Simple Derivation of Phenomenological Browmia otion Theory Sture Nordholml Department of Physical Chemistry, Chalmers University of Technology, S-41296 Goteborg, Sweden
physics and certain fields of engineering, is about to become a standard tool of chemists as well. Chemists generally build their understanding upon observations of int&actingmanymolecule systems while the interpretations are nearly always in terms of the properties of the isolated molecules. It appears certain that increasing attention will have to he given to the effect of the medium upon the properties of the molecules themselves. That is, one will want to continue to interpret the behavior of many-molecule systems as if they were made up of effectively isolated molecules hut must explicitly recognize that both the equilibrium properties and the dynamics of such an effectively isolated molecule are dependent upon the medium in which the molecule finds itself. From the point of view of microscopic first principles, such medium effects present formidable problems. Chemistry, by contrast, is a very practical field of study, in which intuitive physical approximation is not only accepted hut also probably generally preferred over rigorous hut formal arguments. One may, therefore, foresee a growing interest among chemists in phenomenological Brownian motion theory. Brownian motion theory (more specifically the FokkerPlanck and Smoluchowski equations) has, of course, already
of the powerful tools of Brownian motion theory hinges upon
or Smoluchowski eqnation is obtained simply by solving for the velocity in terms of position using a Markovian contraction which limits the validity of the resulting Smoluchowski equation. Particular attention is paid to the flnctuation-dissipation theorems and to the validity of the Smoluchowski equation. The latter aspect is considered on the basis of an extended Smoluchowski equation obtained in a straightforward manner. The fluctuation-dissipation theorems are considered in the absence of relaxation to a known equilibrium and found to collapse, in the case of an exponentially decaying noise correlation, to a single more general relation between the Fokker-Planck and Smoluchowski diffusion coefficients. Derivation of the Fokker-Planck Equation Let us first consider a one-dimensional problem. Suppose that in the absence of any external force the equation of motion is
We now assume that coupling to the environment introduces a force F A t ) = f(a(t))+ 3,(t)
f(a) = ( F A t ) )
numher of derivations of the basic equations of phenomenological Brownian motion theory. An excellent collect,ion of early papers in this field is contained in a hook edited by Wax (10). More recent work of particular relevance can he found in articles by Kuho (11, 12) andvan Kampen (13). Zwanzig (14) and Mori (15) have also initiated the development of a eeneralized Brownian motion theorv on the hasis of micro" scopic first principles and projection operator techniques (16), but the present article will be concerned with the traditional and less ambitious phenomenological theory in which no ath U D t is made to start the derivation from a descri~tionof the dynamics of the entire sysem. Despite the large number of expositions already available, the derivations so tar offered leave room for further improvement. What is needed is a compact, self-contained and simple derivation without recourse to the formal language and ~~~
(2)
where u indicates a dependence on the exact initial state of the environment. Thus, the environmental force can he split into a systematic term f(a) and a random flntuating force F,(t). The systematic force is the average of FJt) over the states of the environment descrihed by some statistical ensemble, (3)
and the random force has a vanishing average, (?At)) = 0 (4) The motion of the system is now descrihed by a Langevin equation,
~
vantage from this point of view. The development starts from a Langevin-type eqnation which seems the simplest means of introducing environmental effects. A simple hut general derivation of the eauation of motion of the corresnondine probability density is then presented to he used repeatedly in later sections on the derivation of the Fokker-Planck and t'rc.el~u r i n n Smcduch.,wnki r.quniim.; l o r .t ~ ) . t r l i , I t . ~~hwin; pu1vnti:d tirld. 'l'hv I..ingrvili equL~liam !.it I ~ 1 1 n g(liit!l&n l~
The corresponding equation of motion of the prohahility density g,(a;t) can he obtained from the continuity equation,
where
Again, g,(a;t) depends on o, which denotes the initial state of the environment. Since we only have a prohability density for o, it is appropriate to solve for the time-development of g(a;t) &a($')
=
(gAa;t))
(8)
However, the equation of motion of g(a;t) cannot be obtained -
' Permanent address: Depallrnent of Theoretical Chemistry. University of Sydney. Sydney. N.S.W. 2006. Auslraiia.
Volume 59
Number 3
March 1982
187
merely by replacing all terms in eqn. (6)by their a-average. The problem is that
Instead, the random force ?,St) has a fundamental effect on the dynamics. This effect may be referred to as fluctuationrenormalization (17). In order to proceed we shall now make certain assumptions about the nature of the random force. The validity of these assumptions should he examined before the theory below is applied to any particular process. 1) The random force fluctuates around zero with atypical amplitude of a and a lifetime of r. In particular, we have
is vew short an the time scale of the relevant dynamics of a(t). 3) The fluctuating force is independent of a ( t )
2)
7
of higher order terms in eqn. (10).The suggestion that the error is of order A2 or less is not strictly correct for an equation of motion containing a term of the character of the fluctuating force T,(t). This can he understood from the fact that the integral on the left in eqn. (10)appears to be of order A hut, in fact, generates a correction of order 0% in eqn. (12). An examination of higher order terms in eqn. (10)reveals that errors of order ah2or a473may arise, hut in all applications here contempleted such errors would also he small [see also refs. (11)and (13)l. The second point concerns the assumption that :T,,(L) is fully independent of a. There is no fundamental reason to suggest that this is always a valid assumption. If we do not make this assumption, the derivation above, extended in a straightforward manner, yields
in the one-dimensional case with
Given these assumptions, we now re-examine the average on the left in eqn. (9).To expose the correlation between 3 J t ) and g,(a;t) we write, using eqn. ( 6 ) ,
If A is much larger than 7 but still very small on the time scale of a ( t ) then we can neglect the terms of order A2. Inserting eqn. (10)on the left in eqn. ( 9 )we get
It is reasonable tu expect that y,,,y02, and yo, will he of order less in typical or traditional applications of phenomenological stochastic theory. These coefficients may themselves depend upon a. They will he neglected helow, but further consideration of such terms seems desirable. 0% or
The Diffusion Equation
We shall now apply the general derivation given in the preceding section to one-dimensional motion in a stochastically simple medium. The equations of motion are then The first term on the right will now vanish because 3,it) and g,(a;t - A ) are uncorrelated. This is a consequence of 3Jt) and 3 J s ) being uncorrelated for s < t - 7.The same argument applied to the second term shows that non-vanishing contributions arise only when t - T < s < t . Thus, g,(a;t - A ) is effectively independent and we get
Here 3,(t) has been assumed fully independent of a at all times. Dropping the term of order A since A is very small on the time scale of g(a;t) we finally get
The derivation ahove generalizes immediately to the case when a is replaced hy a vector of variahles a. We then find
In the absence of the medium the particle would be underaoina free translation. The medium introduces a systematic
with y given hy
If the medium can he treated as a heat bath, i.e., it drives the particle toward the probahility density f~(r,u= )
(2nkTlm)-1/2e-m"2/2kT
(21)
then the fixed condition of f r yields the result
In particular, the ahove equation also applies, given the validitv of the an~roximations 1.. 2.. and 3. to the motion of a .. particle in a medium. Without wishina to subtract from the sim~licitv . " of the derivation we shali point out two relatively more subtle aspects of the theory which have been suppressed ahove. The first concerns the order of the error arising from the neglect 188
Journal of Chemical Education
This result is often referred to as the fluctuation-dissipation theorem. The position prohahility density g(x;t) can be obtained from f(x,u;t)by integration over the velocity,
If we are interested only ing(x;t),it may be inconvenient to first solve the Fokker-Planck equation and then integrate over
u. I t may then be advantageous to obtain an equation of motion for g(x;t) itself. In favorahle circumstances this can he done by a simple extension of the derivation given in the preceding section. Note that eqn. (18) can he formally solved to give " ( t ) = e-%(o)
+
LLdsectS7(t
- s)
124)
The last term can he dropped since there can be no significant correlation between 3 and u(o). Thus, we get from eqn. (36)
This result can be viewed as an extension of the fluctuation dissipation theorem in eqn. (22).
Thus, we get The Smoluchowski Equation
Consider now the case when the particle moves in a potential field V(x). The Langevin equations then become
where
Now we note that 3 ( t ) is a random fluctuating force,
and (exp (-(t))u(o) is a time-dependent systematic force. The derivation still holds as long as assumptions 1,2, and 3 are valid and we get
if u(o) is known. If u (o) is not known we can obtain the solution for a given u(o) = u , gc(x;t) from eqn. (28) and then note that
where V'(x) = dlaxV(x). The Fokker-Planck equation can he ohtained precisely as before and we get
Here ( and y have the same meaning as before and the fluctuation-dissipation theorem in eqn. (22) still holds if the medium can he treated as a heat bath. In order to get an equation of motion for g(x;t), we follow the same procedure as ahove in the absence of a potential and first ohtain a formal solution to eqn. (391, "It) =
where
--L 1
rn
'
ds e-WT(lc(t- s))
+ edlu(o) + (411
piuJ = j d r g,(x;oi
Eix,u:oi
=. ipiui)-'fIx,u;o)
Inserting this result into eqn. (38), we get (30)
One can, of course, also go one step further and set
where = 6(x -z'J d,~,u~(r,oJ
(32)
and the initial velocity is u'. The time-development of 6,,,,,(x;t) can again be obtained from eqn. (28). The gain in using eqn. (28) rather than eqn. (19) is that the number of degrees of freedom has been halved. On the other hand, there is a time-dependent force in eqn. (28). It should be noted, however, that if u ( o ) vanishes, or if we are willing to neglect the initial transient effect produced in the time interval zero to roughly LIE, then we can drop the first term on the right in eqn. (28) and obtain the diffusion equation
The parameter d in eqns. (28) or (33) is called the diffusion coefficient, and it is ohtained as
If the medium is a heat bath, then d can he obtained also as il2l
where u(o) is Maxwell-Boltzmann distributed. Using eqn. (24) we then find
This is a non-Markovian Langevinequation. The first two terms on the right are systematic, the second is an introduction of memory effects, and the last term is a random fluctuating force as before. The memory effects may introduce difficulties that would make it advantageous to work with the Fokker-Planck equation. However, if the motion is slow on the time scale I/[ of the frictional force then we can use a Markovian contraction as follows,
The Langevin equation then simplifies to a -zit) = - (llrn~)V'lx(tJJ + $It)
(45) at This equation is now amenable to the derivation of the Fokker-Planck equation (earlier in this paper) given that the assumptions 1 , 2 , and 3 concerning the random force are satisfied. We ohtain the result
with d still defined by eqn. (37). This is the well-known Smoluchowski eqn. (10). If the medium is a heat bath, then the equilibrium probability density,
must he stationary and the fluctuation-dissipation theorem (37) follows immediately. Volume 59
Number 3
March 1982
189
The validity of the Smoluchowski equat~oncan be assessed by examining the validity of the assumptions 1 , 2 , and 3 in the general derivation and the two Markovian approximations (43) and (44). It is, m fact, also possible to account for at least a minor amount of memory effect by using the approximation V'x(t
-3))
-
V T ( r ( t ) )- su(t)V"(r(t)l
(481
rather than merely using the first term on the right as in eqn. (44). Use of eqn. (48) produces the result
Carrying through the remainder of the derivation as usual we arrive at the equation
This equation is consistent with the fluctuation-dissipation theorem (371, in the case when the medium is a heat bath, only to second order in I/(. Thus, some care has to be exercised in the use of this equation. Its main utility may, in fact, lie in that it offers a means of observine the onset of deviations from the
(1 - (V"(x(t))/mEZ)l-'
which must be close to unity in order for the Smoluchowski equation to hold. In other words, we must have IV"(x(t)ll
> t-' and t >> r-I
+ rll([-2-
= 7 then d
(I/[([ - r l )
s~mpllfiesto
+ (l/r(t -
m2
7)))
= - (A6)
t2r
At thls point we recall that at the Fokker-Planck level the veloeitydiffusion coefficient y is given by
Thus, we have found that
This can he thought of as an extension of the fluctuation-dissipation theorem, which produces the same result when the environment can he treated as a heat bath, to the case when the time-correlation of the random fluctuating force is exponentially decaying.
Literature Cited (1) Kramers, H. A,, Physica, 7,2M (1940). (2) ~ i k i t m E. . E.. ''Theory of Elementary Atomic and Molecular Processes in Gases." Clarendan Press. Oxford. 1974. (3) Doi. M.,Chem. Phys.. 11,107 (19751; 11,115 (1975). (4) Visscher. P. B.. Phys RPU.,B13, a272 (1976): BL4.347 (19761. ( 5 ) Fe1derhofB.U andDeutch. J.M..J. Chem. Phys 64,4551 (1976). (61 Fe1derhof.B.U.. J. Chem. Phys., 66,4386 (19771. (7) Kondon, S. and Takano, Y., Int. J. Chem. Kin., 8.481 (1976). (8) Skinner,J. L. and Wolynes. P. G , J Chrm Phyr.,69,2143 (1Y781. (9) McCaskili, J, and Gilhert,R. G., Chcm. Phy*.,dd,389 (1979). (10) Wax. N., (Editor),,'Selected Papers on Noise and Stochastic P~ocerses,"Duuer,New Ymk, 195b (11) Kubo,R., J. Math. Phys.4.174 (19631. (12) (13) (14) (15) (16) (17) (la)
Kubo.R.,Rep. Pmg.Phys., 29,255 (1966).
vanKampen,N. G.,Phys. Rep.24.171 11976). Zwmip, R.,Phys. R e " , 124,983 (1961). Mod, H.,l?ogi. Thoor Phya., 33,423 (19661; 34,399 (1965). Nordhoim,S. andZwanzig.R., J. S t o t Phys.. 13,347 (1975). Noidholm, S, andZwanzig,R.,J. Stat. Phys , 11,143 11974). Brinkman, H. C., Physica, 22.28 (1956).
Volume 59
Number 3
March 1982
191
