A comparative study of the dynamic critical behavior of the four-state Potts like models

We investigate the short-time critical dynamics of the Baxter-Wu (BW) and n=3 Turban (3TU) models to estimate their global persistence exponent θ_g. We conclude that this new dynamical exponent can be useful in detecting differences between the critical behavior of these models which are very difficult to obtain in usual simulations. In addition, we estimate again the dynamical exponents of the four-state Potts (FSP) model in order to compare them with results previously obtained for the BW and 3TU models and to decide between two sets of estimates presented in the current literature. We also revisit the short-time dynamics of the 3TU model in order to check if, as already found for the FSP model, the anomalous dimension of the initial magnetization x₀ could be equal to zero.     

Short-time dynamics in the 1D long-range Potts model

We present numerical investigations of the short-time dynamics at criticality in the 1D Potts model with power-law decaying interactions of the form 1/r^1+σ. The scaling properties of the magnetization, autocorrelation function and time correlations of the magnetization are studied. The dynamical critical exponents θ' and z are derived in the cases q=2 and q=3 for several values of the parameter σ belonging to the nontrivial critical regime.     

Non-equilibrium phase transition of the fully frustrated square lattice Coulomb gas model

The non-equilibrium phase transitions of the fullyfrustrated (f = 1/2) square lattice Coulomb gas (CG) modeldriven by external electrical fields are studied in the frameworkof the short-time dynamic scaling approach. The criticaltemperature T_c, the static and dynamic critical exponents2β/ν, ν, and z are obtained for several smalldriving fields. The results show that T_c decreases with theincrease of electric field, and 2β/ν and z arestrongly dependent on the external electric field. Interestingly,contrary to the equilibrium case, in the presence of smallelectric field, the calculated exponent ν is close to that inpure 2D Ising model, which provides numerical evidence thatexternal electric field may change the universality class of thef = 1/2 CG system.     

Non-equilibrium behavior at a liquid-gas critical point

Second-order phase transitions in a non-equilibrium liquid-gas model with reversible mode couplings, i.e., model H for binary-fluid critical dynamics, are studied using dynamic field theory and the renormalization group. The system is driven out of equilibrium either by considering different values for the noise strengths in the Langevin equations describing the evolution of the dynamic variables (effectively placing these at different temperatures), or more generally by allowing for anisotropic noise strengths, i.e., by constraining the dynamics to be at different temperatures in d _|| - and d _⊥-dimensional subspaces, respectively. In the first, isotropic case, we find one infrared-stable and one unstable renormalization group fixed point. At the stable fixed point, detailed balance is dynamically restored, with the two noise strengths becoming asymptotically equal. The ensuing critical behavior is that of the standard equilibrium model H. At the novel unstable fixed point, the temperature ratio for the dynamic variables is renormalized to infinity, resulting in an effective decoupling between the two modes. We compute the critical exponents at this new fixed point to one-loop order. For model H with spatially anisotropic noise, we observe a critical softening only in the d _⊥-dimensional sector in wave vector space with lower noise temperature. The ensuing effective two-temperature model H does not have any stable fixed point in any physical dimension, at least to one-loop order. We obtain formal expressions for the novel critical exponents in a double expansion about the upper critical dimension d _c = 4 - d _|| and with respect to d _|| , i.e., about the equilibrium theory. Received 4 April 2002 Published online 13 August 2002     

Short-time behavior of the kinetic spherical model with long-ranged interactions

The kinetic spherical model with long-ranged interactions and an arbitrary initial order m₀ quenched from a very high temperature to T is solved. In the short-time regime, the bulk order increases with a power law in both the critical and phase-ordering dynamics. To the latter dynamics, a power law for the relative order is found in the intermediate time-regime. The short-time scaling relations of small m₀ are generalized to an arbitrary m₀ and all the time larger than . The characteristic functions for the scaling of m₀ and for are obtained. The crossover between scaling regimes is discussed in detail. Received 17 September 1999     

Directed spiral site percolation on the square lattice

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold p _c ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension d _f ≈ 1.733. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as p → p _c with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution P _s(p) show power law behaviour with | p - p _c| in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of P _s(p). The results obtained are in good agreement with other model calculations. Received 10 November 2002 / Received in final form 20 February 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: santra@iitg.ernet.in     

The short-time critical behaviour of the Ginzburg-Landau model with long-range interaction

The renormalisation group approach is applied to the study of the short-time critical behaviour of the d-dimensional Ginzburg-Landau model with long-range interaction of the form in momentum space. Firstly the system is quenched from a high temperature to the critical temperature and then relaxes to equilibrium within the model A dynamics. The asymptotic scaling laws and the initial slip exponents and of the order parameter and the response function respectively, are calculated to the second order in . Received 9 June 2000 and Received in final form 2 August 2000     

Depinning transition of the quenched Mullins-Herring equation: A short-time dynamic method

The depinning phase transition of the Mullins-Herring equation with an external driving force and quenched random noise is studied in a short-time dynamic scaling scheme. Besides the critical driving force, all the critical exponents can be accessed, agreeing well with those in long-time steady-state simulations. The finite size effects on the critical exponents are also discussed. It is found that reasonable results can be achieved with a relatively small system, which highlights the advantage of the present approach.     

A shell-model approach to fractal-induced turbulence

We present a shell-model of fractal induced turbulence which predicts that structure function scaling exponents decrease in absolute value as the fractal dimension of the turbulence-inducing fractal object increases. This qualitative prediction is in agreement with laboratory measurements. Finer details of the fractal induced turbulence statistics and dynamics depend on the fractal force's phases, i.e. on the detailed construction of the fractal stirrer. In a case of deterministic forcing phases, a critical fractal dimension exists below which the average rate of inter-scale energy transfer <T _n> is a decreasing function of the wavenumber k_n and the structure function scaling exponents take close to Kolmogorov values. Above this critical fractal dimension, <T _n> is an increasing function of k_n and the structure function scaling exponents deviate significantly from Kolmogorov values. Received 25 June 2001 / Received in final form 5 April 2002 Published online 19 July 2002     

Front-type solutions of fractional Allen-Cahn equation

Super-diffusive front dynamics have been analysed via a fractional analogue of the Allen-Cahn equation. One-dimensional kink shape and such characteristics as slope at origin and domain wall dynamics have been computed numerically and satisfactorily approximated by variational techniques for a set of anomaly exponents 1<γ<2. The dynamics of a two-dimensional curved front has been considered. Also, the time dependence of coarsening rates during the various evolution stages was analysed in one and two spatial dimensions.     

Renormalization group treatment of the scaling properties of finite systems with subleading long-range interaction

The finite size behavior of the susceptibility, Binder cumulant and some even moments of the magnetization of a fully finite O(n) cubic system of size L are analyzed and the corresponding scaling functions are derived within a field-theoretic ɛ-expansion scheme under periodic boundary conditions. We suppose a van der Waals type long-range interaction falling apart with the distance r as r ^{- (d + σ)}, where 2 < σ < 4, which does not change the short-range critical exponents of the system. Despite that the system belongs to the short-range universality class it is shown that above the bulk critical temperature T _c the finite-size corrections decay in a power-in-L, and not in an exponential-in-L law, which is normally believed to be a characteristic feature for such systems. Received 8 August 2001     

The field theory approach to percolation processes

We review the field theory approach to percolation processes. Specifically, we focus on the so-called simple and general epidemic processes that display continuous non-equilibrium active to absorbing state phase transitions whose asymptotic features are governed, respectively, by the directed (DP) and dynamic isotropic percolation (dIP) universality classes. We discuss the construction of a field theory representation for these Markovian stochastic processes based on fundamental phenomenological considerations, as well as from a specific microscopic reaction-diffusion model realization. Subsequently we explain how dynamic renormalization group (RG) methods can be applied to obtain the universal properties near the critical point in an expansion about the upper critical dimensions d_c = 4 (DP) and 6 (dIP). We provide a detailed overview of results for critical exponents, scaling functions, crossover phenomena, finite-size scaling, and also briefly comment on the influence of long-range spreading, the presence of a boundary, multispecies generalizations, coupling of the order parameter to other conserved modes, and quenched disorder.     

Core percolation in random graphs: a critical phenomena analysis

We study both numerically and analytically what happens to a random graph of average connectivity α when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated vertices plus an induced subgraph we call the core. In the thermodynamic limit of an infinite random graph, we compute analytically the dynamics of leaf removal, the number of isolated vertices and the number of vertices and edges in the core. We show that a second order phase transition occurs at α = e = 2.718 ... : below the transition, the core is small but above the transition, it occupies a finite fraction of the initial graph. The finite size scaling properties are then studied numerically in detail in the critical region, and we propose a consistent set of critical exponents, which does not coincide with the set of standard percolation exponents for this model. We clarify several aspects in combinatorial optimization and spectral properties of the adjacency matrix of random graphs. Received 31 January 2001 and Received in final form 26 June 2001     

Short-time dynamics and critical behavior of three-dimensional bond-diluted Potts model

The short-time dynamics of the three-dimensional bond-diluted 4-state Potts model is investigated with Monte Carlo simulations. A recently suggested nonequilibrium reweighting method is applied, and the tricritical point is determined with the short-time dynamic approach. Based on the dynamic scaling form, both the dynamic and static critical exponents are estimated for the second order phase transition. Dynamic corrections to scaling are carefully considered.     

On the universality class of the 3d Ising model with long-range-correlated disorder

We analyze a controversial topic about the universality class of the three-dimensional Ising model with long-range-correlated disorder. Whereas both theoretical and numerical studies agree on the validity of the extended Harris criterion [A. Weinrib, B.I. Halperin, Phys. Rev. B 27 (1983) 413] and indicate the existence of a new universality class, numerical values of the critical exponents found so far differ considerably. To resolve this discrepancy we perform extensive Monte Carlo simulations of a 3d Ising model with non-magnetic impurities being arranged in a form of lines along randomly chosen axes of a lattice. The Swendsen-Wang algorithm is used alongside with a histogram reweighting technique and finite-size scaling analysis to evaluate the values of critical exponents governing magnetic phase transition. Our estimates for these exponents differ from both previous numerical simulations and are in favor of a non-trivial dependency of the critical exponents on the peculiarities of long-range correlation decay.     

A comprehensive,time-resolved SANS investigation of temperature-change-induced sponge-to-lamellar and lamellar-to-sponge phase transformations in comparison with <Superscript>2</Superscript>H -NMR results

Time-resolved small-angle neutron scattering (TR-SANS) was employed to observe temperature-induced phase transitions from the sponge (L ₃ to the lamellar ( L _α phase, and vice versa, in the water-oil (n -decane)-non-ionic surfactant ( C₁₂E₅ system using both bulk and film contrast. Samples of different bilayer volume fractions φ and solvent viscosities η were investigated applying various amplitudes of temperature jump ΔT . The findings of a previous ²H -NMR study could be confirmed, where the lamellar phase formation was determined to occur through a nucleation and growth process, while it was concluded that the L ₃ -phase develops in a mechanistically different and more rapid manner involving uncorrelated passage formation. Likewise, the kinetic trends of the nucleation and growth transition (decreased transition time with increase of φ and ΔT were witnessed once again. Additionally, NMR and SANS data that demonstrate a strong dependency of that process on solvent viscosity η are presented. Contrariwise, it is made evident via both SANS and NMR results that the L _α -to-L ₃ transition time is independent (within experimental sensitivity) of the varied parameters (φ , ΔT , η . Unusual scattering evolution in one experiment, originating from a highly ordered lamellar phase, intriguingly hints that a major rate determining factor is the disruption of long-range order. Furthermore, the bulk contrast investigations give insight into structure peak shifts/development during the transitions, while the film contrast experiments prove the bilayer thickness to be constant throughout the phase transitions and show that there is no evidence for a change in the short-range order of the bilayer structure. The latter was considered possible, due to the different topology of the L ₃ and L _α phases. Lastly, an unexpected yet consistent appearance of anisotropic scattering is detected in the L ₃ -to- L _α transitions.     

Monte Carlo investigation of phase transitions in the frustrated Heisenberg model on a triangular lattice

The critical properties and phase transitions of the three-dimensional frustrated antiferromagnetic Heisenberg model on a triangular lattice have been investigated using the Monte Carlo method with a replica algorithm. The critical temperature has been determined and the character of the phase transitions has been analyzed using the method of fourth-order Binder cumulants. A second-order phase transition has been found in the three-dimensional frustrated Heisenberg model on a triangular lattice. The static magnetic and chiral critical exponents of the heat capacity α, the susceptibility γ and γ _k , the magnetization β and β _k , the correlation length ν and ν _k , as well as the Fisher exponents η and η _k , have been calculated in terms of the finite-size scaling theory. It has been demonstrated that the three-dimensional frustrated antiferromagnetic Heisenberg model on a triangular lattice forms a new universality class of the critical behavior.     

Magnetic and martensitic transitions in Ni-Fe-Ga alloy

The ferromagnetic shape memory alloy with nominal composition Ni₅₄Fe₁₉Ga₂₇ is investigated by Ac susceptibility and resistivity measurements. The alloy shows long-range ferromagnetic order below 290 K. The anomaly due to the martensitic transition is observed in the susceptibility and resistivity data in the temperature range around 220 K, which is associated with clear thermal hysteresis. Minor hysteresis loop technique was used to investigate the phase coexistence across the martensitic transition, and our analysis indicate that both martensite and austenite phases mutually coexist in the region of hysteresis.