Landau theory relates phase transitions to the minimization of the Landau functional (e.g., free energy functional), which is expressed as a power series of the order parameter. It has been shown that the critical behavior of certain physical systems can be described using Landau functionals that include nonanalytic terms, corresponding to odd or even noninteger powers of the absolute value of the order parameter. In particular, these nonanalytic terms can determine the order of the phase transition and the values of the critical exponents. Here, we show that such terms can also shape the finite-size scaling behavior of fluctuations of observables (e.g., of energy or magnetization) or the response functions (e.g., heat capacity or magnetic susceptibility) at the continuous phase transition point. We demonstrate this on two examples, the equilibrium molecular zipper and the nonequilibrium version of the Curie-Weiss model.
Landau theory relates phase transitions to the minimization of the Landau functional (e.g., free energy functional), which is expressed as a power series of the order parameter. It has been shown that the critical behavior of certain physical systems can be described using Landau functionals that include nonanalytic terms, corresponding to odd or even noninteger powers of the absolute value of the order parameter. In particular, these nonanalytic terms can determine the order of the phase transition and the values of the critical exponents. Here, we show that such terms can also shape the finite-size scaling behavior of fluctuations of observables (e.g., of energy or magnetization) or the response functions (e.g., heat capacity or magnetic susceptibility) at the continuous phase transition point. We demonstrate this on two examples, the equilibrium molecular zipper and the nonequilibrium version of the Curie-Weiss model.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
Phase transitions are typically accompanied by nonanalytic behaviors of the free energy, which can be explained by considering the zeros of the partition function in the complex plane of the control parameter and their approach to the critical value on the real axis as the system size is increased.Recent experiments have shown that partition function zeros are not just a theoretical concept.They can also be determined experimentally by measuring fluctuations of thermodynamic observables in systems of finite size.Motivated by this progress, we investigate here the partition function zeros for the Curie-Weiss model of spontaneous magnetization using our recently established cumulant method.Specifically, we extract the leading Fisher and Lee-Yang zeros of the Curie-Weiss model from the fluctuations of the energy and the magnetization in systems of finite size.We develop a finite-size scaling analysis of the partition function zeros, which is valid for mean-field models and which allows us to extract both the critical values of the control parameters and the critical exponents, even for small systems that are away from criticality.We also show that the Lee-Yang zeros carry important information about the rare magnetic fluctuations as they allow us to predict many essential features of the large-deviation statistics of the magnetization.This finding may constitute a profound connection between Lee-Yang theory and large-deviation statistics.
We discuss the universal scaling laws of order-parameter fluctuations in any system in which a second-order critical behavior can be identified. These scaling laws can be derived rigorously for equilibrium systems when combined with a finite-size scaling analysis. The relation between the order parameter, the criticality, and the scaling law of fluctuations has been established, and the connection between the scaling function and the critical exponents has been found. We give examples in out-of-equilibrium aggregation models such as the Smoluchowski kinetic equations, or at-equilibrium Ising and percolation models.
Non-equilibrium critical phenomena have attracted a lot of research interest in the recent decades. Similar to equilibrium critical phenomena, the concept of universality remains the major tool to order the great variety of non-equilibrium phase transitions systematically. All systems belonging to a given universality class share the same set of critical exponents, and certain scaling functions become identical near the critical point. It is known that the scaling functions vary more widely between different universality classes than the exponents. Thus, universal scaling functions offer a sensitive and accurate test for a system's universality class. On the other hand, universal scaling functions demonstrate the robustness of a given universality class impressively. Unfortunately, most studies focus on the determination of the critical exponents, neglecting the universal scaling functions. In this work a particular class of non-equilibrium critical phenomena is considered, the so-called absorbing phase transitions. Absorbing phase transitions are expected to occur in physical, chemical as well as biological systems, and a detailed introduction is presented. The universal scaling behavior of two different universality classes is analyzed in detail, namely the directed percolation and the Manna universality class. Especially, directed percolation is the most common universality class of absorbing phase transitions. The presented picture gallery of universal scaling functions includes steady state, dynamical as well as finite size scaling functions. In particular, the effect of an external field conjugated to the order parameter is investigated. Incorporating the conjugated field, it is possible to determine the equation of state, the susceptibility, and to perform a modified finite-size scaling analysis appropriate for absorbing phase transitions. Focusing on these equations, the obtained results can be applied to other non-equilibrium continuous phase transitions observed in numerical simulations or experiments. Thus, we think that the presented picture gallery of universal scaling functions is valuable for future work. Additionally to the manifestation of universality classes, universal scaling functions are useful in order to check renormalization group results quantitatively. Since the renormalization group theory is the basis of our understanding of critical phenomena, it is of fundamental interest to examine the accuracy of the obtained results. Due to the continuing improvement of computer hardware, accurate numerical data have become available, resulting in a fruitful interplay between numerical investigations and renormalization group analyzes.
Abstract In this work, we define and calculate critical exponents associated with higher order thermodynamic phase transitions. Such phase transitions can be classified into two classes: with or without a local order parameter. For phase transitions involving a local order parameter, we write down the Landau theory and calculate critical exponents using the saddle point approximation. Further, we investigate fluctuations about the saddle point and demarcate when such fluctuations dominate over saddle point calculations by introducing the generalized Ginzburg criteria. We use Wilsonian renormalization group (RG) to derive scaling forms for observables near criticality and obtain scaling relations between the critical exponents. Afterwards, we find out fixed points of the RG flow using the one-loop beta function and calculate critical exponents about the fixed points for third and fourth order phase transitions.
Non-equilibrium critical phenomena have attracted a lot of research interest in the recent decades. Similar to equilibrium critical phenomena, the concept of universality remains the major tool to order the great variety of non-equilibrium phase transitions systematically. All systems belonging to a given universality class share the same set of critical exponents, and certain scaling functions become identical near the critical point. It is known that the scaling functions vary more widely between different universality classes than the exponents. Thus, universal scaling functions offer a sensitive and accurate test for a system's universality class. On the other hand, universal scaling functions demonstrate the robustness of a given universality class impressively. Unfortunately, most studies focus on the determination of the critical exponents, neglecting the universal scaling functions. In this work a particular class of non-equilibrium critical phenomena is considered, the so-called absorbing phase transitions. Absorbing phase transitions are expected to occur in physical, chemical as well as biological systems, and a detailed introduction is presented. The universal scaling behavior of two different universality classes is analyzed in detail, namely the directed percolation and the Manna universality class. Especially, directed percolation is the most common universality class of absorbing phase transitions. The presented picture gallery of universal scaling functions includes steady state, dynamical as well as finite size scaling functions. In particular, the effect of an external field conjugated to the order parameter is investigated. Incorporating the conjugated field, it is possible to determine the equation of state, the susceptibility, and to perform a modified finite-size scaling analysis appropriate for absorbing phase transitions. Focusing on these equations, the obtained results can be applied to other non-equilibrium continuous phase transitions observed in numerical simulations or experiments. Thus, we think that the presented picture gallery of universal scaling functions is valuable for future work. Additionally to the manifestation of universality classes, universal scaling functions are useful in order to check renormalization group results quantitatively. Since the renormalization group theory is the basis of our understanding of critical phenomena, it is of fundamental interest to examine the accuracy of the obtained results. Due to the continuing improvement of computer hardware, accurate numerical data have become available, resulting in a fruitful interplay between numerical investigations and renormalization group analyzes.