Ch 7. The Approach to Equilibrium 7.1~ 7.4 Adaptive Cooperative Systems, Martin Beckerman, Summarized by Min Su Lee Biointelligence Laboratory, Seoul National University

Contents 7.1 Nonequilibrium Dynamics  Relaxation and Equilibrium Fluctuations  Brownian Motion  Markov Process 7.2 Outline of the Chapter 7.3 The Fokker-Planck Equation  Brownian Motion  Fokker-Plank Equation for Brownian Motion 7.4 The Langevin Equation  Formal Solution  The Mean Square Displacement  Properties of the Random Force 2(C) 2009, SNU Biointelligence Lab,

Nonequilibrium Dyanamics Relaxation and Equilibrium Fluctuations  Spontaneous fluctuations of a dynamic variable A in a system at equilibrium ( δA(t) = A(t) - )  Relaxation of a nonequlilibrium system & Equilibrium fluctuation  The relaxation of a macroscopic nonequilibrium perturbation is governed by the same rules as the regression of spontaneous fluctuations in a system at equilibrium (by Onsager)  Connections between relaxation of a nonequilibrium system and equilibrium fluctuations can be formalized as the regression hypothesis.  Foundation for nonequilibrium dynamics  Fluctuation-dissipation theorems(analysis of brownian motion)  Reciprocity theorems of Onsager (regression hypothesis) 3(C) 2009, SNU Biointelligence Lab,

Nonequilibrium Dyanamics Brownian Motion  A natural phenomenon  Small specs of dust or pollen suspended in a fluid seemed to move about constantly in a random zigzag manner  Observed by botanist Robert Brown (1826)  Explained by Einstein (1905)  Brownian motion arises from interactions of the dust particle with a heat bath, more specifically, from random collisions of the dust particles with the molecules of the fluid  Describe brownian motion in terms of the diffusion equation  Derive a relation connecting the viscous friction of the medium to the diffusion constant  Used to deduce values for Boltzmann’s constant and Avogardo’s number and Milikan’s oil drop experiment 4(C) 2009, SNU Biointelligence Lab,

Nonequilibrium Dyanamics Markov Processes  Notations  X(t): Time-dependent random variable  x 0, x 1, x 2,... values of X(t) at time t 0, t 1, t 2,... s.t. t 0 ≤ t 1 ≤ t 2 ≤...  Systems characterized by these variables will evolve probabilistically in time by means of a stochastic process   ( ∵ Markov assumption)   : Chapman-Kolmogorov equation 5(C) 2009, SNU Biointelligence Lab,

Outline of the Chapter Focus on a class of system that undergoes departures from equilibrium that are linearly related to the perturbations producing the departures Explain the approach to equilibrium in such systems with a derivation of the Fokker-Plank equation (7.3) Introduce Langevin equation for brownian motion (7.4)  Derive its formal solution for the temporal evolution of the velocities  Calculate the mean square displacement of the brownian particle  Two distinct time scale: short-time inertial regime & long-time diffusive regime  Examine the properties of the random force and establish a connection between its properties and those of the system at equilibrium 6(C) 2009, SNU Biointelligence Lab,

Outline of the Chapter Stochastic differential equation (7.5)  Characterize the wiener process as they appear in stochastic differential equation  Provide an interpretative framework that helps us understand the operation of Langevin diffusion method for global optimization Further exploration of the physical principles governing the approach to equilibrium as illustrated by brownian motion (7.6~7.8) 7(C) 2009, SNU Biointelligence Lab,

The Fokker-Planck Equation In Einstein’s theory of brownian motion  τ : small time interval (small compared to observation times but large enough that movements in two successive time interval can be considered to be statistically independent events)  p(x, t+τ): Prob. that the particle is at position x, at time t+τ it depends only on the position at the immediate preceding time t and not on its position at any other earlier time.  The prob. obey the diffusion equation where D is the diffusion coefficient  Diffusion equation is a special case of a class of equations, termed Fokker-Planck equations, that describes systems evolving along continuous paths in sample space 8(C) 2009, SNU Biointelligence Lab,

The Fokker-Planck Equation Fokker-Planck Equation for Brownian Motion  Notation  p(v, t): prob. that a particle has a velocity between v and v+dv at time t.  Modeling a particle as a Markovian process in which the p(v, t) depends on the velocity v o of particle at an earlier time t 0  p(v, t | v 0, t 0 )  General situation  A particle starts at time t a, with velocity v a, at time t’, passes through some intermediate velocity v’, and ends up at time t b with velocity v b  Assume that the conditional prob. are functions of the time differences  9(C) 2009, SNU Biointelligence Lab,

The Fokker-Planck Equation Fokker-Planck Equation for Brownian Motion (cont.)  Assume that the motion occurs through a sequence of small time steps  Expand the left-hand side in a Taylor’s series about τ b small  ( ∵ Smoluchowski equation)  Define Δ = v’ – v b  Assume that in a small time interval the velocity can only change by a small amount (the prob. is appropriate when Δ is small)  Expand the integrand in a Taylor’s series in Δ about the value retaining only the lowest-order terms The resulting expression is 10(C) 2009, SNU Biointelligence Lab,

The Fokker-Planck Equation Fokker-Planck Equation for Brownian Motion (cont.)  The first term (n=0) in the expansion is  Next two terms in the expansion  : drift coefficient (in 1D: drift vector)  : diffusion coefficient (in 1D: diffusion matrix)  And the expansion assumes the form  (1D form of Fokker-Planck equation) 11(C) 2009, SNU Biointelligence Lab,

The Langevin Equation Langevin’s treatment of brownian motion  Equation of motion for the brownian particle in contact with the heat bath  Decompose the forces exerted on the particle and giving rise to the irregular motion into two terms  Slowly varying force operating on a long time scale  velocity-dependent frictional force: -αdx/dt (where, friction coefficient α=6 πηr 0, η : viscosity, r 0 : particle radius)  Rapidly varying fluctuating force F(t) acting on a short time scale  Langevin equation for the velocity v:  Frictional force appears with a minus sign  operates in a manner that restores the velocities to their equilibrium values whenever the brownian particles are perturbed.  Fluctuating force F(t) is completely random; its mean value vanishes independent of position and velocity, and the forces at different times are uncorrelated 12(C) 2009, SNU Biointelligence Lab,

The Langevin Equation Formal Solution  General form of Langevin equation:  Homogeneous equation obtained by setting q(t)=0 with solution where y 0 is a constant of integration  Define so that ye A =y 0  Differentiate the quantity ye A  Replace y’+py with q, can be immediately integrated to give the general solution  Solution of Langevin equation (A=-(α/m)t and q(t)=F(t)) 13(C) 2009, SNU Biointelligence Lab,

The Langevin Equation The Mean Square Displacement  Calculate the mean square displacement of a particle operated on by the two forces  over time the particle executes a diffusive random walk  Rewrite the Langevin equation in terms of positions  Multiply both sides by x   Consider the mean behavior of a large number of particles  Since the mean value of the fluctuating force vanishes  Assume that the brownian particles are in thermal equilibrium with their surroundings, namely with a heat bath, then they must have mean energy given by the equipartition theorem of thermodynamics  For motion in 1D, the equipartition theorem states 14(C) 2009, SNU Biointelligence Lab,

The Langevin Equation The Mean Square Displacement (Cont.)  Upon making these substitutions,  A simple, linear, first-order, inhomogeneous differential equation of the form give by for the variable  Since in terms of this variable  Specialize our solution of Langevin equation to the mean square displacement by setting  Then we have 15(C) 2009, SNU Biointelligence Lab,

The Langevin Equation The Mean Square Displacement (Cont.)  To evaluate the constant of integration, we require that y=0 at t=0  C = -k B T/α   Integration of above  where the constant of integration has been evaluated by setting the mean square position to zero at time t=0  Considering above eq. in the limiting cases of short and long time intervals relative to the characteristic time m/ α,  16(C) 2009, SNU Biointelligence Lab,

The Langevin Equation The Mean Square Displacement (Cont.)  Over a short time interval the brownian particles move inertially with constant thermal velocity (k B T/m) 1/2  For large times the particles move diffusively, and since they undergo a sequence of random collisions, they do not translate as far in a given time interval as they would if moving freely  Utilization of the mean square displacement (by Perrin)  Values for the mean square displacement can be utilized together with values for the radius r 0 of the brownian particles, and the viscosity η of the medium (recall that α=6 πηr 0 ) to deduce values for Boltzmann’s constant K B and given an estimate of the gas constant R, Avogardo’s number. 17(C) 2009, SNU Biointelligence Lab,

The Langevin Equation Properties of the Random Force  The random force term appearing in the Langevin equation is needed to maintain the irregular motion of the Brownian particle  Consider a equation of motion which in the absence of a random term only contains a fore derived from a potential V(v):  A particle moving under the influence of this type of force will settle into a position corresponding to a local minimum of potential associated with the force –Contrary to the correct situation described by a Maxwellian (Gibbs) distribution of velocities of the form exp(-V(v)/k B T)  The random force in the Langevin equation prevents the particle from setting into a minimum position 18(C) 2009, SNU Biointelligence Lab,

The Langevin Equation Properties of the Random Force (Cont.)  Random force in uncorrelated with the position and velocity of a brownian particle, and its values at different times are uncorrelated  where κ is a strength parameter  Time interval t-t’ should be large compared to the time between individual collisions of the brownian particles with the molecules of the fluid 19(C) 2009, SNU Biointelligence Lab,

The Langevin Equation Properties of the Random Force (Cont.)  Exploring the relationship between the random force and the evolution of the system toward equilibrium by making use of our solution to Langevin equation  If the p. d. for the velocities v-v 0 e - α t/m approaches a Maxwellian distribution, then according to the solution of Lagevin equation the p. d. for the quantities must be Maxwellian as well.  Considering small time interval τ during which all physical variables except the fluctuating forces are approximately constant  The net acceleration experienced by a brownian particle during this time interval is given by  Then it can be shown that the distribution of velocities v-v 0 e - α t/m will be Maxwellian provided that the accelerations A(τ) are Gaussian distribution 20(C) 2009, SNU Biointelligence Lab,

The Langevin Equation Properties of the Random Force (Cont.)  In more detail, if the accelerations are Gaussian distributed with Zero mean and variance 2qτ according to where q = α k B T/m 2, then the velocities are governed by the p.d.  and therefore which is the Maxwell velocity distribution  If we take τ to be arbitrarily small, we may also consider the force F(t) to be Gaussian distributed with zero mean and with correlation given by with k=2q 21(C) 2009, SNU Biointelligence Lab,