Transition In Ising Model Research Articles

Incremental learning of phase transition in Ising model: Preprocessing, finite-size scaling and critical exponents

We investigated an unsupervised machine learning to recognize the phase transition in Ising model by using principal component analysis (PCA) with a preprocessing of raw data, and we realized the large-scale calculations and finite-size scaling analysis based on the incremental PCA. It was shown that the two different phases are clearly and reasonably recognized by the k-means clustering of the first and second principal components. Taking the normalized first principal component as the order parameter, we calculated the critical properties near the phase transition point by using the finite-size scaling method. Our results of the critical temperature and critical exponents are consistent with the classical values, which indicates that the first principal component is capable of catching main features of the phase transition. The present scheme provides an alternative way with emphasis on data preprocessing and incremental PCA for the unsupervised machine learning of the phase transition in Ising model and its related spin models.

Abnormal Calcium Leak from Cardiac Sarcoplasmic Reticulum: New Insights Offered by Statistical Physics

Heart muscle contraction is activated by a synchronized systolic Ca release from sarcoplasmic reticulum (SR) via Ca sparks. In disease, Ca sparks fail to terminate, causing a diastolic Ca leak that decreases contraction amplitude and increases the risk of arrhythmia. The mechanisms and treatment of the abnormal Ca leak remain unclear. We have recently shown that spark termination emerges as collective behavior (synchronized closings) of Ca release channels (RyRs) that is identical to synchronization of spin orientation in ferromagnets described by a phase transition in Ising model. We employed the Ising approach to investigate and classify mechanisms of spark termination failure and Ca leak. Methods and Results: The key parameters determining whether spark termination succeeds or fails are SR Ca and RyR opening/closing rates. They define analogues of magnetic field h and the inverse temperature (β) in Ising model. Sparks terminate via a phase transition known as “h polarity reversal” and the leak emerges when h fails to change its sign. This happens when the SR depletes insufficiently and RyR openings remained partially synchronized via Ca-induced-Ca-release, generating long-lasting sparks. Leak can also occur via β, known as Onsager's “order-disorder” transition. This happens at low SR Ca, reducing RyR current and RyRs interactions, resulting in independent RyR openings. The disorder leak is distinguished from synchronized leak by larger Peierls contour lengths reflecting degree of disorder. Abnormal leak results from either a probability imbalance during synchronized firing of RyRs or from disordered RyR firing. Each leak type requires different and balanced treatment to shift RyR operation towards normal spark termination.

Transition matrix cluster algorithms

We demonstrate that a series of procedures for increasing the efficiency of transition matrix calculations can be realized by integrating the standard single-spin flip transition matrix method with global cluster flipping techniques. Our calculations employ a simple and accurate method based on detailed balance for computing the density of states from the Ising model transition matrix.

A projected entropy controller for transition matrix calculations

We define the projected entropy S(T) at a given temperature T in the context of an Ising model transition matrix calculation as the entropy associated with the distribution of Markov chain realizations in energy-magnetization, E-H, space. An even sampling of states is achieved by accumulating the results from multiple Markov chains while decrementing 1/T at a rate proportional to the inverse of the effective number, exp(S(T)), of accessible projected states. Such a procedure is both highly accurate and far simpler to implement than a previously suggested method based on monitoring the evolution of the E-H distribution at each temperature. [1] We further demonstrate a transition matrix procedure that instead ensures uniform sampling in physical entropy.

Strongly anisotropic nonequilibrium phase transition in Ising models with friction

The nonequilibrium phase transition in driven two-dimensional Ising models with two different geometries is investigated using Monte Carlo methods as well as analytical calculations. The models show dissipation through fluctuation induced friction near the critical point. We first consider high driving velocities and demonstrate that both systems are in the same universality class and undergo a strongly anisotropic nonequilibrium phase transition, with anisotropy exponent θ=3. Within a field theoretical ansatz the simulation results are confirmed. The crossover from Ising to mean field behavior in dependency of system size and driving velocity is analyzed using crossover scaling. It turns out that for all finite velocities the phase transition becomes strongly anisotropic in the thermodynamic limit.

Fluctuation cumulant behavior for the field-pulse-induced magnetization-reversal transition in Ising models.

The universality class of the dynamic magnetization-reversal transition, induced by a competing field pulse, in an Ising model on a square lattice, below its static ordering temperature, is studied here using Monte Carlo simulations. Fourth-order cumulant of the order parameter distribution is studied for different system sizes around the phase boundary region. The crossing point of the cumulant (for different system sizes) gives the transition point and the value of the cumulant at the transition point indicates the universality class of the transition. The cumulant value at the crossing point for low temperature and pulse width range is observed to be significantly less than that for the static transition in the same two-dimensional Ising model. The finite-size scaling behavior in this range also indicates a higher correlation length exponent value. For higher temperature and pulse width range, the transition seems to fall in a mean-field-like universality class.

Spin-reversal transition in Ising model under pulsed field

In this communication we report the existence of a dynamic “spin-reversal” transition in an Ising system perturbed by a pulsed external magnetic field. The transition is achieved by tuning the strength ( h p ) and/or the duration ( Δt) of the pulse which is applied in a direction opposite to the existing order. We have studied this transition in the kinetic Ising model in two-dimensions using Monte-Carlo technique, and solved numerically, the mean-field equation of motion. The transition is essentially dynamic in nature and it takes the system from one ordered equilibrium phase to another by means of the growth of opposite spin domains (in the kinetic Ising case) induced during the period when the pulsed field is applied.

The i-δ relations and the spin 1 Ising model

The i-δ relations and the criticality condition are applied to the calculation of the spin 1 Ising model transition temperatures T c for the cubic lattices. An indirect decoupling method is introduced whereby special care is taken of autocorrelations when 〈 S 2 δ 〉 is factored from the relevant correlation functions. The results for T c are consistent with those obtained from series calculations to within 0.33%. An approximate lattice-lattice scaling appears to operate for 〈 S 2 δ 〉, having the value ∼0.72 for all three lattices. Estimates of the multispin correlation functions U 2 n and T 2 n are given for the first time for the 3-d spin 1 case.

Near-equalities for some two-dimensional Ising-model correlation ratios

The i–δ relations and the criticality condition, which are used in recent theoretical calculations of the Ising-model transition temperature and critical exponent η, are tested against exact calculations for the two-dimensional lattices: square, plane triangular, and honeycomb. They are found to be valid to within 2%. They are also tested against series results for the three-dimensional hydrogen peroxide lattice, with which they are shown to be consistent to within 3.3%. A new derivation is given of the i–δ relations.

Nonanalytic features of the first order phase transition in the Ising model

The absence of the analytic continuation for the free energy near the point of the first order phase transition in thed-dimensional Ising model is proved. It is shown that thermodynamic functions in the metastable phase do not have certain values and can be derived only with an uncertaintyδ. The asymptotic expansion near the point of the phase transition yields the values of thermodynamic functions with the same uncertainty.

A geometric interpretation of the roughening transition in Ising models

Progress in the area of the Ising model roughening transition has previously been limited by the lack of a good definition for the interface separating the pure phases. In the present work, a graphical definition is introduced and it is shown that roughening occurs precisely when this interface fluctuates to infinity.

2 × 2 commensurate-incommensurate transition in Ising models: Monte Carlo simulation

Phase diagrams of Ising models with antiferromagnetic nearest-(NN) and next-nearest-neighbor (NNN) interactions are obtained by Monte Carlo simulations. For the triangular lattice a paramagnetic $(P)\ensuremath{-}2\ifmmode\times\else\texttimes\fi{}2$ commensurate ($C$) phase transition is found, which is second order when the NN interaction is small. The exponents are consistent with the ones of the four-state Potts model. For large NNN interactions the transition becomes first order. For three-dimensional stacking of triangular layers an incommensurate ($I$) phase is found in addition. The $P\ensuremath{-}C$ and $I\ensuremath{-}C$ transitions are of first order whereas the $P\ensuremath{-}I$ transition seems to be of second order. The model is used to interpret the $P\ensuremath{-}I\ensuremath{-}C$ transitions in $\ensuremath{\beta}$-eucryptite.

Percolation with long-range interactions

The concentration dependence of the ferromagnetic ordering temperature, Tc(p) in a dilute bond Ising model,, is studied for several long-range interactions near the limit p to 0, Tc to 0. The critical region of the random Ising model transition, occurring at all finite Tc, is shown to shrink to zero as p approaches zero. For dipolar interactions, spin glass ordering may occur at sufficiently small p.