Finite-size corrections to the free energy of Coulomb systems with a periodic boundary condition

Abstract

Classical Coulomb systems ind dimensions (d⩾2) with a periodic boundary condition, periodW, in the directionx ^(d)are considered. With the other directions of the confining volume of lengthL, it is shown that if the system is in a conducting phase, then the “strip” free energykTf _W ,f _W = −lim _L→∞ L ^−(d−1) log Z, has the large-W expansion

$$f_W \sim Wf_\infty + \frac{{(d/2 - 1)\Gamma (d/2 - 1)}}{{\pi ^{d/2} W^{d - 1} }}\zeta (d) + O\left( {\frac{1}{{W^{d + 1} }}} \right)$$

wherekTf _∞ is the bulk free energy per unit volume, ζ(x) denotes the Riemann zeta function, andΓ(x) denotes the gamma function. With 1/W identified askT, this result is precisely the low-temperature behavior of the free energy of a (d−1)-dimensional Debye solid. This fact is explained in terms of an equivalence between the Coulomb gas and quantum fields. Also, the expansion is verified for some exactly solved models of Coulomb systems in two dimensions.