Complex systems of classical physics in certain situations are characterized by intensive fluctuations of the quantities governing their dynamics. These include important phenomena such as (continuous) second-order phase transitions, fully developed turbulence, magnetic hydrodynamics, advective–diffusive processes, the kinetics of chemical reactions, percolation, and processes in financial markets. The theoretical goal of the study of such systems is to determine and predict the temporal and spatial dependence of statistical correlations of fluctuating variables. Modern methods of quantum field theory, originally developed in high-energy physics to describe the properties of elementary particles, allow for quantitative analysis of such correlations. Peculiarities of quantum field theory in solving two paradigmatic statistical problems related to percolation are reviewed, and new results on calculating representative universal parameters such as critical exponents that describe asymptotic behavior are presented.
Complex systems of classical physics in certain situations are characterized by intensive fluctuations of the quantities governing their dynamics. These include important phenomena such as (continuous) second-order phase transitions, fully developed turbulence, magnetic hydrodynamics, advective–diffusive processes, the kinetics of chemical reactions, percolation, and processes in financial markets. The theoretical goal of the study of such systems is to determine and predict the temporal and spatial dependence of statistical correlations of fluctuating variables. Modern methods of quantum field theory, originally developed in high-energy physics to describe the properties of elementary particles, allow for quantitative analysis of such correlations. Peculiarities of quantum field theory in solving two paradigmatic statistical problems related to percolation are reviewed, and new results on calculating representative universal parameters such as critical exponents that describe asymptotic behavior are presented.
In this study we focus on the non-equilibrium phase transition whose asymptotic critical behavior is governed by the dynamic isotropic percolation universality class. In order to quantitatively characterize universal properties associated with this class, we employ a field-theoretic formulation of the model augmented with perturbative renormalization group technique. Utilizing such approach enables us to systematically compute the critical exponents in higher orders of perturbation theory. In particular, we perform an analysis to the three-loop precision. To this order we determine the renormalization constants and present relations for critical exponents in terms of critical dimensions. Finally, we also present numerically estimated values of critical exponents obtained from perturbation theory.
Originally, the term ‘dynamic critical phenomena’ was coined for time-dependent properties near second-order phase transitions in thermal equilibrium . The kinetics of phase transitions in magnets, at the gas–liquid transition, and at the normal- to superfluid phase transition in helium 4 were among the prominent examples investigated already in the 1960s. The dynamic scaling hypothesis , generalizing the scaling ansatz for the static correlation function and introducing an additional dynamic critical exponent, successfully described a variety of these experiments. Yet only the development of the systematic renormalization group (RG) approach for critical phenomena in the subsequent decade provided a solid conceptual foundation for phenomenological scaling theories. Supplemented with exact solutions for certain idealized model systems, and guided by invaluable input from computer simulations in addition to experimental data, the renormalization group now provides a general framework to explore not only the static and dynamic properties near a critical point, but also the large-scale and low-frequency response in stable thermodynamic phases. Scaling concepts and the renormalization group have also been successfully applied to phase transitions at zero temperature driven by quantum rather than thermal fluctuations. It is to be hoped that RG methods may help to classify the strikingly rich phenomena encountered in far-from-equilibrium systems as well. Recent advances in studies of simple reaction-diffusion systems, active to absorbing state phase transitions, driven lattice gases, and scaling properties of moving interfaces and growing surfaces, among others, appear promising in this respect.
It is explained how field-theoretic methods and the dynamic renormalisation group (RG) can be applied to study the universal scaling properties of systems that either undergo a continuous phase transition or display generic scale invariance, both near and far from thermal equilibrium. Part 1 introduces the response functional field theory representation of (nonlinear) Langevin equations. The RG is employed to compute the scaling exponents for several universality classes governing the critical dynamics near second-order phase transitions in equilibrium. The effects of reversible mode-coupling terms, quenching from random initial conditions to the critical point, and violating the detailed balance constraints are briefly discussed. It is shown how the same formalism can be applied to nonequilibrium systems such as driven diffusive lattice gases. Part 2 describes how the master equation for stochastic particle reaction processes can be mapped onto a field theory action. The RG is then used to analyse simple diffusion-limited annihilation reactions as well as generic continuous transitions from active to inactive, absorbing states, which
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.
Overview As discussed in Chapter 16 , when we approach the critical temperature of a phase coexistence line, several fascinating things start to happen which together have come to be known as “critical phenomena.” It is observed that the spatial correlation between relevant order parameters like density, magnetization, etc. becomes infinitely large. This is reflected in the divergence of inverse susceptibility of the respective order parameters. Landau’s theory explains this in terms of the flattening of the free energy surface but fails to unravel many of the other mysteries of critical phenomena that have kept scientists engaged now for about a century. Critical phenomena are characterized by certain singular properties, like the universality of the critical exponents. Several aspects of the critical phenomena are discussed in this chapter with an aim to maintain continuity with earlier chapters. A simple theory (like that of Landau) cannot explain many of the observed features and this problem remained unsolved for a long time until K. G. Wilson could develop a sophisticated theory that explained almost everything. At the core of Wilson’s method is the renormalization group technique. Physically, this technique employs the expected invariance of the Hamiltonian due to a transformation that removes local interactions and replaces them by effective longer range interactions. Because of the existence of long range correlations, the transformation is expected to converge to a fixed point. The procedure is quite mathematical and is the only robust method to obtain the critical exponents. Here we discuss the method briefly, as an introduction to the subject.
Some physical systems, although dierent by their microscopic nature, in the vicinity of critical points can have similar thermodynamic and structural properties governed by universal power laws.Such systems are grouped into the so-called universality classes.The latter are determined by such global characteristics as the type of interaction, space dimensionality, symmetry, and the number of components of the order parameter.Therefore, it is interesting to investigate how the mutual inuence of these characteristics can aect the critical features of the system.The goal of our research is to analyze the changes in the critical behavior of a many-particle magnetic system under the inuence of two competing factors: long-range interaction and weak structural disorder.Using the example of an n-vector model in a d-dimensional space, we will investigate ferromagnetic ordering in a structurally disordered magnet with the long-range interaction decaying with the distance x as J(x) ∼ x -d-σ , with the control parameter σ.The analysis is carried out using the eld-theoretical renormalization group methods, which makes it possible to identify the universality classes of the system under consideration, their change with global parameters, and to determine the universal characteristics of critical behavior (critical exponents and marginal dimensions).It will be shown that there exists such a region of parameters (d, n, σ), where the long-range interaction and structural disorder lead to a synergistic eect and the appearance of a new, random long-range universality class.We obtain the renormalization group functions in the three-loop approximation, and calculate the correlation length critical exponent ν(ϵ ′ , n) as a perturbation theory series in ϵ ′ = 2σ -d.Quantitative estimates are obtained using asymptotic series resummation methods.