Generalizing the Debye-Huckel equation in terms of density functional integral

We discuss the validity of generalized Debye-Huckel (GDH) equation proposed by Fisher et al. [B. P. Lee and M. E. Fisher, Phys. Rev. Lett. 76, 2906 (1 996); Europhys. Lett. 39, 611 (1997); M. N. Tamashiro, Y. Levin, and M. C. Barbosa, Physica A 268, 24 (1999)], from the functional integral point of v iew. The GDH theory considers fluctuations around prescribed densities of p ositive and negative charges. Hence, we first formulate a density functiona l integral expression for the canonical system of Coulomb gas, and also dem onstrate that this is a dual form to the sine-Gordon theory. Our formalism reveals the following: (i) The induced charge distribution around supposed density favors not only the cancellation of additional electrostatic potent ial like the original DH theory, but also the countervailing of chemical po tential difference between imposed and equilibrium value. (ii) As a consequ ence apparent charge, absent in the GDH equation, comes out in our generali zed equation. (iii) That is, the GDH equation holds only in special cases.