The microcanonical partition function is the density of states. The canonical one, from a dimensional point of view, is still a number of states, but does it actually count microstates?

I tried figuring it out from this common derivation: the heat bath B has a density of states deducible from the energy E of the system in contact with it, so the probability of finding the system at energy E is

$P(E) \propto \Omega_B(E_{TOT} - E)$

In the thermodynamic limit the Taylor expansion around E=0 of the log of this has negligible second-order terms:

$\log \Omega_B(E_{TOT} - E) \sim \Omega_B(E_{TOT}) -E/(KT)$

and going back:

$\Omega_B(E_{TOT} - E) \sim \Omega_B(E_{TOT})e^{-E/KT}$

thus the result is

$P(E) \propto e^{-E/KT}$

But, while the first equation I wrote has the clear meaning of counting microstates (bar a constant), I'm not sure the manipulation done to it still retains this meaning in the last equation I wrote. Thus I don't know the same about the caonical partition function. Thanks.


What you have written are all correct. But we should note that $P(E)$ alone is not the partition function. The canonical partition function is $$ Z \propto \Omega_\mathrm{tot}(E_\mathrm{tot}),\qquad (1) $$ which counts the microstates of the univserse, i.e., system and bath. But $P(E)$ only counts the microstates of the bath!

Let us see how to use the definition to continue your argument.

First, since the system and body are weakly interacting, they can be treated as independent. Thus the number of joint microstates with the system energy being $E$ and bath energy being $E_\mathrm{tot} - E$ is given by $$ \Omega_\mathrm{tot}(E_\mathrm{tot}, E) \approx \Omega_\mathrm{sys}(E) \, \Omega_B(E_\mathrm{tot} - E) $$

Now we wish to count all microstates no matter the value of the system energy $E$. So $$ \begin{aligned} Z &\propto \Omega_\mathrm{tot}(E_\mathrm{tot}) \\ &=\int_0^{E_\mathrm{tot}} \Omega_\mathrm{tot}(E_\mathrm{tot}, E) \, dE\\ &\approx \int_0^{E_\mathrm{tot}} \Omega_\mathrm{sys}(E) \, \Omega_B(E_\mathrm{tot} - E) \, dE. \end{aligned} $$ Now using your result, $$ \Omega_B(E_\mathrm{tot} - E) = \Omega_B(E_\mathrm{tot}) \, e^{-E/(KT)} \propto e^{-E/(KT)} , $$ we get $$ \begin{aligned} Z &\propto \Omega_\mathrm{tot}(E_\mathrm{tot}) \\ &\propto \int_0^{E_\mathrm{tot}} \Omega_\mathrm{sys}(E) \, e^{-E/(KT)} \, dE. \end{aligned} $$ This is indeed the usual definition of the canonical partition function.

  • 1
    $\begingroup$ If the equation (1) had an equal sign it would be the microcanonical partition function of the universe system. So if we keep the proportionality factors we could say that, in the thermodynamic limit, the canonical partition function is counting the number of total states of the universe system divided by the number of states the bath would have at total energy: $Z_{microcanonical, universe} = \Omega_{tot}(E_{tot}) = \Omega_B(E_{tot}) Z_{canonical, system}$ $\endgroup$
    – Diri
    Nov 3 '15 at 11:40
  • $\begingroup$ @Diri. Precisely. I cannot put it better myself. $\endgroup$
    – hbp
    Nov 3 '15 at 16:36

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