Narrow Results

This work explores the spectral behavior of interacting many-body systems — gravitating dust solutions (galaxy formations and black hole clusters) and Brownian fluids. The eigenvalue dynamics of these systems are then represented by the two-dimensional Yang–Mills field (i.e. spectral projection). The interacting particles in the many-body systems are associated with random matrices of dimensions, N. The Painlevé II dynamical system is shown to surface at large N ($N\to \infty$); when the mentioned Yang–Mills field is configured in a specific way. Critical phenomena (Douglas–Kazakov phase transition) of the interacting many-body systems at large N were attained via the spectral projection of the mentioned physical systems. In addition, the existence of instantons (spectral Dirac monopoles) in the strong coupling phase was shown during phase transition.

We review Monte Carlo simulations of the Ising model and similar models in three dimensions that were performed in the last decade. Only recently, Monte Carlo simulations provide more accurate results for critical exponents than field theoretic methods, such as the ∊-expansion. These results were obtained with finite size scaling and "improved actions". In addition, we summarize Monte Carlo results for universal amplitude ratios, the interface tension, and the dimensional crossover from three to two dimensions.

We apply two nonparametric methods to further test the hypothesis that log-periodicity characterizes the detrended price trajectory of large financial indices prior to financial crashes or strong corrections. The term "parametric" refers here to the use of the log-periodic power law formula to fit the data; in contrast, "nonparametric" refers to the use of general tools such as Fourier transform, and in the present case the Hilbert transform and the so-called (H, q)-analysis. The analysis using the (H, q)-derivative is applied to seven time series ending with the October 1987 crash, the October 1997 correction and the April 2000 crash of the Dow Jones Industrial Average (DJIA), the Standard & Poor 500 and Nasdaq indices. The Hilbert transform is applied to two detrended price time series in terms of the ln(t_c-t) variable, where t_c is the time of the crash. Taking all results together, we find strong evidence for a universal fundamental log-frequency f=1.02±0.05 corresponding to the scaling ratio λ=2.67±0.12. These values are in very good agreement with those obtained in earlier works with different parametric techniques. This note is extracted from a long unpublished report with 58 figures available at , which extensively describes the evidence we have accumulated on these seven time series, in particular by presenting all relevant details so that the reader can judge for himself or herself the validity and robustness of the results.

Numerical investigation of critical exponents on a hypercubic lattice with L^d random sites with L up to 33 and d up to 7 showed that above the critical dimension the phase transitions in Ising model and percolation are not alike.

We perform extensive molecular dynamics simulations to study the mass size distribution of a two-dimensional fragmentation process. Our model consists of a large number of particles interacting through the Lennard–Jones potential. The fragmentation is induced by suddenly imposing a radial component on the particles' velocities, in order to mimic an explosion phenomenon. We then investigate the effect of the input energy on the resulting mass distribution of fragments.

We investigated a critical short-time relaxation in a lattice gauge theory. A systematic procedure of estimating critical point based on the "short-time scaling" is formulated. It is applied to the (2+1)-dimensional SU(2) lattice gauge theory at finite temperature to deduce its critical point. Finally, we studied the short-time relaxation behavior at the critical temperature starting either from "cold" and "hot" initial configuration, and calculated the dynamic critical exponents θ and z, as well as the static exponents β/ν.

We investigate the critical short-time scaling of the two-dimensional lattice ϕ⁴ field theory with a Langevin dynamics. Starting from a "hot" initial configuration but with a small magnetization, the critical initial increase of the magnetization is observed, through which we determine the critical point. From the short-time relaxation dynamics of various quantities at the critical point obtained, the dynamic critical exponents θ, z and the static exponent β/ν are evaluated. In executing a Langevin simulation, an appropriate discretization method with respect to the time degree of freedom becomes essentially important to be adopted and we show that the well-known second-order form of discretized Langevin equation works well to investigate the short-time dynamics.

A long-standing issue in a three-dimensional Ising spin glass (3DISG) is to settle the critical exponents. Numerical simulations and a conventional finite-size-scaling analysis have given controversial estimates, in particular, for the value of thermal exponent ν. Here a recently proposed extended scaling scheme is reviewed and illustrated for the canonical two-dimensional Ising ferromagnet. By applying the extended scaling to our extensive Monte Carlo data for 3DISG, consistent estimates of critical parameters are obtained in contrast to previous analyses.

Using the mean-field approximation, we study the reorientation transition in a discrete dipolar spin model. We array the d-dimensional spins in the d-1-dimensional hypercubic lattice points, and apply the magnetic anisotropy in the dth direction. Even though the long-range dipolar interaction is truncated to the nearest-neighbor interaction, the model exhibits the reorientation transition. The effects of the exchange interaction on the dipolar system are also studied.

In liquid mixtures or analogous binary systems at low temperatures the pure phases may coexist, separated by an interface. The interface tension vanishes according to σ=σ₀(1−T/T_c)^µ as the temperature T approaches the critical point from below. Similarly the correlation length diverges as ξ=ξ₀(T/T_c−1)^−ν. For three-dimensional systems the dimensionless product R=σ₀(ξ₀)² is universal. This and related universal quantities are investigated by means of a semiclassical expansion in the framework of Φ⁴ theory, as well as by Monte Carlo calculations in the three-dimensional Ising model.

We investigate two three-dimensional Ising models with non-Hamiltonian Glauber dynamics. The transition probabilities of these models can, just as in the case of equilibrium models, be expressed in terms of Boltzmann factors depending only on the interacting spins and the bond strengths. However, the bond strength associated with each lattice edge assumes different values for the two spins involved. The first model has cubic symmetry and consists of two sublattices at different temperatures. In the second model a preferred direction is present. These two models are investigated by Monte Carlo simulations on the Delft Ising System Processor. Both models undergo a phase transition between an ordered and a disordered state. Their critical properties agree with Ising universality. The second model displays magnetization bistability.

With recent improvements in simulation methods and data analysis techniques, Monte Carlo simulations, used in conjunction with finite-size scaling techniques, have become the most powerful and flexible tools for studying critical phenomena. In particular, there has been a dramatic increase in resolution in studies of the three-dimensional Ising model. This paper discusses motivations for performing high-resolution studies, reviews past and current work on the model and points out some potential potholes on the road to higher resolution.

We determine the critical temperature in the four-dimensional nearest-neighbor Ising model as J/k_B T_c=0.149694±0.000002 from kinetic Monte Carlo simulations of up to 576⁴ spins. Here we assume the critical magnetization to decay with time as (t/ log t)^-1/2. However, possible logarithmic additions to this leading scaling behavior could change the estimate beyond these error bars. A reanalyzis of old series expansions for the susceptibility and fourth moment gives 0.149696±0.000004.

The sensitivity on shape and boundaries at the Curie point of the universal critical ratio L^dM²/χ for the Ising model is confirmed by a comparison of square and rectangular lattices. Tests with three different algorithms give the same results for the universal critical ratio on various lattice types and always the same dependence on the system's aspect ratio.

Simulations with up to 112⁵ spins confirm the prediction of Chen and Dohm of three different finite-size exponents for the spontaneous magnetization near the Curie point.

A heuristic optimization algorithm is developed based on the renormalization group idea and applied to the Edwards–Anderson model in two dimensions with symmetric Gaussian bond distribution. Droplets are forced to excite by the boundary condition. We find that the droplets have nontrivial fractal dimension not only for its surface but also for its volume, in contrast to what has been implicitly assumed in conventional droplet arguments.

Nonequilibrium relaxation (NER) process turns out to be an useful method to study phase transition and critical phenomena. The NER method and its applications are reviewed. Characteristic behavior of the local exponent is analyzed with a simple model.

A general introduction into the subject aimed at a general theoretical physics audience. We introduce the sign problem posed by finite density lattice QCD, and we discuss the main methods proposed to circumvent it, with emphasis on the imaginary chemical potential approach. The interrelation between Taylor expansion and analytic continuation from imaginary chemical potential is discussed in detail. The main applications to the calculation of the critical line, and to the thermodynamics of the hot and normal phase are reviewed.

Gravity can play a role in critical phenomena. Topological singularities induce ground state degeneracy and break the continuum symmetry of the vacuum. They also generate momenta oscillations about an average momentum and a positive gravitational susceptibility. Gravitational analogues of the laws of Curie and Bloch have been found for a one-dimensional model. The critical temperature for a change in phase from bound to unbound vortices has also been calculated in a XY-model.

The general dynamics of the spherically symmetric gravitational collapse of a massless scalar field is examined. We apply the Galerkin projection method to transform the partial differential equations ruling the dynamics of the collapse into a finite set of ordinary differential equations for the modal coefficients, after a convenient truncation procedure. We thus generate a dynamical system that not only reproduces the essential features of the dynamics of the gravitational collapse, even for a low order of truncation, but also discloses the structure of the dynamics. Each initial condition in modal space corresponds to a well defined spatial distribution of the scalar field. Numerical experiments with the system show that two main asymptotic configurations are possible: either the formation of a black hole or the dispersion of the scalar field leaving behind flat spacetime. Between the two asymptotic states there is a critical solution represented by a periodic orbit in the modal space with period Δ u ≈ 3.55. This periodic orbit acts as an intermediate attractor for the dynamics, being a limiting configuration for orbits generated from initial conditions in a critical surface of codimension one in the N-dimensional modal space. The compactness and fractality of this surface are briefly discussed.