Fokker-Planck Equation with Detailed Balance

Transcription

1 Appendix E Fokker-Planck Equation with Detailed Balance A stochastic process is simply a function of two variables, one is the time, the other is a stochastic variable X, defined by specifying: a: the set {x} of possible values for X; b: the probability distribution, P X (x), over this set, or briefly P (x) The set of values {x} for X may be discrete, or continuous. If the set of values is continuous, then P X (x) is a probability density so that P X (x)dx is the probability that one finds the stochastic variable X to have values in the range [x, x + dx. An arbitrary number of other stochastic variables may be derived from X. For example, any Y given by a mapping of X, is also a stochastic variable. The mapping may also be time dependent, that is, the mapping depends on an additional variable t: Y X (t) = f(x, t). (E.1) The quantity Y (t) is called a random function, or, since t often is time, a stochastic process. A stochastic process is a function of two variables, one is the time, the other is a stochastic variable X. Let x be one of the possible values of X then y(t) = f(x, t), (E.) 4

2 Appendix E. Fokker-Planck Equation with Detailed Balance 5 is a function of t, called a sample function or realization of the process. In physics one considers the stochastic process to be an ensemble of such sample functions. For many physical systems initial distributions of a stochastic variable y tend to equilibrium distributions: P (y, t) P (y) as t. In equilibrium detailed balance constrains the transition rates: W (y y )P (y ) = W (y y)p (y), (E.3) wehere W (y y) is the probability, per unit time, that the system changes from a state y, characterized by the value y for the stochastic variable Y, to a state y. Note that for a system in equilibrium the transition rate W (y y ) and the reverse W (y y), may be very different. Consider, for instance, a simple system that has only two energy levels ɛ = and ɛ 1 = E. Then we find that W (ɛ 1 ɛ ) exp( ɛ /kt ) = W (ɛ ɛ 1 ) exp( ɛ 1 /kt ). (E.4) Therefore W (ɛ 1 ɛ )/W (ɛ ɛ 1 ) = exp( E/kT ) when E/kT, that is, T tends to zero. Assume that W (y y ) is finite only for small jumps, and that it varies slowly with y. It is convenient to introduce the transition rate R + for positive (or forward) jumps: W (y y) = R + (ρ; y), for y = y + ρ, and ρ >. (E.5) The forward jump rate is assumed to be sharply peaked so that R + (ρ; y) for r > δ, and R + (ρ; y + y) R + (ρ; y) for y < δ. The change in the probability distribution is then given by the Master equation P (y, t) = t + {W (y y ρ)p (y ρ, t) W (y ρ y)p (y, t)} dρ. (E.6) {W (y y + ρ)p (y + ρ, t) W (y + ρ y)p (y, t)} dρ Here we note that the detailed balance equation (E.3) may be written R + (r; y ρ)p (y ρ) = W (y y ρ)p (y ρ) = W (y ρ y)p (y), W (y y + ρ)p (y + ρ) = W (y + ρ y)p (y) = R + (ρ; y)p (y). (E.7)

3 6 Appendix E. Fokker-Planck Equation with Detailed Balance With these expressions equation (E.6) take the form ( ) P (y + ρ, t) P (y, t) P (y, t) = dρ R + (ρ; y)p (y) t P (y + ρ) P (y) ( ) (E.8) P (y ρ, t) P (y, t) + dρ R + (ρ; y ρ)p (y ρ) P (y ρ) P (y) Now we may expand the terms in the parentheses to give ( ) P (y, t) P (y, t) = dρ ρ R + (ρ; y)p (y) t y P (y) y ( ) P (y, t) dρ ρ R + (ρ; y ρ)p (y ρ) y P (y) y ρ (E.9) The two terms in the integral differ only slightly and we expand the last term around y and obtain t P (y, t) = ( D(y)P (y) ) P (y, t). (E.1) y y P (y) Here the generalized diffusion constant D is given by: D(y) = ρ R + (ρ; y)dρ 1 (y y) W (y y)dy, (E.11) where the second expression uses that we assumed that P (y) varies little over the range where R + has a significant value. Equation (E.1) is a consequence of detailed balance. In the case of multivariate stochastic processes we have more than one stochastic variable and if we write r = (y 1, y,..., y n ), then the Fokker- Planck equation for stationary Markov processes with narrow transition rates takes the convenient form: P (r, t) = t ( D(r)P (r) ) P (r, t) P (r) (E.1) where the = ( y 1, y,... y n ). The Fokker-Planck equation in this form makes explicit that there is no time dependence if P (r, t) = P (r). The diffusion tensor D is given in terms of an expression similar to equation (E.11) D(r) = 1 (r r) W (r r) (r r)d n r, (E.13) D ij (r) = 1 (y i y i )W (r r)(y j y j )d n r, (E.14)

4 E.1 The Einstein Relations 7 E.1 The Einstein Relations In a system of Brownian particles undergoing diffusion, the stochastic variable describing particle is its position r. The probability density P (r, t) is proportional to the concentration of particles, c(r, t). Therefore the Fokker- Planck equation (E.1) becomes an equation for the concentration of the Brownian particles: ( ) c(r, t) c(r, t) = D(r)c (r) t c (r) (E.15) We have assumed that the concentration at position r is proportional to P (r, t), that the Brownian particle positions are well approximated by a Markov process, and that the jumps are short ranged. However, the Brownian particles need not be at a low concentration, in fact they may interact strongly. The equilibrium concentration has the general form c (r) exp( G(r)/kT ) (E.16) for a system at constant temperature T, pressure P, and number of particles N. Here, G(r) is the change in Gibbs free energy, from some reference state. For systems at constant volume V, one uses the Helmholtz free energy change F (r) instead of the Gibbs free energy. Other system constrains replaces the appropriate free energy for G. The diffusion tensor D has components D ij (r) = 1 (y i y i )W (r r)(y j y j )d n r, (E.17) If we define the jump rate Γ as Γ = W (r r)d n r (E.18) Then we may define the mean square jump distances as (y (y i y i )(y j i y i )W (r r)(y j y j )d n r y j ) W =, (E.19) W (r r)d n r and we arrive at the Einstein relation for the diffusion constant D ij = 1 Γ (y i y i )(y j y j ) W 1st Einstein relation (E.)

5 8 Appendix E. Fokker-Planck Equation with Detailed Balance That is, the diffusion constant is one half the mean square jump distance times the jump rate. The Fokker-Planck equation for the concentration may also be written as a continuity equation: c(r, t) + J = t (E.1) Where the probability flux, or rather Brownian particle flux, J is given by J(r) = D(r)c (r) [c(r, t)/c (r)] = D(r) c(r, t) + c(r, t)d(r) ln c (r) = D(r) c(r, t) + c(r, t)µ F (E.) Here F is the driving force that generates a drift velocity v = µ F (E.3) The diving force is F = ln c (r) = kt ( G(r)/kT ) = ( G(r)) (E.4) whereas the mobility µ is given by the second Einstein relation µ = 1 D nd. Einstein relation (E.5) kt The driving force is just the negative gradient of the related potential as driving forces should be. The second Einstein relation is a relation between the mobility of a particle and the diffusion constant. One of the best known uses of this relation is for a single spherical particle radius a in a fluid of viscosity µ in this case the Stokes equation gives the mobility of the particle to be µ = (6πµa) 1 (E.6) and therefore, by the second Einstein relation (E.5), we find the diffusion constant of a Brownian particle D = kt 6πµa Stokes-Einstein relation (E.7) This expression for the diffusion constant of Brownian particles, in terms of the Stokes expression for the mobility, is valid only for non-interacting particles. For sedimenting particles at a finite density, which allows the Brownian particles to interact, the Stokes expression (E.6) is no longer valid, however, the second Einstein relation between mobility and Diffusion constant still holds.