Dynamic phase transitions and dynamic phase diagrams of the Blume–Emery–Griffiths model in an oscillating field: the effective-field theory based on the Glauber-type stochastic dynamics

Using the effective-field theory based on the Glauber-type stochastic dynamics (DEFT), we investigate dynamic phase transitions and dynamic phase diagrams of the Blume–Emery–Griffiths model under an oscillating magnetic field. We presented the dynamic phase diagrams in (T/J, h₀/J), (D/J, T/J) and (K/J, T/J) planes, where T, h₀, D, K and z are the temperature, magnetic field amplitude, crystal–field interaction, biquadratic interaction and the coordination number. The dynamic phase diagrams exhibit several ordered phases, coexistence phase regions and special critical points, as well as re-entrant behavior depending on interaction parameters. We also compare and discuss the results with the results of the same system within the mean-field theory based on the Glauber-type stochastic dynamics and find that some of the dynamic first-order phase lines and special dynamic critical points disappeared in the DEFT calculation.     

Convergence to Equilibrium of Random Ising Models in the Griffiths Phase

We consider Glauber-type dynamics for disordered Ising spin systems with nearest neighbor pair interactions in the Griffiths phase. We prove that in a nontrivial portion of the Griffiths phase the system has exponentially decaying correlations of distant functions with probability exponentially close to 1. This condition has, in turn, been shown elsewhere to imply that the convergence to equilibrium is faster than any stretched exponential, and that the average over the disorder of the time-autocorrelation function goes to equilibrium faster than exp[–k(log t) ^d/(d–1)]. We then show that for the diluted Ising model these upper bounds are optimal.     

Shape dependence of the finite-size scaling limit in a strongly anisotropic $\mathsf{O(\infty)}$ model

We discuss the shape dependence of the finite-size scaling limit in a strongly anisotropic O(N) model in the large-N limit. We show that scaling is observed even if an incorrect value for the anisotropy exponent is considered. However, the related exponents may only be effective ones, differing from the correct critical exponents of the model. We discuss the implications of our results for numerical finite-size scaling studies of strongly anisotropic systems.Received: 9 April 2003, Published online: 4 August 2003PACS:   05.70.Jk Critical point phenomena - 64.60.-i General studies of phase transitions     

Interplay between phase ordering and roughening on growing films

We study the interplay between surface roughening and phase separation during the growth of binary films. Renormalization group calculations are performed on a pair of equations coupling the interface height and order parameter fluctuations. We find a larger roughness exponent at the critical point of the order parameter compared to the disordered phase, and an increase in the upper critical dimension for the surface roughening transition from two to four. Numerical simulations performed on a solid-on-solid model with two types of deposited particles corroborate some of these findings. However, for a range of parameters not accessible to perturbative analysis, we find non-universal behavior with a continuously varying dynamic exponent.Received: 23 July 2003, Published online: 23 December 2003PACS: 68.35.Rh Phase transitions and critical phenomena - 05.70.Jk Critical point phenomena - 05.70.Ln Nonequilibrium and irreversible thermodynamics - 64.60.Cn Order-disorder transformations; statistical mechanics of model systems     

Dynamic Dipole and Quadrupole Phase Transitions in the Kinetic Spin-3/2 Model

The dynamic phase transition has been studied, within a mean-field approach, in the kinetic spin-3/2 Ising model Hamiltonian with arbitrary bilinear and biquadratic pair interactions in the presence of a time dependent oscillating magnetic field by using the Glauber-type stochastic dynamics. The nature (first- or second-order) of the transition is characterized by investigating the behavior of the thermal variation of the dynamic order parameters and as well as by using the Liapunov exponents. The dynamic phase transitions (DPTs) are obtained and the phase diagrams are constructed in the temperature and magnetic field amplitude plane and found nine fundamental types of phase diagrams. Phase diagrams exhibit one, two or three dynamic tricritical points, and besides a disordered (D) and the ferromagnetic-3/2 (F_3/2) phases, six coexistence phase regions, namely F _3/2+ F _1/2, F _3/2+ D, F _3/2+ F _1/2+ FQ, F _3/2+ FQ, F _3/2+ FQ + D and FQ + D, exist in which depending on the biquadratic interaction. PACS number(s): 05.50.+q, 05.70.Fh, 64.60.Ht, 75.10.Hk     

Continuous percolation phase transitions of two-dimensional lattice networks under a generalized Achlioptas process

The percolation phase transitions of two-dimensional lattice networks under a generalized Achlioptas process (GAP) are investigated. During the GAP, two edges are chosen randomly from the lattice and the edge with minimum product of the two connecting cluster sizes is taken as the next occupied bond with a probability p. At p = 0.5, the GAP becomes the random growth model and leads to the minority product rule at p = 1. Using the finite-size scaling analysis, we find that the percolation phase transitions of these systems with 0.5 ≤ p ≤ 1 are always continuous and their critical exponents depend on p. Therefore, the universality class of the critical phenomena in two-dimensional lattice networks under the GAP is related to the probability parameter p in addition.     

On the number of contacts of two polymer chains situated on fractal structures

We study the critical behavior of the number of monomer-monomer contacts for two polymers in a good solvent. Polymers are modeled by two self-avoiding walks situated on fractals that belong to the checkerboard (CB) and X family. Each member of a family is labeled by an odd integer b, . By applying the exact Renormalization Group (RG) method, we establish the relevant phase diagrams whereby we calculate the contact critical exponents (for the CB and X fractals with b = 5 and b = 7). The critical exponent is associated with power law of the number of sites at which the two polymers are touching each other.Received: 12 March 2004, Published online: 3 August 2004PACS: 64.60.Ak Renormalization-group, fractal, and percolation studies of phase transitions - 36.20.Ey Conformation (statistics and dynamics)     

Immunization and epidemic dynamics in complex networks

We study the behavior of epidemic spreading in networks, and, in particular, scale free networks. We use the Susceptible-Infected-Removed (SIR) epidemiological model. We give simulation results for the dynamics of epidemic spreading. By mapping the model into a static bond-percolation model we derive analytical results for the total number of infected individuals. We study this model with various immunization strategies, including random, targeted and acquaintance immunization.Received: 3 November 2003, Published online: 14 May 2004PACS: 02.50.Cw Probability theory - 02.10.Ox Combinatorics; graph theory - 89.20.Hh World Wide Web, Internet - 64.60.-i General studies of phase transitions     

Dynamic phase transition in the kinetic spin-1 Blume-Capel model: Phase diagrams in the temperature and crystal-field interaction plane

Within a mean-field approach, the stationary states of the kinetic spin-1 Blume-Capel model in the presence of a time-dependent oscillating external magnetic field is studied. The Glauber-type stochastic dynamics is used to describe the time evolution of the system and obtain the mean-field dynamic equation of motion. The dynamic phase-transition points are calculated and phase diagrams are presented in the temperature and crystal-field interaction plane. According to the values of the magnetic field amplitude, three fundamental types of phase diagrams are found: One exhibits a dynamic tricritical point, while the other two exhibit a dynamic zero-temperature critical point. The text was submitted by the authors in English.     

A Model with Simultaneous First and Second Order Phase Transitions

We introduce a two dimensional nonlinear XY model with a second order phase transition driven by spin waves, together with a first order phase transition in the bond variables between two “bond ordered phases”, one with local ferromagnetic order and another with local anti-ferromagnetic order. We also prove that at the transition temperature the bond-ordered phases coexist with a disordered phase as predicted by Domany, Schick and Swendsen [1]. This last result generalizes the result of van Enter and Shlosman [2] We argue that these phenomena are quite general and should occur for a large class of potentials. PACS number: 64.60.Cn, 75.10.Hk     

The dynamic phase transition in the spin-1/2 Ising system within the path probability method

We study the dynamic phase transitions and present the dynamic phase diagrams of the spin-1/2 Ising system under the presence of a time-varying (sinusoidal) external magnetic field within the path probability method (PPM) of Kikuchi and we observe that the PPM gives exactly the same result as with the Glauber-type stochastic dynamics based on the mean-field theory (DMFT). We also investigate the influence of the rate constant on the dynamic phase diagrams in detail and five new and interesting dynamic phase diagrams are found. We notice that the derivation of the dynamic equations by using the PPM is more clear and easier than within the DMFT and the Glauber-type stochastic dynamics based on the effective-field theory (DEFT). The advantages and disadvantages of the PPM over the DMFT and DEFT are also discussed.     

Breakdown of universality in directed spiral percolation

Directed spiral percolation (DSP), percolation under both directional and rotational constraints, is studied on the triangular lattice in two dimensions (2D). The results are compared with that of the 2D square lattice. Clusters generated in this model are generally rarefied and have chiral dangling ends on both the square and triangular lattices. It is found that the clusters are more compact and less anisotropic on the triangular lattice than on the square lattice. The elongation of the clusters is in a different direction than the imposed directional constraint on both the lattices. The values of some of the critical exponents and fractal dimension are found considerably different on the two lattices. The DSP model then exhibits a breakdown of universality in 2D between the square and triangular lattices. The values of the critical exponents obtained for the triangular lattice are not only different from that of the square lattice but also different form other percolation models.Received: 12 March 2004, Published online: 23 July 2004PACS: 02.50.-r Probability theory, stochastic processes, and statistics - 64.60.-i General studies of phase transitions - 72.80.Tm Composite materials     

Dynamic phase transition in the kinetic Blume--Emery--Griffiths model: Phase diagrams in the temperature and interaction parameters planes

As a continuation of our previously published work, the dynamic phase transitions are studied further, within a mean-field approach, in the kinetic Blume--Emery--Griffiths model in the presence of a time varying (sinusoidal) magnetic field by using the Glauber-type stochastic dynamics. The dynamic phase transitions are obtained and the phase diagrams are constructed in two different planes, namely in the reduced temperature (T) and biquadratic interaction (k) plane and found eight fundamental types of phase diagrams for various values of reduced crystal-field interaction (d) and magnetic field amplitude (h), and in the (T,?d) plane and obtained six distinct topologies for different values of k and h. Phase diagrams exhibit one or two dynamic tricritical points and a dynamic double critical end point, dynamic triple and quadruple points, and besides disordered and ordered phases, three coexistence phase regions exist in which occurring of these strongly depend on the values of d, k and h.     

Frenkel-Kontorova Model with Alternant Coupling Potential

We have studied a modified Frenkel-Kontorova (FK) model with alternant coupling potential. From it, we obtain a coupling conservative map, it shows that the gold-mean number is not the last broken winding number, and the broken critical value is varied with the variance of strength of spire. The phase diagram becomes asymmetric in a period. The Devil's staircase and generalized dimension are different from those of the standard FK model.PACS numbers: 64.60.Fr, 05.70.Fh, 05.70.Jk     

Spin glass and antiferromagnetism in Kondo-lattice disordered system

The competition between spin glass (SG), antiferromagnetism (AF) and Kondo effect is studied here in a model which consists of two Kondo sublattices with a Gaussian random interaction between spins in different sublattices with an antiferromagnetic mean J ₀ and standard deviation J. In the present approach there is no hopping of the conduction electrons between the sublattices and only spins in different sublattices can interact. The problem is formulated in the path integral formalism where the spin operators are expressed as bilinear combinations of Grassmann fields which can be solved at mean field level within the static approximation and the replica symmetry ansatz. The obtained phase diagram shows the sequence of phases SG, AF and Kondo state for increasing Kondo coupling. This sequence agrees qualitatively with experimental data of the Ce₂Au_1-x Co _x Si₃ compound.Received: 9 April 2003, Published online: 9 September 2003PACS: 05.50.+q Lattice theory and statistics; Ising problems - 64.60.Cn Order disorder transformations; statistical mechanics of model systems     

The kinetic spin-1 Blume-Capel model with Glauber dynamic

We study the dynamic phase transitions (DPT), within a mean-field approach, in the kinetic spin-1 Blume-Capel model by using the Glauber-type stochastic dynamics. The nature of the transition is characterized by investigating the behavior of the thermal variation of the dynamic order parameter and the Lyapunov exponent. The phase diagram is constructed in the temperatures (T) and single-ion anisotropy amplitude (D) plane. Our results predict first-order transitions at low temperature and large anisotropy strengths, which correspond in the phase diagram to the existence of a nonequilibrium tricritical point (TCP). We compare our results with the equilibrium phase diagram.     

Mixed Ising system designed with integer and half-integer spins: dynamic behaviors under oscillating magnetic field

By using the simplified effective-field theory based on Glauber-type stochastic dynamics, namely the dynamic simplified effective-field theory, the dynamic phase diagrams of a two-dimensional mixed spin-1 and spin-3/2 Blume–Capel model are studied in an oscillating external magnetic field. The dynamic equations are derived for two interpenetrating square lattices. The time variations of average magnetizations and the temperature dependence of the dynamic magnetizations are examined, and the dynamic phase diagrams are presented in the two different planes. The dynamic phase diagrams illustrate several ordered phases, the coexistence phase region and critical points as well as a re-entrant behavior depending on the interaction parameters. Finally, the discussion and comparison of the dynamic phase diagrams are given briefly.     

Condensation in the Zero Range Process: Stationary and Dynamical Properties

The zero range process is of particular importance as a generic model for domain wall dynamics of one-dimensional systems far from equilibrium. We study this process in one dimension with rates which induce an effective attraction between particles. We rigorously prove that for the stationary probability measure there is a background phase at some critical density and for large system size essentially all excess particles accumulate at a single, randomly located site. Using random walk arguments supported by Monte Carlo simulations, we also study the dynamics of the clustering process with particular attention to the difference between symmetric and asymmetric jump rates. For the late stage of the clustering we derive an effective master equation, governing the occupation number at clustering sites.     

Dynamical phase transition in two-dimensional fully frustrated Josephson junction arrays with resistively shunted junction dynamics

The dynamical phase transitions in two-dimensional fully frustrated Josephson junction arrays at zero temperature are investigated numerically with the resistively shunted junction model through the fluctuating twist boundary condition. The model is subjected to a driving current with nonzero orthogonal components i _x , i _y parallel to both axes of the square lattice. We find a roughly lattice size independent phase diagram with three dynamical phases: a pinned vortex lattice phase, a moving vortex lattice phase and a moving plastic phase. The phase diagram shows a direct transition from the pinned vortex to the moving vortex phase and the separation of the pinned vortex and the moving plastic phases. The time-dependent voltages v _x and v _y are periodic in the moving vortex lattice phase. But they are aperiodic in the moving plastic phase, resulting in non-monotonic characteristics and hysteresis in the current-voltage curves. It is found that the characteristic frequency is twice the time-averaged voltage in the moving vortex phase and around the time-averaged voltage in the plastic flow regime.Received: 29 May 2003, Published online: 2 October 2003PACS: 64.60.Ht Dynamic critical phenomena - 74.25.Sv Critical currents - 74.25.Fy Transport properties     

Lennard-Jones fluids confined in nanoscopic slits: Evidence for reentrant filling transitions

By using grand canonical and canonical ensemble Monte Carlo simulations, the structure and phase behavior of a Lennard-Jones (LJ) fluid confined between the parallel (100) planes of a face-centered cubic crystal are studied. Slit pores with a width which allows three adsorbate layers to form are used. It is shown that the filled pore consists of three commensurate layers over a wide range of the surface potential strength, while the pore-filling mechanism and the topology of the phase diagram change when the strength of this fluid-wall potential is varied. Condensation may occur in one step or via two layering-like transitions. The structure of monolayer films depends on the strength and corrugation of the surface potential, and the condensation of the middle layer may induce a reentrant first-order transition.Received: 16 January 2004, Published online: 23 March 2004PACS: 64.70.Nd Structural transitions in nanoscale materials - 64.60.Cn Order-disorder transformations; statistical mechanics of model systems - 68.35.Rh Phase transitions and critical phenomena