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Exact relations which connect the order parameters, k = 0 Fourier transforms of the direct correlation function harmonics and parameters of the single-particle distribution function for the uniaxial uids are proposed. These relations are an algebraic represetation of the Lovett equation in anisotropic uids. It is shown that in the anisotropic (ori-entationally ordered) phase the transverse correlations of spins in magnetic uids and the correlations of the director transverse uctuations in nematics become of long-range character. Anisotropic uids have no long-range positional order, but they do exhibit orientational order. As a consequence, the single-particle distribution function (1) does depend only on the molecular orientation ! = (; ') and (1) = f(! 1), where is the density of molecules. These systems are polar liquids, magnetic uids and the simplest of the liquid crystals | nematics. Anisotropic uids without the presence of an external eld are the systems with spontaneously broken symmetry, because the direction of the orien-tational ordering is not predetermined. The spontaneous ordering breaks the continuous symmetry (rotational invariance that is intrinsic in isotropic systems) and leads to orientational excitations, or Goldstone modes, which rotate the direction of ordering without any energy cost. This results in the peculiar physical properties of anisotropic uids. 1. Stability analysis and the Lovett equation in anisotropic uids From the density functional theory the following stability condition of equilibrium systems is known 1] Z Z d(1)d(2) (1)(2) > 0; (1.1) where c(1; 2) is the direct pair correlation function, 1 (or 2) being spatial and orientational coordinates of particle 1 (or 2). The equation (1.1) follows directly from the condition of the free energy minimum with respect to arbitrary variations in the particle distribution (i): Z Z 2 F (1)(2) (1)(2)d(1)d(2) > 0: (1.2) But in the anisotropic uids without the presence of external elds some variations (connected to rotations of the director: (1) r !1 (! 1)) keep c T.

EXACT RELATIONS IN THE THEORY OF ANISOTROPIC LIQUIDS