Growth of distributions for a model of diffusion in a random potential

Solid State Communications, Vol. 75, No. 8, pp. 683-686, 1990. Printed in Great Britain.

0038-1098/90 $3.00 + .00 Pergamon Press plc

G R O W T H OF DISTRIBUTIONS F O R A M O D E L OF D I F F U S I O N IN A R A N D O M POTENTIAL J. Heinrichs Institut de Physique B5, Universit6 de Lirge, Sart Tilman, B-4000 Li+ge, Belgium

(Received 26 April 1990 by S. Amelinckx) Using known results for the average density of localised states in a one-dimensional gaussian white-noise potential we calculate exactly the long time growth of the distribution of multiplying diffusing systems in a statically random medium. Our result agrees with a previous leading order expression. We also study the space and time dependence of the distribution for the case where the random potential has a finite shortrange correlation. Results for three-dimensional systems are also discussed. C H E M I C A L and physical reactions with randomly distributed nucleation centers, chain reactions with random fissile distributions, biological multiplication (reproduction) with random nutrient concentrations are examples of systems involving proliferation and diffusion of specified entities in a random medium. The random distribution of a species produced in such reaction processes has recently been modelled [1-3] by a phenomenological linear stochastic equation. In a simple one-dimensional case this equation reads

~P O2p & = D + U(x)P,

U(x) =

(1)

where ~ dxP(x, t) is not a conserved (i.e. constant) quantity because of the exponential local growth or decay of P(x, t) described by the random potential term. Here 2 and the diffusion coefficient D are constants and U(x) is chosen to be a static gaussian whitenoise potential, namely

(U(x))

= O, (U(x)U(x'))

= 2 2 6 ( x - x').

(2)

One wishes to find the configuration averaged distribution (P(x, t)), which is of interest for experimental studies. This problem has recently been studied by path-integral and variational formulations [4, 5]. The correct asymptotic behaviour for t ~ ~ obtained by Rosenbluth [5] may be written ,,~ exp (t3),

(3)

where 0 is an arbitrary origin at which the initially assumed distribution,

P(x, O) = 6(x),

(4)

The purpose of this letter is twofold. Firstly, we present exact results for arbitrary t and arbitrary distances x from the origin for a special gaussian model for which the correlation function (2) is finite for x = x', that is for 2 2 = ~2/6(0), where ~-~'lsa finite constant. We anticipate, of course, that P(x, t) depends on 22 and on the infinite constant 6(0) only through the quantity ~2 since, otherwise, the model would not be meaningful. A significant advantage of this model is that it allows exact solutions in various physical situations, as was shown by Kumar and the author [6]. These results suggest a qualitative difference between gaussian models with finite correlation functions [5] and the true white-,oise model with infinite correlation at x = x'. Secondly, we obtain an exact expression for the configuration-averaged distribution of(l) at x = 0, in terms of the density of negative energy states (localised states) for a quantum particle in a random potential. This expression permits a straightforward analysis of the exact asymptotic behaviour of (P(0, t)), using well-known results for the density of states for an electron in a one-dimensional gaussian white-noise potential [7]. Finally, we briefly study (P(0, t)) for three-dimensional systems, using the above-mentioned expression which is valid in higher dimensions as well. Consider first the solution of equation (1) in the finite correlation case of equation (2), namely 22 = ~2/6(0). An exact equation for (P(x, t)) may be obtained by averaging (1) using Novikov's theorem [8] for averaging functionals of gaussian random variables such as the random term in equation (1). Using equation (2) we thus obtain

is centered. However, due to the linearity of equation (1) the asymptotic behaviour (3) remains valid under more general boundary conditions as well [5]. 683

O(P(x, t)) _ ot

O

2
/rP(x, t)\

x

(5)

684

G R O W T H OF DISTRIBUTIONS FOR A MODEL OF DIFFUSION

where the functional derivative on the r.h.s, may be related to the higher order derivatives through a hierarchy of equations obtained by successively differentiating (1) with respect to U(x) and averaging the resulting equations, using Novikov's theorem together with (2). The resulting hierarchy takes the form of a set of recursion relations which may be solved exactly. By defining

,_)\

B, = \ 6U(x)" / '

(6)

Bo = ,

(7)

the average of the equation obtained by functionally differentiating (1) n times takes the form

OB. ot

632Bn O~2 D-~-x~ + ~ B.+, + n6(O)B._,,

-

n =

1,2,3 ....

(8)

The form (13) for the exponential growth in the case of potentials with a finite correlation for x -, x' is similar to results obtained previously for this case [2, 5], using different methods. While this growth does not involve diffusion, we obtain, in addition, the form of the spreading of the distribution [equation (12)] clue to diffusion away from an arbitrary location of the initial distribution. We now turn to the study of the growth of the averaged distribution (P(x, t)> in the general case of a white-noise potential of arbitrary strength [2 [. For this purpose we shall first derive a general formula for (P(0, t)> [equation (17) below] from the connection between equation (1) and the SchrSdinger equation for a particle in a random potential 1/2D U(x). We expand P(x, t) in terms of eigenfunctions of a Schr6dinger equation to be obtained from (1), as follows [9]

P(x, t) :

~ ~ ( x ) ~(0) e - ~ ,

i dx ei~XB,(x, t),

(9)

and defining the dimensionless quantities -

6(0)"

B.,

T

=

e6(O)t,

k -

(10)

D6(O)

6(0)'

[~ =

the Eqs. (8) reduce to

~c~ _

~c.

(14)

Note that in this form the expansion automatically incorporates the boundary condition (4). For equation (14) to be a solution of (1) the functions ~bi(x) must obey the one-dimensional SchrSdinger equation 1 d2¢i 2 dx 2

an

C.

z = 2Dt.

i

By introducing Fourier transforms

B,(k, t) =

Vol. 75, No. 8

n c._~, + ~ 1 c.+, + 6-~

n = 0, 1,2 . . . . .

(11)

This recursion equation is similar to the ones which have been obtained in [6] for situations where mobile systems may diffuse out to infinity rather than remaining trapped in a region near the origin. The solution of the equations (11) subject to the boundary conditions C0(r = 0 ) = 1 [equation (4)] and C,(z = 0 ) = 0, n = 1, 2 . . . . follows the same pattern as discussed in detail in the first of Refs. [6]. After transforming the exact solution for Co(x, r) back to x-space and reverting to our original parameters we get

1 U(x)Oi(x) = cAke(x). 2D

(15)

For the quantum Hamiltonian H = /'2/2 - 1/2D U(x) with a gaussian white-noise potential we expect negative eigenvalues corresponding to states localised by fluctuations of U(x) and positive eigenvalues ~ > (I/2D) J-U(x)I corresponding to extended (or diffusion-) states. Clearly, the asymptotic growth of
l

-- - Im

<(

(xl

1 e - H+

= ~ <(¢~(x)6(~ - e;)))

iO+

)> Ix> (16)

i



=

(47cDt)

112 e x p

= e x p ( - ~ ' } . \z/

-

~

{P(0,

(12)

(where the averaged Green's function on the r.h.s, is translationally invariant and hence independent of x), it follows indeed that

(13)

=

t)>,

; de p(e) e ~, oo

(17)

G R O W T H OF DISTRIBUTIONS FOR A M O D E L OF D I F F U S I O N

Vol. 75, No. 8

which is formally similar to analogous relations discussed in other contexts [10]. This expression remains valid for systems of higher dimensions if one uses for p(e) the densities of states for the corresponding higher-dimensional Schrfdinger equations. This may be readily seen by starting from the d-dimensional form [2, 3] of the governing equation (1). In applying equation (17) in the case of the potential (2) we may use the detailed results for the averaged density of states derived long ago by Halperin [7]. The limiting expressions of interest, when reexpressed in terms of our parameters, are [7] 1 ( ~2 x~2/3 p ( e ) "" ~ x / ~ , e >> \ - ~ ] , (18a)

p(e)'~(-2e)exp

( - ~ (8n- 22g )

)

3/2 ,

e-+-oo.

685

As noted above, the equation (17) is valid for arbitrary random potentials and for any space dimensionality. For 3-dimensional systems approximate analytical forms of the density of states are known in two limiting cases [1 l] for random potentials, U(r)/2D, having gaussian distributions,

1 exp [

Pu (V(r)[2D) = V~----~o

l

8D2 W°2j,

(22)

which implies, in particular, that their correlation function (U(r)U(r')) is finite for r --+ r . When the range of individual atomic potentials giving rise to U(r) is large compared to a typical de Broglie wavelength one has [11] O(e) -~ 2~2(-e)-3/2exp

2

,

e~

-oo.

(18b)

(23)

The equation (18a) enables us to study the distribution in the limit of a vanishing random potential (2 --} 0). From (17) we obtain in this case

In the opposite case, where the range of the individual atomic potentials is small compared to a de Broglie wavelength one has

(P(0, t))

~ -

1 ~ e -~T de0 N//~' 1

4,/a7 '

p(e) -

(19)

exp ( - X 0 x/-Z--g), ~ ~ - oo,

(24)

where X0 is related to W0 and to some other microscopic parameters of U(r). A saddle-point evaluation of (P(0, t)) for t ~ oo using equation (23) yields

namely diffusion behaviour, as expected. (P(O, t)) ~(2DWo2t)-3/2 exp (2DWo2ta), In the presence of the random potential the ,}" % dominant contribution in (17) arises from negative (25) energies corresponding to localised eigenstates. By substituting (18b) into (17) we get while equation (24) leads to an unbounded result. In conclusion, using well-known results for the 8D 2 0 average density of states at negative energies for a (P(0, t)) ~ ,a.2~z f d e ( - 2 e ) --o0 gaussian white-noise potential, we have obtained the long-time growth of an initial distribution of diffusing x exp - - ~ - ( - 2 e ) 3 a - ez , (20) systems in a one-dimensional medium with static randomness. A similar discussion for 3-dimensional whose evaluation, for z ~ oo, by the saddle-point systems has also been given. The simplicity of our method [saddle point at e - e0 ~- -(,PP)/(32D2)] approach based on equation (17), which is valid in any yields dimension and for arbitrary random potentials, ~4t,/2 ( ~4 t 3 should be noted. On the other hand, we have presented (P(0, t)) ~ 8x/~D3/2 exp \ 4 - ~ . ] ' t ~ oo. (21) an exact treatment of the spatial- and time-dependence of the distribution for a situation where the gaussian .I The leading exponential growth of (P(0, t)) involving correlation function of the potential is finite for both diffusion and static randomness coincides with X -+ X'. the result obtained by Rosenbluth [5], using a variational principle. The equation (21) includes, in REFERENCES addition, the dominant time-dependence of the exponential prefactor.* 1. W. Ebeling, A. Engel, B. Esser & R. Feistel, J. Stat. Phys. 37, 369 (1984). 2. Y.B. Zel'dovich, S.A. Molchanov, A.A. Ruz* The Erratum to Tao's Letter [4], which appeared maikin & D.D. Sokolov, Zh. Eksp. Teor. Fiz 89, after this work was completed, gives an expression 2061 0985) [Soy. Phys. JETP 62, 1188 (1985)]. for (P(O, t)) which coincides with equation (21) up 3. F.C. Zhang, Phys. Rev. Lett. 56, 2113 0986). to an overall numerical factor.

686 4. 5. 6. .

GROWTH OF DISTRIBUTIONS FOR A MODEL OF DIFFUSION R. Tao, Phys. Rev. Lett. 61, 2405 (1988); 63, 2695 (Erratum) (1989). M.N. Rosenbluth, Phys. Rev. Lett. 63, 467 (1989). J. Heinrichs & N. Kumar, J. Phys. C 17, 769 (1984); J. Heinrichs, Phys. Rev. Lett. 52, 1261 (1984). B.I. Halperin, Phys. Rev. 139A, 104 (1965).

8. 9. 10. 11.

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E.A. Novikov, Zh. Eksp. Teor. Fiz. 47, 1919 (1964) [Soy. Phys. JETP 20, 1290 (1965)]. See e.g.N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, North Holland, 1981. See e.g.S. Alexander, J. Bernasconi & R. Orbach, Phys. Rev. B17, 4311 (1978). J. Ziman, Models of Disorder, Cambridge University Press, 1979, chap. 13.