We consider entropy and relative entropy in field theory and establish relevant monotonicity properties with respect to the couplings. The relative entropy in a field theory with a hierarchy of renormalization-group fixed points ranks the fixed points, the lowest relative entropy being assigned to the highest multicritical point. We argue that as a consequence of a generalized $H$ theorem Wilsonian RG flows induce an increase in entropy and propose the relative entropy as the natural quantity which increases from one fixed point to another in more than two dimensions.

There is an ongoing debate in cosmology about the value of the length scale at which ``homogeneity'' in the matter distribution is reached or even if such a scale exists. In the wake of this debate, we intend in this letter to clarify the meaning of the statement transition to homogeneity and of the concept of correlation length. We show that there are two scales each associated to the fractal and to the homogeneous regimes of the matter distribution, respectively. The distinction between both scales has deep consequences, for there can be a regime which despite having small fluctuations around the average density exhibits large clusters of galaxies.Comment: 10 pages, 2 large PS figures, AAS macros. Substantial additions as regards the implications for cosmology, including the prospective observation of critical phenomena, and new references. Version to appear in ApJ

On the one hand, the large scale structure of matter is arguably scale invariant, and, on the other hand, halos and voids are recognized as prominent features of that structure. To unify both approaches, we propose to model the dark matter distribution as a set of fractal distributions of halos of different kinds. This model relies on the concept of multifractal as the most general scaling distribution and on a plausible notion of halo as a singular mass concentration in a multifractal. Voids arise as complementary to halos, namely, as formed by regular mass depletions. To provide halos with definite size and masses, we coarse-grain the dark matter distribution, using the length given by the lower cutoff to scaling. This allows us to relate the halo mass function to the multifractal spectrum. Hence, we find that a log-normal model of the mass distribution nicely fits in this picture and, moreover, the Press-Schechter mass function can be recovered as a bifractal limit. To support our model of fractal distributions of halos, we perform a numerical study of the distribution produced in cosmological N-body simulations. In the Virgo L-CDM GIF2 simulation, we indeed find fractal distributions of halos with various dimensions and a halo mass function of bifractal type. However, this mass function is not of Press-Schechter's type, and we interprete it instead as caused by the undersampling of the distribution at the scale of halos, due to discretization.

The cosmic web structure is studied with the concepts and methods of fractal geometry, employing the adhesion model of cosmological dynamics as a basic reference. The structures of matter clusters and cosmic voids in cosmological N-body simulations or the Sloan Digital Sky Survey are elucidated by means of multifractal geometry. A nonlacunar multifractal geometry can encompass three fundamental descriptions of the cosmic structure, namely, the web structure, hierarchical clustering, and halo distributions. Furthermore, it explains our present knowledge of cosmic voids. In this way, a unified theory of the large-scale structure of the universe seems to emerge. The multifractal spectrum that we obtain significantly differs from the one of the adhesion model and conforms better to the laws of gravity. The formation of the cosmic web is best modeled as a type of turbulent dynamics, generalizing the known methods of Burgers turbulence.