Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his ype of problem is given by he Fokker-Planck equaion. We can eiher formulae he quesion in erms of he evoluion of a nonsaionary probabiliy disribuion from a defined iniial condiion, or in erms of he evoluion of he condiional probabiliies for a saionary random process. I will choose he laer approach. Remember he condiional probabiliy P n (y, ; y, ;... y n, n )dy n () is he probabiliy (in he ensemble sense) ha if y() akes on he values y a, y a... y n a n hen i will lie beween y n and y n + dy n a ime n (where > > 3... > n ). Noe he noaion for condiional probabiliy is disinguished from he probabiliies p n hrough he case, and he noaion. Also noe in P (y, y, ) he noaion is y, implies y, wih > ; unforunaely Reif uses he reverse order, i.e. y, implies y,. (For higher order P n here is less possibiliy of confusion.) The probabiliy disribuions p n and he condiional probabiliies P n are relaed hrough p n (y, ;... ; y n, n ) = p n (y, ;... ; y n, n ) P n (y, ;... ; y n, n y n, n ). () A Markoff process is one for which fuure probabiliies are deermined by he mos recenly known value, and do no depend on he previous hisory P n (y, ; y, ;... ; y n, n y n, n ) = P (y n, n y n, n ). (3) Saionary Markoff processes are herefore compleely characerized by p (y) and P (y y, ) = p (y, 0; y )/p (y ) (where we use he convenien noaion P (y y, ) for P (y, y, + )). The imporance of Markoff processes is no ha all physical processes are Markoff, bu ha he analysis of Markoff processes is considerably simpler. For example in he descripion of Brownian moion in erms sharp molecular kicks he x-velociy of he paricle is Markoff (he probabiliy of u( + δ) depends only on u() and he molecular collisions in ime δ; on he oher hand he posiion x() is no, because x( + δ) depends on x() and u()δ x() x( δ), as well as he molecular collisions. I is of course easy o formulae he problem in erms of u() and hen derive properies of x() from his. The quesion of wheher a random process is Markoff or no migh depend on he level of he descripion. For example, in he coarse grained descripion of Brownian moion where we discuss x() as a random walk (i.e. on a ime scale large compared o he relaxaion ime of he velociy), he random process x() becomes a Markoff one. Time Evoluion From general probabiliy heory he wo poin condiional probabiliy disribuion saisfies he Chapman- Kolmogorov equaion P (y, y 3, 3 ) = dy P (y, y, )P 3 (y, ; y, y 3, 3 ) (4) where < < 3. This corresponds o inegraing over all possibiliies a he inermediae ime.
For a Markoff process he P 3 is given by P and he equaion reduces o he Smoluchowski equaion P (y, y 3, 3 ) = This is effecively an inegral equaion for he ime evoluion of P. dy P (y, y, )P (y, y 3, 3 ). (5) If in a small ime inerval only small changes in y can occur, his ime evoluion of a Markoff random process can be rewrien as a differenial equaion known as he Fokker-Planck equaion. An example where his is he case is he Brownian moion of a heavy paricle: in a small ime inerval he velociy of he heavy paricle is only changed by a small amoun by he small number of molecular collisions. On he oher hand for he velociy disribuion of molecules in a gas, each binary collision can change he velociy by a large amoun, and he Fokker-Planck equaion does no apply. Thus he Fokker-Planck equaion is appropriae for he flucuaions of macroscopic degrees of freedom. Wriing in erms of P (y 0 y, ) for saring a y 0 and ending a y a ime laer he Fokker-Planck equaion is P = y A(y)P + y B(y)P (6) which is o be solved wih he iniial condiions P (y 0 y, 0) = δ(y y 0 ). In his expression A and B are given by he rae of growh of he mean and sandard deviaion A(y) = lim B(y) = lim (y y)p (y y, )dy, (7a) (y y) P (y y, )dy. (7b) The firs erm is a drif of he disribuion corresponding o a sysemaic bias, and he second o a diffusion of he disribuion, corresponding o he residual average effec of posiive and negaive jumps. (Higher momens of P increase less rapidly han, and do no give higher order derivaive erms in he equaion.) We have already encounered a Fokker-Planck equaion: he diffusion equaion for he probabiliy disribuion of a random walk is a simple example. Derivaion The derivaion from he Smoluchowski equaion is elemenary, albei a lile ricky. Wrie Eq. (5) in he form (wih y y 0, y 3 y, y y ξ and, 3 τ) P (y 0 y, + τ) = dξ P (y 0 y ξ, )P (y ξ y, τ), (8) where τ will be a small ime incremen. In his equaion we are sudying he condiional probabiliy of geing o y a ime + τ in erms of he probabiliy of geing nearby o y ξ a ime and hen o y in he small ime incremen τ. Now expand he lef hand side in a Taylor expansion in τ o give P (y 0 y, + τ) P (y 0 y, ) + P (y 0 y, ) τ (9) P (y 0 y, ) τ = P (y 0 y, ) + dξ P (y 0 y ξ, )P (y ξ y, τ). (0)
Noe ha he firs erm on he righ hand side could be wrien dξ P (y 0 y, )P (y y ξ, τ) () since dξ P (y y ξ, τ) =. The form of he equaion is hen a scaering ou erm and a scaering in erm, as in he Bolzmann equaion. This sor of equaion is known as a Maser Equaion. The difference from he Bolzmann equaion is ha here we assume ha only small changes in y are possible in small τ, and we expand he dependence on he from value z = y ξ in he scaering-in erm in small ξ. Noe ha we are no expanding P (y ξ y, τ) in small ξ for fixed y his funcion decreases rapidly wih ξ, and i is precisely because of his ha we only need o know P (y 0 z, )P (z z + ξ, τ) for z near y. Thus wrie This gives P (y 0 y ξ, )P (y ξ y, τ) = P (y 0 z, )P (z z + ξ, τ) z=y ξ () ( ) n = ξ n n n! y P (y n 0 y, )P (y y + ξ, τ). (3) P (y 0 y, ) τ = P (y 0 y, ) + n=0 ( ) n n=0 The zeroh order erms cancel, leaving (aking τ 0) P (y 0 y, ) = n= n! n P y n (y 0 y, ) dξξ n P (y y + ξ, τ). (4) ( ) n n P n! y n (y 0 y, ) lim dξξ n P (y y + ξ, τ). (5) τ 0 τ If he random process evolves hrough he effec of many small changes, only he firs wo momens n =, pf P will conribue, wih higher momens increasing as τ p wih p > giving no conribuion as τ 0. Fokker-Plank Equaion for he Brownian Velociy Derivaion The Fokker-Planck approach Eq. (6) and Eq. (7) is really quie independen of he Langevin equaion, and in many ways is preferable from a formal poin of view, since he Langevin equaion is raher pahological and needs careful reamen. We should herefore calculae A, B direcly from hinking abou he molecular collisions, and indeed his can be done. Bu here I will jus ge hem by inegraing he Langevin equaion from a known velociy u over a small ime inerval Using F = 0 gives u A = lim u B = lim M u + γ u = + F ( )d. (6) = u τ r, τ r = M γ, (7) = lim + + F ( M )F ( ) d d. (8) 3
You should be familiar wih his sor of double inegral by now! Noe ha alhough is small, we assume i o be large compared o he molecular collision ime, so ha he force correlaion funcion is nonzero only over ime differences shor compared o. In he usual way we have lim + + F ( )F ( ) d d = lim + d F (0)F (τ) dτ (9) 0 = G F(0) = kt γ (0) so ha B = γ kt M. () Soluion To sudy he decay of he velociy o he equilibrium Maxwellian we solve he Fokker-Planck for P (u 0 u, ) wih he iniial condiion P (u 0 u, = 0) = δ(u u 0 ). You can find a forward soluion in Reif 5.. I will jus show he soluion and le you verify ha i saisfies he differenial equaion by subsiuion: P (u 0 u, ) = πσ u () exp (u ū()) σ u () where he ime dependen mean ū() and variance σu () are wha we found before () ū() = u 0 e /τ r (3) σ u () = kt M ( e /τ r ). (4) Example As an example of how o use he soluion, consider he calculaion of he correlaion funcion C u (τ) = u(0)u() = For he saionary random process u() we have and so ha C u (τ) = p (u, 0; u, )u u du du. (5) p (u, 0; u, ) = p (u )P (u u, ) (6) πσ m e u /σ m p (u) = lim P (u 0 u, ) (7) πσ u () exp (u ū()) σu () u u du du, (8) wih σ m = kt/m and ū() = u e /τ r. The u inegraion jus gives ū(), so ha C u (τ) = e /τ r From his he specral densiy is given by Fourier ransform πσ m e u /σ mu du (9) = kt M e /τ r. (30) G u ( f ) = 4kT Mτ r (π f ) + (/τ r ). (3) 4
Saionary, Gaussian, Markoff Processes Resuls very similar o he ones we have jus calculaed acually apply o a large class of random processes namely hose ha are saionary, Gaussian, and Markoff. For such a process y() all he saisical properies are deermined by he mean ȳ, he variance σ y, and he wo poin correlaion ime τ r. This is known as Doob s heorem. Remember ha a saionary Markoff process is characerized by p (y) and P (y y, ) or p (y, 0; y, ). For a Gaussian process p is characerized by ȳ and σ y, and for p we need in addiion he α parameer of he Gaussian disribuion, which is deermined by he wo poin correlaion funcion C y (). Furher, since he behavior in each successive ime incremen does no depend on he hisory, he decay of correlaions mus be a simple exponenial The specral densiy is hen Lorenzian G y ( f ) = C y (τ) = σ y e /τ r. (3) (4/τ r )σ y (π f ) + (/τ r ). (33) This appears whie for f τr, and falls off as f for high frequencies. The complee characerizaion also involves (y ȳ) p (y) = exp (34) πσy σy and using he noaion P (y y, ) = πσ () exp (y ȳ()) σ () ȳ() = ȳ + e /τ r (y ȳ) σ () = ( e /τ r )σ y (35) (36a) (36b) showing how he mean and he variance of he condiional probabiliy relaxes from he iniial value se by he iniial condiion y = y o he long ime asympoic forms se by p. I will no derive hese general resuls here. Revised Sepember, 04 5