A sum rule for interaction measurements

Thermodynamic sum rules provide the necessary tests for global equilibrium. A particularly convenient form may be derived from Eq. (7) if the pair potential is radially symmetric. In this case,

$$k_B T \left< \nabla_r^2 u(r) \right> = \left< \vert \nabla_r u(r) \vert ^2 \right>,$$ (29)

and we can explicitly calculate the thermodynamic averages of both sides of this equation:

$$\left< \nabla^2_r u(r) \right> = 2\pi n N \, \int_0^{\infty} \left( \frac{1}{r}\, \frac{du}{dr}+\frac{d^2u}{dr^2} \right) \, rg(r) \, dr$$

$$= 2\pi n N \, \left\{ \int_0^{\infty} \frac{du}{dr} \, g(r) \, dr + \left. \frac{du}{dr} \, rg(r) \right\vert _0^{\infty} \right.$$

$$\left. \quad - \int_0^{\infty} \frac{du}{dr} \, \frac{d}{dr} \left(rg(r)\right) \, dr \right\}$$

$$= 2\pi n N \, \int_0^{\infty} \frac{du}{dr} \, \frac{dg(r)}{dr} \, r \, dr,$$ (30)

and

$$\left< \vert \nabla_r u(r) \vert ^2 \right> = 2\pi n N \, \int_0^{\infty} \left(\frac{du}{dr} \right)^2 \, g(r) r \, dr.$$ (31)

Combining these results yields the sum rule

$$\int_0^\infty \left( \beta \frac{du(r)}{dr} - \frac{d \ln{g(r)}}{dr} \right) \, \frac{du(r)}{dr} \, g(r) \, r \, dr = 0.$$ (32)

This sum rule should apply at arbitrary areal densities for any system whose interactions can be described by a pairwise-additive central potential, $u(r)$. A similar result was obtained in Ref. (22) for three-dimensional systems.

Using the radial distribution function to average over pairs of particles removes any sensitivity to local structural variations, and thus focuses attention on global properties such as the degree of equilibration. Consequently, Eq. (32) complements the hierarchy of configurational temperature consistency checks.