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zi-lao-lan
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Homework Statement
Show how the Boltzmann entropy is derived from the Gibbs entropy for systems in equilibrium.
Homework Equations
Gibbs entropy S= - [tex]\int[/tex] [tex]\rho[/tex](p,q) (ln [tex]\rho[/tex](p,q)) dpdq
where [tex]\rho[/tex](p,q) is the probability distribution
Boltzmann entropy S= ln[tex]\Omega[/tex]
where [tex]\Omega[/tex] is the number of microstates in a given macrostate.
The Attempt at a Solution
1. Well, when the system is in equilibrium (ie when the Boltzmann entropy can be used) all microstates have equal probability. So this means that each microstate has a probability of 1/[tex]\Omega[/tex] and the probability distribution [tex]\rho[/tex] will have a constant value regardless of what p and q are.
2. I tried putting [tex]\rho[/tex]=1/[tex]\Omega[/tex] and subbing it into the Gibb's equation
S= - [tex]\int[/tex] 1/[tex]\Omega[/tex] (ln [tex]\1/[tex]\Omega[/tex]) d[tex]\Omega[/tex]
using d[tex]\Omega[/tex] since we want to add up over all the microstates and there are
[tex]\Omega[/tex] of them. But I can see that this won't give me the Boltzmann entropy.
Any ideas?