Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 181

Letting we rewrite(S.4.70.3) as

(S.4.70.4)defines aparametric equation for thechemical potential The

decrease ofvolume (ortemperature)willincrease the value of theintegral,

and therefore thevalue of (which is always negative inBosestatistics) will

increase. Thecritical parameters or correspond to the point where

(i.e., if you decreasethe volume or temperature anyfurther, should

increase evenfurther to provide asolution to(S.4.70.4),whereas it cannot

becomepositive). So we canwrite at acertain temperature:

Therefore,

b) In two dimensions the integral(S.4.70.3) becomes

and there is noBose condensation(seeProblem4.71).

4.71 Bose Condensation (Princeton, Stony Brook)

For Boseparticles,

where is thetemperature inenergyunits. The total number ofparticles
in a Bose distribution is

Substituting into the

integralgives