| dbp:proof
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- There are multiple approaches to deriving the partition function. The following derivation follows the more powerful and general information-theoretic Jaynesian maximum entropy approach.
According to the second law of thermodynamics, a system assumes a configuration of maximum entropy at thermodynamic equilibrium. We seek a probability distribution of states that maximizes the discrete Gibbs entropy
subject to two physical constraints:
# The probabilities of all states add to unity :
# In the canonical ensemble, the system is in thermal equilibrium, so the average energy does not change over time; in other words, the average energy is constant :
Applying variational calculus with constraints , we write the Lagrangian as
Varying and extremizing with respect to leads to
Since this equation should hold for any variation , it implies that
Isolating for yields
To obtain , one substitutes the probability into the first constraint:
where is a number defined as the canonical ensemble partition function:
Isolating for yields .
Rewriting in terms of gives
Rewriting in terms of gives
To obtain , we differentiate with respect to the average energy and apply the first law of thermodynamics, :
Thus the canonical partition function becomes
where is defined as the thermodynamic beta. Finally, the probability distribution and entropy are respectively (en)
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