Derivation of the Boltzmann Distribution

The Boltzmann distribution describes the probability of a system being in a particular energy state at thermal equilibrium. It arises from statistical mechanics by maximizing entropy subject to constraints.


1. Entropy and Probability Distribution

The entropy of a system is defined as:

where:


2. Constraints

Two constraints are imposed:

  1. Probabilities must sum to 1:

This ensures the probabilities form a valid distribution. Without this, the solution wouldn’t represent meaningful probabilities for the system's states.

  1. 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 , maximize the entropy under the constraints using the method of Lagrange multipliers. Define the function:

where and are Lagrange multipliers.


4. Solving the Optimization Problem

Take the derivative of with respect to :

Set :

Simplify to express :

Exponentiate both sides:

Define , known as the partition function:

Thus:


5. Interpretation of Parameters


6. Final Result

The Boltzmann distribution is:

This describes the probability of the system occupying the energy state at temperature .