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Physical review. B, Condensed matter

Non-Gaussian models for critical fluctuations

Jack A. Tuszyński, M. J. Clouter, Harry Kiefte

Published Mar 1, 1986• 17 citations
Publisher

Abstract

This paper investigates the role of statistical fluctuations about the equilibrium value of the order parameter in the vicinity of a phase-transition temperature. The systems under consideration are assumed to be of finite size, but the behavior in the thermodynamic limit is also described. Our objective is to examine the simplest exactly solvable model of non-Gaussian static fluctuations for samples of finite size. To this end a Landau form is employed for the model Hamiltonian and various expansions are analyzed in order to describe first- and second-order phase transitions, field-induced transitions, and liquid-vapor critical phenomena. The approach differs from previous treatments primarily in the application of a quartic non-Gaussian model for statistical fluctuations which redresses some of the shortcomings of the standard Gaussian approximation. In particular, it is found that for finite values of V (volume) no singularities exist even at the critical temperature. The treatment is generalized by examining a class of non-Gaussian distribution functions, and particular emphasis is placed on strict adherence to the uniform convergence criterion so that general conclusions can be drawn.

DOI
10.1103/physrevb.33.3423

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Physical review. B, Condensed matter•Apr 1, 1991•10 citations
This paper investigates the role of statistical fluctuations about the equilibrium value of the order parameter in the vicinity of a phase-transition temperature. The systems under consideration are assumed to be of finite size, but the behavior in the thermodynamic limit is also described. Our objective is to examine the simplest exactly solvable model of non-Gaussian static fluctuations for samples of finite size. To this end a Landau form is employed for the model Hamiltonian and various expansions are analyzed in order to describe first- and second-order phase transitions, field-induced transitions, and liquid-vapor critical phenomena. The approach differs from previous treatments primarily in the application of a quartic non-Gaussian model for statistical fluctuations which redresses some of the shortcomings of the standard Gaussian approximation. In particular, it is found that for finite values of V (volume) no singularities exist even at the critical temperature. The treatment is generalized by examining a class of non-Gaussian distribution functions, and particular emphasis is placed on strict adherence to the uniform convergence criterion so that general conclusions can be drawn.
First-order phase transitions are ubiquitous in nature but their presence is often uncertain because of the effects which finite size has on all transitions. In this article we consider a general treatment of size effects on lattice systems with discrete degrees of freedom and which undergo a first-order transition in the thermodynamic limit. We review recent work involving studies of the distribution functions of the magnetization and energy at a first-order transition in a finite sample of size N connected to a bath of size N′. Two cases: N′ = ∞ and N′ = finite are considered. In the former (canonical ensemble) case, the distributions are approximated by a superposition of Gaussians corresponding to the different phases; all finite-size effects then vary as N or 1/N. The latter case involves the Gaussian ensemble where the entropy of the bath has a convenient form; for small N′, first-order transitions are characterized by van der Waals' loops in (for example) the energy vs. temperature curves. Results from extensive Monte Carlo simulations of Ising and Potts models in d = 2 are presented to confirm the predictions. Comparison is made with data from second-order transitions to show that the order of a transition can be distinguished through such studies, although problems still occur for first-order transitions very close to critical points.
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.
We present a detailed investigation of the probability density function (PDF) of order parameter fluctuations in the finite two-dimensional XY (2d XY ) model. In the low-temperature critical phase of this model, the PDF approaches a universal non-Gaussian limit distribution in the limit T → 0. Our analysis resolves the question of temperature dependence of the PDF in this regime, for which conflicting results have been reported. We show analytically that a weak temperature dependence results from the inclusion of multiple loop graphs in a previously derived graphical expansion. This is confirmed by numerical simulations on two controlled approximations to the 2d XY model: the harmonic and ‘harmonic XY ’ models. The harmonic model has no Kosterlitz–Thouless–Berezinskiĭ (KTB) transition and the PDF becomes progressively less skewed with increasing temperature until it closely approximates a Gaussian function above T ≈ 4π. Near to that temperature, we find some evidence of a phase transition, although our observations appear to exclude a thermodynamic singularity.
Finite-size and Fluctuation Effects on Phase Transition and Critical Phenomena using Mean-Field Approach Based on Renormalized ϕ4ϕ^{4}ϕ4 Model: I. Theory
An investigation of the spatial fluctuations and their manifestations in the vicinity of the quantum critical point within the framework of the renormalized ϕ4ϕ^{4}ϕ4 theory is proposed. Relevant features are reported through the Ginzburg-Landau-Wilson (GLW)-based calculations, combined with an efficient non perturbative technique. Both the dimension and size, but also microscopic details of the system, leading to critical behavior, and strongly deviating from the classical mean-field approach far from the thermodynamic limit, are taken into account. Further, the important role that harmonic and anharmonic fluctuations and finite-size effects can play in the determination of the characteristic properties of corresponding various systems, involving phase transitions and critical phenomena, is then discussed in detail with emphasis on the qualitative validity of the analysis
This paper investigates the application of the renormalized Gaussian approach (RGA) on spatial fluctuations and their manifestations in the vicinity of the critical point. The basis for calculation is the Landau–Ginzburg–Wilson (LGW) Hamiltonian ( H LGW ). Within the framework of this approach, the model predictions are examined in some detail and the importance of fluctuations is established. Analytic expressions for the correlation length, susceptibility, correlation function as well as the specific heat are calculated and an approximate Ginzburg criterion is deduced. The approach also establishes the limitations of the mean-field theory description, since it is found that the mean-field critical temperature T c is a characteristic scale of temperature linked to the thermal fluctuations rather than the transition temperature. By applying the RGA, the concept of dimensionality appears essential in the manifestation and analysis of phase transitions since the approach suggests the dependence of some critical parameters and exponents on the dimensionality, as well as the quantum character. Also, it is shown that at low temperature, the correlation length of 1D systems becomes infinite at T = 0 K in accord with the Landau and Mermin–Wagner theorems. Finally, three illustrative examples are provided as analysis support in order to demonstrate the usefulness of the approach for a variety of phase transitions: (1) non-conventional superconductors, (2) localized magnetism, and (3) ferroelectricity.
This paper deals with applications of non-Gaussian method to treat spatial fluctuations and their manifestations in the near-critical region. The basis for calculations is the celebrated Landau-Ginzburg-Wilson Hamiltonian. The results obtained are largely analytical, but numerical computations have also been done for further support and illustration. We have calculated expressions for the partition function, dynamic susceptibility, and total scattering intensity, as well as the correlation function and the correlation range. It is demonstrated that, in the limit of the volume V\ensuremath{\rightarrow}\ensuremath{\infty} or the reduced temperature \ensuremath{\tau}==(T-Tc{\mathit{T}}_{\mathit{c}}Tc​)/Tc{\mathit{T}}_{\mathit{c}}Tc​ being large enough, the obtained results all tend to their Gaussian estimates. However, for finite volumes or small but nonzero \ensuremath{\epsilon}, asymptotic behaviors are found that are mostly governed by exponential functional relations. Two illustrative examples are provided for experimental support: (1) Brillouin scattering in ferroelastics and (2) Raman scattering in simple fluids.