We study the short-time dynamics of a mean-field model with non-conserved order parameter (Curie–Weiss with Glauber dynamics) by solving the associated Fokker–Planck equation. We obtain closed-form expressions for the first moments of the order parameter, near to both the critical and spinodal points, starting from different initial conditions. This allows us to confirm the validity of the short-time dynamical scaling hypothesis in both cases. Although the procedure is illustrated for a particular mean-field model, our results can be straightforwardly extended to generic models with a single order parameter.
We study the short-time dynamics of a mean-field model with non-conserved order parameter (Curie–Weiss with Glauber dynamics) by solving the associated Fokker–Planck equation. We obtain closed-form expressions for the first moments of the order parameter, near to both the critical and spinodal points, starting from different initial conditions. This allows us to confirm the validity of the short-time dynamical scaling hypothesis in both cases. Although the procedure is illustrated for a particular mean-field model, our results can be straightforwardly extended to generic models with a single order parameter.
Given an arbitrary finite dimensional Hamiltonian H 0 , we consider the model H = H 0 +Δ H , where Δ H is a generic fully connected interaction. By using the strong law of large numbers, we easily prove that, for all such models, a generalized Curie-Weiss mean-field equation holds. Unlike traditional mean-field models, the term H 0 gives rise to short-range correlations and, furthermore, when H 0 has negative couplings, first-order phase transitions and inverse transition phenomena may take place even when only two-body interactions are present. The dependence from a non-uniform external field and finite-size effects are also explicitly calculated. Partially, these results were derived long ago by using min-max principles but remained almost unknown.
We investigate the short-time quantum critical dynamics in the imaginary-time relaxation processes of finite-size systems. Universal scaling behaviors exist in the imaginary-time evolution. In particular, the system undergoes a critical initial slip stage characterized by an exponent $\ensuremath{\theta}$, in which an initial power-law increase emerges in the imaginary-time correlation function when the initial state has a zero order parameter and a vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo (QMC) simulations. In the one-dimensional case, we locate the quantum critical point at (h/J)c=1.00003(8) in the thermodynamic limit, and we estimate the critical initial slip exponent $\ensuremath{\theta}=0.3734(2)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)$ by analyzing data on chains of length L=32−−256 and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by 3.04451(7) in the thermodynamic limit, while the critical exponents are $\ensuremath{\theta}=0.209(4)$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)$ estimated by data on systems of size L=24--64 and 32--64, respectively. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts due to the essential differences between classical and quantum dynamics. The short-time critical dynamics and the imaginary-time relaxation QMC approach can be readily adapted to various models.
In the companion Letter we have highlighted the dynamical universality class of the Nagle-Kardar model, where a mean-field interaction is added to the one-dimensional nearest-neighbor Ising model. Starting from the microscopic Glauber dynamics, this paper provides a complete derivation of the Fokker-Planck equation that describes the time evolution of the macroscopic variables (magnetization and defect density) appearing in the model's Hamiltonian. The study of the Langevin equation, associated with the Fokker-Planck equation, allowed us to prove that the model belongs to the universal class of systems with diffusive dynamics and non-conserved order parameter (model A). To this goal, we used several features of the model at equilibrium, including the phase diagram and the fluctuations of macroscopic variables, which are here discussed for completeness. The derivation of the Fokker-Planck equation requires the solution of some combinatorial problems that appear in the counting of configurations at an intermediate level between the microscopic Glauber dynamics and the macroscopic Fokker-Planck one. Finally, we apply both the Glauber and the Langevin dynamics to the study of the average first passage time between local equilibrium states. We confirm that this time obeys an exponential Arrhenius law in terms of system's size, offering a direct link between microscopic energy landscapes and macroscopic relaxation mechanisms.
Understanding the dynamical universality classes of systems with long-range interactions remains a key challenge in statistical physics. In this Letter, we analytically and numerically investigate the non-equilibrium critical dynamics of the one-dimensional spin-1/2 Nagle-Kardar model, which is characterized by the competition between short- and long-range interactions and the presence of a tricritical point. We focus on the slowing-down of the magnetization m at criticality under Glauber dynamics. Starting from the corresponding master equation, we perform a coarse-graining procedure to obtain a Fokker-Planck equation for the macroscopic variables. Then, the asymptotic decay of the magnetization is derived using central manifold theory. We find that m decays as t−1/2 along the critical line and as t−1/4 precisely at the tricritical point. This finding confirms that the dynamical critical exponent is z=2 as for mean-field models, proving that the macroscopic critical dynamics of the Nagle-Kardar model falls within the dynamical universality class of purely relaxational, non-conserved order parameters (model A). While Kardar proved that the equilibrium Curie-Weiss theory extends to Ising models where nearest-neighbor interactions are included, we here show that such result is valid also for critical dynamics. Our work provides the semi-analytical solution for the critical dynamics of a model with mixed-range interactions, assigning its universality class.
The time evolution of the spin distribution for the XY model in mean-field theory is described by the Fokker–Planck transport equation. This equation is Fourier transformed to generate an infinite-dimensional nonlinear dynamical system. The otherwise well-known behavior of the XY model is then explained using a fully dynamical-systems approach. The evolution of the phase-space trajectories of this dynamical system from initial nonequilibrium states is obtained. Regardless of the initial spin distribution, its attractors (all are point attractors) determine the equilibrium magnetic state of the model. The phase transition at a critical temperature is accompanied by a supercritical bifurcation. Numerical results for the time dependence of the phase-space trajectories and the order parameter are presented.
We have analyzed the dynamic phase transition of the kinetic Ising model in mean-field approximation by means of an analytical approach. Specifically, we study the evolution of the system under the simultaneous influence of time-dependent and time-independent magnetic fields. We demonstrate that within the approximate analytical treatment of our approach, the dynamic phase transition exhibits power-law dependencies for the order parameter that have the same critical exponents as the mean-field equilibrium case. Moreover we have obtained an equation of state, with which we can prove that the time-independent field component is effectively the conjugate field of the order parameter. Our analysis is limited to the parameter range, in which only second-order phase transitions occur, i.e., for small applied field amplitudes and temperatures close to the Curie point. In order to ensure the reliability of our analytical results we have corroborated them by comparison to numerical evaluations of the same model.