boltzmann weight factor and statistical ensembles i am working on a project about in-equivalence between statistical ensembles ( micro-canonical and canonical to be more precise ).
how can we show that the in the canonical ensemble the system is weighed by $ e^{-\beta x} $ ?
Also can any one suggest me some interesting articles on the subject.
 A: There is a really neat way to prove this.
The equilibrium probability distribution $P_{e}$ for the canonical ensemble is that which minimizes the free energy functional $F[P]$, i.e.
$$F[P] \geq F[P_e]$$
using the method of lagrangian multipliers, we impose that the functional derivative of the free energy with the constraint of normalization
$$\int P(C) dC =1$$ 
is zero ($C$=a configuration of the system -we integrate in configuration space-). So, we have to take the functional derivative of
$$F_{\lambda}[P]=E[P]-TS[P]=$$
$$=\int P(C) H(C) dC+ k T \int P(C) \ln P(C) dC+\lambda \left(\int P(C) dC -1\right)$$
We obtain
$$\frac{\delta F_{\lambda}}{\delta P}[P_e]=H(C)+kT(\ln P_e+1)+\lambda=0$$
from which
$$\ln P_e=-\frac{H(C)}{kT}-\frac{\lambda}{kT}-1$$
That is to say,
$$P_e (C) = e^{\frac{-H(C)}{kT}} e^{-1-\frac{\lambda}{kT}}$$
Normalizing we finally obtain the canonical distribution:
$$P_e (C) = \frac{e^{-\beta H(C)}}{\int e^{-\beta H(C)} dC}$$
For an alternative demonstration, try K.Huang: Statistical Mechanics. It goes like this:
Consider a system composed of two sub-systems of hamiltonians $H_1$, $H_2$ and number of particles $N_1$, $N_2$, with $N_1>>N_2$. Consider a microcanonical ensamble of the composite system with energy between $E$ and $E+2\Delta$. This includes a large set of energies, but only one set of values $\bar E_1$, $\bar E_2$ will be important (the values which maximize the entropy). We assume $\bar E_2 >> \bar E_1$. If we are only interested in the state of system 1, we can integrate over the degrees of freedom of system 2, so that the probability density of system 1 verifies
$$P_1(p_1, q_1) \propto \Gamma_2 (E_2) = \Gamma_2 (E-E_1)$$
where $\Gamma(E) = \exp \left(\frac {S(E)} k\right)$. Since only the values $\bar E_1$, $\bar E_2$ are important, we expand $k \ln \Gamma_2 (E_2)$:
$$k \ln  \Gamma_2 (E-E_1) = S_2(E-E_1) \simeq S_2(E)-E_1 \left( \frac{\partial S_2(E_2)}{\partial E_2} \right)_{E_2=E} = S_2(E)-\frac{E_1}{T}$$
where $T$ is the temperature of system 2 (it is now clear that system 2 is a heat reservoir).
Hence 
$$P_1(p_1,q_1) \propto \Gamma_2 (E_2) \propto \exp \left(-\frac{E_1}{kT}\right)$$
Since $E_1 = H_1(p_1,q_1)$, the demonstration is complete.
You can find a slightly different demonstration on M.E.Tuckerman: Statistical Mechanics Theory and Molecular Simulation
