- #1
Abigale
- 56
- 0
Hi,
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!
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!