- #1
nonequilibrium
- 1,439
- 2
Hello,
I thought the statistical definition of entropy for an isolated system of energy E (i.e. microcanonical ensemble) was [itex]S=k \ln \Omega[/itex] where [itex]\Omega[/itex] is the volume in phase space of all the microstates with energy E.
However, if you take a look here http://en.wikipedia.org/wiki/Equipartition_theorem#The_microcanonical_ensemble
there is the line
so they use the volume in phase space where energy < E instead of the surface where energy = E. Do these notions coincide? I would think they'd conflict. Why do they say "by the usual definitions", I'm confused.
I thought the statistical definition of entropy for an isolated system of energy E (i.e. microcanonical ensemble) was [itex]S=k \ln \Omega[/itex] where [itex]\Omega[/itex] is the volume in phase space of all the microstates with energy E.
However, if you take a look here http://en.wikipedia.org/wiki/Equipartition_theorem#The_microcanonical_ensemble
there is the line
[tex]\textrm{... Similarly, $\Sigma(E)$ is defined to be the total volume of phase space where the energy is less than $E$ ...}[/tex] [tex]\textrm{By the usual definitions of statistical mechanics, the entropy $S$ equals $k_B \log \Sigma(E)$ ...}[/tex]
so they use the volume in phase space where energy < E instead of the surface where energy = E. Do these notions coincide? I would think they'd conflict. Why do they say "by the usual definitions", I'm confused.