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| In [[physics]], '''Langevin dynamics''' is an approach to the mathematical modeling of the [[dynamics (physics)|dynamics]] of molecular systems, originally developed by the French physicist [[Paul Langevin]]. The approach is characterized by the use of simplified models while accounting for omitted [[Degrees of freedom (physics and chemistry)|degrees of freedom]] by the use of [[stochastic differential equation]]s.
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| A molecular system in the real world is unlikely to be present in vacuum. Jostling of solvent or air molecules causes friction, and the occasional high velocity collision will perturb the system. Langevin dynamics attempts to extend [[molecular dynamics]] to allow for these effects. Also, Langevin dynamics allows controlling the temperature like a thermostat, thus approximating the [[canonical ensemble]].
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| Langevin dynamics mimics the viscous aspect of a solvent. It does not fully model an [[implicit solvent]]; specifically, the model does not account for the [[Poisson-Boltzmann equation|electrostatic screening]] and also not for the [[hydrophobic effect]]. It should also be noted that for denser solvents, hydrodynamic interactions are not captured via '''Langevin dynamics'''.
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| For a system of <math>N</math> particles with masses <math>M</math>, with coordinates <math>X=X(t)</math> that constitute a time-dependent [[random variable]], the resulting [[Langevin equation]] is <ref>{{cite book
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| | first=Tamar | last=Schlick | year=2002 | title=Molecular Modeling and Simulation | publisher=Springer | isbn=0-387-95404-X | page = 480
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| }}</ref>
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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 process|stationary]] [[Gaussian Process|Gaussian process]] with zero-mean, satisfying
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| :<math>\left\langle R(t) \right\rangle =0</math> | |
| :<math>\left\langle R(t)R(t') \right\rangle = \delta(t-t')</math>
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| Here, <math>\delta</math> is the [[Dirac delta]].
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| If the main objective is to control temperature, care should be exercised to use a small damping constant <math>\gamma</math>. As <math>\gamma</math> grows, it spans the inertial all the way to the diffusive ([[Brownian motion|Brownian]]) regime. The Langevin dynamics limit of non-inertia is commonly described as [[Brownian dynamics]].
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| The Langevin equation can be
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| reformulated as a [[Fokker–Planck equation]] that governs the [[probability distribution]] of the random variable ''X''.
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| ==See also==
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| * [[Hamiltonian mechanics]]
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| * [[Molecular dynamics]]
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| * [[Brownian dynamics]]
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| * [[Statistical mechanics]]
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| * [[Implicit solvation]]
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| * [[Stochastic differential equations]]
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| * [[Langevin equation]]
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| ==References==
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| <references />
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| ==External links==
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| * [http://cmm.cit.nih.gov/intro_simulation/node24.html Langevin Dynamics (LD) Simulation]
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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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| [[Category:Symplectic geometry]]
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Secondary School Teacher Merle from Osoyoos, has several interests which include aeromodeling, como ganhar dinheiro na internet and operating on cars. Gets encouragement by making a vacation in Group of Monuments at Mahabalipuram.