Particle in a 1-d potential/ Fokker-Planck(FPE)

[Abigale]

Hi guys,
I have a Problem.
I regard a brownian particle in a potential $U(z)=gz$ with $f=-\frac{d}{dz}U$ and it is connected to a heat reservoir.
The particle is in the interval $z>0$.

I regard the equilibrium-state.
So I got the FPE.

$$
\frac{\partial}{\partial t} P_{1}(z,t)
=
-\frac{\partial}{\partial z}[\Gamma~f(z)P_{1}(z,t)]
+
\frac{D}{2}
\frac{\partial^{2}}{\partial z^{2}} P_{1}(z,t)
=0
$$

$$
\Rightarrow -\Gamma~f(z)P_{1}(z,t)
+
\frac{D}{2}
\frac{\partial}{\partial z} P_{1}(z,t)
=0
$$

The solution is:
$$

P_{eq}(z,t)=\frac{2\Gamma g}{D}\exp{\lbrace\frac{-2\Gamma g z}{D} \rbrace}
$$

Please check if the following callculations are correct?

Now I regard the situation:

At a certain time $t_{0}$ opens a door, which increases the first volume.
The volume ranges now from $-z_{0}$ to $+ \infty$. ($z_{0}>0$)
But I am just interested in the equilibrium state of this situtaion.

I think the FPE must have the same form like the first case. So again:

$$
\frac{\partial}{\partial t} P_{2}(z,t)
=
-\frac{\partial}{\partial z}[\Gamma~f(z)P_{2}(z,t)]
+
\frac{D}{2}
\frac{\partial^{2}}{\partial z^{2}} P_{2}(z,t)
=0
$$

$$
\Rightarrow -\Gamma~f(z)P_{2}(z,t)
+
\frac{D}{2}
\frac{\partial}{\partial z} P_{2}(z,t)
=0
$$

$$
\Rightarrow \Gamma~gP_{2}(z,t)
+
\frac{D}{2}
\frac{\partial}{\partial z} P_{2}(z,t)
=0
$$

$$
\Rightarrow \Gamma~g dz
=
-
\frac{D}{2}
\frac{1}{P_{2}(z,t)} d(P_{2}(z,t))

$$
This Solution leads to:
$$
P_{2}=C\exp{\lbrace \frac{-2\Gamma g z}{D} \rbrace}
$$

Now for normalisation:
$$
\int \limits_{-z_{0}}^ {+ \infty} P_{2}(z,t) ~dz =1
$$

Can this be true? Is the range of integration, the only thing i have to change for this situation?

THX
and sorry for the long letter.
Bye
Abby

 

[jvicens]

Hi Abby,

Thank you for reaching out with your question. Your calculations seem to be correct. When the door opens at time t0, the equilibrium state of the system will change, but the form of the Fokker-Planck equation will remain the same. Therefore, your solution for P2 is also correct. And yes, you are correct that the only thing that needs to be changed for this situation is the range of integration. In the first case, the range is z>0, while in the second case, it is -z0 to +∞.

I hope this helps. Let me know if you have any other questions or concerns. Good luck with your research!