A classical degree of freedom is a coordinate ranging over a constant

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 classical degree of freedom is a coordinate

x

ranging over

-\infty < x < \infty

whose energy is

E(x) = cx^{2}

with

c > 0

a constant. The system is in contact with a reservoir at temperature

T

. The states of

x

and its momentum are counted as one state per area

h

of the

x

-momentum plane, and the states are dense on the scale of

k_BT

, so sums over them become integrals. You may use

\int_{-\infty}^{\infty}e^{-u^{2}}\,\mathrm{d}u = \sqrt{\pi},\qquad \int_{-\infty}^{\infty}u^{2}e^{-u^{2}}\,\mathrm{d}u = \frac{\sqrt{\pi}}{2}.

((a)) By substituting

u = x\sqrt{\beta c}

with

\beta = 1/k_BT

, show that

\langle cx^{2}\rangle = \tfrac12 k_BT

, whatever the value of

c

. Say where

h

went. ((b)) A small mirror hangs on a fine torsion fibre in a vacuum at

T = 295\,\mathrm{K}

. Its angular deflection

\theta

from equilibrium has potential energy

\tfrac12\kappa\theta^{2}

with torsion constant

\kappa = 3.0\times10^{-13}\,\mathrm{N\,m\,rad^{-1}}

, and

\theta

is a classical coordinate of the kind treated in (a). Find the root-mean-square deflection

\theta_{\rm rms}

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