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| '''Brownian dynamics (BD)''' can be used to describe the motion of [[molecule]]s in molecular [[simulation]]. It is a simplified version of [[Langevin dynamics]] and corresponds to the limit where no average acceleration takes place during the simulation run. This approximation can also be described as '[[overdamping|overdamped]]' Langevin dynamics, or as Langevin dynamics without [[inertia]].
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| In [[Langevin dynamics]], the equation of motion is
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| :<math>M\ddot{X} = - \nabla U(X) - \gamma M\dot{X} + \sqrt{2 \gamma k_B T M} R(t)</math> | |
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| where <math>U(X)</math> is the particle interaction potential; <math>\nabla</math> is the gradient operator such that <math>- \nabla U(X)</math> is the force calculated from the particle interaction potentials; the dot is a time derivative such that <math>\dot{X}</math> is the velocity and <math>\ddot{X}</math> is the acceleration; T is the temperature, k<sub>B</sub> is Boltzmann's constant; and <math>R(t)</math> is a delta-correlated stationary Gaussian process with zero-mean, satisfying
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| :<math>\left\langle R(t) \right\rangle =0</math>
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| :<math>\left\langle R(t)R(t') \right\rangle = \delta(t-t').</math>
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| In Brownian dynamics, no acceleration is assumed to take place. Thus, the <math>M\ddot{X}(t)</math> term is neglected, and the sum of these terms is zero.
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| :<math>0 = - \nabla U(X) - \gamma M\dot{X}+ \sqrt{2 \gamma k_B T M} R(t)</math>
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| Defining <math>\zeta = \gamma M</math>, and using the [[Einstein relation (kinetic theory)|Einstein relation]], <math>D = k_B T/\zeta</math>, it is often convenient to write the equation as,
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| :<math>\dot{X}(t) = - \nabla U(X)/\zeta + \sqrt{2 D} R(t).</math> | |
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| ==See also==
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| * [[Brownian motion]]
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| * [[Immersed boundary method]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://www.csc.fi/english/research/sciences/chemistry/courses/gmx2004/Berendsen4.pdf/download Course on Brownian and Langevin Dynamics]
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| {{physical-chemistry-stub}}
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| [[Category:Classical mechanics]]
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| [[Category:Statistical mechanics]]
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| [[Category:Dynamical systems]]
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