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i regard a Brownian Particle connectet to a Spring and there is a heat-reservoir.

The distribution of the x-coordinate of the particle follows the Diffusion-Equation (Fokker-Planck-Equation):

[itex]\partial_{t}P(x,t)=\frac{D}{2}

\partial_{x}^{2}P(x,t)-

\Gamma\partial_{x}[f(x)P(x,t)]

[/itex]

A deterministic Force is given by [itex]f(x)=-\frac{d}{dx}U(x) [/itex].

Whereby

[itex]

U(x)=

\frac{1}{2}(x-x_{0})^{2}

[/itex] is a Potential.

Also i know that the equilibrium-equation is a Gaussian-Function.

[itex]

P_{eq}=(\frac{\beta}{2\pi})^{1/2}\exp[{-\frac{\beta}{2}(x-x_{0})^2}]

[/itex]

I want to determine the Expected Value of the (potential) internal Energy.

But i don't know how I can get it. ;-(

Please help me ;) and

thank you a lot!!!

Bye Abigale

Sorry for my bad english!

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# Brownian Particle bound by a Spring / internal Energy

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