Boltzman Distribution

statistics

Definition

Boltzman Distribution

The state occupation probability of a thermodynamical system within fixed temperature $T$:

$$p(x)=\alpha e^{-\frac{E(x)}{kT}}$$

where:

• $x$ is the state
• $\alpha$ is the normalisation denominator
• $E(x)$ is the energy
• $k$ is the Boltzmann constant

If $T$ decreased slowly enough: reach best (lowest energy) state $x^{\ast }$ with probability approaching $1$, as:

$$e^{-\frac{E(x^{\ast })}{kT}}/e^{-\frac{E(x)}{kT}}=e^{\frac{E(x)-E(x^{\ast })}{kT}}\gg 1\text{for small }T$$