(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a thermal system at temperature T where the probability of finding the system

in a microstate r with energy Er is given by an arbitrary probability distribution pr that is

normalised so that Sum(pr) = 1.

Let kB denote Boltzmann’s constant and consider the Boltzmann distribution

[tex]p^{B}_{r}= \frac{e^{-\beta E_{r}}}{Z}[/tex]

where Z is the partition function and beta = 1/kB.T

Now we want to compare an arbitrary probability distribution

[tex]p_{r}[/tex] with the Boltzmann distribution [tex]p^{B}_{r}[/tex]. Let

[tex]S = - k_{B} \sum p_{r} ln p_{r}[/tex] denote the entropy of the system characterised by an arbitrary probability distribution and

[tex]S^{B} = - k_{B} \sum p^{B}_{r} ln p^{B}_{r}[/tex]

denote the entropy of the system characterized by the Boltzmann distribution. By adding and subtracting

[tex]k_{B} \sum p_{r} ln p^{B}_{r}[/tex]

we can write:

[tex]S - S^{B} = k_{B}\sum (-p_{r} ln p_{r} + p_{r} ln p^{B}_{r} - p_{r} ln p^{B}_{r} + p^{B}_{r} ln p^{B}_{r} )[/tex]

Assuming that the two probability distributions pr and pBr yield the same mean energy <E>, show that

[tex]S - S^{B} = k_{B} \sum p_{r} ln \frac{p^{B}_{r}}{p_{r}} [/tex]

2. Relevant equations

[tex]<E> = \sum p_{r} E_{r} [/tex]

3. The attempt at a solution

[tex] <E> = \sum p_{r} E_{r} = \frac{1}{Z} \sum E_{r} e^{-\beta E_{r}}[/tex]

I was thinking you can cancel the summation and conclude the arbitrary distribution = the boltzmann distribution? I would then use this result and show the last two parts of the S - SB expression = 0.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Statistical Physics

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**