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Contents 1 Introduction 1 2 Dynamics under the Influence of Stochastic Forces 3 2.1 Newton's Equation and Langevin's Equation . . . 3 2.2 Stochastic Differential Equations .4 2.3 How to Describe Noise . . . . 5 2.4 Ito calculus . . . . . . . . . . . . . 21 2.5 Fokker-Planck Equations . . . . . . . . . . . . . . 29 2.6 Stratonovich Calculus . . . . . . . . . . 31 2.7 Appendix: Normal Distribution Approximation . . . 34 2.7.1 Stirling's Formula . .. . . . . 34 2.7.2 Binomial Distribution . . . .. . . 34 3 Einstein Diffusion Equation 37 3.1 Derivation and Boundary Conditions . . . . . . . . 37 3.2 Free Diffusion in One-dimensional Half-Space . . 40 3.3 Fluorescence Microphotolysis . . . . . . . . . 44 3.4 Free Diffusion around a Spherical Object . . . . . . 48 3.5 Free Diffusion in a Finite Domain . . . . . . . . . . .57 3.6 Rotational Diffusion . . . . . . . . . 60 4 Smoluchowski Diffusion Equation 63 4.1 Derivation of the Smoluchoswki Diffusion Equation for Potential Fields . . . . . . . 64 4.2 One-Dimensional Diffusion in a Linear Potential . . . . . . . . . . . . . . . . . . . . . 67 4.2.1 Di usion in an infinite space 67 4.2.2 Difusion in a Half-Space 70 4.3 Diffusion in a One-Dimensional Harmonic Potential . 74 5 Random Numbers 79 5.1 Randomness . . . . . .. 80 5.2 Random Number Generators . . . . . . . . 83 5.2.1 Homogeneous Distribution . . . . . 83 5.2.2 Gaussian Distribution . . . 86 5.3 Monte Carlo integration . . . . . .. . . 88 i ii CONTENTS 6 Brownian Dynamics 91 6.1 Discretization of Time . . . . 91 6.2 Monte Carlo Integration of Stochastic Processes . . . . . . . . . . . . . . . . . . . . . 93 6.3 Ito Calculus and Brownian Dynamics . . . . . 95 6.4 Free Diffusion . .. . . . . . . . 96 6.5 Reflective Boundary Conditions . . .100 7 The Brownian Dynamics Method Applied 103 7.1 Diffusion in a Linear Potential . 103 7.2 Diffusion in a Harmonic Potential . . . 104 7.3 Harmonic Potential with a Reactive Center . . . . . 107 7.4 Free Diffusion in a Finite Domain . . . . 107 7.5 Hysteresis in a Harmonic Potential . . . 108 7.6 Hysteresis in a Bistable Potential . . . 112 8 Noise-Induced Limit Cycles 119 8.1 The Bonhoefer--van der Pol equations . . . . . . . . 119 8.2 Analysis . 121 8.2.1 Derivation of Canonical Model . . 121 8.2.2 Linear Analysis of Canonical Model . . . 122 8.2.3 Hopf Bifurcation Analysis . . . . 124 8.2.4 Systems of Coupled Bonhoeffer{van der Pol Neurons . . 126 8.3 Alternative Neuron Models .. . . . 128 8.3.1 Standard Oscillators . . 128 8.3.2 Active Rotators . . . . . . . 129 8.3.3 Integrate-and-Fire Neurons . . . 129 8.3.4 Conclusions . .. . . . . . . . . . 130 9 Adjoint Smoluchowski Equation 131 9.1 The Adjoint Smoluchowski Equation . . . . . 131 9.2 Correlation Functions . . . .. . . . . . . 135 10 Rates of Diffusion-Controlled Reactions 137 10.1 Relative Diffusion of two Free Particles . . . . . 137 10.2 Diffusion-Controlled Reactions under Stationary Conditions . 139 10.2.1 Examples . . . . . . . . . . . . 141 11 Ohmic Resistance and Irreversible Work 143 12 Smoluchowski Equation for Potentials: Extremum Principle and Spectral Expansion 145 12.1 Minimum Principle for the Smoluchowski Equation . . . . 146 12.2 Similarity to Self-Adjoint Operator . . 149 12.3 Eigenfunctions and Eigenvalues of the Smoluchowski Operator . 151 12.4 Brownian Oscillator . . . . . . . . . . . . 155 13 The Brownian Oscillator 161 13.1 One-Dimensional Diffusion in a Harmonic Potential . . 162 http://www.ks.uiuc.edu/Services/Class/PHYS498/LectureNotes.html http://www.ks.uiuc.edu/Services/Class/NSM.pdf

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