Blundell Thermal Physics · Problem 4.3

Temperature and the Boltzmann factor

Physics — Statistical Mechanics · microstates-temperature-two-energy-levels

Problem

A system comprises

N

states, which can have energy 0 or

\Delta

. Show that the number of ways

\Omega(E)

of arranging the total system to have energy

E = r\Delta

(where

r

is an integer) is given by

\Omega(E) = \frac{N!}{r!(N-r)!}. \tag{4.33}

Now remove a small amount of energy

s\Delta

from the system, where

s \ll r

. Show that

\Omega(E - \epsilon) \approx \Omega(E) \frac{r^s}{(N-r)^s}, \tag{4.34}

and hence show that the system has temperature

T

given by

\frac{1}{k_\mathrm{B}T} = \frac{1}{\Delta} \ln \left(\frac{N-r}{r}\right). \tag{4.35}

Sketch

k_\mathrm{B}T

as a function of

r

from

r = 0

r = N

and explain the result.