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Ising model
About: Ising model is a research topic. Over the lifetime, 25508 publications have been published within this topic receiving 555000 citations.
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TL;DR: In this article, the critical indices for the two-dimensional Ising model were derived from an assumed form of an operator algebra which describes how the product of two fluctuating variables may be reduced to a linear combination of the basic variables.
Abstract: A previous publication showed how the critical indices for the two-dimensional Ising model could be derived from an assumed form of an operator algebra which describes how the product of two fluctuating variables may be reduced to a linear combination of the basic fluctuating variables. In this paper, the previously used algebra is derived from the Onsager solution of the two-dimensional Ising model. The calculation makes use of a "disorder" variable which is mathematically the result of applying the Kramers-Wannier transformation to the Ising-model spin variable. The average of products of spin and disorder variables are evaluated at the critical point for the special case in which all the variables lie on a single straight line. The ordering of these variables on the line determines a "quantum number" $\ensuremath{\Gamma}$ such that the average is nonzero only for $\ensuremath{\Gamma}=0$. Composition rules for this quantum number are derived and used to develop an algebra for the multiplication of complex variables at the critical point. Arguments are given to suggest the identifications of elements of the algebra as the spin, the energy density, the Kaufman spinors, and a stress density. The result of this calculation is the operator algebra which formed the starting point of the previous paper.
421 citations
TL;DR: In this paper, a simple dynamical model of interacting Ising spins is discussed, where each spin flips spontaneously with a transition probability which depends on the temperature and the configuration of surrounding spins, but its functional form is assumed to be the simplest.
Abstract: A simple dynamical model of interacting Ising spins is discussed. Each spin is assumed to flip spontaneously with a transition probability which depends on the temperature and the configuration of surrounding spins, but its functional form is assumed to be the simplest. The frequency-wave number dependent susceptibility χ( q , ω) is given exactly in the one-dimensional case. In two-and three-dimensional cases the model is treated in the molecular field and the generalized approximations.
419 citations
01 Feb 1997
TL;DR: In this article, Redner et al. proposed a scaling theory of diffusion-controlled and ballistically-controlled bimolecular reactions with competing dynamics: mappings, correlations, steady states, and phase transitions.
Abstract: Part I. Reaction-Diffusion Systems and Models of Catalysis 1. Scaling theories of diffusion-controlled and ballistically-controlled bimolecular reactions S. Redner 2. The coalescence process, A+A->A, and the method of interparticle distribution functions D. ben-Avraham 3. Critical phenomena at absorbing states R. Dickman Part II. Kinetic Ising Models 4. Kinetic ising models with competing dynamics: mappings, correlations, steady states, and phase transitions Z. Racz 5. Glauber dynamics of the ising model N. Ito 6. 1D Kinetic ising models at low temperatures - critical dynamics, domain growth, and freezing S. Cornell Part III. Ordering, Coagulation, Phase Separation 7. Phase-ordering dynamics in one dimension A. J. Bray 8. Phase separation, cluster growth, and reaction kinetics in models with synchronous dynamics V. Privman 9. Stochastic models of aggregation with injection H. Takayasu and M. Takayasu Part IV. Random Sequential Adsorption and Relaxation Processes 10. Random and cooperative sequential adsorption: exactly solvable problems on 1D lattices, continuum limits, and 2D extensions J. W. Evans 11. Lattice models of irreversible adsorption and diffusion P. Nielaba 12. Deposition-evaporation dynamics: jamming, conservation laws and dynamical diversity M. Barma Part V. Fluctuations In Particle and Surface Systems 13. Microscopic models of macroscopic shocks S. A. Janowsky and J. L. Lebowitz 14. The asymmetric exclusion model: exact results through a matrix approach B. Derrida and M. R. Evans 15. Nonequilibrium surface dynamics with volume conservation J. Krug 16. Directed walks models of polymers and wetting J. Yeomans Part VI. Diffusion and Transport In One Dimension 17. Some recent exact solutions of the Fokker-Planck equation H. L. Frisch 18. Random walks, resonance, and ratchets C. R. Doering and T. C. Elston 19. One-dimensional random walks in random environment K. Ziegler Part VII. Experimental Results 20. Diffusion-limited exciton kinetics in one-dimensional systems R. Kroon and R. Sprik 21. Experimental investigations of molecular and excitonic elementary reaction kinetics in one-dimensional systems R. Kopelman and A. L. Lin 22. Luminescence quenching as a probe of particle distribution S. H. Bossmann and L. S. Schulman Index.
419 citations
TL;DR: In this paper, the authors show that directed percolation is equivalent to a cellular automata, and thus to an Ising model, and that the time development of cellular Automata in $d$ dimensions is mapped onto equilibrium statistical mechanics of Ising models in $n+1$ dimensions.
Abstract: Time development of cellular automata in $d$ dimensions is mapped onto equilibrium statistical mechanics of Ising models in $d+1$ dimensions. Directed percolation is equivalent to a cellular automaton, and thus to an Ising model. For a particular case of directed percolation we find ${\ensuremath{
u}}_{\ensuremath{\parallel}}=2$, ${\ensuremath{
u}}_{\ensuremath{\perp}}=1$, ${\ensuremath{\eta}}_{\ensuremath{\perp}}=0$.
417 citations
TL;DR: In this article, a network of four degenerate optical parametric oscillators (OPOs) is employed to find the ground state of the Ising Hamiltonian, and a small non-deterministic polynomial time-hard problem is solved on a 4-OPO Ising machine.
Abstract: A network of four degenerate optical parametric oscillators (OPOs) is employed to find the ground state of the Ising Hamiltonian. The good performance of the network reveals the potential of OPOs for many similar problems. Finding the ground states of the Ising Hamiltonian1 maps to various combinatorial optimization problems in biology, medicine, wireless communications, artificial intelligence and social network. So far, no efficient classical and quantum algorithm is known for these problems and intensive research is focused on creating physical systems—Ising machines—capable of finding the absolute or approximate ground states of the Ising Hamiltonian2,3,4,5,6. Here, we report an Ising machine using a network of degenerate optical parametric oscillators (OPOs). Spins are represented with above-threshold binary phases of the OPOs and the Ising couplings are realized by mutual injections7. The network is implemented in a single OPO ring cavity with multiple trains of femtosecond pulses and configurable mutual couplings, and operates at room temperature. We programmed a small non-deterministic polynomial time-hard problem on a 4-OPO Ising machine and in 1,000 runs no computational error was detected.
414 citations