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Hemmer, Per Chr.
1961
Physica 27
79-02
ON A GENERALIZATION OF SMOLUCHOWSKIS
DIFFUSION EQUATION
by PER CHR. HEMMER
l:ysisk Institutt, Norges Tekniske Hsgskole, Trondheirn,
Norge.
Brinkman and Sack have recently modified the Smoluchowski
equation for the
position distribution of a Brownian particle in order to
describe the initial stage of the
process too.
In this paper the Brinkman-Sack equation is solved for a free
Brownian particle,
and the solution is compared with Ornstein and Uhlenbecks exact
solution. The
obtained distribution has the correct variance, but does not
approximate the exact
distribution in the initial stage.
1. Introdztction. Smoluchowskis equation for the probability
density p(r, t) for a Brownian particle in an external field of
force F(r) reads 1)
3P - = 5 div (kT grad p - Fp), at
where ,3 is the friction coefficient and T the temperature. A
well-known short-coming of this equation is its prediction of an
infinite spreading velocity at the outset. This is so because the
derivation of (1) from the full Fokker-Planck equation in phase
space is valid only when a time t > 1 /p (the relaxation time)
has elapsed after the process was started.
Brinkman 2) and Sack 3) have recently attempted to generalize
Smoluchowskis equation to include times nearer the start of the
process than the relaxation time. Sacks modified equation is of
hyperbolic type:
1 a2p 1 f+- __ = ~ div(kT grad p - Fp).
p 32 rnp (2)
Since the opposite statement has been made a), it is worth while
to point out that Brinkmans (first) approximation for the Laplace
transform of the density can be shown to be identical with Sacks
equation.
The purpose of this note is to discuss the physical implications
of this generalized equation. In order to compare with an exactly
soluble case, (2) is solved for a free Brownian particle. Anyhow,
external forces are not expected to produce so great accelerations
that their inclusion has much effect in the initial stage where (1)
fails.
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80 PER CHR. HEMMER
2. Initial conditions. Since the Brinkman-Sack equation is of
second order in the time, the initial value problem is determined
by two initial conditions, e.g. p and api% at t = 0. However, the
interpretation of p as a probability density puts the
restriction
on the initial values. This is so because integration of (2)
over all space yields
i, .A -3t + -i G p(r, t) daX = 0 ; (4)
the right-hand side has been transformed to a vanishing surface
integral. By integration,
Jp dsx = cl + cs e-ot, (5)
and (3) is just the condition for this to be time independent.
If we make the assumption that the initial velocity is
isotropically
distributed, then the probability current density vanishes
everywhere at t = 0, and in order to conserve probability locally
it is necessary to replace (3) by the stronger condition
3. The free Brownian particle. In the free particle case the
Fokker-Planck equation for the simultaneous distribution of
position and velocity has been solved by Uhlenbeck and Ornstein 5).
When only the (one-dimensional) position x0 is observed at t = 0,
and the probability distribution of the initial velocity is assumed
Maxwellian, the displacement x at a later time t is distributed
according to
pex (x, t) = [4nkT(fit - 1 + +t)/rn/?pg exp P?qP(x - x0)2
- 4kT(pt - 1 + e-fit) I . (7)
For t > 1,p this exact solution may be approximated by the
fundamental solution of Smoluchowskis equation (1) for a free
particle:
ps(x, t) = [4nkTt;@-i exp m$(x - x0)2 1 - pkTt--m (8)
In the appendix is shown that the fundamental solution (with the
con- dition (6)) of the Brinkman-Sack equation is
+ e(ut - lx - Xoi) [&IO (& [.U2t2 - (X - xO,Ll') -I
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ON A GENERALIZATION OF SMOLUCHOWSKIS DIFFUSION EQUATION 8 1
+ Bt
2[uW - (x - x0)2] I1 (-& [z&2 - (x - xp,i,]) (9)
where u denotes the r.m.s. thermal velocity (kT/m)*. I, is the
Bessel function of imaginary argument and of order PZ. The step
function e(x) is 1 for positive x, 0 for other values.
By means of the asymptotic expressions for the Bessel
functions,
la(x) M II(X) m (27c+ ex (x > I),
one easily verifies that for large values of the time
parameter,
(10)
PBS m PS m pex (11)
The probability density (9) vanishes for jx - x01 > ut, and
thus the r.m.s. thermal velocity plays the role of the maximum
particle velocity. Therefore the description of the initial stage,
before the friction has become effective, must be unsatisfactory.
For t < l/,4,
PBS C% +[s(% - x 0 - tit) + 6(x - x0 + 41, (14
similar to the spreading of a sharp pulse along a string, in
contrast to
pex w (2nu2t2)-6 exp [
- Gg] , (t < l/B). (13)
These features are clearly exhibited in the figure, where the
distributions at the relaxation time, t = l/B, are shown.
Nevertheless, the probability density predicted by the
Brinkman-Sack equation has the correct variance for all times,
(x - x~)~ = (2kT/q!12)(Pt - 1 + e&t). (4)
,.f . . .
0 ./ /
A . _c /
./ ./ ./ . ..
Fig. 1. The probability distribution functions (7), (8) and (9)
at the relaxation time.
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82 ON A GENERALIZATION OF SMOLUCHOWSKIS DIFFUSION EQUATION
APPENDIX
The fundamental solution of the Brinkman-Sack equation. We seek
the solution of
which for t = 0 satisfies
po = b(x - x0)
(Zppi%)o = 0.
Introduce instead of the density its Fourier transform
~(s, t) = jp(n-, t) e-is(c-sa) dx,
which fulfills the equation
(15)
(6)
(7)
(18)
The solution of (18) which corresponds to the initial values
(16) is
+, t) = e-tfit { cosjql2s2 _ @2)&] + p %?~t(U2S2 - WY1 1
2
( 1125.2 _ $82); J . (19)
By Fourier inversion we obtain the density:
If we write the trigonometric functions as Bessel functions of
half integer
order, the integral has the form of Sonine and Gegenbauers
discontinuous
integral 6).
(21)
Performing the differentiation, we obtain (9).
Rewired 4-1 O-60.
1)
2) 3) 4) 5) 6)
\. s111o1ucho\fski, \I., AIIII. lhy, m (1915) 1103; Kramers, H.,
Physica 7 (1940) 284. Urinkn~an, H. C., Physica 82 (1956) 29. Sack,
R. A., Physica Zt (1956) 917. Davies, R. O., Phys. Abstr. BU (1957)
826. Uhlenbeck, G. E. and Ornstein, L. S., Phys. Rrv. Ni (1930)
823. Watson, G. K., Theory of I&se1 Iunctions (Cambridge 1958),
p, 415.
