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, PX (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 PX (x) is a probability density so that PX (x)dx is the probability that one finds the stochastic variable X to have values in the range [x, x + dxi. 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: YX (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 therefore simply 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 Yx (t) = f (x, t) , 224 (E.2) Fokker-Planck Equation with Detailed Balance 225 Appendix E. 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) → P0 (y) as t → ∞. In equilibrium detailed balance that constrains the transition rates: W (y|y 0 )P0 (y 0 ) = W (y 0 |y)P0 (y) , (E.3) ehere W (y 0 |y) is the probability, per unit time, that the system changes from a state | y i, characterized by the value y for the stochastic variable Y , to a state | y 0 i. Note that for a system in equilibrium the transition rate W (y|y 0 ) and the reverse W (y 0 |y), may be very different. Consider, for instance, a simple system that has only two energy levels 0 = 0 and 1 = ∆E. Then we find that W (1 |0 ) exp(−0 /kT ) = W (0 |1 ) exp(−1 /kT ) . (E.4) Therefore W (1 |0 )/W (0 |1 ) = exp(−∆E/kT ) → 0 when ∆E/kT → ∞, or when the temperature, T , tends to zero. Assume that W (y|y 0 ) 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 0 |y) = R+ (r; y) , for y 0 = y + r , and r > 0 . (E.5) The forward jump rate is assumed to be sharply peaked so that R+ (r; y) ' 0 for r > δ, and R+ (r; y + ∆y) ' R+ (r; y) for ∆y < δ. The change in the probability distribution is then given by the Master equation ∂ P (y, t) = ∂t Z ∞ {W (y|y − r)P (y − r, t) − W (y − r|y)P (y, t)} dr Z 0∞ . (E.6) {W (y|y + r)P (y + r, t) − W (y + r|y)P (y, t)} dr + 0 Here we note that the detailed balance equation (E.3) may be written R+ (r; y − r)P0 (y − r) = W (y|y − r)P0 (y − r) = W (y − r|y)P0 (y) , W (y|y + r)P0 (y + r) = W (y + r|y)P0 (y) = R+ (r; y)P0 (y) . (E.7) 226 Appendix E. Fokker-Planck Equation with Detailed Balance With these expressions equation (E.6) take the form Z ∞ ∂ P (y + r, t) P (y, t) P (y, t) = dr R+ (r; y)P0 (y) − ∂t P0 (y + r) P0 (y) 0 (E.8) Z ∞ P (y − r, t) P (y, t) + dr R+ (r; y − r)P0 (y − r) − P0 (y − r) P0 (y) 0 Now we may expand the terms in the parentheses to give Z ∞ ∂ ∂ P (y, t) P (y, t) = dr r R+ (r; y)P0 (y) ∂t ∂y P0 (y) y 0 Z ∞ ∂ P (y, t) dr r R+ (r; y − r)P0 (y − r) − ∂y P0 (y) y−r 0 (E.9) The two terms in the integral differ only slightly and we expand the last term around y and obtain ∂ ∂ P (y, t) ∂ P (y, t) = D(y)P0 (y) . (E.10) ∂t ∂y ∂y P0 (y) Here the generalized diffusion constant D is given by: Z Z ∞ 1 (y 0 − y)2 W (y 0 |y)dy 0 , D(y) = r2 R+ (r; y)dr ' 2 0 (E.11) where the second expression uses that we assumed that P0 (y) varies little over the range where R+ has a significant value. Equation (E.10) is a consequence of detailed balance (see also van Kampen [151], page 214). In the case of multivariate stochastic processes we have more than one stochastic variable and if we write r = (y1 , y2 , . . . , yn ), then the FokkerPlanck equation for stationary Markov processes with narrow transition rates takes the convenient form: P (r, t) ∂ (E.12) P (r, t) = ∇· D(r)P0 (r)·∇ ∂t P0 (r) where the ∇ = ( ∂y∂ 1 , ∂y∂ 2 , ∂y∂n ). The Fokker-Planck equation in this form makes explicit that there is no time dependence if P (r, t) = P0 (r). The diffusion tensor D is given in terms of an expression similar to equation (E.11) Z 1 D(r) = (r 0 − r)·W (r 0 |r)·(r 0 − r)dn r 0 , (E.13) 2 Z 1 Dij (r) = (yi0 − yi )W (r 0 |r)(yj0 − yj )dn r 0 , (E.14) 2 E.1 E.1 227 The Einstein Relations 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 to the concentration of particles, c(r, t). Therefore the Fokker-Planck equation (E.12) becomes an equation for the concentration of the Brownian particles: ∂ c(r, t) c(r, t) = ∇· D(r)c0 (r)·∇ (E.15) ∂t c0 (r) 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 c0 (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 Z 1 Dij (r) = (yi0 − yi )W (r 0 |r)(yj0 − yj )dn r 0 , (E.17) 2 If we define the jump rate Γ as Z Γ= W (r 0 |r)dn r 0 Then we may define the mean square jump distances as Z (yi0 − yi )W (r 0 |r)(yj0 − yj )dn r 0 0 0 Z h(yi − yi )(yj − yj ) iW = , W (r 0 |r)dn r 0 (E.18) (E.19) and we arrive at the Einstein relation for the diffusion constant Dij = 1 Γh(yi0 − yi )(yj0 − yj ) iW 2 1st Einstein relation (E.20) 228 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 = 0 ∂t (E.21) Where the probability flux, or rather Brownian particle flux, J is given by J (r) = − D(r)c0 (r)·∇[c(r, t)/c0 (r)] = − D(r)·∇c(r, t) + c(r, t)D(r)·∇ ln c0 (r) = − D(r)·∇c(r, t) + c(r, t)µ·F (E.22) Here F is the driving force that generates a drift velocity v = µ·F (E.23) The diving force is F = ∇ ln c0 (r) = kT ∇(−∆G(r)/kT ) = −∇(∆G(r)) (E.24) whereas the mobility µ is given by the second Einstein relation µ= 1 D kT 2nd. Einstein relation (E.25) 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.26) and therefore, by the second Einstein relation (E.25), we find the diffusion constant of a Brownian particle D= kT 6πµa Stokes-Einstein relation (E.27) 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.26) is no longer valid, however, the second Einstein relation between mobility and Diffusion constant still holds.