.14͒ because now KϭD/2 and EϭD/2k B T. Since, in this case, we know exactly the time evolution of the variances for all times, we can identify several regimes.We first distinguish three different time scales involved in the motion of the Brownian particle.Namely, the observation time t, the relaxation time r ͑the inverse of E), and the correlation time of the input noise ͓which is related with the correlation function C(t)͔.Note that for internal driving noise the correlation time and the relaxation time are essentially of the same order of magnitude ͓cf.Eq. ͑1.3͔͒, that is, r ϳ c ϵ. Hence, the behavior of the variances will depend on whether t and are similar or not comparable.In this case we have only two extreme situations ͑i͒ tӶ, and ͑ii͒ tӷ.Case ͑i͒ obviously corresponds to t→0 while case ͑ii͒ is equivalent to t→ϱ.From Eqs. ͑4.5͒ and ͑4.6͒, we see that Let x and v be exponents defined by xx 2 (t)ϳt x and vv 2 (t)ϳt v .We may then summarize the above results with the following transitions between exponents ͑as t increases͒ x ϭ4→ x ϭ1, v ϭ2→ v ϭ0. ͑4.11͒

B. Markovian relaxation
Suppose now that the driving noise is external.In this case C(t) and ␤(t) are not related.Let us further assume that the relaxation is Markovian, that is, ␤(t)ϭ␤␦(t).In this case, h͑t ͒ϭe Ϫ␤t , H͑t ͒ϭ͑ 1Ϫe Ϫ␤t ͒/␤.

͑4.12͒
The substitution of Eq. ͑4.12͒ and Eq.͑4.1͒ into Eqs.͑2.8͒-͑2.10͒yields and Let us analyze the asymptotic properties of the variances.We first observe that in this case there are three different time scales: the correlation time c , the relaxation time r ϭ␤ Ϫ1 , and the observation time t.We will now show that the behavior of the variances depends on whether these three time scales are similar or not comparable.For t smaller than any other time scale, it follows from Eqs. ͑4.13͒ and ͑4.14͒ that On the other hand, if t is larger than any time scale, then

͑4.17͒
Let us now investigate the intermediate-time behavior of the variances between these two extreme situations.We first assume that c and r are not comparable.In this situation we have two different cases.
͑1͒ c Ӷ r .In this case we distinguish the time regime where the observation time is much longer than the correlation time but still smaller than the relaxation time: c ӶtϽ r .In this regime F(t) acts as white noise, with C(t)ϭD␦(t), and from Eqs. ͑2.9͒-͑4.12͒we get ͑recall that ␤tϽ1) Therefore, when c Ӷ␤ Ϫ1 , we have the following transition ͑as t increases͒ ͑2͒ c ӷ r .In this case we distinguish the time regime where the observation time is much longer than the relaxation time but smaller than the correlation time: r ӶtϽ c .In this regime the inertial term may be neglected and the dynamical equation can be approximated by X ˙Ϸ␤ Ϫ1 F(t).As a consequence the variances are given by ͑recall that t/ c Ͻ1) and the transitions between exponents are given by where now c ӷ␤ Ϫ1 .We also observe that when the time scales are similar, c ϳ r ͑as is the case of internal driving noise͒, we are only able to distinguish two extreme asymptotic regimes: t→0 and t→ϱ.This results in the transition given by Eq. ͑4.11͒.

V. PROBABILITY DENSITY FUNCTIONS
We now assume that the input noise is Gaussian.Due to linearity of Eq. ͑1.1͒, we see that the joint probability density function p(x,v,t) for a displacement x and a velocity v at time t must be Gaussian.As a consequence, the joint characteristic function of the process can be written as

͑5.1͒
Using the method of the characteristic function ͓8͔ or the method of functional derivative ͓6͔, one can easily see that the joint probability density function p(x,v,t) obeys the following Fokker-Planck equation: Let us now obtain the marginal probability density functions of the velocity p(v,t) and the displacement p(x,t).Setting ϭ0 in Eq. ͑5.1͒ and following the method of the characteristic function, we can see that p(v,t) obeys the following Fokker-Planck equation: where The marginal density of the displacement p(x,t) obeys the Fokker-Planck equation where D x (t) is the time derivative of the variance, i.e.,

VI. CONCLUSIONS
We have analyzed the long-time behavior of processes obeying a generalized Langevin equation.When the driving noise is external several behaviors arise depending on the memory kernel and the correlation function.The common diffusive behavior appears when both the relaxation time and the correlation are finite and nonvanishing.
In the presence of long time tails, either in the memory kernel or in the correlation function, the system diffuses anomalously.The exact nature of the leading behavior depends on the exponents of the long-time tails of both the friction kernel and the correlation function.Thus, the possible behaviors include stationary state, logarithmic growth, subdiffusion, normal diffusion, and superdiffusion.Figure 1 summarizes the limiting states as a function of the exponents of the long-time tails.However, we have shown that when the driving noise is internal the final state is always subdiffusive.
At this point we mention that the problem of anomalous diffusion has been studied mainly for particles moving on fractals and other disordered media.In this context subdiffusion appears due to trapping sites where particles disappear, or to excursions of the particles along blobs or dead ends.Most of the results in this area are known by a combination of scaling arguments and simulation results ͓1,2͔.However, superdiffusion seems to be more difficult to explain although models based on Levy flights have successfully explained it for some systems ͓19͔.A superdiffusive behavior with exponent x ϭ3 has been proved nonetheless for free inertial particles driven by Gaussian white noise ͓4͔.In any case it is quite difficult to obtain general anomalous diffusion behaviors with exponents x Ͼ1.On the other hand, subdiffusion appears for particles moving on a percolation cluster for which the dynamics can be described by a generalized Langevin equation with a slow relaxation kernel ͓9,18͔.In such a case the noise is internal and we have proved that the behavior of the system is subdiffusive.Therefore, our main objective in this paper has been to extend this last result to external driving noise where the system not only shows subdiffusion but all kinds of diffusive behavior ͑e.g., logarithmic, subdiffusion, normal diffusion, and superdiffusion͒.
Finally, when the noise is Gaussian, the expressions for the joint and marginal probability densities can be derived.These densities satisfy the Fokker-Planck equation with time dependent coefficients.The time dependence is the signature of the non-Markovian character of the process.As an example, we have exactly solved two cases and study the behavior of the mean square displacement at all times.When the time scales of the process are completely separated, there is a new transition from the ballistic behavior to the final diffusive behavior.This intermediate state is characterized by an exponent x that takes the value 2 or 3 depending on the fundamental time scales r and c .

APPENDIX A: VELOCITY CORRELATION FUNCTION FOR INTERNAL NOISE
We assume that the velocity X ˙(t) relaxes to a stationary state.The correlation function of this process will be

͑A1͒
From Eq. ͑2.5͒ and by taking into account the correlation function C() of the driving noise, we write where h ˆ(i) is the Laplace transform of h(t) evaluated at sϭi.When the driving noise is internal, C()ϭk B T␤() and where Re(z) is the real part of z and ␤ ˆ(s) is the Laplace transform of ␤(t).Substituting Eq. ͑2.6͒ into Eq.͑A4͒ and using the symmetry property of C() yield On the other hand, we know that ͗X ˙2(t)͘→k B T as t→ϱ.

APPENDIX B: RELAXATION FUNCTIONS FOR A LONG-TIME TAIL KERNEL
As an example, the asymptotic expansions of functions h(t) and H(t), Eq. ͑3.12͒, can be checked for the following relaxation kernel: ␤͑t͒ϭ1/ͱt.͑B1͒ The Laplace transform of this reads ␤ ˆϭ1/ͱs ͑B2͒ and from Eqs. ͑2.3͒ and ͑2.6͒ we have The Laplace inversion yields which agrees with previous results ͓18͔.The long-time behavior can be obtained from these expressions.In effect, when t→ϱ, the first term of the expression for h(t) decays exponentially while the maximum contribution of the second term comes from r around zero.Therefore, which agrees with Eq. ͑3.19͒.
When the friction kernel is of type ͑b͒, there are more possibilities.Let us discuss the case of a driving noise of finite intensity, K, and a power law kernel of exponent ␥ ͓cf.Eq. ͑3.7͔͒.The Laplace transform of function (t) behaves for small s as ˆ͑s ͒ϳK s Ϫ␥ B⌫͑1Ϫ␥ ͒ . ͑C7͒ Let us obtain the long-time behavior of (t).From our assumptions we know that h(t), and consequently H(t), becomes monotonic when tуt 0 ͑for some t 0 у0).As a consequence H(t) possesses an asymptotic expansion ͑as t goes to infinity͒ that allows us to apply a convenient Tauberian theorem to the expression for ˆ(s) given by Eq. ͑C7͒.Therefore, the long-time behavior of (t) reads ͓16͔ ͑t ͒ϳK sin͑␥ ͒ B t ␥Ϫ1 , t→ϱ.͑C8͒ Once this expansion is substituted into Eq.͑3.1͒ along with the asymptotic expansion of H(t) ͓cf.Eq. ͑3.12͔͒ the leading behavior of xx 2 (t) is given by the integral ͵ t tЈ 2␥Ϫ2 dtЈ, ͑C9͒ where the lower limit of integration is irrelevant.The reason for different limiting behaviors of the mean square displacement now becomes clear.In effect, the integral above grows as t→ϱ in three different ways.Thus, if 2␥Ϫ2ϽϪ1, the integral converges to a finite value.In this case, xx 2 (t) reaches a constant value and the process X(t) becomes stationary.On the other hand, when 2␥Ϫ2ϾϪ1, the integral diverges as t 2␥Ϫ1 , and The derivation when both the friction kernel and the correlation function present a long-time tail is obtained in a similar way.
where C v (tЈ,tЉ)ϵ͗X ˙(t)X ˙(tЈ)͘ is the velocity autocorrelation function.When t→ϱ, the velocity becomes a stationary random process and C v (tЈ,tЉ) will be a function of tЈϪtЉ.Hence from Eq. ͑3.23͒ we get xx of internal noise we have shown in Appendix A that C v ()ϭk B Th(t), ͓cf.Eq. ͑1.5͔͒.Therefore,
t→ϱ.Likewise, it can be demonstrated from Eq. ͑B4͒ thatThe Laplace transform of this equation reads ˆ͑s ͒ϭH ˆ͑s ͒C ˆ͑s ͒. ͑C3͒The long-time behavior of (t) can be obtained from its Laplace transform because it is the convolution of two functions, H(t) and C(t), both having an asymptotic expansion ͓16͔.Let us now prove Eq. ͑3.19͒.Assume that ␤(t) is of type ͑a͒ (E is finite͒ and that C() shows a long tail ͓cf.Eq. ͑3.3͔͒.Then, ͑B7͒We can now test that the latter results are in agreement with the general expansion Eq. ͑3.12͒.The kernel of this example is of type ͑b͒ ͑see Sec.III͒ with ␥ϭ1/2 and Bϭ1/ͱ.Then, Eq. ͑3.12͒ for these values results in expansions ͑B6͒ and ͑B7͒.
Now, the integrant in Eq. ͑C9͒ decays as t Ϫ1 for large t and the integral diverges as a logarithm.Hence