Temperature For Ising Model Research Articles

Probabilistic Cellular Automata for Low-Temperature 2-d Ising Model

We construct a parallel stochastic dynamics with invariant measure converging to the Gibbs measure of the low temperature Ising model. The proof of such convergence requires a polymer expansion based on suitably defined Peierls-type contours.

Fast Mixing for the Low Temperature 2D Ising Model Through Irreversible Parallel Dynamics

We study tunneling and mixing time for a non-reversible probabilistic cellular automaton. With a suitable choice of the parameters, we first show that the stationary distribution is close in total variation to a low temperature Ising model. Then we prove that both the mixing time and the time to exit a metastable state grow polynomially in the size of the system, while this growth is exponential in reversible dynamics. In this model, non-reversibility, parallel updatings and a suitable choice of boundary conditions combine to produce an efficient dynamical stability.

Simulation of Ising Model by Monte Carlo Method

Simulating Ising model to calculate magnetization intensity by Monte Carlo method. The Ising model was introduced simply, sampled importantly, and calculated with programming. It shows the dependency relationship between the magnetization intensity and the size of dot-square line in different temperatures for Ising model. It cans edulcorate the approximation of analytic method by computer simulating. It obtains a method to appraise a model right or wrong by comparing the model and the experimental data.

A variance inequality for Glauber dynamics applicable to high and low temperature regimes

A variance inequality for spin-flip systems is obtained using comparatively weaker knowledge of relaxation to equilibrium based on coupling estimates for single site disturbances. We obtain variance inequalities interpolating between the Poincare inequality and the uniform variance inequality, and a general weak Poincare inequality. For monotone dynamics the variance inequality can be obtained from decay of the autocorrelation of the spin at the origin i.e. from that decay we conclude decay for general functions. This method is then applied to the low temperature Ising model, where the time-decay of the autocorrelation of the origin is extended to arbitrary quasi-local functions.

Non-equilibrium phase transitions in the two-temperature Ising model with Kawasaki dynamics

Phase transitions of the two-finite temperature Ising model on a square lattice are investigated by using a position space renormalization group (PSRG) transformation. Different finite temperatures, T x and T y , and also different time-scale constants, α x and α y for spin exchanges in the x and y directions define the dynamics of the non-equilibrium system. The critical surface of the system is determined by RG flows as a function of these exchange parameters. The Onsager critical point (when the two temperatures are equal) and the critical temperature for the limit when the other temperature is infinite, previously studied by the Monte Carlo method, are obtained. In addition, two steady-state fixed points which correspond to the non-equilibrium phase transition are presented. These fixed points yield the different universality class properties of the non-equilibrium phase transitions.

The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model

We define a new percolation model by generalising the FK representation of the Ising model, and show that on the triangular lattice and at high temperatures, the critical point in the new model corresponds to the Ising model. Since the new model can be viewed as Bernoulli percolation on a random graph, our result makes an explicit connection between Ising percolation and critical Bernoulli percolation, and gives a new justification of the conjecture that the high temperature Ising model on the triangular lattice is in the same universality class as Bernoulli percolation.

Exponential Distribution for the Occurrence of Rare Patterns in Gibbsian Random Fields

We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice $\mathbb{Z}^d$, $d\geq 2$. A typical example is the high temperature Ising model. This distribution is shown to converge to an exponential law as the size of the pattern diverges. Our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns. A similar result holds for the repetition of a rare pattern. We apply these results to the fluctuation properties of occurrence and repetition of patterns: We prove a central limit theorem and a large deviation principle.

Freezing transition in the Ising model without internal contours

We consider the low temperature Ising model in a uniform magnetic field h > 0 with minus boundary conditions and conditioned on having no internal contours. This simple contour model defines a non-Gibbsian spin state. For large enough magnetic fields (h >: h c ) this state is concentrated on the single spin configuration of all spins up. For smaller values (h≤h c ), the spin state is non-trivial. At the critical point h c ≠ 0 the magnetization jumps discontinuously. Freezing provides also an example of a translation invariant weakly Gibbsian state which is not almost Gibbsian.

Exponential Decay of Truncated Correlation Functions Via the Generating Function: A Direct Method

We consider statistical mechanics lattice models where the external field dependent partition function can be represented as a standard polymer system. Using this polymer representation and elementary complex analytic arguments, we obtain upper bounds and give a simple proof on the uniform (in n) exponential decay of the n-point truncated correlation function. We illustrate the method by applying it to the high and low temperature Ising model and to contour models.

Finite-size scaling of the mass-gap for first-order phase transitions

I summarize recent work on finite-size scaling in long cylinders, partially obtained in collaboration with J. Imbrie. It is sketched how a system in a long cylinder can be mapped onto a one dimensional spin system. For the low temperature Ising model, one obtains the finite-size scaling of the mass-gap as a function of the cross-section A = L d−1 and the magnetic field h. I discuss a recent controversy stemming from the numerical work of Shen and Jansen on the finite-size scaling of this mass-gap for d = 4 and h = 0, and argue that the observed discrepancy is a crossover effect. In a second part, I consider the q states Potts model. I discuss both the finite-size scaling of the mass-gap in long cylinders and the numerical determination of the interface tension. I present an exact formula for the interface tension in d = 2 which has been recently obtained by W. Janke and myself.

Low temperature Ising model series and the ratio method

A transformation is applied to low temperature series for the FCC Ising model, which enables the ratio method to be used in their extrapolation, and apparently more precise estimates to be made of the critical exponents than hitherto. These estimates are consistent with the scaling value of alpha '=1/8, and strongly favour the value of gamma '=5/4 over the previous estimates of around 1.28; they thus counter previous evidence for a possible violation of scaling in this model.

Determination of critical behaviour in lattice statistics from series expansions II

Low temperature Ising model series in the usual variable z = exp(−4J/kT) are analysed. The positions of the singularities in the disk|z| <or=|zc| and the values of the corresponding critical exponents are determined for the simple cubic, body-centred cubic and face-centred cubic lattices. We obtain the following estimates of the physical critical exponents: β = 0312 ± 0002 for the simple cubic and face-centred cubic lattices, and β = 0312 ± 0004 for the body-centred cubic lattice. γprime = 131 ± 002 for the simple cubic lattice, γprime = 128 ± 004 for the body-centred cubic lattice, and γprime = 13 ± 01 for the face-centred cubic lattice. For the low temperature specific heat exponent we obtain fraction one-sixteens <or= αprime <or= ¼.

Determination of critical behaviour in lattice statistics from series expansions III

Low temperature Ising model series in the usual variable z = exp(−4J/kT) are analysed. The positions of the singularities in the disk|z| <or=|zc| and the values of the corresponding critical exponents are determined for the simple cubic, body-centred cubic and face-centred cubic lattices. We obtain the following estimates of the physical critical exponents: β = 0312 ± 0002 for the simple cubic and face-centred cubic lattices, and β = 0312 ± 0004 for the body-centred cubic lattice. γprime = 131 ± 002 for the simple cubic lattice, γprime = 128 ± 004 for the body-centred cubic lattice, and γprime = 13 ± 01 for the face-centred cubic lattice. For the low temperature specific heat exponent we obtain fraction one-sixteens <or= αprime <or= ¼.

Low temperature susceptibility exponent for the three dimensional Ising model

It is pointed out that in the variable x = 1 − tanh ( J kT ) , low temperature Ising model series converge up to the physical singularity. The low temperature susceptibility series in x for a s.c. Ising lattice is analysed and it is suggested that the exponent γ′ = 21 16 rather than 5 4 , the value implied by scaling and homogeneity arguments.