Deterministic Equations of Motion and Dynamic Critical Phenomena

Taking the two-dimensional ${\phi }^4$ 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 $\theta$, and the static exponents $\beta$ and $\nu$ are estimated. The deterministic dynamics with random initial states is in the same universality class as the Monte Carlo dynamics.