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Canonical ensemble

  • Thread starter ehrenfest
  • Start date
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1. Homework Statement
In the canonical ensemble, the probability that a system is in state r is given by

[tex] P_i = \frac{g_i \exp (-\beta E_i)}{\sum_i g_i \exp( -\beta E_i)} [/tex]

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

[tex] P_i = \frac{g_i}{\sum_i g_i} = [/tex] 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?


2. Homework Equations



3. The Attempt at a Solution
 

Answers and Replies

kdv
336
1
1. Homework Statement
In the canonical ensemble, the probability that a system is in state r is given by

[tex] P_i = \frac{g_i \exp (-\beta E_i)}{\sum_i g_i \exp( -\beta E_i)} [/tex]

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

[tex] P_i = \frac{g_i}{\sum_i g_i} = [/tex] 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?


2. Homework Equations



3. 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.
 

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