Langmuir 1992,8, 2947-2951
2947
Nature of the Random Force in Brownian Motion P. Mazur Instituut-Lorentz, University of Leiden, P.O. Box 9506,2300 RA Leiden, The Netherlands
D.Bedeaux’ Department of Physical and Macromolecular Chemistry, Gorlaeus Laboratories, University of Leiden, P.O. Box 9502, 2300 RA Leiden, The Netherlands Received May 15,1992. In Final Form: August 27, 1992 We study the stochastic properties of the random force in a nonlinear Langevin equation for the time dependenceof a variable a whose equilibriumdistribution function P (a)is known. We assume that this random force is of a multiplicative character and consists of a factor%(a(t - e)) multiplied by a random function f d t ) independent of a(t). We prove that in this case f d t )is a Gaussian white process. We show that the function C(a) is the solution of a differential equation which involves Pw(a) and can easily be solved when this last function is Gaussian. 1. Introduction
J. Perrin published in 1908 and 1909 in the Comptes Rendus’ the results of his experiments on Brownian motion. A detailed review “MovementBrownien et rbalit.4 moleculaire” (Brownian motion and the objective reality of molecules) of these results and of his considerations appeared in 1909,in Annales de Chimie et de Physique.2 His experiments enabled him to conclude that “il devient assez difficile de nier la rbalit.4objective des molbcules”(it becomes rather difficult to deny the objective reality of molecules),a conclusionwhich would definitivelybring to an end a long and at that time still existing controversy. Perrin’s Brownian motion experiments, which contributed to a large extent to his fame for elucidating the structure of matter, were based in part on Einstein’s formula for the mean square displacement of a particle. Einstein3 obtained this formula by considering, as did Smoluchowski4who obtained essentially the same expression, Brownian motion as a diffusive process. But already in 1908Langevin5had proposed a simpler deviation based, as all modern theory of Brownian motion, on the stochastic equation of motion for the velocity of the Brownian particle, which is now known under his name. The theory of Langevin’s equation, which is a strictly linear equation in the velocity, was subsequently further developed, in particular, by Uhlenbeck and Ornateinswho assumed Langevin’s “complementary (random) force” to be a Gaussian process with a white frequency spectrum. Later this theory was generalized to apply to other fluctuation-or noise-phenomena, all findingtheir origin in the corpuscularstructure of matter, mostly to situations such that the systematic forces were still linear in the variables describing the state of the system considered. For cases, however, in which the equation of motion is nonlinear in the variable considered, the properties of the complementaryforce were in general not well established so that the application of Langevin’s method of analysis in such instances was often problematic.‘ In this paper (1)Perrin, J. Compt. Rend., a number of short contributions between M a y 1908 and September 1909. (2)Perrin, J. Ann. Chim. Phys. 1909,18, 5. (3)Einstein, A. Ann. Phys. 1905,17,549; 1906,19,371. (4)Smoluchowski, M.v. Ann. Phys. 1906,21,756. (6)Langevin, P.Compt. Rend. 1908, 146, 530. (6)Uhlenbeck, G.E.; Omstein, L. S. Phys. Reu. 1930,36, 823. (7)van Kampen, N.G.StochasticPropertiesinPhysics and Chemistry; North-Holland: Amsterdam, 1981.
we shall show that nevertheless a coherent method can be developed for that purpose for a nonlinear Langevin equation. In sections 2 and 3 we shall consider a Brownian particle which has a momentum-dependent friction coefficient 8@) = 8(-p) and investigate if and under which circumstances its motion can consistently be described by the following stochastic differential-or generalized Langevin-equation for the stationary process p ( t ) : dp(t)/dt = -8@(t)) p ( t ) + f ( t ) (1.1) where f ( t )is the random or Langevin force which is due to collisions with the particles of the fluid in which the Brownian particle is suspended. The stationary (equilibrium) distribution for the momentum is given by the Maxwell distribution
P,@) = (2?rmk,~)-’/~ exp(-p2/2mkB~ (1.2) Here m is the mass of the particle, k g is Boltzmann’s constant, and Tisthe temperature. Using the symmetric nature of the friction coefficient, it follows that the stationary average, indicated by (...), of the systematic force is zero:
(8@)P) =0
(1.3)
Since the average of the left-hand side of eq 1.1vanishes due to stationariness, it also follows that the average of the random force is zero:
(fwj= o
(1.4)
In a recent papeP we have shown that if one demands that f ( t )be independent of the momentump, aproperty which will henceforth be indicated by a subindex zero, it then followsfrom causality and from microscopic time reversal invariance that (i) the Langevin force is Gaussian and white, with (f&) f&’ 1) = 2 L w
- t’ 1
(1.5)
and (ii) the friction force is related to the equilibrium distribution and the constant L, which according to eq 1.5 determines the strength of the random force, by (8).Mazur,P.;Bedearu,D.Physica 1991,A173,156;eeealeoanerratum Physic0 1992, A188,693;Biophys. Chem. 1991,41,41.
0743-7463/92/2408-2947$03.QQ/Q @I 1992 American Chemical Society
Mazur and Bedeaux
2948 Langmuir, Vol. 8, No. 12, 1992 b@)p = -L d In f',@)ldp = (L/mkBT)P
-
b@) =
(1.6)
Clearly eq 1.6 implies that the friction coefficient must be a constant if the random force is independent of the state p ( t ) . The Langevin equation is then strictly linear. But if the friction coefficient is momentum dependent, the theorem quoted implies that the random force cannot be independent of the state p(t), and therefore f ( t )= f@(t),t)
(1.7) In the next section we shall discuss this nonlinear case assumingthe random forceto be of amultiplicative nature' with the objective to determine ita stochastic properties consistently. In the third section we show that the Stochastic differential equation derived in the second section can be written in two alternative forms. One of these forms contains an It&like@multiplicative random force. The other form contains a multiplicative random force of the Stratonovich type.l0 In the fourth section we extend the analysisto a variablewhich has a non-Gaussian equilibrium distribution. In the last section we formulate our conclusions. 2. Nonlinear Langevin Equation The question which now arises is whether it is possible to derive the properties of the variable-dependentrandom force if one knows explicitly the dependenceof the friction coefficient on the momentum as well as the equilibrium distribution P,@). As in the linear case knowledge of these properties is of course necessary if the stochastic differentialequation (eq1.1)is to have meaningful content. In the analysis which follows, we will use the theorem which we quoted above and which we now state in ita general form. Consider a Langevin equation for a stationary random process y(t) d
y(t) = -B(y(t)) + f o ( t )
(2.1)
in which, as indicated, the random force is independent of the state y(t) and has a zero mean value; furthermore, the process is c a d (i.e., y(t) is uncorrelatedto the random force at a later time) and the system is microscopically reversible, with y either odd or even under time reversal. It was then shown in ref 8 that (a) the noise is Gaussian and white, with (fo(t)
f&' 1) = %6(t
- t' 1
(2.2)
and (b) the systematic force is given in terms of the equilibrium distribution and the constant L by B(y) = -L d In P,(y)ldy
P,W dy = P,@) dp (2.4) We shall show that such a function y@) can indeed be found and leads to a consistent description for the Brownian motion of a particle with a momentum-dependent friction. From eqs 2.1, 2.3, and 2.4 one finds a stochastic differential equation for p(y(t)) of the following form:
Here -
-
-Lc2@)
[f In (Po@)c@))](2.6)
where e is infinitesimally small and positive. The expression for the first term on the right-hand side of eq 2.5, A @ ) , is self-evident. In view of the fact that lim C,@(t)) = C@W) 40
(2.9)
the prefactor in the second term on the right-hand side ofeq2.5wouldbeequalto C@(t))iff&) wereacontinuous function. As f o ( t ) is in general not continuous, however, the more complicated prefactor given above is needed as is shown in Appendix A. An important aspect of the derivation of eq 2.5 is that no use is made of the statistical properties of the random force f o ( t ) . As is clear from the above analysis the requirement, or assumption, that the random force f ( t ) = f o ( t ) in eq 2.1 be independent of y(t) results, for the stochastic equation forp(t),eq 2.5, in, and is tantamount to,a random force of a given multiplicative nature. We shall now first relate the functiony@)to the friction coefficient @@I. To this end, we observe that
(2.3)
The right-hand side of eq 2.3 is proportional to what one often calls the thermodynamicforce. The systematicforce in eq 2.1 is therefore linear in this thermodynamic force. In view of this linear relationship a Langevin equation in which the random force is independent of the state of the system shall be called quasi-linear. If the random force depends on the variable,we shall refer to the corresponding Langevin equation as genuinely nonlinear.' (9) It&, K.R o c . Imp. Acad. (Tokyo) 1944,20,519;Mem. Am. Math. SOC.1961, 4, 51. (10) Stratonovich, R. L. SZAM J. Control 1966,4,362; Topics in Theory of Random noise; Gordon and Breach New York, 1963; Vol. I, Chapter
4.
In order then to describe Brownian motion in the genuinely nonlinear case, we assume that there exista an invertible function y@) which obeys the stochastic differential equation (eq 2.1) and satisfiesthe abovetheorem. The normalized equilibrium distribution of this new variable, P,(y), is given in terms of the Maxwell distribution by
where we used the fact that due to stationariness the average of the left-hand side of eq 2.5 is equal to zero. Equation 2.5 may now be rewritten in the following form: (2.11) with, using also eq 2.4
Langmuir, Vol. 8, No.12,1992 2949
Nature of the Random Force in Brownian Motion
D@) A @ ) - L dC/dy = -LC(y)$
In [P&J) CCy)I/ =
f@(t),t) E C,@(t))f o ( t ) - L dC/dy =
- i1j L dC2@(t))/dp(t) (2.13)
C,@(t))
It follows from eq 2.10 that the averages of the last two expressions are equal to zero. Comparing eq 2.11 with eq 1.1, one may now identify
B@)P = D@> = -LC2@)$ In [P,@) C2@)l) (2.14) Upon substitution of the Maxwell distribution this equation reduces after some algebra to a differential equation for the product LC2@)
m k ~ T @ @=)LC2@)- d[LC2@)]/d[p2/2mkBT](2.15) which has the solution
Lc2@)= mkBT
dp(t)/dt = -B@(t)) p ( t ) + C@(t - €1) fo(t)
E
-B@(t)) PW + f-@(t),t) (3.1) The proof of this equivalence is given by verifying that the stochastic properties of f-@(t),t)and f@(t),t)are the same. For this purpose we rewrite f@(t),t)by Taylor expansion of p(y(t + e)) around p(y(t - e)) in the following way:
L[d2p/dy21, (3.2) It follows from eq 2.1 by integration that y(t
+ e) - y ( t - e) = -JTdt'
[B(y(t' )) - fo(t' )] =
where we used the fact that the physical process is such that the function B(y(t)) is at most discontinuous as a function oft but not singular. Substitution of this equation in the random force gives
2
dnB@)/d[p2/2mk,T]" (2.16)
The quantities L and C2are therefore determined up to a constant multiplicative factor. This indeterminancy arises from the fact that the scale of y may be chosen arbitrarily. A convenient choice for L is, cf. eq 1.6 L = mk,Tfl(O)
(2.17)
One then finds, if the friction coefficient is independent of p, that C = 1, and therefore y = p. In general one has
Using the fact that f o ( t ) is Gaussian and white, we show in Appendix B that the momenta of f n ( t ) are given by (3.5) (fn(tl) f n ( t 2 ... ) f n ( t l )= ) 0 for n 1 1 which implies in the context of the statistical description that all f n except fo may be taken to be zero:
OD
c2@) = Cdn[B@)/8(0)l/d[p2/2mkBT]n (2.18) n=O
and the function y is found to be
f n ( t )= 0 for n 1 1 It then follows that
(3.6)
f @ ( t ) , t )= (dP/dy)t-, f o ( t ) = C@(t - e)) f o ( t ) = f-@(t),t) (3.7)
This concludes the explicit construction in terms of the friction coefficientB@) of the variabley@) which satisfies the stochastic differential equation (eq 2.1) and the theorem given below that equation. At the same time we have therefore shown that the fluctuations of the momentum p are consistently described by the stochastic differential equation
This completes the proof of the form of the Langevin equation given in eq 3.1. It should be noted that this form corresponds to the prescription given by It69 for the interpretation of the product C@(t))fo(t). Using again the Gaussian white nature off&), one may show in an analogous way that the form of the Langevin equation given in eq 2.5 is equivalent to
-dp(t) - -A@(t)) + i [ C @ ( t- e)) + C@(t + e))3fo(t)= dt
1 -B@(t)) p ( t ) - 5 L dC2@(t))/dp(t)+ '2[ C @ ( t - 4) +
C@(t + 4))lfO(t) (3.8)
where Ct is given by eq 2.8 together with eq 2.19 and f o ( t ) is Gaussian and white (cf. eq 2.2). Furthermore, L is defined by eq 2.17, and C2is given by eq 2.18. It should be noted again that the assumption that the random force is of a multiplicative nature, cf. eq 2.5, is necessary to make the above derivation of the properties of the random force possible. 3. Equivalent Forms of the Langevin Equation In this section we shall show that the above stochastic differential equation (eq 2.20) for p ( t ) is equivalent to
The last term in this equation has Stratonovich's formlo for the interpretation of the product C@(t))fo(t). One could of course also verify the equivalence of eqs 3.1 and 3.8 directly. 4. A Non-Gaussian Equilibrium Distribution
In the above analysiswe have restricted ourselvesto the case of a Brownian particle for which the equilibrium distribution of the momentum is given by the Maxwell distribution. For much of the above analysis this restriction to a Gaussian equilibrium distribution of the variable is not needed, and we will give the appropriate modifications for a variable with a non-Gaussian equilibrium
Mazur and Bedeaux
2950 Langmuir, Vol. 8,No. 12, 1992 distribution in this section. For such a variable a(t)one wants to know under what circumstancesthe fluctuations can consistently be described by a stochastic differential equation of the form
Here L(a) is an a-dependent so-calledOnsager coefficient and f ( t ) the Langevin force which is assumed to be of a multiplicative nature. The analysis is completely analogous to the one given above for the Gaussian equilibrium distribution, and one finds that the random force is given by f(t> = f(a(t),t)= C,(a(t))fo(t)
This gives
LJda a rdP , , ( a ) C2(a) -LJda P,(a) C2(a) a
-L( @(a)) (4.9) The third equality follows by partial integration. Using this identity, one finds for the two-point correlation function of the random force
+ L dC(y(t))/dy(t) =
+
C,(a(t))fo(t)
L dC2(a(t))/da(t) (4.2)
where C ( a )s daldy = [dy/dal-’
(4.3)
a(t + €1 - a(t - t) YMt + 4) - YMt - €1) a(y(t + €1) - .(yo - 4) (4.4) Y(t + €1 - Y(t - €1 The above equations are equivalent to eqs 2.13,2.7,and 2.8,respectively. The variable y(a) must now be determined from the following equation: C,(a(t))
-
For the proof of the factorization in the second equality we refer to Appendix C. This expression for the twopoint correlation of the a-dependent Langevin force was obtained by one of usl1 using only causality and without the use of time reversal invariance. 5. Conclusions
which is analogousto eq 2.14and can be simplified to read
(4.6) which is to be compared with eq 2.15. The problem at this point of the analysis is that it is not possible to solve this differential equation without further knowledge of the equilibrium distribution. Again this equation determines only the product Lc2(a) and not L and C2(a)separately due to the fact that the scale ofy may be chosen arbitrarily. We shall not be concerned here with the existence of physically acceptable solutions of the above equation for C(a). The formulation of criteria for this existence,which depend on the equilibrium distribution,is outaide the scope of the present paper. Therefore,we shall restrict ourselves to systems for which such a solution can be constructed. One may then also show the equivalence of eq 4.1 with the simpler form, corresponding to IWsBprescription
and with the Stratonovich formlo
Can one describe the fluctuations of a variable with a nonlinear Langevin equation? This is the question which we considered in this paper. The origin of such a nonlinearity in for instance Brownian motion may be due to a dependence of the friction coefficient on the momentum. Though the dependence of the friction coefficient on the momentum will in practice always be weak and therefore in the description of Brownian motion not really of measurable importance, this case serves as a convenient illustration of the general problem and is therefore use in the text. However, the rotation of a nonsperical molecule and chemical reactions with fluctuating rate coefficientsare examplesof nonlinear system whose random properties should be discussed within the frameworkdeveloped in this paper.12 With regards to the general problem the main results derived in this paper may be summerised in the following theorem. Consider a stationary process d t ) ,which is either even or odd under time reversal, described by the stochastic differential equation
where B(a),withB(0) = 0 and ( B ( a ) )= 0, is the so-called systematic force and where fo is independent of a,and then it follows using causality and microscopic time reversal invariancethat fo is a Gaussian and white process with (fo(t) ) = 0
There is one useful result which may be derived for the general case of a non-Gaussian equilibrium distribution without the explicit construction of the function C(a).To obtain this result, we multiply eq 4.5 by a and average.
(fo(t)
fo(t’ 1) = Z&t- t’
(5.2)
and that the function C ( a ) is a solution of the following differential equation: (11) Mazur, P.Phys. Reo. 1992, A15,8957. (12) Lindenberg, K.and Weat, B.Y . The Nonequilibrium Statistical Mechnics of Open and Closed System; VCH New York, 1990.
Nature of the Random Force in Brownian Motion
Langmuir, Vol. 8, No. 12,1992 2951 Using the Gaussian nature of fo (cf. eq 2.2), this gives, 1) upon averaging, a sum of products of (1/2)l(n &functionsof time. Each f n contains n time integrations over the time. For n > 1&functionswhich depend on the integration variables may be integrated leaving (1/2)l(n1)or more integrations still to be performed. These final integrations than lead to a sum in which all contributions are proportional to a finite power of e and can therefore be neglected. This shows that, for n > 1,f n satisfies the conditions in eq 3.3 and is therefore zero in the context of the statistical description. For the remaining n = 1 case one has
+
In the above process we have chosen the variable such that its average value is zero. In order to obtain an explicit expression for the function C ( a )in terms of the systematic force B(a), one must know the equilibrium distribution. In this paper we have in particular considered the case that this equilibrium distribution is Gaussian: ~ , ( a ) = (g/27rkg)1’2exp(-ga2/2kB) (5.4) In that case the differential equation for LC2(a)can be solved, and one obtains
f l ( t )= L r d t ’ fo(t’)fo(t)- 2L fl’(t)- 2L
m
LC2@) = (2k$g) E d ”[B(a)/al/d [gar2/ 2kgI
(5.5)
n=O
The factor L is a scale factor which can be chosen arbitrarily. If one defines
L
W$g) 1 2 [B(a)/aI = (2k$g) [dB(a)/daI,,o (5.6)
one finds that C ( a ) = 1if the systematic force is linear in the variable a. Since one does not normally introduce a constant C into the linear case, eq 5.6 seems a natural choice.
Appendix A Integration of eq 2.5 together with eq 2.8 gives p ( t + €1 - p ( t - e) =
(B.1)
If one now considers the first term on the right-hand side, fl’(t),one may go through the similar argument as above for the n > 1 case, and show that (fi’(t1)ti’@,) ...fi’(tl)) = (2L)’= (fi’(t1)) (fI’(t2)) (fi’(tJ) (B.2) Therefore, f l ’ ( t ) has no correlations but has a nonzero average. Substracting from fi’(t)its average, one obtains f l ( t )which then also satisfies eq 3.5 and is consequently zero.
Appendix C In this appendix we will prove that ( C W - e)) f o W C(a(t’ - €1) f&’ 1) =
1) (C.1) It is sufficientto verify this property for t It’. We consider first the case that t S t’ - E. Then one may write, cf. eq 4.1 ( C2(a)) ( f o ( t ) fo(t’
(A.1) We may now use the fact that the variable p may have a discontinuity in the domain of integration but no singularity. fo(t’ 1, however, may have a singularity. The fiist term in the integrand thereforegivesat most a contribution of the order e and may be neglected. In the second term one may to the same order replace the expression between square brackets by ita value in t’ = t. In this way we obtain p(t
+ e) - p ( t - €1 =
Together with eq 3.3 this completes the proof that eq 2.5 with eq 2.8 is correct.
Appendix B Consider the following average: ( f n ( t 1 ) f n ( t 2 ) ...f n ( t l ) ). If one substitutes the definition of fn, one finds for n > 1 an expression with a product of l(n + 1)random forces fo.
( C(cr(t - 4) f&) C M t ’
- 4 )f o ( t ’ ) )
=
(C.2) where we used causality, cf. ref 8, in the second equality. Next we consider the case that t’ - e C t C t’. Using again causality, one then has
C(&’ - €1) to@’ 1) = ( C M t - €1) C W ’ - 4 ) f o W f&’ ) ) = ( - €1) C(a(t’ - €1)) ( f o ( t ) f o ( t ’ ) ) =
( C(a(t- 0 )f o W
( C 2 ( a ) ()f o ( t )
f&’ 1) ((2.3)
Equations C.2 and C.3 together prove eq C.1.
