## Brownian Dynamics at Boundaries and Interfaces: In Physics, Chemistry, and BiologyBrownian dynamics serve as mathematical models for the diffusive motion of microscopic particles of various shapes in gaseous, liquid, or solid environments. The renewed interest in Brownian dynamics is due primarily to their key role in molecular and cellular biophysics: diffusion of ions and molecules is the driver of all life. Brownian dynamics simulations are the numerical realizations of stochastic differential equations that model the functions of biological micro devices such as protein ionic channels of biological membranes, cardiac myocytes, neuronal synapses, and many more. Stochastic differential equations are ubiquitous models in computational physics, chemistry, biophysics, computer science, communications theory, mathematical finance theory, and many other disciplines. Brownian dynamics simulations of the random motion of particles, be it molecules or stock prices, give rise to mathematical problems that neither the kinetic theory of Maxwell and Boltzmann, nor Einstein’s and Langevin’s theories of Brownian motion could predict. This book takes the readers on a journey that starts with the rigorous definition of mathematical Brownian motion, and ends with the explicit solution of a series of complex problems that have immediate applications. It is aimed at applied mathematicians, physicists, theoretical chemists, and physiologists who are interested in modeling, analysis, and simulation of micro devices of microbiology. The book contains exercises and worked out examples throughout. |

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### Contents

1 | |

Chapter
2 Eulers Scheme and Wieners Measure | 35 |

Chapter
3 Brownian Simulation of Langevins | 89 |

Chapter
4 The First Passage Time to a Boundary | 111 |

Chapter
5 Brownian Models of Chemical Reactions in Microdomains | 135 |

Chapter
6 Interfacing at the Stochastic Separatrix | 164 |

### Other editions - View all

Brownian Dynamics at Boundaries and Interfaces: In Physics, Chemistry, and ... Zeev Schuss No preview available - 2013 |

Brownian Dynamics at Boundaries and Interfaces: In Physics, Chemistry, and ... Zeev Schuss No preview available - 2013 |

Brownian Dynamics at Boundaries and Interfaces: In Physics, Chemistry, and ... Zeev Schuss No preview available - 2013 |

### Common terms and phrases

absorbing boundary absorbing window approximation assume asymptotic backward Kolmogorov bath bound boundary conditions boundary layer boundary value problem Brownian dynamics Brownian motion Brownian particle Brownian trajectory calcium calcium ions calculate channel concentration constant convergence defined dendritic spine denote differential equations diffusion coefficient diffusion process distribution domain eigenvalue Euler scheme Exercise exit finite flux density Fokker–Planck equation function given Holcman independent initial condition integral equation interval Itô Langevin equation leading order leading-order Lemma limit membrane MFPT molecules narrow escape problem neck neuronal normal oblique reflection obtain one-dimensional partially reflecting potential probability density Proof proteins radius random reactant recrossings reflecting boundary Robin boundary condition satisfies Schuss Sect separatrix solution spine head stochastic stochastic differential equation stochastic process surface synaptic t)At Theorem trajectories transition variables vector velocity