Kinetic models for nonequilibrium phase transitions have been extensively studied via mean-field theory, series expansions, and Monte Carlo simulation. In this paper we present a different approach based on studying the normal-mode solutions of the master equation for the system to extract information on the static and dynamic behavior of the system in the vicinity of the phase transition. This method provides insight into the physics of the transition and offers an alternative technique for analyzing kinetic phase transitions. We demonstrate the method by extracting the critical behavior of a simple model from finite-size scaling.
We explore the critical properties of the recently discovered finite-time dynamical phase transition in the nonequilibrium relaxation of Ising magnets after a temperature quench. The transition is characterized by a sudden switch in the relaxation dynamics and it occurs at a sharp critical time. While previous works have focused either on mean-field interactions or on investigating the critical time, we analyze the critical fluctuations at the phase transition in the nearest-neighbor Ising model on a square lattice using Monte Carlo simulations. By means of a finite-size scaling analysis, we extract the critical exponents for the transition. In two spatial dimensions, we find that the exponents are consistent with those of the two-dimensional Ising universality class when the system is initially in the vicinity of the critical point. For initial temperatures below the critical one, however, the critical exponents differ from the Ising-exponents, indicating a distinct dynamical critical phenomenon.
The current state of Monte Carlo work on phase transitions and critical phenomena is reviewed. Both classical and quantum Monte Carlo results are discussed with emphasis on statistical lattice-model studies involving the highly accurate calculation of critical exponents. It is shown that finite-size scaling is an effective method whereby not only simple lattice models but also those of complex practical magnetic materials can be treated. For models allowing crossover phenomena, it is shown that both finite-size scaling theory and conventional power-law functions must be used in carrying out a Monte Carlo analysis.
We report a Monte Carlo study on the critical relaxation of the three-dimensional kinetic Ising model from a nonequilibrium initial state. It is shown that the details of the method, namely, a choice of the sampling procedure and a functional form of the transition probability do not affect an asymptotic power-law relaxation. The finite-size effects on the critical relaxation is also studied. Cross-over from the power-law relaxation to the exponential one is found, which is discussed in view of the dynamical finite-size scaling theory.
The nonequilibrium Ising model on a restricted scale-free network has been studied with one- and two-spin flip competing dynamics employing Monte Carlo simulations. The dynamics present in the system can be defined by the probability
q in which the one-spin flip process simulate the contact with a heat bath at a given temperature
T, and with a probability (
1−q) the two-spin flip process mimics the system subjected to an external flux of energy into it. The system network is described by a power-law degree distribution in the form
P(k)∼k−α, and the restriction is made by fixing the maximum,
km, and minimum,
k0, degree on distribution for the whole network size. This restriction keeps finite the second and fourth moment of degree distribution, allowing us to obtain a finite critical point for any value of
α. For these critical points, we have calculated the thermodynamic quantities of the system, such as, the total
mNF and staggered
mNAF magnetizations per spin, susceptibility
χN, and reduced fourth-order Binder cumulant
UN, for several values of lattice size
N and exponent
1≤α≤5. Therefore, the phase diagram was built and a self-organization phenomena is observed from the transitions between antiferromagnetic AF to paramagnetic P, and P to ferromagnetic F phases. Using the finite-size scaling theory, we also obtained the critical exponents for the system, and a mean-field critical behavior is observed, exhibiting the same universality class of the system on the equilibrium and out of it.