Derivation of the Boltzmann Distribution

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

1. Entropy and Probability Distribution

The entropy $S$ of a system is defined as:

$$S = -k_B \sum_i p_i \ln p_i$$

where:

• $k_B$ is Boltzmann's constant,
• $p_i$ is the probability of the system being in state $i$,
• The sum is over all possible states $i$.

2. Constraints

Two constraints are imposed:

1. Probabilities must sum to 1: $$\sum_i p_i = 1$$

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

2. Energy conservation: $$\langle E \rangle = \textnormal{constant} = \sum_i p_i E_i$$ where $E_i$ is the energy of state $i$. 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 $\{p_i\}$, maximize the entropy $S$ under the constraints using the method of Lagrange multipliers. Define the function:

$$\mathcal{L} = -k_B \sum_i p_i \ln p_i + \alpha \left( \sum_i p_i - 1 \right) + \beta \left( \sum_i p_i E_i - \langle E \rangle \right)$$

where $\alpha$ and $\beta$ are Lagrange multipliers.

4. Solving the Optimization Problem

Take the derivative of $\mathcal{L}$ with respect to $p_i$:

$$\frac{\partial \mathcal{L}}{\partial p_i} = -k_B (1 + \ln p_i) + \alpha + \beta E_i$$

Set $\frac{\partial \mathcal{L}}{\partial p_i} = 0$:

$$-k_B (1 + \ln p_i) + \alpha + \beta E_i = 0$$

Simplify to express $p_i$:

$$\ln p_i = -\frac{\alpha}{k_B} - \frac{\beta E_i}{k_B} - 1$$

Exponentiate both sides:

$$p_i = \exp\left(-\frac{\alpha}{k_B} - 1\right) \exp\left(-\frac{\beta E_i}{k_B}\right)$$

Define $Z = \exp\left(-\frac{\alpha}{k_B} - 1\right)$, known as the partition function:

$$Z = \sum_i \exp\left(-\frac{\beta E_i}{k_B}\right)$$

Thus:

$$p_i = \frac{\exp\left(-\frac{\beta E_i}{k_B}\right)}{Z}$$

5. Interpretation of Parameters

• $\beta$ is related to the temperature $T$ by $\beta = 1/T$ (in natural units).
• The partition function $Z$ ensures normalization of probabilities: $$\sum_i p_i = 1$$

6. Final Result

The Boltzmann distribution is:

$$p_i = \frac{\exp\left(-\frac{E_i}{k_B T}\right)}{\sum_i \exp\left(-\frac{E_i}{k_B T}\right)}$$

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