Deterministic Equations of Motion and Dynamic Critical Phenomena

Taking the two-dimensional $\phi^4$ theory as an example, 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 already from the short-time dynamic behavior. Initial increase of the magnetization and critical slowing down are observed. The dynamic critical exponent z, the new exponent $\theta$ and the static exponents $\beta$ and $\nu$ are estimated. Interestingly, the deterministic dynamics with random initial states is in a same dynamic universality class of Monte Carlo dynamics.