PX284 - C3 - boltzmann distribution

considering a system with an energy, $E_{i}$
it is connected to an extremely large heat bath (reservoir) with: $E_{b a t h} = E - E_{i}$ , $Ω_{b a t h} (E - E i)$ , and $T$
for the system, temperature $T$ , number of particles $N$ , and volume $V$ are fixed
assuming that the probability of the system being in a particular state, $E_{i}$ , is proportional to the number of microstates available to the bath in the state:

P (E_{i}) \propto Ω (E - E_{i})

taylor series expansion of $\ln Ω (E - E_{i})$ around $E_{i} = 0$

\ln Ω (E - E_{i}) = \ln Ω (E) - \frac{d}{d E} [\ln Ω (E)] E_{i} + \dots \approx \ln Ω (E) - \frac{E_{i}}{k_{B} T}

where, $\frac{d \ln Ω}{d E} = \frac{1}{k_{B} T}$ from PX284 - C2 - temperature

the additional terms are not relevant as $E ≫ E_{i}$

∴ Ω (E - E_{i}) \approx Ω (E) \exp (- \frac{E_{i}}{k_{B} T})

therefore, the probability:

P (E_{i}) \propto \exp (- \frac{E_{i}}{k_{B} T})

for normalization:

\sum_{i} P (E_{i}) = \frac{1}{Z} \sum_{i} e^{- β E_{i}} = 1

the boltzmann distribution:

P (E_{i}) = \frac{e^{- β E_{i}}}{Z}

$Z$ is the partition function such that:

Z = \sum_{i} e^{- β E_{i}}

example

a two state system with energies: $0$ or $E$
the expected energy:

\begin{aligned} ⟨ E ⟩ & = \sum_{i} E_{i} P_{i} \\ = 0 \times P (0) + E \times P (E) \\ = E \times \frac{e^{- β E}}{Z} \end{aligned}

where, $Z = e^{- β \cdot 0} + e^{- β E}$

∴ ⟨ E ⟩ = \frac{E e^{- β E}}{1 + e^{- β E}} = \frac{E}{e^{β E} + 1}