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  Obtaining the canonical distribution from Fokker-Planck equation?

+ 6 like - 0 dislike

First I will provide a summary of the problem. Subsequently, I will provide more detail regarding the problem. Please note that entropy is in units of the Boltzmann constant.


I have a Fokker-Planck equation which I have derived from the internal energy EOS. The Fokker-Planck equation describes the distribution of the internal energy in the mesoscopic system. The reason it only describes internal energy is because the system is a perfect solid. Therefore, there is no momentum EOM: nothing moves. The expression is shown below:

$$\frac{\partial P}{\partial t} = L^{HC}P$$ where $$L^{HC} = \sum_i \sum_{j \neq i} W_R \frac{\partial}{\partial \epsilon_i} \left[ \frac{1}{T_j} - \frac{1}{T_i} + K_B \left( \frac{\partial}{\partial \epsilon_i} - \frac{\partial}{\partial \epsilon_j}\right) \right] K_{ij} $$ $L^{HC}$ is the heat conduction operator. It is claimed that the canonical & microcanonical equations for statistical weight can be obtained as equilibrium solutions to the Fokker-Planck equation shown above. By 'equilibrium solution' it is meant that one should set the change in the statistical weight with respect to time to zero. I.E $$\frac{\partial P}{\partial t}=0.$$ The expression is shown below:

$$\rho_{mic}(\epsilon_i,..., \epsilon_N) = \frac{1}{\Omega(E_o, N)}e^{k_B^{-1}\sum_is(\epsilon_i)}\sigma(\sum_i(\epsilon_i)-E_o)$$

$$\rho_{can}(\epsilon_i,..., \epsilon_N) = \frac{1}{Z(\beta, N)}e^{k_B^{-1}\sum_i(s(\epsilon_i)-\beta\epsilon_i)}$$

How can I obtain these expressions?


I have a Fokker-Planck equation which I have obtained for a 'perfect solid'. That is all the degrees of freedom are internal to the system. As such, I have only an equation of motion for internal energy shown below:

$$d\epsilon_i=\sum_j K_{ij}\omega(r_{ij})(\frac{1}{T_i}-\frac{1}{T_j})dt+\sum_j \sqrt{k_B}\alpha_{ij}\overline\omega(r_{ij})dW_{ij}^{\epsilon} \, .$$

I won't define terms at this point, because this equation isn't relevant to the computations. I will, however, note that the first term is due to conduction between mesoscopic particles in the system of interest; the second is due to thermal fluctuations. There is an equivalent Fokker Planck equation

$$\frac{\partial P}{\partial t} = L^{HC}P$$ where $$L^{HC} = \sum_i \sum_{j \neq i} W_R \frac{\partial}{\partial \epsilon_i} \left[ \frac{1}{T_j} - \frac{1}{T_i} + K_B \left( \frac{\partial}{\partial \epsilon_i} - \frac{\partial}{\partial \epsilon_j}\right) \right] K_{ij} \, . $$

Here Wr is the lucy weighting function with some constant coefficient. $\epsilon_i$ is the internal energy of particle i. $T_i$ is the temperature of particle i. $K_{ij}$ is the the coefficient for the generalized driving force of internal energy exchange. It can be thought of as similar to thermal conductivity, except the driving force is inverse temperature. $K_B$ is Boltzmann's constant.

We are also given expressions for the EOS & entropy of a perfect solid

EOS: $T(\epsilon) = \epsilon/C_v$ where $T(\epsilon)$ is read as 'T is a function of epsilon'.

Entropy: s(ϵ) = Cv ln(ϵ)

We are given an expression for the faux-thermal Conductivity

$K_{ij}=\frac{C_V \overline K} {\lambda^2} T^2 (\frac{\epsilon_i+\epsilon_j}{2})=\frac{C_V \overline K} {4\lambda^2}(T_i+T_j)^2$

$\overline K$ is a constant with units of a diffusion coefficient $(length^2/time)$. Lambda is the lattice spacing or $(n^{-0.333} = (N/V)^{-0.333})$ where N is the # of particles and V is volume. The following relation can also be derived.

$[\frac{\partial} {\partial \epsilon_i} - \frac{\partial} {\partial \epsilon_j}]K_{ij}=0$

Using the Fokker-Planck equation for the distribution of internal energy (which I presented earlier) I am supposed to be able to obtain these expressions for the canonical & microcanonical ensembles shown below:

$\rho_{mic}(\epsilon_i,..., \epsilon_N) = \frac{1}{\Omega(E_o, N)}e^{k_B^{-1}\sum_is(\epsilon_i)}\sigma(\sum_i(\epsilon_i)-E_o$)

$\rho_{can}(\epsilon_i,..., \epsilon_N) = \frac{1}{Z(\beta, N)}e^{k_B^{-1}\sum_i(s(\epsilon_i)-\beta\epsilon_i)}$

Note that $\beta = \frac{1}{k_B T}$.

They provide definitions for the partition functions shown in the image...

$\Omega(E_o, N) = \int_0 ^{E_o}d\epsilon_1...d\epsilon_N e^{k_B^{-1}\sum_i s(\epsilon_i)}\sigma(\sum_i \epsilon_i-E_o)$

$Z(\beta, N) = z^N (\beta)$ & $z(\beta)=\int_0^{\infty} d\epsilon e^{k_B^{-1}s(\epsilon)-\beta\epsilon}$

Any help would be appreciated. Apologies for the silly formatting. I am new to Stack Exchange.

Source: I couldn't find a non-institutional access version of the paper. the paper which I was using has the title "Heat conduction modelling with energy conservation dissipative particle dynamics" by Marisol Ripoll & Pep Espanol. However, I did find a paper which covers exactly the same material. It has nearly all of the same equations & doesn't require institutional access. The link is provided below:

Link to the piece for more information

This post imported from StackExchange Physics at 2016-02-10 14:11 (UTC), posted by SE-user Nick L.

asked Jun 15, 2015 in Theoretical Physics by Nick L. (30 points) [ revision history ]
edited Feb 10, 2016 by Dilaton
Please type the equations instead of pasting images.

This post imported from StackExchange Physics at 2016-02-10 14:11 (UTC), posted by SE-user DanielSank
@DanielSank I would be happy work out how & to rewrite them. The only reason I didn't before is because I don't know how on this site. However, it doesn't look like I'm really going to get a response at this point.

This post imported from StackExchange Physics at 2016-02-10 14:11 (UTC), posted by SE-user Nick L.
Don't give up yet on getting a response. This is a long post and that usually means just the right person has to see it before you get a good answer. You can even ask someone to put a bounty on it for you!

This post imported from StackExchange Physics at 2016-02-10 14:11 (UTC), posted by SE-user DanielSank
I reworked the first equation to use the site's mathjax feature. Mathjax is basically TeX support in the browser. If you're familiar with TeX already then you can just go ahead and use it! If not you can find lots of great tutorials and references online, or you can hit the "edit" button on your post and see how I did the first equation.

This post imported from StackExchange Physics at 2016-02-10 14:11 (UTC), posted by SE-user DanielSank
@DanielSank Thanks for the helpful suggestions and for reworking that first equation. I will give the other equations a shot.

This post imported from StackExchange Physics at 2016-02-10 14:11 (UTC), posted by SE-user Nick L.

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