Langevin equation

Langevin equation

http://scai.fraunhofer.de/HuPSON#SCAIVPH_00000123

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Term information

PMID

20836037

definition

In statistical physics, a Langevin equation (Paul Langevin, 1908) is a stochastic differential equation describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation. ... Let A={Ai} denote the slow variables. The generic Langevin equation then reads \frac{dA_{i}}{dt}=k_{B}T\sum\limits_{j}{\left[ {A_{i},A_{j}}\right] \frac{{d}\mathcal{H}}{{dA_{j}}}}-\sum\limits_{j}{\lambda _{i,j}\left( A\right) \frac{d\mathcal{H}}{{dA_{j}}}+}\sum\limits_{j}{\frac{d{\lambda _{i,j}\left(A\right) }}{{dA_{j}}}}+\eta _{i}\left( t\right). source: http://en.wikipedia.org/wiki/Langevin_equation

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\frac{dA_{i}}{dt}=k_{B}T\sum_{j}{[ {A_{i},A_{j}}] \frac{{d}\mathcal{H}}{{dA_{j}}}}-\sum_{j}{\lambda _{i,j}\left( A\right) \frac{d\mathcal{H}}{{dA_{j}}}+}\sum_{j}{\frac{d{\lambda _{i,j}\left(A\right) }}{{dA_{j}}}}+\eta _{i}\left( t\right)

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<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <msub> <mi>A</mi> <mi>i</mi> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <msub> <mi>k</mi> <mi>B</mi> </msub> <mi>T</mi> <munder> <mo>&Sum;</mo> <mi>j</mi> </munder> <mrow> <mfenced close="]" open="["> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>A</mi> <mi>j</mi> </msub> </mrow> </mfenced> <mfrac> <mrow> <mi>d</mi> <mi mathvariant="script">ℋ</mi> </mrow> <mrow> <mi>d</mi> <msub> <mi>A</mi> <mi>j</mi> </msub> </mrow> </mfrac> </mrow> <mo>-</mo> <munder> <mo>&Sum;</mo> <mi>j</mi> </munder> <mrow> <msub> <mi>λ</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mfenced close=")" open="("> <mi>A</mi> </mfenced> <mfrac> <mrow> <mi>d</mi> <mi mathvariant="script">ℋ</mi> </mrow> <mrow> <mi>d</mi> <msub> <mi>A</mi> <mi>j</mi> </msub> </mrow> </mfrac> <mo>+</mo> </mrow> <munder> <mo>&Sum;</mo> <mi>j</mi> </munder> <mfrac> <mrow> <mi>d</mi> <mrow> <msub> <mi>λ</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mfenced close=")" open="("> <mi>A</mi> </mfenced> </mrow> </mrow> <mrow> <mi>d</mi> <msub> <mi>A</mi> <mi>j</mi> </msub> </mrow> </mfrac> <mo>+</mo> <msub> <mi>η</mi> <mi>i</mi> </msub> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> <annotation encoding="SnuggleTeX">\[ \frac{dA_{i}}{dt}=k_{B}T\sum_{j}{[ {A_{i},A_{j}}] \frac{{d}\mathcal{H}}{{dA_{j}}}}-\sum_{j}{\lambda _{i,j}\left( A\right) \frac{d\mathcal{H}}{{dA_{j}}}+}\sum_{j}{\frac{d{\lambda _{i,j}\left(A\right) }}{{dA_{j}}}}+\eta _{i}\left( t\right) \]</annotation> </semantics> </math>

related synonym

Langevin

LD simulation

LD simulations

Langevin Dynamics Simulation

Langevin dynamics

LD propagation

Langevin Dynamics Simulations

Term relations

Subclass of:

differential equation
describes some motion and has_quality some stochastic quality
has_part_equation some Poisson bracket equation and has_variable some (
variable and
is_about some force)