Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2309.01815 · Physical review. E

Effective Hamiltonian approach to kinetic Ising models: Application to an infinitely long-range Husimi-Temperley model

V. I. Tokar

Published Apr 9, 2024• 1 citation
arXivHTMLPDF

Abstract

The probability distribution (PD) of spin configurations in kinetic Ising models has been cast in the form of the canonical Boltzmann PD with a time-dependent effective Hamiltonian (EH). It has been argued that in systems with extensive energy EH depends linearly on the number of spins N leading to the exponential dependence of PD on the system size. In macroscopic systems the argument of the exponential function may reach values of the order of the Avogadro number which is impossible to deal with computationally, thus making unusable the linear master equation (ME) governing the PD evolution. To overcome the difficulty, it has been suggested to use instead the nonlinear ME (NLME) for the EH density per spin. It has been shown that in spatially homogeneous systems NLME contains only terms of order unity even in the thermodynamic limit. The approach has been illustrated with the kinetic Husimi-Temperley model (HTM) evolving under the Glauber dynamics. At finite N the known numerical results has been reproduced and extended to broader parameter ranges. In the thermodynamic limit an exact nonlinear partial differential equation of the Hamilton-Jacobi type for EH has been derived. It has been shown that the average magnetization in HTM evolves according to the conventional kinetic mean field equation.

DOI
10.1103/physreve.109.044123

Related papers

Hamiltonian approach to kinetic Ising models

V. I. Tokar

In the framework of the quantum approach to stochastic dynamics, the master equations governing the temporal evolution of the kinetic Ising models (KIM) are cast in the form of imaginary-time Schr\"odinger equations with second-quantized Hermitian Hamiltonians. On the basis of the quantum formalism a classical evolution equation is derived for an effective time-dependent Ising-type Hamiltonian. The grand ensemble corresponding to the latter describes the statistics of the spin configurations evolving under the dynamics of the KIM. The latter approach was used to qualitatively explain the results of recent Monte Carlo simulations of the Kawasaki KIM. \textcopyright{} 1996 The American Physical Society.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Feb 1, 1996•4 citations

Fock-space approach to the kinetic Ising model

Michael Schulz, Steffen Trimper

The Master equation of the kinetic Ising model with spin-flip and spin-exchange dynamics is written in a second quantized in terms of Pauli operators. The introduction of weighted transition rates allows a general formulation in accordance with the principle of detailed balance. Static and dynamic correlation functions are calculated. The structure factor reveals three different relaxation times already in an extended mean-field approximation. Additionally, we derive a nonlinear evolution equation for the magnetization under the influence of nonthermal external noise. Already in lowest order, the correlation function offers a stretched-exponential behavior with a universal exponent \ensuremath{\sigma}=4(d+2) below the critical dimensionality ${\mathit{d}}_{\mathit{c}}$=2. A one-loop renormalization group approach is used to calculate the critical exponents for the growth of the magnetization. The value of the growth exponent \ensuremath{\beta}=2(2-d)(8+d) is supported by simulations in one dimension. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matter•Apr 1, 1996•33 citations

Derivation of finite-size scaling for mean-field models from the Burgers equation

J.G. Brankov

The magnetization per spin for a class of models with the Husimi-Temperley type interaction satisfies the Burgers equation with a diffusion coefficient 1/2N, where N is the number of particles. The role of time is played by the dimensionless interaction parameter, and the role of spatial coordinate by the dimensionless field variable. Both thermodynamic scaling and finite-size scaling for the magnetization near the critical point are derived from a family of self-similar solutions of the corresponding Burgers equation. The models are specified by the initial conditions which are chosen to correspond in the high-temperature region to a vanishing interaction constant and in the low-temperature region to an infinitely large interaction constant.

Journal of Physics A Mathematical and General•Dec 7, 1990•3 citations

Phase transitions in a three-dimensional kinetic spin-1/2 Ising model with random field: Effective-field-theory study

Emanuel Costabile, J. Ricardo de Sousa

The dynamical phase transitions of the kinetic Ising model in the presence of a random magnetic field with a bimodal probability distribution is studied by using effective-field theory (EFT) with correlations. We have used a Glauber-type stochastic dynamic to describe the time evolution of the system, where the system strongly depends on the H≡√<H(i)(2)>(c) root mean square deviation of the magnetic field. The EFT dynamic equation is given for the simple cubic lattice (z=6), and the dynamic order parameter is calculated. The system presents ferromagnetic and paramagnetic states for low and high temperatures, respectively. Our results predict first-order transitions at low temperatures and large disorder strengths, which corresponds to the existence of a nonequilibrium tricritical point (TCP) in a phase diagram in the T-H plane. We compare the results with the equilibrium phase diagram, where only the first-order line is different. Our qualitative results are compatible with recent Monte Carlo simulations.

Physical Review E•Jan 13, 2012•15 citations

One-dimensional kinetic Ising model with competing spin-flip and spin-exchange dynamics: Ordering in the case of long-range exchanges

Michel Droz, Zoltàn Ràcz, P. Tartaglia

Ordering of a one-dimensional stochastic Ising model evolving by a combination of spin flips and spin exchanges is investigated. The spin-flip rates satisfy detailed balance for the equilibrium state of the Ising model at temperature T, while the spin exchanges are random and of arbitrary range. Analytical methods and Monte Carlo simulations are used to show that, depending on the details of the spin-flip rate, finite-temperature phase transition may or may not occur in the system. When ordering occurs, it is of mean-field type and the scaling function describing the finite-size scaling of the magnetization fluctuations is found to be indistinguishable from that of an equilibrium Ising model with infinite-range interaction.

Physical Review A•Jun 1, 1990•35 citations

Theory for the dynamics of clusters near the critical point. I. Relaxation of the Glauber kinetic Ising model

Kurt Binder, Dietrich Stauffer, Heiner Muller-Krumbhaar

A semiphenomenological cluster theory is developed for dynamic critical properties, which is not limited to small deviations from equilibrium. Explicit numerical expressions are derived for linear and nonlinear response functions of the kinetic Ising model, which are compatible with dynamic scaling and recent calculations of dynamic critical exponents. An uniaxial magnet (or Ising system) and its critical fluctuations are approximated by a "droplet" or "cluster" model, the critical part of the free energy being the sum of the contributions of clusters each containing $l$ spins. These clusters are assumed to grow and shrink with a phenomenological rate $\ensuremath{\propto}{l}^{r}$, $r\ensuremath{\approx}1$. These rates are compatible with direct Monte Carlo simulations in two and three dimensions. The resulting cluster reaction and diffusion equation is an approximation to the master equation of the Glauber kinetic Ising model (in which the magnetization is not conserved), and has the same structure as used in nucleation theories. In linear response to space- and time-dependent fields the relaxation times of energy and magnetization are expressed as triple integrals. All relaxation times diverge as ${|1\ensuremath{-}\frac{T}{{T}_{c}}|}^{\ensuremath{-}(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}}$. This treatment is consistent with dynamic scaling and universality, and is more general than the conventional (Van Hove) theory of critical slowing down. Also the (smaller) exponents found for the response to localized variations ("autocorrelation functions") agree with dynamic scaling. The wave-vector dependence of the relaxation times arises from the static correlation function only, and not from the cluster diffusion term. If the equilibrium cluster distribution is assumed to be that of the Fisher droplet model, and for special values of $r$, the (discrete) eigenvalue spectrum and eigenfunctions (generalized Laguerre polynomials) of the cluster reaction equation are found explicitly. Relaxation functions and frequency-dependent susceptibilities are expressed by hypergeometric series, even in the case of nonlinear response. At the critical point, the spectrum becomes continuous and the exponential decay for large times is replaced by power-law behavior. Similarly, it is found that dynamic scaling applies also to the nucleation rate for this model, and the scaling exponent of the nucleation rate is $2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$.

Physical review. B, Solid state•Dec 1, 1975•98 citations
The probability distribution (PD) of spin configurations in kinetic Ising models has been cast in the form of the canonical Boltzmann PD with a time-dependent effective Hamiltonian (EH). It has been argued that in systems with extensive energy EH depends linearly on the number of spins N leading to the exponential dependence of PD on the system size. In macroscopic systems the argument of the exponential function may reach values of the order of the Avogadro number which is impossible to deal with computationally, thus making unusable the linear master equation (ME) governing the PD evolution. To overcome the difficulty, it has been suggested to use instead the nonlinear ME (NLME) for the EH density per spin. It has been shown that in spatially homogeneous systems NLME contains only terms of order unity even in the thermodynamic limit. The approach has been illustrated with the kinetic Husimi-Temperley model (HTM) evolving under the Glauber dynamics. At finite N the known numerical results has been reproduced and extended to broader parameter ranges. In the thermodynamic limit an exact nonlinear partial differential equation of the Hamilton-Jacobi type for EH has been derived. It has been shown that the average magnetization in HTM evolves according to the conventional kinetic mean field equation.
In the framework of the quantum approach to stochastic dynamics, the master equations governing the temporal evolution of the kinetic Ising models (KIM) are cast in the form of imaginary-time Schr\"odinger equations with second-quantized Hermitian Hamiltonians. On the basis of the quantum formalism a classical evolution equation is derived for an effective time-dependent Ising-type Hamiltonian. The grand ensemble corresponding to the latter describes the statistics of the spin configurations evolving under the dynamics of the KIM. The latter approach was used to qualitatively explain the results of recent Monte Carlo simulations of the Kawasaki KIM. \textcopyright{} 1996 The American Physical Society.
The Master equation of the kinetic Ising model with spin-flip and spin-exchange dynamics is written in a second quantized in terms of Pauli operators. The introduction of weighted transition rates allows a general formulation in accordance with the principle of detailed balance. Static and dynamic correlation functions are calculated. The structure factor reveals three different relaxation times already in an extended mean-field approximation. Additionally, we derive a nonlinear evolution equation for the magnetization under the influence of nonthermal external noise. Already in lowest order, the correlation function offers a stretched-exponential behavior with a universal exponent \ensuremath{\sigma}=4(d+2) below the critical dimensionality dc{\mathit{d}}_{\mathit{c}}dc​=2. A one-loop renormalization group approach is used to calculate the critical exponents for the growth of the magnetization. The value of the growth exponent \ensuremath{\beta}=2(2-d)(8+d) is supported by simulations in one dimension. \textcopyright{} 1996 The American Physical Society.
The magnetization per spin for a class of models with the Husimi-Temperley type interaction satisfies the Burgers equation with a diffusion coefficient 1/2N, where N is the number of particles. The role of time is played by the dimensionless interaction parameter, and the role of spatial coordinate by the dimensionless field variable. Both thermodynamic scaling and finite-size scaling for the magnetization near the critical point are derived from a family of self-similar solutions of the corresponding Burgers equation. The models are specified by the initial conditions which are chosen to correspond in the high-temperature region to a vanishing interaction constant and in the low-temperature region to an infinitely large interaction constant.
The dynamical phase transitions of the kinetic Ising model in the presence of a random magnetic field with a bimodal probability distribution is studied by using effective-field theory (EFT) with correlations. We have used a Glauber-type stochastic dynamic to describe the time evolution of the system, where the system strongly depends on the H≡√<H(i)(2)>(c) root mean square deviation of the magnetic field. The EFT dynamic equation is given for the simple cubic lattice (z=6), and the dynamic order parameter is calculated. The system presents ferromagnetic and paramagnetic states for low and high temperatures, respectively. Our results predict first-order transitions at low temperatures and large disorder strengths, which corresponds to the existence of a nonequilibrium tricritical point (TCP) in a phase diagram in the T-H plane. We compare the results with the equilibrium phase diagram, where only the first-order line is different. Our qualitative results are compatible with recent Monte Carlo simulations.
Ordering of a one-dimensional stochastic Ising model evolving by a combination of spin flips and spin exchanges is investigated. The spin-flip rates satisfy detailed balance for the equilibrium state of the Ising model at temperature T, while the spin exchanges are random and of arbitrary range. Analytical methods and Monte Carlo simulations are used to show that, depending on the details of the spin-flip rate, finite-temperature phase transition may or may not occur in the system. When ordering occurs, it is of mean-field type and the scaling function describing the finite-size scaling of the magnetization fluctuations is found to be indistinguishable from that of an equilibrium Ising model with infinite-range interaction.
A semiphenomenological cluster theory is developed for dynamic critical properties, which is not limited to small deviations from equilibrium. Explicit numerical expressions are derived for linear and nonlinear response functions of the kinetic Ising model, which are compatible with dynamic scaling and recent calculations of dynamic critical exponents. An uniaxial magnet (or Ising system) and its critical fluctuations are approximated by a "droplet" or "cluster" model, the critical part of the free energy being the sum of the contributions of clusters each containing lll spins. These clusters are assumed to grow and shrink with a phenomenological rate $\ensuremath{\propto}{l}^{r}$, $r\ensuremath{\approx}1$. These rates are compatible with direct Monte Carlo simulations in two and three dimensions. The resulting cluster reaction and diffusion equation is an approximation to the master equation of the Glauber kinetic Ising model (in which the magnetization is not conserved), and has the same structure as used in nucleation theories. In linear response to space- and time-dependent fields the relaxation times of energy and magnetization are expressed as triple integrals. All relaxation times diverge as ${|1\ensuremath{-}\frac{T}{{T}_{c}}|}^{\ensuremath{-}(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}}$. This treatment is consistent with dynamic scaling and universality, and is more general than the conventional (Van Hove) theory of critical slowing down. Also the (smaller) exponents found for the response to localized variations ("autocorrelation functions") agree with dynamic scaling. The wave-vector dependence of the relaxation times arises from the static correlation function only, and not from the cluster diffusion term. If the equilibrium cluster distribution is assumed to be that of the Fisher droplet model, and for special values of rrr, the (discrete) eigenvalue spectrum and eigenfunctions (generalized Laguerre polynomials) of the cluster reaction equation are found explicitly. Relaxation functions and frequency-dependent susceptibilities are expressed by hypergeometric series, even in the case of nonlinear response. At the critical point, the spectrum becomes continuous and the exponential decay for large times is replaced by power-law behavior. Similarly, it is found that dynamic scaling applies also to the nucleation rate for this model, and the scaling exponent of the nucleation rate is $2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$.