Statistical Mechanics · Problem 3.20

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · virial-theorem-homogeneous-potential

Problem

Show that, for a statistical system in which the interparticle potential energy

u(\mathbf{r})

is a homogeneous function (of degree

n

) of the particle coordinates, the virial

\mathcal{Y}

is given by

\mathcal{Y} = -3PV - nU

and, hence, the mean kinetic energy

K

K = -\frac{1}{2}\mathcal{Y} = \frac{1}{2}(3PV + nU) = \frac{1}{n+2}(3PV + nE);

here,

U

denotes the mean potential energy of the system and

E = K + U

. Note that this result holds not only for a classical system but for a quantum-mechanical one as well.