Derivation of the Boltzmann Distribution
The Boltzmann distribution describes the probability of a system being in a particular energy state
1. Entropy and Probability Distribution
The entropy
where:
is Boltzmann's constant, is the probability of the system being in state , - The sum is over all possible states
.
2. Constraints
Two constraints are imposed:
- Probabilities must sum to 1:
This ensures the probabilities
- Energy conservation:where
is the energy of state . This constraint ensures that while individual particles may exchange energy, the total energy across all particles in the system remains constant.
3. Maximizing Entropy
To find the probability distribution
where
4. Solving the Optimization Problem
Take the derivative of
Set
Simplify to express
Exponentiate both sides:
Define
Thus:
5. Interpretation of Parameters
is related to the temperature by (in natural units). - The partition function
ensures normalization of probabilities:
6. Final Result
The Boltzmann distribution is:
This describes the probability of the system occupying the energy state