Statistical Mechanics · Problem 3.28

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · quantum-oscillator-probability-state

Problem

(a) When a system of

N

oscillators with total energy

R

is in thermal equilibrium, what is the probability

p_n

that a particular oscillator among them is in the quantum state

n

? [Hint: Use expression (3.8.25).] Show that, for

N \gg 1

and

R \gg n

p_n \approx (\bar{n})^n/(\bar{n}+1)^{n+1}

, where

\bar{n} = R/N

. (b) When an ideal gas of

N

monatomic molecules with total energy

E

is in thermal equilibrium, show that the probability of a particular molecule having an energy in the neighborhood of

\varepsilon

is proportional to

\exp(-\beta\varepsilon)

, where

\beta = 3N/2E

. [Hint: Use expression (3.5.16) and assume that

N \gg 1

and

E \gg \varepsilon