Dissipation and fluctuation for a randomly kicked particle: Normal and anomalous diffusion

Chemical Physics

ELSEVIER Chemical Physics 212 (1996) 69-88

Dissipation and fluctuation for a randomly kicked particle: Normal and anomalous diffusion

E. Barkai, V. Fleurov School of Physics and Astronomy, Raymanod and Beverley Sackler Faculty of Exact Science, Tel Aviv University, Ramat-Aviv 69978, Tel

Aviv 69978, Israel

Received 25 March 1996

Abstract

A stochastic model is considered which describes the motion of a one-dimensional classical particle kicked at random by heat bath particles. The effect of the distribution of the time intervals between the collision events on the motion of the test particle is studied. It is shown that the general pattern strongly depends on the long time behavior of the waiting time distribution function. The Brownian or normal diffusive type of motion is obtained when the distribution function has finite moments. If, however, all these moments diverge the diffusion has an anomalous character. The relaxation is characterized by power law dependences, while the mean square displacement is dominated by the ballistic contribution. The second, superdiffusive term can be related to the momentum relaxation by a relation differing from the conventional Einstein relation. It is also important to emphasize that the information on the initial state of the test particle is essential for the long time behavior of the mean square displacement for the proper formulation of the dissipation-fluctuation relation.

1. Introduction

A model of a Brownian type of motion is discussed which consists of a test particle with a mass M, which moves deterministically (without friction) in a one-dimensional space with an external potential V ( x )

according to the Newton law of motion. At random times the velocity of the particle is randomly changed due to elastic impacts of other particles with a mass m constituting the bath. In this kind of a model two independent probability density functions are assumed to be defined. One of these, Q(~'), is the probability density function of the waiting times between the collision events, whereas the second function, F ( p ) , gives the probability density function of the momenta of the colliding particles. These two functions determine the stochastic evolution of the test particle.

Such a model can mimic different phenomena related to the dissipation and fluctuation processes the test particle undergoes. A widely spread approach is to take the waiting times from a Poisson distribution and the momentum of the "gas" particles from the Maxwell distribution at a temperature T [ 1 ]. This case was investigated in our previous publication [2] for an arbitrary mass ratio e = m / M . The case of equal masses e = 1, when the collisions are strong and may be rare, is considered in the chemical reaction rate theory (see, e.g., Ref. [3] ). Probably the best known limit of this model, sometimes called diffusion limit, is reached at e ~ 0 when the collisions are weak but frequent. This limit under certain conditions leads to a Fokker-Planck

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70 E. Barkai, V. Fleurov/Chemical Physics 212 (1996) 69-88

equation or to the equivalent Langevin equation [4]. Another approach is to consider the collisions following one after the other with a constant time interval elapsing between collision events (see, e.g., Ref. [5] and references therein).

Generalizations of the Brownian motion to a case when the mean square displacement of the test particle behaves as

(x2(t)) ,-~ t a, (1)

with B ~ I are of a special interest in many fields [6-10]. Different stochastic models are used to generate such a non-Brownian behavior, we shall mention but a few. The continuous time random walk model (CTRW) assumes that particles hop from site to site with a distribution of waiting times which has a long tail. In the L6vy flight approach (the mean square displacement diverges) one assumes long tails for the jump length probability. In this case the particle hops from site to site and visits only the end points of each jump. Considering L6vy walks a velocity is introduced to each jump. In these kind of (position) hoping models there is no place for velocity relaxation which is due to many weak impacts (see Eqs. ( 1 ) - (4 ) in Ref. [11]) .

Such type of a model was discussed by Zanette and Alemany [ 12] who considered a particle executing a random walk of a L6vy flight type. It can be shown that the model discussed here may be mapped on their model in the case of e = 1, V ( x ) = const and a constant time interval r between the collision events. If e = 1 the test and gas particle exchange momenta at each collision. Then the ith displacement is xi = (pi/M'r) where pi is the test particle momentum in the ith interval. Choosing Pi from a L6vy distribution will obviously lead to the corresponding jump probability in the x space. In this case the equilibrium distribution function would not be necessarily the Maxwell-Boltzmann distribution. It would rather coincide with a generalized form suggested by Tsallis [ 13].

The model considered in this paper bears also some similarities with the well known continuous time random walk (CTRW) model. Consider again the case of e = 1 and V ( x ) = const. Then the random duration of the ith flight is ri and its length is xi = (p i /Mri ) . This differs from the CTRW model where xi and "/'i are statistically independent, while in our model ri and Pi are statistically independent. When ~ :# 1 the similarities seem to break down completely. In this case our model incorporates effects of the momentum relaxation in addition to the spatial diffusion. However, mathematical techniques developed in studying the CTRW model [ 10] appear to be extremely useful for our model as well.

The approach pursued in this paper stems from the 1D model proposed by Rayleigh (see, e.g., Refs. [ 14,15] ) more than one hundred years ago. The so-called Rayleigh piston was introduced to approach the kinetic theory of gases from a microscopical point of view. It gives also an insight on the classical Brownian motion which is usually described by means of a Langevin equation including the Einstein fluctuation-dissipation relation.

This paper will mainly address the problem of various waiting time distributions and their influence on the dissipation and fluctuation processes. A particular emphasize will be laid upon distributions with diverging moments. These types of the waiting time distribution lead to an anomalous diffusion with a nonexponential relaxation of the test particle velocity. In order to illustrate differences between two types of diffusion processes simulations of the test particle motion are presented in Figs. 1 and 2. They show realizations of the particle trajectory for two cases. In the first case (Fig. l) the waiting times are Poisson distributed. In the second case (Fig. 2) a distribution for which the average time between the kicks diverges. This distribution will be discussed in more detail in Section 4.3.1 (Eq. (50)) .

Although our model may, in principle, incorporate an external potential acting on the test particle, we will restrict ourselves in this paper by considering a free particle moving in a constant potential and kicked at random by the bath particles.

E. Barkai, V. Fleurov/Chemical Physics 212 (1996) 69-88 71

~ ' ' T

k

!

I ~ I I I L I I ~ I L

t = O [

i'

~: I I

• i

L. L

Fig. 1. A trajectory (in arbitrary units) of the test particle is plotted for a Poisson waiting time distribution which corresponds to a normal diffusion. The small squares give the locations of the test particle at the moments of each collision event. The mass ratio e = 0.1 is chosen to be rather small so that the momentum relaxes slowly.

2. Model

A classical test particle with a mass M moving in a 1D space and coupled to a bath of one-dimensional particles with a mass m is considered. No external force is supposed to act on the test particle. At random, the test particle is elastically kicked by a bath particle with a Maxwell distributed momentum/7,

F( f i ) = ( 1 / ~ ) e x p { - ~ 2 / 2 m T } . (2)

Here and below, T is the temperature of the bath, measured in energy units, (kB = 1 ). The change of the test particle momentum due to a kick is described by the equation

P+ = P, l P - + / z2p , (3)

where

/ Z l = ( l - ~ ) / ( l + e ) , / . , 2 = 2 / ( 1 + e ) .

Here - and + indicate the values of the momentum just before and after the collision. The coordinate of the test particle is not changed by the kick. Here we also assume that the duration of a collision event is much shorter than any other time appearing in the problem.

The time ri, which elapses between the ( i - 1)-th and ith collision events, is a random quantity described by a yet unspecified probability density function, Q(r) , defined for 0 <_ r < c~. This function is assumed to be independent of the mechanical state of the test particle, and does not change in the course of the systems evolution. This approximation is expected to work well in real physical systems if the velocity of the test particle is smaller than those of the bath particles.

Now a sample sequence {ffi} and {ri} describing s collision events is considered. The free motion between collision events with the momentum defined by Eq. (3) give a possibility to find the coordinate, Xs(t), and momentum, ps(t), of the test particle at any time t.

It is clear that these two quantities

xs(t; (pi},(ri};xo,PO), ps(t; {pi},{zi};xO,PO) (4)

72 E. Barkai, V. Fleurov/Chemical Physics 212 (1996) 69-88

are unambiguously determined by the initial values, x0 and P0, and the particular choice of the sequences {,if,.}, and {7i}. Summing over all possible sequences (with the correct weights) allows us to obtain averaged characteristics of the test particle as a function of the time t.

3. Definitions

The following definitions and mathematical tools will be used below. The sample space consists of:

(i) a nonnegative integer s = 0, 1 . . . . which describes the number of collision events during the time t; (ii) for each s there exists a set of s + 1 real time intervals or waiting times ri (s + 1 >~ i >1 1) obeying

0 < "/'i < oo. '/'s+l is the time between the last collision in the sequence and the observation time t; the waiting times are related to the dots on the time axis at which collision events occur by 7i = ti - ti-l (here to = 0);

(iii) a real momentum, - o o < / ~ < o~, of the kicking particle is assigned to each collision event. The domain for which the waiting times are defined is determined by the equality

S

Z '/'i+1 ---- t. i=0

A function Qs(t; {7"i}]i<~s+l) is defined which yields the probability density that s collision events during a time t which are distributed according the sequence {ri}li<~s+l. Using the waiting time probability density Q(r) , this function can be explicitly written in the form

Q,(t;{ri}]i<.,+l)= ~ Q(ri) W(r,+l)exp ig r i+l- t , (5) _ ~ \ i=O

where r s + !

W(rs+l) = 1 - / Q(~') d~" (6) , #

0

is the probability that during the time interval between the last collision event and the observation time t no collision event has occurred;

Qo(t) = 6(rl - t)w(rl).

The representation

{ s+l t ) 1 _ oo - t ) 8~i~=l ~ - i - = - ~ f e x p { i g ( t - ~ r i } d g (7)

of the delta function ensures that the sum of all time intervals equals the observation time. The function (5) is normalized according to the condition

[oJOiJ ] odrl + drl dr2Ql(rl,r2) + . . . . 1. (8)

o o

E. Barkai, V Fleurov / Chemical Physics 212 (1996) 69-88 73

In order to calculate the average value of a physical quantity A(t, x, p) one has to consider the following sequence

{A0, A| (~'l, Pl, "r2) . . . . . As(r l . . . . ) . . . . }

of functions over the state space. Then the average value is determined by the equation oo

( A ( x , p ) ) = A o W ( t ) + Z ( a s ( x , p ) ) , (9) s=l

with oo

( A , ( x , p ) ) = ~ i=1 --00

oo oo

/ O(tli) d'l'i f w(~sq-1) dT"s-b|

o o

l.= )} x __ F ( ~ ) d ~ e x p ig r i - - t As{ . . . . Pi,~i . . . . rs+|}, (10) --oo \ i=l

which implies summation over the number s of the collision events during the observation time t, as well as integrations over all distributions of the time intervals between the kicks and over the momenta of the kicking particles.

Our analysis is restricted by the nonstationary processes when the observation starts just after a collision event. A stationary condition can be achieved by considering a process which has started at t = -cxz and choosing the start of the observation at random. This procedure relates to the initial conditions and the information about these decays if the waiting time distribution function has nondivergent moments. The latter means, in particular, that there is a finite average time between the collision events. We plan here to address processes with nonstationary conditions for which the waiting time distribution functions have diverging first moments. The problem of the first waiting time distribution in the CTRW model was addressed by Tunaley [ 16] (see also Ref. [9] ) and later by Zumofen and Klafter [ 17,18]. The latter authors discussed, in particular, how a stationary process could be defined in this case and what were the differences between the stationary and nonstationary conditions.

3.1. Stochastic map fo r a free particle

A particle moving freely between the collision events is considered. Its evolution is described by the following stochastic map,

+ ) + + q- Pk Tk+l/M, Xk+ 1 : X k P~I /z ip[ +/x2/~k+l •

Here + Xk+ l (pk~+l) is the coordinate (mechanical momentum) just after the (k + 1)-th collision event. The coordinate and mechanical momentum after s collisions are given by

s ~ s--i Ps = t / , lPO -+" / z I tz2Pi,

i=1 s $ s

Xs ~ (Ti+I/M)/'~P Oq- ~-~Z].Zi l -k~2(Ti+l /m)f fk • i=O k= l i=k

Here the evolution during the (s + 1 )-th interval elapsing between the sth collision and the observation time is included.

74 E. Barkai, V. Fleurov/Chemical Physics 212 (1996) 69-88

4. Averages of the mechanical momentum

4.1. Generating functions

The averaging procedure defined by Eq. (9) is applied to the calculation of the characteristic function

o o

( eihl' >t = ~-~ < eihps ) t (11 )

,r=0

of the mechanical momentum. First, the integrations over the times between the collisions are carried out in Eq. (10). Then the complex variable Laplace transforms

O¢3 OO

Q(g)= f Q(r)e igr dr, ~"(g) = i W(r)eigr dr (12) o o

of the probability functions Q(r) and W(r) are used. The sth collision term in Eq. (10) is proportional to the probability

o o ' I Ps(t) = ~ dgQ_.(g)SW(g)e -ig' (13)

- - 0 0

that s collisions have happened during the time interval of the length t. This kind of weighting will hold for any variable which does not depend explicitly on the waiting times, i.e., which is a constant of motion between successive collision events. Now integrating (see Eq. (10)) over the momenta of the colliding particles and using the identity

mtz~l (1 - / x 2) = M

one obtains the momentum characteristic function

0<3

<eihP)t = ~ Ps(t) exp[ihlz~P0] exp[-½MTh2( 1 - /x2s) ] (14) s=0

for a given initial momentum P0 of the test particle. Differentiating the characteristic function (14) with respect to h, and then taking h = 0, one obtains the first

two moments o o

(P) = Po ~ Ps(t)btsl, (15) s=O

o o

(p2) $ - ~ P s ( t ) r 2 2s = tP0lXl + MT( 1 - ~2s) ] (16) .r=0

of the mechanical momentum of the test particle. The relaxation of mechanical momentum, according to Eqs. (15) and (16), depends directly on the prob-

ability function Ps(t) that s collisions have occurred during the observation time t. This is so due to the fact that the momentum is a conserved quantity between the collisions, when no interaction takes place. This would not be necessarily the case for any other quantity varying between the collision events, e.g., for the coordinate of the test particle.

E. Barkai, V. Fleurov/ Chemical Physics 212 (1996) 69-88 75

Now the generating function

oo

O(z , t ) = ~ zSps(t) (17) s=0

for the number of collisions, s( t ) , the particle encounters during a time t is considered. The first (15) and second (16) moments of the mechanical momentum are readily given by two generating functions

(p) = p0,&(/Zl, t), (18)

(t 92) = P~¢(I~ , t) + MT[1 - ~b(IX~, t)]. (19)

The functions ~,(/z~, t) will be used below to study the relaxation of the test particle momentum. Substituting Eq. (13) in Eq. (17) and summing the geometric series results in

oo

¢ ( z , t ) = ~ / dg ~'(g) exp(- igt) . (20) 2rr 1 - zO.(g)

- - 0 0

Introducing a new integration variable - i g = u and using the Laplace transform

~V(iu) = ( 1 - Q . ( i u ) ) /u (21)

of the function (6), the generating function (20) is rewritten as

ioo

~/'(Iz~,t) = --/ du 1 - Q(iu) .eU," (22) 2¢ri u[ l - /x~0( iu ) ]

- - ioo

This integral is the inverse Laplace transform of the function

( 1 - Q _ . ( i u ) / u [ 1 - iz] '0( iu)]) , (23)

whose form coincides with that of the expression investigated by Montroll and Scher [ 19] (see their Eq. (44a)) in the context of the CTRW model. There the expression analogous to ¢(/z]', t) generates the moments of the probability function of a walker to be in a given location at time a t. The similarity of the functions will allow us to use some of the results obtained in Ref. [ 19] when describing the momentum relaxation of our test particle.

It is worthwhile to discuss here our identification of Eq. (22) with the inverse Laplace transform of Eq. (23). It requires that the function (23) has no singularities to the right from the integration path in the integral (22). In order to understand that this is really the fact one should take into account that according to its definition the function Q(r ) (as well as the other functions in which it appears) is defined only for positive times and is zero otherwise. It means that they have properties of the response functions whose Fourier transforms do not have singularities in the lower half of the complex plane. By introducing the new integration variable - i g = u the lower semi-plane is converted into the right semi-plane, which justifies the inverse Laplace transformation (22).

4.2. Waiting time function Q (7") with finite moments

Here the momentum relaxation will be considered in the case when the waiting time probability density function Q(r ) decays (at 7- ~ cx~) rapidly enough and has well defined moments

76 E. Barkai, V. Fleurov / Chemical Physics 212 (1996) 69-88

O 0

(7 "/) = / T I Q ( r ) dr, (24) , /

0

connected by the conditions

(Tl> = CI<T> l, (25)

where {G} is a set of finite nondivergent coefficients for all l. The generating function ¢(/z~, t) in the e ~ 0 limit is considered. Each collision in this case becomes

infinitely weak which may result in an infinitely large relaxation time. In order to avoid this, the standard procedure usually applied in master equations is to take another free parameter, e.g., the mean time between the collisions, tending to zero under the condition that the relaxation coefficient

/3 = lim 2e/ (r ) (26) e ~ o

(~)~0

remains finite. The Laplace transformation of any waiting time distribution function for which all moments exist can be

Taylor expanded and written in the form

OO

~)(iu) = Z (--1)l(rl)ul / l] . (27)

I--O

Using Eqs. (23), (24) and the expansion

/x~ ~ 1 - 2ne (28)

for small e one can notice that the terms (rt)U t with l > 2 do not contribute to the generating function for a finite time t in the e ~ 0 limit. Therefore

lim O(/z~ t ) = lim L - l { 1 } °40 ' ~-o u + 2en/<r) = exp(-n/3t) . (29) if> --o (,) ~o

Here the operator L-I denotes the inverse Laplace transformation. This equation demonstrates an important feature of the e --, 0 limit. A simple exponential relaxation controlled

by a single relaxation coefficient/3 holds for all moments of the test particle momentum and is not sensitive to other details of the waiting time distribution function. These details may become of importance for finite mass ratios e and may result (see examples below) in a multimode decay characterized by a set of decay times. Another important case is the waiting time distribution function which, being normalized, does not nevertheless have finite moments. Such a situation will be considered in Section 4.3.

4.2.1. Example 1 Quite a general class of waiting time functions is considered here. For these the Laplace transform of the

probability density function is assumed to be a ratio of two polynomials

~9(iu) = g ( u ) / S ( u ) . (30)

Then using Eqs. (20) - (23) the generating function reads

~b(l~7, t) = ~ - I {rl ( u ) / r 2 ( u ) } , (31)

E. Barkai, V Fleurov/Chemical Physics 212 (1996) 69-88 77

where the numerator

r t ( u ) = S (u) - R ( u ) (32)

is of a lower degree than the denominator

r2(u) = u [ S ( u ) - / z ~ ' R ( u ) ] . 33)

Assuming that the denominator has a form

r2(u) = u ( u - a2) . . . ( u - an), 34)

then (see Ref. [20] p. 130)

~(/z~, t) = ~ rl (a~) exp(a~t). 35) ~:2 r2(a~)

Here the normalization condition Q(0) = 1 (or R(0) = S(0)) has been used which means that the polynomial rl (u) must have a zero at u = 0. This condition excludes the first zero, al = 0, of Eq. (34) from the sum (35).

It follows from the discussion of the inverse Laplace transformation applicability that Re a~ < 0 for all l, = 2 . . . . . n and, hence,

iim ¢ ( ~ 7 , t ) = 0.

Now the set of the relaxation times for the chosen waiting time distribution function is determined by the finite set of solutions of the equation

S(u) - txTR(u) = 0. (36)

and the whole relaxation pattern is controlled by a finite number of decaying exponentials. It is clear that the zero with the smallest real part corresponding to the largest decay time will be responsible

for the long time behavior of the system. There remains only a single decay time in the e ~ 0 limit which can be found by expanding Eq. (33) for small u,

S' (O)u - 2enR(0) - R ' (0) = 0.

One can readily see now that the characteristic relaxation time is

dQ(iu) ,=0 S ' (0) - R'(0) (r) = du = S ( 0 ) ( 3 7 )

and the nth moment of the test particle momentum relaxes with the characteristic time n(z) (cf. Eq. (29)) . A remark is in order. Although the zeros of the denominator (33) were assumed to be simple, they might be

/-fold degenerate as well. In this case multiples of t t would appear in the pre-exponential factors in Eq. (35) but they would not change qualitatively any conclusion done in this subsection.

4.2.2. Example 2 - Gamma distribution The Gamma function

Q( ~) = O( Or)°IT~-~exp(-Or) / F ( O(r) ), (38)

where (r) is its first moment is an example of a set of the waiting time distribution functions with finite moments.

78 E. BarkaL V. Fleurov/Chemical Physics 212 (1996) 69-88

Inserting the Laplace transforms

0( iu) = [1 + u/0] - ° +

of Eq. (38) into Eq. (23) (e.g., Ref. [ 19]) the following is obtained:

(39)

(i) I f 0(~') = 1 the times between collision events are completely uncorrelated and described by the Poisson distribution. Then the generating function

~Pa (/x~', t) = exp[ - tO( 1 - /x ] ' ) ] (40)

leads to the relaxation pattern discussed in our paper [2]. (ii) The Gamma function with 0(r) = 2 corresponds to anticorrelations between the successive kicks when the

probability of a pair of kicks following shortly one after another with a small waiting time is suppressed. One can see it from the fact that Q(~" = 0) -- 0. Then

~b (/27, t) = exp(--0t) [cosh(0 tv / -~) + ( 1 / V / - ~ sinh(0tv/-~-) ]. (41)

(iii) The Gamma function with 0(~-) = 1 corresponds to correlations between the successive kicks when the probability of a pair of kicks following shortly one after another with a very small waiting time is enhanced. This follows from the fact that lin~__.0 Q ( r ) = oc. Then

Oc (/z~, t) = ( 1 +/x~) - I {Erfc [ (Ot)1/2] _.[_ ].L~ exp[ - ( 1 - IX2n)Ot] Erfc[- /z~ (Ot)1/2] }. (42)

(iv) The limit 0(~-) ~ (x) corresponds to very strong anticorrelations when the kicks follow in fixed intervals (r) one after the other. This case has been considered in detail in our paper [51.

In the e ~ 0 limit one obtains the general Eq. (29) with fl defined by Eq. (26) as it should be according to the general statement discussed above.

4.3. Waiting time function Q (7") with diverging moments

This subsection considers the relaxation of the test particle momentum when the waiting time probability density function Q ( r ) decays very slowly so that even its first momentum diverges. A class of such functions can be represented by means of the expansion

~)(iu) = 1 - Au ~ + c2A2u 2a + . " ", ,(43)

where 0 < a < I. Such distributions are normalized and have asymptotic behavior

Q( 7-) ~ aar - ( l+a ) / F(1 - a) (44)

at 7" ---, c~. Two examples of such functions originally proposed by Montroll and Scher [ 19] in the context of CTRW model will be discussed in the next subsection.

The relaxation of the test particle momentum is calculated by substituting the Laplace transform (43) into Eq. (23) and considering the u ~ 0 limit. Then the Tauberian theorem [21] yields a generating function with the asymptotics

~9(tz~,t) ~ A t - ~ / [ (1 - ~')r(1 - a ) ] (45)

at t ~ oo whose power law dependence on time characterizes the relaxation of various moments of the test particle momentum.

The e ~ 0 limit is now considered. This means that the strength of each kick tends to zero. In our above consideration the average time between the kicks was assumed also to tend to zero, thus leading to a finite

E. Barkai, V. Fleurov/Chemical Physics 212 (1996) 69-88 79

relaxation coefficient (26) . But now we are dealing with the waiting time distribution functions for which the average time between the kicks is not defined, meaning that some modifications are to be done in the definition of the e ~ 0 limit.

Using Eqs. (23) , (28) and (43) and keeping only the terms proportional to Au ~ one obtains

lim g,(/x]', t) = lim ~-1 {UCt-l/ (U a -t- n B ) } , (46) ~ o ~ o A ~ O A ~ O

where the limit

B = l i m 2 e / A (47) ~ o A ~ 0

is assumed to be finite. This is now the definition of the generalized relaxation coefficient B which replaces the definition (26) applied in the normal diffusion. An exponential type of decay is recovered for a = 1.

The asymptotic behavior

!ira ¢( t z~ , t) ~ t - ~ / B n F ( 1 - a ) (48) A ~ 0

of the generating function with various values of n at t --* cx~ can be obtained either directly from Eq. (45) or by applying the Tauberian theorem to Eq. (46). In particular, Eq. (48) yields the power law

!imo (p) ~ pot-"/BF(1 - a ) (49) A ~ 0

for the relaxation of the test particle mean momentum.

4.3.1. Two examples Two waiting time probability density functions

Q ( z ) = - 4 a 2 exp(ra2)Erfc(av/-~) ~ (x/-~a,r3/2) - I at "r ~ c~z

and

Q(~-) = a( 'n ' r ) -1/2 - a 2 exp (aZr )E r f c ( av~ ) ~ (X/r-~a73/2) -1 at r ~

(50)

introduced in Ref. [ 19] are briefly discussed in this subsection. Time evolutions of the test particle coordinate and momentum generated by means of the waiting time function (50) are shown in Figs. 2 and 3.

These two examples are chosen since the Laplace transforms of the functions (50) and (51) are also explicitly known:

~)(iu) = (1 + x/-u/a) -1 "~ 1 - v/-ff/a at u ~ 0 (52)

and

~)(iu) = (1 + x/ -u /a) -2 ~ 1 - 2v/-ff/a at u ~ 0, (53)

meaning that a = 1 in both cases whereas A = 1 /a for the first function and A = 2 / a for the second function.

Both functions decay as t -3/2 for large t, however, their small t behaviors differ. The first function diverges as t-1/2 at t ~ 0 indicating a rather strong correlation between the kicks arrivals, whereas the second function approaches a constant value in this limit.

(51)

80 E. Barkai, V. Fleurov/Chemical Physics 212 (1996) 69-88

r T 7 - - - ~ r 7 I ' ' '

J , , , , t , , , , I , , , , [ , J

t - O t

Fig. 2. A trajectory (in arbitrary units) of the test particle is plotted for the waiting time function (50) which corresponds to an anomalous diffusion. One sees bursts of collisions separated by long time intervals during which the particle moves ballistically.

p=0 -

S I

j , , r ' ] ' ' ' ' T - ' , 1

, I , , ~ , L , I , , , , J , , ,

t-O t

Fig. 3. The test particle momentum for the same realization as in Fig. 2. The momentum changes discontinuously at each kick since e is finite whereas the duration of each collision event is zero. The function (50) with a = 1 is chosen when M = 1, mT = 1; the initial conditions are x0 = P0 = 0. generating this process by means of the numerical procedure described in Ref, [23]. The parameters of the system are e = 0.1, M = 1, m T = 1; the initial

conditions are xo = P0 = 0.

Now the generating function for the function (52) becomes

[ ( ~ - ~ ) 2 t 1 ~pd (/z~, t) = exp Erfc [ ( ~ - ~ ) v/t] (54)

and for large x/7/A can be represented by means of the asymptotic expansion (see, e.g., Ref. [22] )

r(,/2)xl [ m ] + (55) m=l

where x = (1 - t z~)x , / t /A. The first term of the expansion (55) (when n = 1 ) corresponds to Eq. ( 4 5 ) . The second function (53) yields the generating function

d/e (/x~, t)= 2---~ (1 + V/~)exp [4/( 1-~V/-~) 2- Erfc [2x/~ (1 - S ) ]

1 I + 2 V ~ ( 1 - v ~ ) e x p 4t I+Ax/~-' Erfc 2v~ • (56)

Both functions at large t have the same asymptotic expansions as can be seen in Fig. 4. According to the more general result (45 ) the large t behavior of these functions is controlled by only by the parameters A / ( 1 - I~1 ) and ot and such a convergence is to be expected. These functions, as shown in Fig. 4, are normalized to unity at t = 0 and give the same result

E. Barkai, V. Fleurov/Chemical Physics 212 (1996) 69-88 81

2

T \

e=O.1

4 , , , i , , , I ,_ , , I

- 4 - 2 0 2 log(x)

Fig. 4. ~b vs. ~ = V'7/A in the double logarithmic scale, log(Oa) (Eq. ( 54 ) - solid l ine) and log(Oe) (Eq. ( 56 ) - dashed l ine) are plotted for different mass ratios ~. The power law relaxation for the large times T is same for both functions Q(r), In the ~ --~ 0 limit the generating functions Oe and ~b d converge; the difference between them becomes more noticeable at larger values of the mass ratio, see, e.g., the curves for s = 0.5.

[ i~ Od (#~', t) = H~ Oe (#7, t) = exp[ (nB)2tlErfc[nB,, / t] (57) a ~ 0 A~0

in the e ~ 0 limit which depends only on two parameters B and a (a = ½ in this case). The same result can be achieved by performing the inverse Laplace transform (46) (see Laplace transform tables [24] ).

5. Mean square displacement

The time evolution of the mean square displacement of the test particle is discussed in this section. Studying moments of the coordinate presents a more complicated problem than the study of the time evolution of the test particle momentum carried out in the previous section. The principal reason is that the test particle coordinate changes during the intervals between the kicks and its general time evolution depends not only on the number of kicks, as in the case of the momentum, but on the durations of the intervals as well.

The coordinate of the test particle by the observation time t after s kicks is represented as a sum Xs(t) = Xis(t) q- Xfs( t ) o f t w o terms:

xi.( ,) = } xf,(,) = E Z ,

k=l i=k

where only the first term contains information on the initial momentum of the test particle. The averaging is carried out using Eqs. (9) and (10) . Then

( x 2 ( t ) ) = ( x 2 ( t ) > + (xi2(t)>, ( 5 9 )

where

82

oo

( x 2 f ( t ) ) = ~--~(Xf2s( t ) ) s

s--O

E. Barkai, V. Fleurov/Chemical Physics 212 (1996) 69-88

(60)

and

oo

( x 2 ( t ) ) = Z ( X i 2 s ( t ) ) s . s=O

(61)

A mixed term, depending linearly on the initial momentum P0 of the test particle, is absent since (P0Pk) = 0. In the case of a normal diffusion the long time behavior of (x 2) is not influenced by the initial conditions

since the corresponding terms decay exponentially rapidly with time. That is why the consideration of the terms containing Xis (t) will be postponed till the problem involving waiting time distribution functions with diverging moments is addressed.

Using ( ~ j ) = mT6ij and the notation

s i - k T Gk~ = 2..~/zl i+l, (62)

i=k

one obtains

s

(Xfs(t)2)s = ( k e2 /M)2mTZ G~s. k=l

(63)

This equation implies averaging over the distributions of kicks according to

- - dg G~ s = ~ Q('l'i) dT"i W(rs+l) drs+l exp ig( ri - t) G2~.

--cx~ i=1 0 0

Eq. (64) is rewritten as

(64)

S S S G~ s ~ 2 ( i - k ) _ 2 ~ ~ i + j - 2 k

= ~_..,t IZl 7i+1 "~ ~ ,~,¢ ] ' / "1 T i+ITj+ 1' i=k i=k j=k

(65)

where j 4= i in the second term of Eq. (65). It is easy to see that four types of averages are to be considered: (i) diagonal terms 7i+17"i+1 with i 4: s which appear in the first addend of Eq. (65); for these we define

7-TD = L-l {O_(iu)S-l~'(iu)O.(2)(iu) }, (66)

where O~n) (iu) is the nth derivative of ~)(iu) with respect to u; (ii) offdiagonal terms ri+l~'k+l for k ~ i, k 4= s and i ~ s which appear in the second addend in Eq. (65);

for these we define

T2D = ~--1 {O.(iu)S--2~(iu) [O(l)(iu) ]2}; (67)

(iii) mixed terms which include the s + 1 (boundary) interval only once; for these we define

"l's+lTi+l]iq~ s ---- ~-l {O(iu)S-l~'(l) (iu)O_(1)(iu) ] } ; (68)

E. Barkai, V. Fleurov/ Chemical Physics 212 (1996) 69-88 83

(iv) a single term

~s+lr~+l = E-1 {O(iu)S~2~ (iu) }. (69)

Now using these definitions Eq. (65) becomes

aks-"oD [ \ i - -7 , ,' - i--1.7 + "g i - - 7

s - k - - ( 1 - _ +21., r,+:i+lli,s - i --'~1 / -e 1., r,+,.

Summation over k in Eq. (63) yields

(x2( t ) )s=(- -~) mT{ I* 2

(70)

1 - 1.~] Z~D [ 1 -- 1.~ + 1 --/-I,12s ] i ~ J + (1 _-~,)2 ~ s - 2 1 - 1 . 1 1 _ - - ~ 2 j

l-,4'1 _1.2'1, z: . . . . . ---~--- 1 ( 1 - 1 . 1 ) 1.1 1 _ 1 . 2 j + r s + 1 l _ / z 2 j . (71)

square displacement the result (71) should be substituted into Eq. (60) and In order to obtain the mean summation over all s is to be carried out. This will be done in the next two subsections in the limit of large t.

This calculation has been carried out (for all t) for the exponential waiting time function Q(r) . The resulting equation for the mean square displacement (x 2) is identical to the one derived directly in our paper [2].

5.1. (x~( t) ) for Q(r) with finite moments

Considering the large time limit for the mean square displacement of the test particle we may neglect the exponentially decaying contributions of the terms containing the initial momentum P0 of the test particle. Carrying out the summation over s we keep only the leading terms in Eq. (71) which are proportional to s. Then the mean square displacement in the large time (t ~ c~) limit becomes

(xf2) ~ ~s=l \ M ' ] mT 1.2 + (l "Z~l)2 j s. (72)

The quantity "/'OD is not defined for s = 1, however, ignoring this fact when summing in Eq. (72) over s from 1 to c~ does not influence the results at t ~ ~ .

Using definitions (66) and (67) the summation over s in Eq. (72) results in

(X 2) ,~ M ~_-I Q(2)(iu) 21.1 ~_-1 [Q(I)( iu)] 2 (73)

i, u[ 1 - - - ~ u ) ] 1-1 .1 u [ 1 - ~)(iu) ] g

Assuming that all the moments of the waiting time distribution function exist and using the expansion (27) and condition (25) Eq. (73) may be expanded for small u. Then the inverse Laplace transform yields the asymptotic mean square displacement in the form

(x 2) ---- (TIM(r)) [(7 -2) + (21. , / (1 - 1.1 )) ('r) 2] t. (74)

AS is expected for a normal diffusion process the mean square displacement increases linearly with time t and is proportional to the squared thermal velocity v 2 = TIM.

8 4 E. Barkai, V Fleurov/Chemical Physics 212 (1996) 69-88

Generally, the diffusion constant

D = lim (xZ)/2t = (T/2M(r)) [ (~) + (2 /z l / (1 - / z l ) ) (7- ) 2] t -"* OO

(75)

produced by Eq. (74) is connected with the relaxation coefficient fl = 2e/(r) by a relation deviating from the standard Einstein relation. The latter is recovered only in the e --+ 0 limit,

lim D = T/Mfl. (76) (r) ~ o

However, for the special case of the exponential waiting time distribution function the Einstein relation holds for any value of the mass ratio e (see discussion in Ref. [2] ).

Attention should paid to the role played by the first term in Eq. (75) in the e ---* 0 limit. This term does not contain any dependence on the mass ratio e which is one of the principal parameters controlling the relaxation processes in the system. That is why we should not expect that the Einstein relation should hold when this term is not vanishingly small. It may be neglected only under the condition

(77)

which is usually achieved for small enough values of the mass ratio e and for waiting time distribution functions with well defined moments. However, for any finite value of the mass ratio e special attention should be paid to this anomalous term giving an important contribution to the deviation from the Einstein relation.

It is clear now that one may expect that the Einstein relation, reflecting the connection between the fluctuation and dissipation processes in its standard form, will collapse completely if the moments of the waiting time distribution function diverge. This problem will be addressed in the next two subsections.

5.2. (x2f ) for Q( 7") with diverging moments

This subsection considers the mean square displacement of the test particle when the waiting time probability density function Q(7) decays so slowly for large ~" that all its moments diverge, e.g., functions of the type (44). The calculation of (x2(t)) is carried out along the same lines as in the previous subsection. However, the waiting time distribution function with diverging moments makes this procedure much more involved. In particular, actually all the terms contribute now to the sums and there is no way to choose the leading ones. The details of this calculations are outlined in the appendix. Results for the large time limit are presented here.

The summation of all the necessary terms results in

(x2(t))~--~ ( 1 - a ) t 2+2 / Z l a ( l + / z l ) - l + a At 2-a I T t 2-" 1 - / z~ F(3 - a ) +2AeeC2MF(3-ot)" (78)

One can easily see that this equation has not too much in common with Eq. (74) describing the normal diffusion. Here we obviously deal with an anomalous diffusion. First of all the leading term in the mean square displacement (78) is proportional to t 2 corresponding to a ballistic motion of the test particle. A ballistic motion is present also in the normal diffusion processes but there it may be of importance only for short times which are on the order of the relaxation time (i.e., 1/fl).

The fact that now the ballistic type of motion appears to be important even for large times is connected with the properties of the waiting time distribution functions with diverging moments. There is no time scale characterizing the momentum relaxation. This means that the weight of the neighboring pairs of the collision events separated by very long time intervals when the test particle moves ballistically appears to be large and yields this t 2 dependence of the mean square displacement.

E. Barkai, E Fleurov/Chemical Physics 212 (1996) 69-88 85

Notice also that the t 2 term does not depend on the mass m of the bath particle and therefore cannot be related to the dissipation controlled by the mass ratio e. Expression (78) suggests that the ballistic particle travel is performed with an average velocity

2 Uav = (1 - a)T/M. (79)

Since the thermal velocity is connected with the temperature by means of the standard equation v 2 = TIM the factor ( 1 - a ) reflects the expectancy of very long intervals between the kicks whose length may be comparable with the observation time t.

A ballistic behavior, including the prefactor 1 - a, was also found in Refs. [18,25] for a different model in which a test particle was considered whose velocity might be either +1 or - 1 .

There is also a superdiffusive term in the mean square displacement (78) proportional to t 2 - ' . Its coefficient can be both positive or negative depending on the relations between the parameters tzl and c~. This term contains a dependence on the mass ratio e and hence is related to the dissipation processes.

The last term is linear in the coefficient c2 and hence requires more information on the shape of the function Q(~') (beyond the parameters a and A). On the other hand it does not depend on the mass ratio e and, hence, cannot be related to the dissipation. Fortunately, it becomes negligible in the e ~ 0 limit for the mean square displacement which takes now the form

f 2 - a lim(xf2(t)) .-~ Va2t 2 + 2D M , (80) A ~ 0 e ~ l }

where the generalized diffusion constant D f is related to the relaxation coefficient B according to

D f = (r/M)(3o~ - l ) / ( B F ( 3 - a ) ) . (81)

Similarly to the normal diffusion coefficient the anomalous diffusion coefficient D f is proportional to the ratio of the temperature T to the test particle mass M and inversely proportional to the relaxation coefficient B. But the value of the numerical factor differs from the one which stands in the normal Einstein relation (76). As mentioned above this coefficient may even change sign.

The expression (78) is obtained using the expansion (43) including its third term proportional to u2% However, there are waiting time probability density functions Q ( r ) (see Ref. [6] ) with diverging moments in which this third term is proportional to u. Then the last term in Eq. (78) will change, but this would not contribute in the e ~ 0 limit.

Eqs. (80) and (81) are not complete since they do not contain terms connected with the initial momentum of the test particle. In other words one may say that these equations describe only the experiments when this initial momentum is chosen to be zero. An important feature of the anomalous diffusion is that the memory about the starting momentum P0 is kept for a long time and the corresponding contribution should be taken into account when calculating the superdiffusive tenn. This issue will be addressed below.

5,3. (x~) for Q(r) with diverging moments

The part (xi 2) of the mean square displacement explicitly depends on the initial momentum, P0, of the test particle. This term is usually neglected when considering normal diffusion processes as, e.g., in Section 5.1, due to its exponentially rapid decay with time. However, when addressing an anomalous diffusion process one should be more cautious, since relatively slow power laws are typical for the problem. It will be shown here that the initial momentum dependent part of the mean square displacement, (x~), being smaller than the ballistic term in Eq. (78), remains, nevertheless, important and is of the same order as the superdiffusive term.

Using Eqs. (61) and (70) one writes

86 E. Barkai, V. F leurov /Chemica l Physics 212 (1996) 6 9 - 8 8

oo

(x~(t)) = ( p o / M ) 2 Z G2os . (82) s=O

The sum in this equation is handled in the way similar to that used when calculating the quantity (x~(t)). As a result, we have for large t

(po~ 2 2A t 2-~ (x2) ~ ~MJ 1---S-~12(1-'~)r(3-o0 (83)

or in the e ~ 0 limit

( p o ) 2 t 2-~ !ira0 (x2) ,,~ ~ (I - a ) B F ( 3 _ or). (84) A~0

The slow decay of the initial conditions leads to their rather strong contribution to the mean square displace- ment at all times. The superdiffusive term in Eq. (78) has the same time dependence as the initial momentum dependent term considered here. The e ~ 0 limit for the total mean square displacement reads

(x2)=(x~)+(x2 f ) , ,~ - -~(1-a) t2+ ( l - a ) + (3 o r - 1) (85)

meaning that being measured in any specific experiment it will explicitly depend on the initial momentum P0 of the test particle.

We may now consider an ensemble of stochastic processes generated with various initial momenta p0. Then the mean square displacement (85) should be averaged over this ensemble and the result may be different for different types of ensembles. An interesting possibility is to consider a thermal equilibrium ensemble. Then the equation

( (po/M)2) = T/M (86)

may be used while averaging Eq. (85) over this ensemble of the initial momenta. Comparing the resulting superdiffusive term with the generalized relaxation coefficient (47) yields the equation

lim (x2(t)) 2 z = Vavt + 2D~t 2-~ (87) t --'~ 00

for the mean square displacement in which the diffusion coefficient

O~ = ( T / M ) a / ( O F ( 3 - ,~)) (88)

is related to the generalized relaxation coefficient B. This relation characterizes the connection between the dissipation and fluctuation processes for the anomalous diffusion considered here. Notice that in this case D , is positive for all allowed values of the parameters.

6. Summary

A stochastic model has been considered in which a test particle is randomly kicked by thermalized bath particles. The principal attention is paid to the distribution of the waiting times between the collision events. Different types of diffusion can be obtained assuming different shapes of these distributions. The e ~ 0 or diffusion limit when the strength of each kick tends to zero is discussed in more details.

The general diffusion pattern depends strongly on the long time behavior of the waiting time distribution function. First, the waiting time distribution functions which have finite moments are considered. For these the

E. Barkai, V. Fleurov/Chemical Physics 212 (1996) 69-88 87

test particle momentum relaxation has an exponential time dependence whereas the mean square displacement grows linearly with time which is typical of a normal diffusion. Deviations from the standard Einstein relation between the diffusion and the relaxation coefficients are possible at finite intervals between the kicks. However, the e ~ 0 limit always yields the Einstein relation regardless of the detailed shape of the waiting time distribution function.

The second possibility is the waiting time distribution function which decays very slowly with time so that all its moments diverge. It means that now no characteristic time appears which can characterize the typical time interval between the kicks. This makes the diffusion of the test particle anomalous and completely different from the normal diffusion. First, all the relaxation processes are now characterized by power law time dependences rather than exponential dependences typical of the normal diffusion. Second, the ballistic contribution to the mean square displacement, which rapidly decays in the normal diffusion, becomes now the leading one. The second important contribution is the superdiffusive one which can be related to the momentum relaxation. Special care should be taken when considering the e --* 0 limit due to the absence of a well defined time scale. Moreover, the information on the initial momentum of the test particle does not decay with time and should be incorporated both in the superdiffusive term and in the formulation (88) of the relation between the fluctuation and dissipation characteristics which substitutes now the conventional Einstein relation. It is emphasized also that all these results in the e --, 0 limit depend only on the long time asymptotics of the waiting time distribution function and are not sensitive to details of its short time behavior.

Acknowledgements

The authors are grateful to J. Klafier and E. Pollak for fruitful discussions. V.E is indebted to the Institute for Complex Systems, Dresden, for the hospitality and financial support.

Appendix A

According to Eqs. (60) and (71 ) the expression for the mean square displacement (x 2 (t)) includes the sum

l ,=~- -~ -~os=~- -~_ , 1 - ( i u ) O ( i u ) S _ , ~ ( 2 ) ( i u ) s . (A.1)

s=O s=O

Performing the summation over s gives

I , = ' L - ' {Q_(2>( iu> / ( [1 -O( iu ) ]u ) } . (A.2)

Using now Eq. (43) the asymptotic behavior for small u becomes

I1 = (1 - Or')oiL -1 {U -3 } -~- c2Act(3a - 1)£- I{u a-3 } q- O ( L - I {g2Ct-3}). (A.3)

In a similar way one can find

0 o

12 = ~ r2D s = a2AL -1 {u a-3 } + O(L -1 {u2a-3}), (A.4) s=0 o o

13 = ~--~D( I --/-t 2s) = o~(I -- o<)AL-' {u ̀~-3 } + O ( L - ' {u2a-3}), (A.5) s=O

88

/4 = ~ - - ] ~ - - s=O

15 = Z - - s ~

E. Barkai, V. Fleurov/Chemical Physics 212 (1996) 69-88

r~o( l --/z~') = (_9(L-' {u : ' -3 }) ,

~ i ~ s ( l --/z~ s) = 0~(I -- ol)AL-l{u a-3} .-[- O(Z - l {u2a-3}) ,

( A . 6 )

(A .7 )

A

- ~ t , l{u'~ 3} 1 6 = Z T z + l ( 1 - - t z ~ s ) = ( 1 - a ) ( 2 - a ) L - ' { u - 3 } ( l - a ) ( 2 - a ) ^ - -

s=O l t~ 1

+ ( 1 - a ) 3 a c 2 a L - l { u '~-3} + O ( L - l { u 2 ' ~ - 3 } ) . (A .8 )

The Tauberian theorem al lows one to t ransform all above equat ions into the t ime domain and obtain the

cor responding asymptot ics in the large t ime limit. The terms conta in ing

~-t {u-3} = t2/2 (A.9)

produce the ball ist ic term. Not ice that the 16 term (A.8 ) which accounts for the last t ime interval ~'s+l be tween

the last co l l i s ion and the observat ion t ime t contr ibutes also to the t 2 term. This has to do with the fact that

there is a high probabi l i ty that the last t ime interval appears to be long. The terms propor t ional to

~,-t {u ~-3 } = tz-u ( r ( 3 - a ) ) . (A.10)

correspond to the superdiffusion.

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