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Journal of Physics A Mathematical and General

Time coarse graining near critical points

Ann Bruce, David B. Nicolaides

Published Dec 21, 1987• 0 citations
Publisher

Abstract

The authors investigate the statistical properties of local variables, coarse grained in time, in systems undergoing continuous phase transitions. A general theory is developed which suggests universal scaling behaviour in the limit of large coarse-graining times. The predictions are substantiated with Monte Carlo studies of two-dimensional scalar models. A simple fractal model of the time profile of the ordering variable is developed which captures and illuminates its timescale-invariant character. The theory is used to study the critical behaviour of local resonance lineshapes. In the two-dimensional case studied explicitly the critical slowing down is shown to drive a crossover to a split slow-motion lineshape.

DOI
10.1088/0305-4470/20/18/036

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The authors investigate the statistical properties of local variables, coarse grained in time, in systems undergoing continuous phase transitions. A general theory is developed which suggests universal scaling behaviour in the limit of large coarse-graining times. The predictions are substantiated with Monte Carlo studies of two-dimensional scalar models. A simple fractal model of the time profile of the ordering variable is developed which captures and illuminates its timescale-invariant character. The theory is used to study the critical behaviour of local resonance lineshapes. In the two-dimensional case studied explicitly the critical slowing down is shown to drive a crossover to a split slow-motion lineshape.
The author reports some universal and non-universal aspects of the critical dynamics of the three-dimensional Ising model, obtained in extensive Monte Carlo finite-size simulations. The author shows that the time-dependence of the magnetization of finite lattices is composed of two kinds of fluctuations at the critical point: (i) phase fluctuations from one metastable minimum of the free energy to the other, dominating the long-time behaviour of the magnetization; (ii) critical fluctuations inside each minimum, decaying on a comparatively short timescale. Both kinds of fluctuations show up the same critical exponent z=2.10+or-0.02, which is also in excellent agreement with the exponent z of energy fluctuations.
Abstract Time evolution, near a phase transition in the critical domain of critical systems not far from equilibrium, using a Langevin-type evolution is studied. Typical quantities of interest are relaxation rates towards equilibrium, time-dependent correlation functions and transport coefficients. The main motivation for such a study is that, in systems in which the dynamics is local (on short time-scales, a modification of a dynamic variable has an influence only locally in space) when the correlation length becomes large, a large time-scale emerges, which characterizes the rate of time evolution. This phenomenon called critical slowing down leads to universal behaviour and scaling laws for time-dependent quantities. In contrast with the situation in static critical phenomena, there is no clean and systematic derivation of the dynamical equations governing the time evolution in the critical domain, because often the time evolution is influenced by conservation laws involving the order parameter, or other variables like energy, momentum, angular momentum, currents and so on. Indeed, the equilibrium distribution does not determine the driving force in the Langevin equation, but only the dissipative couplings are generated by the derivative of the equilibrium Hamiltonian, and directly related to the static properties. The purely dissipative Langevin equation specifically discussed, corresponding to static models like the f4 field theory and two-dimensional models. Renormalization group (RG) equations are derived, and dynamical scaling relations established.
Abstract Phase transitions, as one of the most intriguing phenomena in nature, are divided into first-order phase transitions (FOPTs) and continuous ones in current classification. While the latter shows striking phenomena of scaling and universality, the former has recently also been demonstrated to exhibit scaling and universal behavior within a mesoscopic, coarse-grained Landau–Ginzburg theory. Here we apply this theory to a microscopic model—the paradigmatic Ising model, which undergoes FOPTs between two ordered phases below its critical temperature—and unambiguously demonstrate universal scaling behavior in such FOPTs. These results open the door for extending the theory to other microscopic FOPT systems and experimentally testing them to systematically uncover their scaling and universal behavior.
We employ the machinery of smooth scaling and coarse-graining of observables, developed recently by us in the context of so-called fluctuation operators (inspired by prior work of Verbeure et al) to make a rigorous renormalisation group analysis of the critical regime. The approach appears to be quite general, encompassing classical, quantum, discrete and continuous systems. One of our central topics is the analysis of the famous `scaling hypothesis', that is, we make a general investigation under what conditions on the l-point correlation functions a scale invariant (non-trivial) limit theory can be actually attained. Furthermore, we study in a rigorous manner questions like the quantum character of the system in the scaling limit, the phenomenon of critical slowing down etc.