The mutual influence of singularities of the dielectric permittivitty ε(q, ω) in a Coulomb system in two limiting cases ω → 0, q → 0 and q → 0, ω → 0 is established. It is shown that the dielectric permittivity ε(q → 0, ω) satisfies the Kramers-Kronig relations, which possesses the a singularity due to a finite value of the static conductivity. This singularity is associated with the long "tails" of the time correlation functions. PACS number(s): 05.20.Dd, 05.30.Fk, 52.25.Mq, 72.10.Bg I. INTRODUCTIONThe dielectric function ε(q, ω) of a Coulomb system is one of the most important characteristics of matter [1-3]. Exact relations for the dielectric function (dielectric permittivity, abbreviated below as DP), which determine general behavior of the DP as a function of the wave vector q (in the present paper we consider only the isotropic Coulomb system, therefore the vector q is replaced by its modulus q) and the frequency ω, have a special significance in the theory of Coulomb systems (CS). At first, it is necessary to mention the Kramers-Kronig relations (KKR). According to the theory of linear response [2,3], the DP ε(q, ω) of a homogeneous and isotropic Coulomb system in volume V at temperature T is

It is shown that the common expression for the static dielectric permittivity of the homogeneous and isotropic Coulomb system (CS) for small wave vectors q describes both metallic and dielectric states of the CS. The states of the 'true' dielectric and 'ideal' conductor are introduced and have a limiting nature. It is established that their thermodynamic parameters are defined by the critical point parameters. The second sum rule is found for the dielectric function in the thermodynamic limit. An exact relation for the pair correlation function of the two-component homogeneous and isotropic system of electrons and nuclei g ec (r) is obtained for arbitrary thermodynamic parameters.

It was found that the equivalence of the grand canonical and canonical ensembles for the Coulomb systems is possible only when charged particles of different types in calculating the physical quantities are considered as formally “independent” ones, and the quasi-neutrality condition is used in the final stage of calculations. The phase equilibrium condition is obtained and the expression is derived for the isothermal compressibility of matter as a two-component Coulomb system, which corresponds to the known limit relations for static structure factors. On this basis, it is demonstrated that the critical point of matter, considering as the Coulomb system is determined from the condition of vanishing mean square of fluctuations of the total charge per unit volume.

The development of the theory of thermal radiation is important for studies of many physical phenomena (see, e.g., [1]). The derivation of exact relations between the characteristics of equilibrium radiation and the electromagnetic properties of matter is of great significance. In this work, we calculate the spectral distribution of the energy of equilibrium radiation in the presence of matter, which obviously differs from the well known Planck formula corresponding to the blackbody model (see, e.g., [2]). This is topical because the Planck distribution corresponds to an equilibrium ideal photon gas, whereas the presence of at least a small amount of matter is necessary for the possibility of obtaining equilibrium radiation, because the direct interaction between photons is absent in nonrelativistic theory [2]. In fact, it is implicitly accepted that the equilibrium properties of an ideal photon gas are limiting properties of a real system of the electromagnetic field and matter in thermody namic equilibrium.To solve the formulated problem, we consider a sys tem of nonrelativistic particles and photons in the vol ume V. The Hamiltonian of such a system in the sec ond quantization representation has the form [3] (1)and are the field creation and annihilation operators, respectively, for charged parti cles of type a, which have the mass m a , charge z a e, and operator of the intrinsic magnetic moment ; andis the Hamiltonian of the free field of radiation:,where the creation ( ) and annihilation ( ) operators for photons with momentum បk and polar ization λ = 1, 2 satisfy the commutation relations [ , ] = δ k, k' δ λ,λ' ; is the vector potential corresponding to the quantized electromagnetic field:(3)where are the polarization vectors of photons sat isfying the conditions ,and is the Hamiltonian of the Coulomb interac tion between charged particles:An exact relation has been obtained for the spectral distribution of the energy of radiation in thermodynamic equilibrium with a system of charged nonrelativistic particles. The difference from the Planck formula is unambiguously determined by the transverse permittivity of a medium, which takes into account not only fre quency dispersion but also spatial dispersion.