MIT OpenCourseWare http://ocw.mit.edu 8.334 Statistical Mechanics II: Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.VIII. Dissipative Dynamics VIII.A Brownian Motion of a Particle Observations under a microscope i ndicate that a dust part icle in a liquid drop under-goes a random jittery motion. This is because of the random impacts of t he much smaller fluid particles. The theory of such (Brownian) motion was developed by Einstein in 1905 and starts with the equation of mot ion for the particle. The displacement ~x(t), of a particle of mass m is governed by, m ~x ¨= − ~µx˙− ∂∂~V x + f~random(t). (VIII.1) The three forces acting on t he particle are: (i) A friction force due to the viscosity of the fluid. For a spherical particle of radius R, the mobili ty in the low Reynolds number limit is given by µ = ηR)−1, where ¯(6π¯ η is the specific viscosity. (ii) The force due to the external potential V(~x), e.g. gravity. (iii) A random force of zero mean due to the impacts o f fluid particles. The viscous term usually dominates the inertial one (i.e. the motion is overdamped), and we shall henceforth ignore the acceleration term. Eq.(VIII.1) now reduces to a Langev in equation, ~x˙= ~v(~x) + ~η( t ), (VIII.2) where ~v(~x) = −µ∂V/ ∂~x is the deterministic velocity. The stochastic velocity, ~η(t) = µf~random(t), has zero mean, �~η(t)� = 0. (VIII.3) It is usually assumed that the probability distribution for the noise in velocity is Gaussian, i.e. Z η(τ )2  P [~η(t)] ∝ exp − dτ 4D. (VIII.4) Note that different components of the noi se, and at different times, are indep endent , and the covariance is �ηα(t)ηβ(t ′ )� = 2Dδα,βδ(t − t ′ ). (VIII.5) 141The parameter D is related to diffusion of particles in t he fluid. In the a bsence of any potenti al, V(~x) = 0, the position of a particle at ti me t is given by Z t ~x(t) = ~x(0) + dτ~η(τ). 0 Clearly the separation ~x(t) −~x(0) w hich is the sum of random Gaussian variables is itself Gaussian distributed with mean zero, and a varaince D 2EZ t (~x(t) − ~x(0)) = dτ1dτ2 �~η(τ1) ~η(τ2)� = 3 × 2Dt. · For an ensemble of particles released at ~x(t) = 0, i.e. with P (~x, t = 0) = δ3(~x), the particles at time t are distributed according to 0 3/2  2 1 xP (~x, t) = √4πDt exp −4Dt , which is the solutio n to the diffusion equation ∂P D∇2P.= ∂t A si mple exa mple is provided by a particle connected to a H ookian spring, with V(~x) = Kx2/2. The deterministic velocity is now ~v(~x ) = −µK~x, and the Langevi n equation, ~x˙= −µK~x + η~(t), can be rearranged as d  eµKt~x(t) = eµKt~η(t). (VIII.6) dt Integrating the equation from 0 to t yields Z t eµKt~x(t) −~x( 0) = dτeµKτη~(τ), (VIII.7) 0 and Z t ~x(t) = ~x(0)e −µKt + dτe−µK(t−τ )~η(τ ). (VIII.8) 0 Averag ing over the noise indicates that the mean position, �~x(t)� = ~x(0)e −µKt, (VIII.9) 142Z decays with a characteristic relaxation time, τ = 1/(µK). Fluctuations around the mean behave as 2Dδ(τ1−τ2)×3 D 2EZ t z }| { ~x(t) − �~x(t)� = 0 dτ1dτ2e −µK(2t−τ1−τ2) �~η(τ1) · ~η(τ2)� Z t =6D dτe−2µK(t−τ) (VIII.10) 0 3D −2µKt t→∞ 3D = e . µK 1 − −→ µK However, once the dust particle reaches equilibrium with the fluid at a temperature T , its probability distribution must sati sfy the normalized Boltzmann weight  K 3/2  Kx2  Peq.(~x) =2πkBT exp −2kBT, (VIII.11) yielding  x2 = 3kBT/K. Since the dynami cs is expected to bring the particle to equilib-rium with the fluid at temperat ure T , eq.(VIII.10) implies the condition D = kBT µ . (VIII.12) This is the Einstein relation connecting the fluctuations of noise to the dissipation in the medium. Clearly the Langevin equation at long times reproduces the correct mean and variance for a particle in equilibrium at a t emperature T in the potential V(~x) = Kx2/2, provided that eq.(VIII.12) is satisfied. Can we show that the whole probability distribution evolves to the Boltzmann weight for any po tential? Let P(~x, t) ≡ �~x|P(t)|0� denote the probability density of finding the particl e at ~x at time t, g iven that it was at 0 at t = 0. This probability can be constructed recursively by noting that a particle found at ~x at time t + ǫ must have ′ arrived from some o ther point ~x at t. Adding up all such probabilities yields P(~x, t + ǫ) = d3~x ′ P(~x ′ , t) �~x|Tǫ|~x ′ �, (VIII.13) ′ ′ where �~x|Tǫ|~x � ≡ �~x|P(ǫ)|~x � is the transition probability. For ǫ ≪ 1, ′ ~x = ~x + ~v (~x ′ )ǫ + ~ηǫ , (VIII.14) 143D   E    where ~ηǫ = Rtt+ǫ dτ~η(τ). Clearly, �η~ǫ� = 0, and ηǫ 2 = 2Dǫ × 3, and following eq.(VIII.4),  1 3/2  ηǫ 2  p(~ηǫ) = exp . (VIII.15) 4πDǫ −4Dǫ The transiti on rate i s simply the probability of finding a noise of the right magnitude according to eq.(VIII.14), and ′  1 3/2 " (~x − ~x ′ −ǫ~v(~x ′ ))2 # �~x |T (ǫ)|~x � = p(ηǫ) =4πDǫ exp − 4Dǫ 3/2 ~˙v(~x) 2  (VIII.16) 1  x −~= exp .4πDǫ −ǫ 4D By subdividing the time interval t , into infinitesimal segments of size ǫ, repeated application of the above evolution operator yields P(~x, t) = ~x T (ǫ)t/ǫ0  2  = Z (~x,t) D~x(τ)exp Z t dτ ~x˙−~v(~x) . (VIII.17) N − 4D(0,0) 0 The integral is over all paths connecting the initial and final points; each path’s weig ht is related to its deviation from the classical t rajectory, ~x˙= ~v(~x). The recursion relation (eq.(VIII.13)), Z3/2 " ′ 2 # P(~x, t) = d3~x ′ 1 exp (~x − ~x −ǫ~v(~x ′ ))P(~x ′ , t −ǫ), (VIII.18) 4πDǫ − 4Dǫ can be simplified by the change of variables, ~y