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:
∂P∂t=LHCP
where
LHC=∑i∑j≠iWR∂∂ϵi[1Tj−1Ti+KB(∂∂ϵi−∂∂ϵj)]Kij
LHC 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
∂P∂t=0.
The expression is shown below:
ρmic(ϵi,...,ϵN)=1Ω(Eo,N)ek−1B∑is(ϵi)σ(∑i(ϵi)−Eo)
ρcan(ϵi,...,ϵN)=1Z(β,N)ek−1B∑i(s(ϵi)−βϵi)
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:
dϵi=∑jKijω(rij)(1Ti−1Tj)dt+∑j√kBαij¯ω(rij)dWϵij.
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
∂P∂t=LHCP
where
LHC=∑i∑j≠iWR∂∂ϵi[1Tj−1Ti+KB(∂∂ϵi−∂∂ϵj)]Kij.
Here Wr is the lucy weighting function with some constant coefficient. ϵi is the internal energy of particle i. Ti is the temperature of particle i. Kij 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. KB is Boltzmann's constant.
We are also given expressions for the EOS & entropy of a perfect solid
EOS: T(ϵ)=ϵ/Cv where T(ϵ) 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
Kij=CV¯Kλ2T2(ϵi+ϵj2)=CV¯K4λ2(Ti+Tj)2
¯K is a constant with units of a diffusion coefficient (length2/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.
[∂∂ϵi−∂∂ϵj]Kij=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:
ρmic(ϵi,...,ϵN)=1Ω(Eo,N)ek−1B∑is(ϵi)σ(∑i(ϵi)−Eo)
ρcan(ϵi,...,ϵN)=1Z(β,N)ek−1B∑i(s(ϵi)−βϵi)
Note that β=1kBT.
They provide definitions for the partition functions shown in the image...
Ω(Eo,N)=∫Eo0dϵ1...dϵNek−1B∑is(ϵi)σ(∑iϵi−Eo)
Z(β,N)=zN(β) & z(β)=∫∞0dϵek−1Bs(ϵ)−βϵ
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.