arXiv subjects

R. B. Stinchcombe

Publications and source records attributed to R. B. Stinchcombe.

36 records · Page 2Linked to original sources

arXiv2005

Quantum approach to nucleation times of kinetic Ising ferromagnets

Low temperature dynamics of Ising ferromagnets under finite magnetic fields are studied in terms of quantum spin representations of stochastic evolution operators. These are constructed for the Glauber dynamic as well as for a modification of this latter, introduced by K. Park {\it et al.} in Phys. Rev. Lett. {\bf 92}, 015701 (2004). In both cases the relaxation time after a field quench is evaluated both numerically and analytically using the spectrum gap of the corresponding operators. The numerical work employs standard recursive techniques following a symmetrization of the evolution operator accomplished by a non-unitary spin rotation. The analytical approach uses low temperature limits to identify dominant terms in the eigenvalue problem. It is argued that the relaxation times already provide a measure of actual nucleation lifetimes under finite fields. The approach is applied to square, triangular and honeycomb lattices.

cond-mat.stat-mech↗

arXiv2004

Cluster growth in far-from-equilibrium particle models with diffusion, detachment, reattachment and deposition

Monolayer cluster growth in far-from-equilibrium systems is investigated by applying simulation and analytic techniques to minimal hard core particle (exclusion) models. The first model (I), for post-deposition coarsening dynamics, contains mechanisms of diffusion, attachment, and slow activated detachment (at rate epsilon<<1) of particles on a line. Simulation shows three successive regimes of cluster growth: fast attachment of isolated particles; detachment allowing further (epsilon t)^(1/3) coarsening of average cluster size; and t^(-1/2) approach to a saturation size going like epsilon^(-1/2). Model II generalizes the first one in having an additional mechanism of particle deposition into cluster gaps, suppressed for the smallest gaps. This model exhibits early rapid filling, leading to slowing deposition due to the increasing scarcity of deposition sites, and then continued power law (epsilon t)^(1/2) cluster size coarsening through the redistribution allowed by slow detachment. The basic (epsilon t)^(1/3) domain growth laws and epsilon^(-1/2) saturation in model I are explained by a simple scaling picture. A second, fuller approach is presented which employs a mapping of cluster configurations to a column picture and an approximate factorization of the cluster configuration probability within the resulting master equation. This allows quantitative results for the saturation of model I in excellent agreement with the simulation results. For model II, it provides a one-variable scaling function solution for the coarsening probability distribution, and in particular quantitative agreement with the cluster length scaling and its amplitude.

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arXiv2004

Disordered asymmetric simple exclusion process: mean-field treatment

We provide two complementary approaches to the treatment of disorder in a fundamental nonequilibrium model, the asymmetric simple exclusion process. Firstly, a mean-field steady state mapping is generalized to the disordered case, where it provides a mapping of probability distributions and demonstrates how disorder results in a new flat regime in the steady state current--density plot for periodic boundary conditions. This effect was earlier observed by Tripathy and Barma but we provide treatment for more general distributions of disorder, including both numerical results and analytic expressions for the width $2Δ_C$ of the flat section. We then apply an argument based on moving shock fronts to show how this leads to an increase in the high current region of the phase diagram for open boundary conditions. Secondly, we show how equivalent results can be obtained easily by taking the continuum limit of the problem and then using a disordered version of the well-known Cole--Hopf mapping to linearize the equation. Within this approach we show that adding disorder induces a localization transformation (verified by numerical scaling), and $Δ_C$ maps to an inverse localization length, helping to give a new physical interpretation to the problem.

cond-mat.stat-mech↗

arXiv2003

Correlation--function distributions at the Nishimori point of two-dimensional Ising spin glasses

The multicritical behavior at the Nishimori point of two-dimensional Ising spin glasses is investigated by using numerical transfer-matrix methods to calculate probability distributions $P(C)$ and associated moments of spin-spin correlation functions $C$ on strips. The angular dependence of the shape of correlation function distributions $P(C)$ provides a stringent test of how well they obey predictions of conformal invariance; and an even symmetry of $(1-C) P(C)$ reflects the consequences of the Ising spin-glass gauge (Nishimori) symmetry. We show that conformal invariance is obeyed in its strictest form, and the associated scaling of the moments of the distribution is examined, in order to assess the validity of a recent conjecture on the exact localization of the Nishimori point. Power law divergences of $P(C)$ are observed near C=1 and C=0, in partial accord with a simple scaling scheme which preserves the gauge symmetry.

cond-mat.stat-mech↗

arXiv2001

Correlation functions, free energies and magnetizations in the two-dimensional random-field Ising model

Transfer-matrix methods are used to calculate spin-spin correlation functions ($G$), Helmholtz free energies ($f$) and magnetizations ($m$) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature $T_{c 0}$, on long strips of width $L = 3 - 18$ sites, for binary field distributions. Analysis of the probability distributions of $G$ for varying spin-spin distances $R$ shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large $R$. Connections between field-- and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed $R/L$, the fractional widths of correlation-function distributions saturate asymptotically with $L^{-2.2}$. Considering an added uniform applied field $h$, a connection between $f(h)$, $m(h)$, the Gibbs free energy $g(m)$ and the distribution function for the uniform magnetization in zero uniform field, $P_0(m)$, is derived and first illustrated for pure systems, and then applied for non-zero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of $P_0(m)$ varies against (non-vanishing, but small) random-field intensity $H_0$ as $H_0^{-3/7}$.

cond-mat.stat-mech↗

arXiv1999

Nagel scaling, relaxation and universality in the kinetic Ising model in an alternating isotopic chain

The dynamic critical exponent and the frequency and wave-vector dependent susceptibility of the kinetic Ising model with Glauber dynamics on an alternating isotopic chain are examined. The analysis provides to our knowledge the first connection between a microscopic model and the Nagel scaling curve originally proposed to describe dielectric susceptibility measurements of several glass-forming liquids. While support is given to the hypothesis relating the Nagel scaling to multiple relaxation processes, it is also found that the scaling function may also exhibit plateau regions and does not hold for low temperatures.

cond-mat.stat-mech↗

arXiv1999

Correlation functions in the two-dimensional random-field Ising model

Transfer-matrix methods are used to study the probability distributions of spin-spin correlation functions $G$ in the two-dimensional random-field Ising model, on long strips of width $L = 3 - 15$ sites, for binary field distributions at generic distance $R$, temperature $T$ and field intensity $h_0$. For moderately high $T$, and $h_0$ of the order of magnitude used in most experiments, the distributions are singly-peaked, though rather asymmetric. For low temperatures the single-peaked shape deteriorates, crossing over towards a double-$δ$ ground-state structure. A connection is obtained between the probability distribution for correlation functions and the underlying distribution of accumulated field fluctuations. Analytical expressions are in good agreement with numerical results for $R/L \gtrsim 1$, low $T$, $h_0$ not too small, and near G=1. From a finite-size {\it ansatz} at $T=T_c (h_0=0)$, $h_0 \to 0$, averaged correlation functions are predicted to scale with $L^y h_0$, $y =7/8$. From numerical data we estimate y=0.875 \pm 0.025$, in excellent agreement with theory. In the same region, the RMS relative width $W$ of the probability distributions varies for fixed $R/L=1$ as $W \sim h_0^κ f(L h_0^u)$ with $κ\simeq 0.45$, $u \simeq 0.8$ ; $f(x)$ appears to saturate when $x \to \infty$, thus implying $W \sim h_0^κ$ in $d=2$.

cond-mat.stat-mech↗

arXiv1999

Diffusion in disordered lattices and related Heisenberg ferromagnets

We study the diffusion of classical hard-core particles in disordered lattices within the formalism of a quantum spin representation. This analogy enables an exact treatment of non-instantaneous correlation functions at finite particle densities in terms of single spin excitations in disordered ferromagnetic backgrounds. Applications to diluted chains and percolation clusters are discussed. It is found that density fluctuations in the former exhibit a stretched exponential decay while an anomalous power law asymptotic decay is conjectured for the latter.

cond-mat.stat-mech↗

arXiv1999

Length and time scale divergences at the magnetization-reversal transition in the Ising model

The divergences of both the length and time scales, at the magnetization- reversal transition in Ising model under a pulsed field, have been studied in the linearized limit of the mean field theory. Both length and time scales are shown to diverge at the transition point and it has been checked that the nature of the time scale divergence agrees well with the result obtained from the numerical solution of the mean field equation of motion. Similar growths in length and time scales are also observed, as one approaches the transition point, using Monte Carlo simulations. However, these are not of the same nature as the mean field case. Nucleation theory provides a qualitative argument which explains the nature of the time scale growth. To study the nature of growth of the characteristic length scale, we have looked at the cluster size distribution of the reversed spin domains and defined a pseudo-correlation length which has been observed to grow at the phase boundary of the transition.

cond-mat.stat-mech↗

arXiv1998

Non-universal dynamics of staggered non-equilibrium particle systems and Ising chains

Non-universal dynamics is shown to occur in a one-dimensional non-equilibrium system of hard-core particles. The stochastic processes included are pair creation and annihilation (with rates e and e') and symmetric hopping rates which alternate from one bond to the next (Pa, Pb). A dynamical scaling relation between the relaxation time and the correlation length in the steady state is derived in a simple way for the case e' > Pa >> Pb >> e. We find that the dynamical exponent takes the non-universal value z = 2 ln(e'/e)/ln[(Pb e')/(Pa e)]. For the special condition e + e'= Pa + Pb, where the stochastic system is in principle soluble by reduction to a free fermion system, the model is mapped to the Glauber dynamics of an Ising chain with alternating ferromagnetic bonds of values J1 and J2, in contact with a quantum thermal bath. The full time dependence of the space-dependent magnetization and of the equal time spin-spin correlation function are studied by writing the master equation for this system in the quantum Hamiltonian formalism. In particular we obtain the dispersion relations and rigorously confirm the results obtained for the correlation length and for the dynamical exponent.

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arXiv1998

Connectivity-dependent properties of diluted sytems in a transfer-matrix description

We introduce a new approach to connectivity-dependent properties of diluted systems, which is based on the transfer-matrix formulation of the percolation problem. It simultaneously incorporates the connective properties reflected in non-zero matrix elements and allows one to use standard random-matrix multiplication techniques. Thus it is possible to investigate physical processes on the percolation structure with the high efficiency and precision characteristic of transfer-matrix methods, while avoiding disconnections. The method is illustrated for two-dimensional site percolation by calculating (i) the critical correlation length along the strip, and the finite-size longitudinal DC conductivity: (ii) at the percolation threshold, and (iii) very near the pure-system limit.

cond-mat.stat-mech↗

arXiv1997

Particle-hopping Models of Vehicular Traffic: Distributions of Distance Headways and Distance Between Jams

We calculate the distribution of the distance headways (i.e., the instantaneous gap between successive vehicles) as well as the distribution of instantaneous distance between successive jams in the Nagel-Schreckenberg (NS) model of vehicular traffic. When the maximum allowed speed, $V_{max}$, of the vehicles is larger than unity, over an intermediate range of densities of vehicles, our Monte Carlo (MC) data for the distance headway distribution exhibit two peaks, which indicate the coexistence of "free-flowing" traffic and traffic jams. Our analytical arguments clearly rule out the possibility of occurrence of more than one peak in the distribution of distance headways in the NS model when $V_{max} = 1$ as well as in the asymmetric simple exclusion process. Modifying and extending an earlier analytical approach for the NS model with $V_{max} = 1$, and introducing a novel transfer matrix technique, we also calculate the exact analytical expression for the distribution of distance between the jams in this model; the corresponding distributions for $V_{max} > 1$ have been computed numerically through MC simulation.

cond-mat.stat-mech↗

arXiv1996

Correlation functions in the two-dimensional random-bond Ising model

We consider long strips of finite width $L \leq 13$ sites of ferromagnetic Ising spins with random couplings distributed according to the binary distribution: $P(J_{ij})= {1 \over 2} ( δ(J_{ij} -J_0) + δ(J_{ij} -rJ_0) ) ,\ 0 < r < 1 $. Spin-spin correlation functions $ <σ_{0} σ_{R}>$ along the ``infinite'' direction are computed by transfer-matrix methods, at the critical temperature of the corresponding two-dimensional system, and their probability distribution is investigated. We show that, although in-sample fluctuations do not die out as strip length is increased, averaged values converge satisfactorily. These latter are very close to the critical correlation functions of the pure Ising model, in agreement with recent Monte-Carlo simulations. A scaling approach is formulated, which provides the essential aspects of the $R$-- and $L$-- dependence of the probability distribution of $\ln <σ_{0} σ_{R}>$, including the result that the appropriate scaling variable is $R/L$. Predictions from scaling theory are borne out by numerical data, which show the probability distribution of $\ln <σ_{0} σ_{R}>$ to be remarkably skewed at short distances, approaching a Gaussian only as $R/L \gg 1$ .

cond-mat↗

arXiv1995

Domain scaling and marginality breaking in the random field Ising model

A scaling description is obtained for the $d$--dimensional random field Ising model from domains in a bar geometry. Wall roughening removes the marginality of the $d=2$ case, giving the $T=0$ correlation length $ξ\sim \exp\left(A h^{-γ}\right)$ in $d=2$, and for $d=2+ε$ power law behaviour with $ν= 2/εγ$, $h^\star \sim ε^{1/γ}$. Here, $γ= 2,4/3$ (lattice, continuum) is one of four rough wall exponents provided by the theory. The analysis is substantiated by three different numerical techniques (transfer matrix, Monte Carlo, ground state algorithm). These provide for strips up to width $L=11$ basic ingredients of the theory, namely free energy, domain size, and roughening data and exponents.

cond-mat↗

arXiv1995

Scaling treatment of the random field Ising model

Analytic phenomenological scaling is carried out for the random field Ising model in general dimensions using a bar geometry. Domain wall configurations and their decorated profiles and associated wandering and other exponents $(ζ,γ,δ,μ)$ are obtained by free energy minimization. Scaling between different bar widths provides the renormalization group (RG) transformation. Its consequences are (1) criticality at $h=T=0$ in $d \leq 2$ with correlation length $ξ(h,T)$ diverging like $ξ(h,0) \propto h^{-2/(2-d)}$ for $d<2$ and $ξ(h,0) \propto \exp[1/(c_1γh^γ)]$ for $d=2$, where $c_1$ is a decoration constant; (2) criticality in $d = 2+ε$ dimensions at $T=0$, $h^{\ast}= (ε/2c_1)^{1/γ}$, where $ξ\propto [(s-s^{\ast})/s]^{-2ε/γ}$, $s \equiv h^γ$. Finite temperature generalizations are outlined. Numerical transfer matrix calculations and results from a ground state algorithm adapted for strips in $d=2$ confirm the ingredients which provide the RG description.

cond-mat↗

arXiv1995

Monte Carlo simulation and global optimization without parameters

We propose a new ensemble for Monte Carlo simulations, in which each state is assigned a statistical weight $1/k$, where $k$ is the number of states with smaller or equal energy. This ensemble has robust ergodicity properties and gives significant weight to the ground state, making it effective for hard optimization problems. It can be used to find free energies at all temperatures and picks up aspects of critical behaviour (if present) without any parameter tuning. We test it on the travelling salesperson problem, the Edwards-Anderson spin glass and the triangular antiferromagnet.

cond-mat↗

arXiv1994

Replica Symmetry Breaking in the Critical Behaviour of the Random Ferromagnet

We study the critical properties of the weakly disordered $p$-component random Heisenberg ferromagnet. It is shown that if the specific heat critical exponent of the pure system is positive, the traditional renormalization group (RG) flows at dimensions $D=4-\e$, which are usually considered as describing the disorder-induced universal critical behavior, are {\it unstable} with respect to replica symmetry breaking (RSB) potentials as found in spin glasses. It is demonstrated that the RG flows involving RSB potentials lead to fixed points which have the structure known as the 1 step RSB, and there exists a whole spectrum of such fixed points. It is argued that spontaneous RSB can occur due to the interactions of the fluctuating fields with the local non-perturbative degrees of freedom coming from the multiple local minima solutions of the mean-field equations. However, it is not clear whether or not RSB occurs for infinitesimally weak disorder. Physical consequences of these conclusions are discussed.

cond-mat↗

arXiv1994

Transfer-matrix scaling from disorder-averaged correlation lengths for diluted Ising systems

A transfer matrix scaling technique is developed for randomly diluted systems, and applied to the site-diluted Ising model on a square lattice in two dimensions. For each allowed disorder configuration between two adjacent columns, the contribution of the respective transfer matrix to the decay of correlations is considered only as far as the ratio of its two largest eigenvalues, allowing an economical calculation of a configuration-averaged correlation length. Standard phenomenological-renormalisation procedures are then used to analyse aspects of the phase boundary which are difficult to assess accurately by alternative methods. For magnetic site concentration $p$ close to $p_c$, the extent of exponential behaviour of the $T_c \times p$ curve is clearly seen for over two decades of variation of $p - p_c$. Close to the pure-system limit, the exactly-known reduced slope is reproduced to a very good approximation, though with non-monotonic convergence. The averaged correlation lengths are inserted into the exponent-amplitude relationship predicted by conformal invariance to hold at criticality. The resulting exponent $η$ remains near the pure value (1/4) for all intermediate concentrations until it crosses over to the percolation value at the threshold.

cond-mat↗