Thermodynamics, Statistical Physics, and Quantum Mechanics

44 PROBLEMS

b)From (a), find the relation between and the meansquare fluctua-

tion of N in the grand canonical ensemble. How does this fluctuation

depend on the number of particles?

4. 93 Energy Fluctuation in Canonical Ensemble
   (Colorado, Stony Brook)

Showthat for acanonical ensemble thefluctuation of energyin a system

of constant volume is related to the specific heat and, hence, deduce that

the specific heat at constant volume is nonnegative.

4.94 Number Fluctuations (Colorado (a,b), Moscow

Phys-Tech (c))

Show that for a grand canonical ensemble the number of particles N and

occupationalnumber in an ideal gas satisfy the conditions:

4.95 Wiggling Wire (Princeton)

A wire of length and mass per unit length is fixed at both ends and

tightened to a tension What is the rms fluctuation, in classical statistics,

of the midpoint of the wire when it is inequilibrium with a heat bath at

temperature A useful series is

4.96 LC Voltage Noise (MIT, Chicago)

The circuit in Figure P.4.96 consists of a coil of inductance and a ca-
pacitor of capacitance C. What is the rms noise voltage across AB at
temperature in the limit where

a) is very large?

b) is very small?

a) quantum statistics

b) classical statistics

For an electron spin Fermi gas attemperature

c) Find