The scaling properties of pair correlations at criticality are reproduced through an equivalence between random walk distributions and order parameter correlations. The shift from Gaussian to fractal walks with self-similar clusters corresponds to the changeover from a Gaussian to a nontrivial fixed point with nonvanishing dimensional anomaly. We show that the renormalization group trajectories lead to fixed points of minimum entropy, and use the Tsallis entropy index $q$ to measure nonextensivity as behavior departs from Gaussian.
The scaling properties of pair correlations at criticality are reproduced through an equivalence between random walk distributions and order parameter correlations. The shift from Gaussian to fractal walks with self-similar clusters corresponds to the changeover from a Gaussian to a nontrivial fixed point with nonvanishing dimensional anomaly. We show that the renormalization group trajectories lead to fixed points of minimum entropy, and use the Tsallis entropy index q to measure nonextensivity as behavior departs from Gaussian.
Critical phenomena, characterized by abrupt changes in macroscopic properties near phase transition points, exhibit universal scaling behavior independent of microscopic details. This paper explores the hypothesis that entropy serves as a fundamental universal scaling law governing these phenomena. We review the theoretical foundations of critical phenomena, including Landau theory and the renormalization group, highlighting the role of critical exponents and scaling functions. We then propose a framework where the configurational entropy, entanglement entropy, and other entropic measures exhibit universal scaling, linking them directly to the underlying critical exponents. Our methodology involves analyzing how entropy behaves under coarse-graining transformations near critical points, identifying a potential entropic fixed point in the renormalization group flow. We present theoretical arguments suggesting that the information content, as quantified by various entropy definitions, reflects the diverging correlation length and the emergent scale invariance at criticality. The discussion extends to how different entropic forms, such as Rényi and Tsallis entropies, might provide a more generalized framework for understanding critical behavior in complex and non-extensive systems. By demonstrating entropy's role as a universal scaling law, this work offers a unifying perspective on critical phenomena, deepening our understanding of phase transitions and the emergence of macroscopic order from microscopic fluctuations.
Self-Similarity and Scale Invariance. Renormalization Group Approach to Chaos. Renormalization Approach to Percolation. Renormalization Group and Critical Phenomena. The Ising Model. Renormalization Group for the Ising Model. Other Real-Space Renormalization Group Methods for the Ising Model. Mean Field Theory and the Gaussian Fixed Point. The epsis Expansion. The Spherical Model and the 1/n Expansion. The Two-Dimensional X-Y Model and the Kosterlitz-Thouless Transition. Appendices. Author Index. Subject Index.
The renormalization-group treatment of the critical behavior of random systems is augmented by including all the cumulants of the disorder distribution. The results substaniate Lubensky's argument that the $n\ensuremath{\rightarrow}0$ isotropic fixed point is "unphysical," and that the phase transition of random m-component systems which have a positive specific-heat exponent will be sharp, with exponents determined by the "random" fixed point. The renormalization-group transformations of a random Gaussian model lead to a "runaway," which is shown to be unrelated to a first-order transition. This is related to the problem of a particle in a random potential with and without an absorptive part.
Abstract This chapter analyzes systems with emergent scale invariance -- fractal, self-similar behavior -- by developing the renormalization group. The renormalization group is an amazing abstraction. It describes the flow of the laws governing the system as one coarse-grains -- blurring out the short-distance or short-time details. In the huge space of possible systems (experimental and theoretical), a fixed point of the renormalization group will be the same after blurring and shrinking -- implying emergent scale invariance, a fractal self-similarity. The points which flow into the fixed point share its properties under rescaling -- implying universality, with behavior shared by theory and a wide variety of different experimental systems. The renormalization group also predicts universal power laws and universal functions, describing all behavior on long length and/or time scales. Exercises explore applications to the Ising model, the onset of lasing, superconductors, the onset of chaos, percolation, crackling noise and avalanches, earthquakes, random walks and diffusion, chemical reaction rate theory, and extreme value statistics.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].
We show that the functional renormalization group (FRG) allows for the generalization of the central limit theorem to strongly correlated random variables. On the example of the three-dimensional Ising model at criticality and using the simplest implementation of the FRG, we compute the probability distribution functions of the order parameter or equivalently its logarithm, called the rate functions in large deviations theory. We compute the entire family of universal scaling functions, obtained in the limit where the system size L and the correlation length of the infinite system xiinfty diverge, with the ratio zeta=L/xiinfty held fixed. It compares very accurately with numerical simulations.