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The nonlinear dielectric relaxation spectra of ferroelectric liquid crystals (FLCs) have been studied in the chiral smectic-C phase. The linear and third-order nonlinear dielectric spectra show the relaxation corresponding to the fluctuation in the azimuthal angle of directors called the Goldstone mode. We calculated the nonlinear dielectric spectra of the Goldstone mode theoretically by the torque balance equation which describes the dynamics of FLCs under the electric field. The calculated spectra make good agreement with the measured ones. We also evaluated the material constants of FLCs from the best-fitted values of the linear and nonlinear dielectric increment and relaxation time.

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 (1996); 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 view. The GDH theory considers fluctuations around prescribed densities of positive and negative charges. Hence, we first formulate a density functional integral expression for the canonical system of Coulomb gas, and also demonstrate 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 potential like the original DH theory, but also the countervailing of chemical potential difference between imposed and equilibrium value. (ii) As a consequence apparent charge, absent in the GDH equation, comes out in our generalized equation. (iii) That is, the GDH equation holds only in special cases.

To go beyond Gaussian approximation to the Hohenberg-Kohn free energy playing the key role in the density functional theory (DFT), the density functional integral representation would be relevant, because the field theoretic approach to perturbative calculations becomes available. Then we first derive the associated Hamiltonian of the density functional, explicitly including the logarithmic entropy term, from the grand partition function expressed by configurational integrals. Moreover, two things are done so that the efficiency of the obtained form may be revealed: we demonstrate that this representation facilitates the field theoretic treatment of the perturbative calculation and, further, compare our perturbative formulation with that of the DFT.