Canonical Ensemble Homework: Equal Probabilities Postulate

[ehrenfest]

Homework Statement

In the canonical ensemble, the probability that a system is in state r is given by

P_i = \frac{g_i \exp (-\beta E_i)}{\sum_i g_i \exp( -\beta E_i)}

where g_i is the multiplicity of state i. This is confusing me because I thought

P_i = \frac{g_i}{\sum_i g_i} = states consistent with i / total number of states

was always true by the equal probabilities postulate. What am I missing? Are those two expressions the same?

Homework Equations

The Attempt at a Solution

 

[kdv]

Homework Statement

In the canonical ensemble, the probability that a system is in state r is given by

P_i = \frac{g_i \exp (-\beta E_i)}{\sum_i g_i \exp( -\beta E_i)}

where g_i is the multiplicity of state i. This is confusing me because I thought

P_i = \frac{g_i}{\sum_i g_i} = states consistent with i / total number of states

was always true by the equal probabilities postulate. What am I missing? Are those two expressions the same?

Homework Equations

The Attempt at a Solution

The basic idea of stat mech is that the configurations of different energies are not equally likely. The higher the energy of a state is, the least likely the system will be in that state. This is reflected in the Boltzmann distribution you cite at the top. Your second equation would be valid if all states (irrespective of their energy) would be equally probable.