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arXiv:cond-mat/9909322 · Physical Review Letters

Deterministic Equations of Motion and Dynamic Critical Phenomena

Bo Zheng, M. Schulz, Steffen Trimper

Published Mar 1, 1999• 68 citations
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Abstract

Taking the two-dimensional ${\ensuremath{\varphi}}^{4}$ theory, we numerically solve the deterministic equations of motion with random initial states. Short-time behavior of the solutions is systematically investigated. Assuming that the solutions generate a microcanonical ensemble of the system, we demonstrate that the second-order phase transition point can be determined from the short-time dynamic behavior. An initial increase in the magnetization and a critical slowing down are observed. The dynamic critical exponent $z$, the new exponent $\ensuremath{\theta}$, and the static exponents $\ensuremath{\beta}$ and $\ensuremath{\nu}$ are estimated. The deterministic dynamics with random initial states is in the same universality class as the Monte Carlo dynamics.

DOI
10.1103/physrevlett.82.1891

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Taking the two-dimensional \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 theory, we numerically solve the deterministic equations of motion with random initial states. Short-time behavior of the solutions is systematically investigated. Assuming that the solutions generate a microcanonical ensemble of the system, we demonstrate that the second-order phase transition point can be determined from the short-time dynamic behavior. An initial increase in the magnetization and a critical slowing down are observed. The dynamic critical exponent zzz, the new exponent \ensuremathθ\ensuremath{\theta}\ensuremathθ, and the static exponents \ensuremathβ\ensuremath{\beta}\ensuremathβ and \ensuremathν\ensuremath{\nu}\ensuremathν are estimated. The deterministic dynamics with random initial states is in the same universality class as the Monte Carlo dynamics.
Non-equilibrium critical dynamics of the two-dimensional XY model is investigated with Hamiltonian equations of motion. Critical relaxation starting from both ordered and random states is carefully analyzed, and the short- time dynamic scaling behavior is revealed. Logarithmic corrections to scaling are detected for relaxation with a random initial state, while power-law corrections to scaling are observed for relaxation with an ordered initial state. The static exponent η and dynamic exponent z are determined around and below the Kosterlitz-Thouless phase transition temperature. Our results show that the deterministic dynamics described by Hamiltonian equations is in the same universality class as the stochastic dynamics described by Monte Carlo algorithms and Langevin equations.
The short-time critical dynamics of the two-dimensional eight-state random-bond Potts model is investigated with large-scale Monte Carlo simulations. Dynamic relaxation starting from a disordered and an ordered state is carefully analyzed. The continuous phase transition induced by disorder is studied, and both the dynamic and static critical exponents are estimated. The static exponent beta/nu shows little dependence on the disorder amplitude r, while the dynamic exponent z and static exponent 1/nu vary with the strength of disorder.
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.
We employed Monte Carlo simulations and short-time dynamic scaling to determine the static and dynamic critical exponents for the generalized two-dimensional Blume-Capel model of spin-3/2. We showed that the critical behavior at the second-order phase-transition line between the paramagnetic and ferromagnetic phases is in the same universality class of the two-dimensional Ising model. However, at the double critical end point, which is present in the phase diagram of the model, the critical exponent beta , associated to the order parameter, is different from that of the Ising model.
Monte Carlo simulations of the short-time critical dynamics are reviewed. The short-time universal scaling behavior of the dynamic Ising model and Potts model are discussed in detail, while extension and application to more complex systems as the XY model, the fully frustrated XY model and other dynamic systems are also presented. The investigation of the universal behavior of the short-time dynamics not only enlarges the fundamental knowledge on critical phenomena but also, more interestingly, provides possible new ways to determine not only the new critical exponents θ and θ1, but also the traditional dynamic critical exponent z as well as all static critical exponents.