1. 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

2. (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
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3. (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.
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4. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Annealing as MCMC
(Pincus, 1968)
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5. Brownian Motions (Robert Brown, 1827)
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6. 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, 

7. 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
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8. 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)
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9. 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
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10. 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
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11. (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.
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12. (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
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13. 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)
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14. 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
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15. Regression of spontaneous fluctuations
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