(, BROWNIAN MOTION IN A FIELD OF FORCE AND THE

a'r".r,.: :' Brinkman, H. C. 1956 Physica XXII 2934 (, BROWNIAN MOTION IN A FIELD OF FORCE AND THE DIFFUSION THEORY OF CHEMICAL REACTIONS by H. C. BRINKMAN Centraal Laboratorium T.N.O.,Delft, Nederland

Summary Kramers t) hasderivedadiffusionequation inphasespace, describingthemotion of a particle subject to an external force and to the shuttling action of the Brownian forces caused by a surrounding medium in temperature equilibrium. In this paper a solution of Kramers'equation is obtained by systematic developinent oftheparticledensitywithrespecttothesmoluchowskidiffusioncurrentoperator. ft is shown that Smoluchowski's diffusion equation in coordinate space is a first approximation to Kramers' equation, valid if all but the first power of the Smoluchowski current operator may be neglected. The region of validity of the Smolucho*'ski equation is determined for a particle in a potential well. l. Introd,uction. ln 1940 H. A. Kramers attacked the problem o{ the applicability of the transition state method for calculating the rate of chemical reactions 1). The following model was studied. A particle moves in is subject to the ir an external field of force, but in addition to this regular forces of a surrounding medium in temperature equilibrium (Brownian motion). Originally the particle is caught in a potential hole, but it may escape in the course of time by passing over a potential barrier. K r a m e r s calculated the probability of escape under the assumption of a stationary diffusion current and obtained the result that for such a situation the transition state method has a limited region of applicability. The basic problem of Kramers' treatment the Brownian movement of a particle in a field of force was attacked as a diffusion problem in phase space. Assuming the validity of Einstein's theory of the Brownian motion, Kramers derived a diffusion equation in phase space. This equation, however, presented averydifficultmathematicalproblemand K r a m e r s limited his treatment to two extreme cases, viz Brownian forces that are much larger or on the other hand much smaller than the external force. In the first case Kramers showed that Smoluchowski's diffusion equation in coordinate space may be derived from Kramers' equation in phase space under the introduction of certain approximations. Several questions remain open for discussion, however. A first question is the region of validity of the Smoluchowski equation. A second question is 29

30 H. C. BRINKMAN the possible extension of Kramers' treatment of the escape problem for nonstationary states (diffusion current not constant). A solution of this second problem would be of interest for the study of chemical reactions, where the reaction may proceed in both directions (shuttling forward and back between two potential holes). An answer to these questions might be given if a general formalism for the treatment of Kramers' equation were available. In this paper such a formalism is set up and the first question (validity of the Smoluchowski equation as a first approximation) is answered precisely. 2. Kramers' dillusion equation in phase space. For the derivation of Kramers' equation (to be called the "Kequation") the reader is referred to Kramers' original paper 1). Here we shall give a short survey. A particle of mass m in a onedimensional extension is acted upon by a conservative external field of force F(q) and an irregular force X(l), due to the medium: 0) b:r@)+x(t), mt:f The force.f(q) is derived from a potential U(q) Fduldq (2) while the force X(l) has properties which lead to Brownian motion of the Einstein pattern. Referring to Kramers for a discussion of these properties we immediately write down the resulting diffusion equation in phasespace, derived by Kramers in a manner analogous to the derivation of the FokkerPlanck diffusion equation #:+("n +, T#)*+(*,) (3) In this equation p is the particle density in phase space (p, q). The K equation (3) has a form analogous to the well known Gibbs equation of continuity in phase space. It is completed with the terms rnht1glfip and ppfu dtrc to the Brownian motion. The term PSl, originates from the friction between particles and surrounding medium. The constant z may be called a relaxation time, shortly to be discussed (cf. (10)). The term mktaplr?p represents the shuttling action of the Brownian motion, tending to set up a Boltzmann distribution. For the applications the particle distribution o(q) along the q coordinate is of interest. It may be defined as o(q, t) : lls e@,q, t)dp Kramers showed that for large viscosity 4 (4) (small relaxation time z)

BROWNIAN MOTION IN A FIELD OF FORCE 31 the qdistribution o approximately obeys the Smoluchowski diffusion equation (to be called the Sequation) 0o 0 /Ft ktt Ao\ t_r(5) u E\;" * aql after a certain time which will be shown to be of the order of the relaxation time z. For his derivation of the Sequation (5) Kramers introduced a definition of o differing from (4). According to this definition o is obtained by integration along a straight line q f fulm: qo in the p, q plane o(qo, t) : I[::3 p(p, q, t)df (6) @+ltltu:qol Obviously definition (6) tends to (4) for r > O. However definition (4) has a clear physical meaning for any value of z. Therefore ( ) will be consistently used in our calculations. The Sequation may be very simply derived from macroscopic considerations. Indeed a spherical particle (radius a) subject to a force F in a medium of viscosity 4 will obtain a velocity u u: Fl$nqa) (7) The particle current za resulting from ordinary diffusion (diffusion constant D) and the velocity u will then be equal to u D@olaq) (Fl6nqa)o (8) Putting D: htl$nr1a) and writing down the equation of continuity results in 0o A/ kt 0o F \ u :,, \,"* E + 6"11.') equivalent to the Sequation (5) if the relaxation time z is put equal to t : ml6nqa (10) 3. Formal solution ol the Kequation. The Kequation (3) gives the change in the particle density g if one starts at time I : 0 with a certain initial distribution po(f, q).equation (3) is therefore most easily treated by using a Laplace transformation, defined by e(b, q' s) :/; e*'t s(p, q, t)dt yielding the transformed Kequation mkt AzD /b \ a; 7F+\;F) 'e *+*(+') A solution of this equation is obtained by developing the (e) (11) P: so 02) deviations from

32 H. C. BRINKMAN the Boltznrann distribution along the y' coordinate in a system of orthogonal Hermite polynomials He*(plt/kTm). The solution then reads b2 A: e 2nhr2* ohe,lpl{nf4*,(q,s) where t}ire g, obey a recursion system obtained by substituting (13) in (12) and using the recursion formulae f.or the He,. The initial distribution is chosen as (13) Qo ep'l'o'!o@) (14) The recursion system for the g, reads and f.or n>o sqo : lo (+.,),.: Lr,,l t/mkt (ktlnfittz 1Vrl1q kt )E,t _ rn 0q @+r)l hr lyr:t (rs) f/1, oq However, not all qn are of interest for our purpose. Indeed, from the orthogonality properties of the Hermite polynomials the following expressions for the Laplace transforms of the density 6 and the current Dr along the q coordinate are obtained (cf. (13)) ; : /JS e dp : {zn*kt q, A 6t : I:S LA ap : {2n kt qr (16) (17) In analogy currents of higher order are defined as **: {2n ht q, For the discussion of (15) rve introduce the Smoluchowski current operator (18) f s: (htrtm) 0l0q t Frlm (19) f'he recursion system (15) may now be written in a simple form bv the introduction of the following dimensionless operators I : (*1krYr' J' f o: (htlm)ltzv212, (20) L: kl{mht rvhere I : J, * "I, is the Scurrent operator divided by the gas kinetic velocity \/htl*. Introduction of (16)...(20) in (15) and of the initial density os: fis Qodf : t/a*nf 6 (2t\

BROWNIAN MOTION IN A FIELD OF FORCE 33 vields the following recursion r]*"* sd: oo _ 0d, Aq_ (l f sz)d1 : {@Tlm) I" + 2Iod, (22a) (22b) and for n :, I (n I st)w,: ID,r * (n * l) IoD,+, (22c) The system (22) may be solved in n'h approximation by neglecting all wr for fr > n. In this case the system consists of. n { I equations for the functions 6,d,t...6,. Sufficient assumptions for this procedure to be valid are that the S curreats are small as compared to the gas kinetic currents l/"1 < lol and ll*"1 4l*,1 while the diffusion currents are not larger than the gas kinetic currents lloal < lol and llotu) <lu,l Then the uo f.or k < n are of the order I *o o A Io'frr. In third approximation the elimination of.ur, u, and rz, yields the following equation for 6 sd oe : (l/(l I n)t)loja I + (21$ + s")' (2 t st)t) I'" 'a + + (4/(l + s")t (2 { sr)2r) I'oIIoI'a + + (6/(1 + sr)' (2 a sr)2 (3asr)t)J3oJ3a! I l'.'. (lst order) 1 (2nd order) ) t.,u order) I t*' (higher orderl I Limited to first order terms the following equation is obtained sd os: (1i(1 sr)r)lrjo (24) This is the Laplace transform of the Sequation (5) containing a small correction. Indeed, after introduction of the / operators the transform of (5) reads sd oo : t IoIo (2s) This is identical to Qa) if sz ( 1. The physical meaning of this condition is that the Sequation obtains its validity after a time which is large as compared to the relaxation time z. The region of validity of the Sequation may be determined by substitution of the solution ol Qa) in the second order term of (23). This term should be small as compared to the first order term.

34 BROWNIAN MOTION IN A FIELD OF FORCE a. Appl,ication to a parabol,ic potential, well. In this paragraph an illustration of the application of the higher order terms (23) in the Sequation will be given. We limit ourselves to the simple case of a parabolic potential well F:aQ (26) The general solution of the Sequation (5) may then be written as where o : ELo cro*enbtlm)' Q7) o,: so42l2h' ne,6/a1kf q) and the cn are arbitrary constants to be determined from the initial conditions. The Sequation completed with second order terms is found from (23) oolot : tt I o Jo + ;t J2o I2o. (28) From (28) it is easily inferred that the second order correction is of the order of magnitude of fi2ol(0r)2. The relative correction is of the order of fu a2ola*)lqolaz). This may be verified by substituting (n) in (28). Now from (27) 8o lao a* 2, n2 cror rnlarlnlt (2e) nt, En n cn of en@tlnl' If in this expression some constant c, is large for large n, the second factor in the righthand member is large at small l. The conclusion is that the Sequation is not valid for an initial distribution os which shows steep gradients (c, large for large n). On the other hand (29) is large if. ar2fmis large. The magnitude of at2fm may be ascertained by observing that,: {"1* is the frequency of the oscillation of the particle in its potential well. Therefore 'Fl * o2o I oo T_l_N_y"t' arl 0t We conclude that the Sequation is a good approximation if the relaxation time z is small as compared to the period of oscillation of the particle. This is the case if the particle is subject to many Brownian perturbations during one period. The author is indebted to many friends in and outside his laboratory for clarif ying discussions. Received 23955. (30) REFERENCE l) H. A. Kramers, Physica Z (1940) 284,