210 SOLUTIONS4.91 Quivering Mirror (MIT, Rutgers, Stony Brook)
a) When themirror is in thermalequilibriumwith gas in the chamber, one
may again invoke theequipartition theorem andstate that there is
of energy in the rotationaldegree offreedom of the torsionalpendulum,
where the torque isgiven by The mean squarefluctuation in the
angle would then be given by(seeChapter 13, Fluctuations, in Pathria)
So,Now, Avogadro’s number andwe obtain
b) Even if the gas density werereduced in the chamber, themean square
fluctuation would notchange. However, inorder to determine whether
individual fluctuations might have larger amplitudes, we cannot rely on the
equipartition theorem. Weinstead will examine the fluctuations in the
frequency domain. may bewritten
where is thepowerspectraldensity of At high gas density, is
broader andsmaller inamplitude, while theintegral remainsconstant.This
corresponds to morefrequentcollisions and smaller amplitudes, whereas,
at low density, is more peaked around the natural frequency of the
torsional pendulum where Iis its moment of inertia, still keeping
the integralconstant. Itthen appearsthat by reducing the density of the
gas we actuallyincrease theamplitude offluctuations!
4.92 Isothermal Compressibility and Mean Square
Fluctuation (Stony Brook)
a) Let us use the Jacobiantransformation for thermodynamicvariables: