Biointelligence Laboratory, Seoul National University Ch 7. The Approach to Equilibrium 7.5 ~ 7.9 Adaptive Cooperative Systems, Martin Beckerman, 1997. Summarized by J. Yang Biointelligence Laboratory, Seoul National University http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Contents 7.5 Stochastic Differential Equations 7.5.1 The Wiener Process 7.5.2 Global Optimization 7.6 Fluctuations and Dissipation 7.7 The Regression of Spontaneous Fluctuations 7.7.1 Stationary Processes 7.7.2 The Velocity Correlation Function 7.8 The Velocity Increments 7.8.1 The Mean Incremental Change in Velocity 7.8.2 The Mean Square Incremental Change in Velocity (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Stochastic Process (xt : random variable, T: infinite sequence or interval) Stationary process: the joint distribution of is the same as the joint distribution of for any positive h and k points. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Annealing as MCMC (Pincus, 1968) (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 4

Brownian Motions (Robert Brown, 1827) (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 5

Stochastic Differential Equation Differential equation in which one or more terms is a stochastic process resulting in a solution which is also of a stochastic process. Brownian motion (Wiener process) is non-differentiable and most algorithms for ODE works very poor in SDE SDE needs its own method of calculus: Ito stochastic calculus , Stratonovich stochastic calculus. Examples of SDE Langevin equation (ODE with white noise) Fokker-Planck equation (PDE ) (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

Stochastic Differential Equation SDE in Physics: Langevin equation SDE in probability/financial mathematics: (interpretation) In a small time interval, the stochastic process Xt changes its value by an amount (a r.v.) that is normally distributed with mean and variance and independent of the past behavior of the process. Noise term Wiener process (Brownian motion) Ito integral diffusion process diffusion coefficient drift coefficient (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 7

Describing Brownian motion Langevin equation: by expressing force of irregular Brownian motion into slowly varying force and rapidly fluctuating force Diffusion equation: under Markovian assumptions of Brownian motion, probability of a particle in Brownian motion obey the diffusion equation (Einstein) Cf. Fick’s 2nd law of diffusion (1855) (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 8

Describing Brownian motion Diffusion equation is a special case of the Fokker-Planck equation Probability of the (stationary) Markov process can be described as Fokker-Planck equation (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 9

Describing Brownian motion Diffusion equation is a special case of the Fokker-Planck equation Probability of Markov process can be described as Fokker-Planck equation Drift coefficient and diffusion coefficient are computed from the solution of Langevin equation By solving the Fokker-Planck equation with these coefficient values, the probability of the velocity of a particle in Brownian motion at certain time can be obtained as a Gaussian approaching a Maxwellian distribution over time (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 10

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Wiener Process Langevine equation: W(t) is a Markov process (a solution form) satisfy the Fokker-Plank equation Drift coeff. is 0 and diffusion coeff. is 1 (direct computation) The variables are Gaussian distributed with mean w0 and variance t-t0 after solving the above Fokker-Plank equation Markov process of this form are known as Weiner process. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 11

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Global optimization SDE in terms of the Weiner process SDE for global optimization (Note: gradient descent procedure without W) Derivation of the associated Fokker-Plank equation Thus the continuous state chains generated by Langevin diffussions converge to Gibbs distributions as do the Markov chains in MCMC Function to optimize (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 12

Fluctuation-Dissipation theorem There is an explicit relationship between molecular dynamics at thermal equilibrium and the macroscopic response that is observed in a dynamic measurement. (the inverse relation between friction constant and diffusion constant) (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 13

Regression of spontaneous fluctuations For the Fokker-Plank equation Friction coefficient  controls the rate of approach to the equilibrium Greater the fluctuation, faster to the equilibrium Fluctuations are generated by the coupling interactions between the brownian particles and the molecules of heat bath Equilibration will occur more rapidly if the particles are strongly coupled to the heat bath than they are weakly coupled (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 14

Regression of spontaneous fluctuations (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 15