A small system

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles · Statistical foundations and the equilibrium ensembles

Thermal Physics, Ch The Boltzmann factor, the canonical ensemble and the partition function

Problem

A small system, whose states have energies from

0

up to

E_{\max} = 2.0\times10^{-20}\,\mathrm{J}

, is weakly coupled to a block of copper of mass

0.50\,\mathrm{kg}

T = 320\,\mathrm{K}

, of constant specific heat capacity

385\,\mathrm{J\,K^{-1}\,kg^{-1}}

. System plus block are isolated, their energies add (weak coupling), and every joint microstate of the isolated whole is equally likely. Write

S_R(U_R)

for the block's entropy, with

(\partial S_R/\partial U_R)_{V,N} = 1/T

and

C_R = (\partial U_R/\partial T)_V

. ((a)) Starting from the equal likelihood of joint microstates, show that the probability of the small system's state

s

obeys

P(s) \propto \exp\!\left[-\frac{E_s}{k_BT} - \frac{E_s^{\,2}}{2k_BT^{2}C_R}\right],

and hence that the ratio of the second term in the exponent to the first is

E_s/2TC_R

. ((b)) Evaluate that ratio at

E_s = E_{\max}

for this block, and state what it licenses. ((c)) Find the change in the block's temperature when it hands over

E_{\max}

to the small system.

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