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.

# Summary

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?

# Detail

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.