Ising spins

More recently suggested models for bulk systems treat oil, water and amphiphiles on equal footing and place them all on lattice sites. They are thus basically lattice models for ternary fluids, which are generalized to capture the essential properties of the amphiphiles. Oil, water, and amphiphiles are represented by Ising spins 5 = -1,0 and +1. If one considers all possible nearest-neighbor interactions between these three types of particle, one obtains a total number of three independent interaction parameters, and... 

An is the chemical potential difference between sohd and gas. Also, from the equivalence of the system to an Ising spin system, one often uses the terminology spin for the variable s, although it has nothing really to do with a quantum-mechanical spin. 

For high temperatures, the spin-glass system behaves essentially the way conventional Ising-spin systems behave namely, a variety of different configurations are accessible, each with some finite probability. It is only at low enough tempera tures that a unique spin-glass phase - characterized chiefly by the appearance of a continuum of equilibrium states - first appears. 

For conventional Ising-spin systems, Pising(o ) takes on the expected simple forms namely, either Puingiq) = < (0) in the (high-temperature, zero magnetization) paramagnetic phase or the double-peaked Pising(o ) = 5(q + M ) + 6 q — M ) for temperatures below the Curie critical temperature, T < Tc. 

There is a fundamental relationship between d-dimensional PCA and d + 1)-dimensional Ising spin models. The simplest way to make the connection is to think of the successive temporal layers of the PCA as successive hyper-planes of the next higher-dimensional spatial lattice. Because the PCA rules (at least the set of PCA rules that we will be dealing with) are (1) Markovian (i.e. the probability of a state at time t + T depends only on a set of states at time t, and (2) local, one can always define a Hamiltonian on the higher-dimensioned spatial lattice such that the thermodynamic weight of a configuration 5j,( is equal to the probability of a corresponding space-time history Si t). ... 

In both of these cases, represents the energy of two noninteracting Ising spin systems, one system having spins s over one sublattice and the other spins s over the other sublattice. Indeed, the Q2R CA actually provides an efficient microcanonical algorithm for performing parallel simulation of the Ising model (see discussion in section 7.1.5). 

This tells us immediately that, just as for Ising spins, we have a spontaneous magnetization and that there is an effective phase transition for T = 1 stored patterns will only be stable for temperatures T < 1. 

Figure 2 Fragments of a randomly disordered and an ordered arrangement of one-dimensional Ising spins. The long arrows identify equivalent symmetry...

It was shown by Wilson [131] that the Kadanoff procedure, combined with the Landau model, may be used to identify the critical point, verify the scaling law and determine the critical exponents without obtaining an exact solution, or specifying the nature of fluctuations near the critical point. The Hamiltonian for a set of Ising spins is written in suitable units, as before... 

Figure 3.3. Equilibrium linear susceptibility (x/Xiso) versus temperature for an infinite spherical sample on a simple cubic lattice. The dotted lines are the results for independent spins, while the solid lines show the results for parallel and random anisotropy calculated with thermodynamic perturbation theory, as well as for Ising spins calculated with an ordinary high-temperature expansions. We notice in this case that the linear susceptibility for systems with random anisotropy is the same as for isotropic spins calculated with an ordinary high-temperature expansion. The dipolar interaction strength is hj = a/2a = 0.004.

The Hamiltonian of an Ising spin glass, given by Edwards and Anderson (EA) [78], is... 

As an application of this theorem let us calculate in leading order the decay of the Ising spin correlation function r(0, N) to its limit r(oo) below Tc. From Eq. (12) and the definitions (42) we find easily... 

According to RG theory [11, 19, 20], universality rests on the spatial dimensionality D of the systems, the dimensionality n of the order parameter (here n = 1), and the short-range nature of the interaction potential 0(r). In D = 3, short-range means that 0(r) decays as r p with p>D + 2 — tj = 4.97 [21], where rj = 0.033 is the exponent of the correlation function g(r) of the critical fluctuations [22] (cf. Table I). Then, the critical exponents map onto those of the Ising spin-1/2 model, which are known from RG calculations [23], series expansions [11, 12, 24] and simulations [25, 26]. For insulating fluids with a leading term of liquid metals [27-29] the experimental verification of Ising-like criticality is unquestionable. 

G. J. M. Koper and H. J. Hilhorst, Nonequilibrium and aging in a one-dimensional Ising spin... 

Leeuwen theorem. Marathe and Dhar study an Ising spin in a time-varying magnetic field. They verify the Crooks FR and JE, and confirm that time reversibility is required for the transient ES FR. They also observe that the steady state ES FR is not obeyed, confirming that this is an asymptotic result and indicating that observation times were insufficient for convergence. 

For the number of shells in both structures, each lattice is related to the radius (R) of the nanoparticle [27-29]. Therefore, the value of R contains a number of shells and the size of a nanoparticle increases as the number of shells increases. The shells (R) and their numbers are only bounded to the nearest-neighbour pair exchange interactions (J) between spins. To provide the magnetization of the whole particle, each of the spin sites, which stand for the atomistic moments in the nanoparticle, are described by Ising spin variables that take on the values S1-= l, 0. For a core/surface (C/S) morphology, all spins in the nanoparticle are organized in three components that are core (C, filled circles), interface (or core-surface) (CS) and surface (S, empty circles) parts. The number of spins in these parts within the C/S-type nanoparticle are denoted byNc, Ncs and Ns, respectively. But, the total number of spins (N) in a C/S nanoparticle covers only C and S spin numbers, i.e. N =NC + Ns. On the other hand, the numbers of spin pairs for C, CS and S regions in 2D are defined by N [,=(N (.y(. /2)-Ncs,... 

In 1987, Swendsen and Wang (SW) [3] introduced a new Monte Carlo algorithm for the Ising spin model, which constituted a radical departure from the Metropolis or single-spin flip method used until then. Since the recipe is relatively straightforward, it is instructive to begin with a description of this algorithm. 

A. B. Bortz, M. H. Kalos, J. L. Lebowitz (1975) New algorithm for monte-carlo simulation of ising spin systems. J. Comput. Phys. 17, p. 10... 

F. Barahona (1982) On the computational complexity of Ising spin glass models. J. Phys. A 15, p. 3241... 

S. Alder, S. Trebst, A. K. Hartmaim, and M. Troyer (2004) Dynamics of the Wang-Landau algorithm and Complexity of rare events for the three-dimensional bimodal Ising spin glass. J. Stat. Mech. P07008... 

P. Werner, K. Volker, M. Troyer, and S. Chakravarty (2005) Phase diagram and critical exponents of a dissipative Ising spin chain in a transverse magnetic field. Phys. Rev. Lett. 94, p. 047201... 

Bortz, A.B. Kalos, M.H. Lebowitz, J.L. A new algorithm for Monte Carlo simulations of Ising spin systems. J. Comp. Phys. 1975, 17, 10-18. 

The Ising spin model does not consider the coexistence of the ordered (ferromagnetic) and non-ordered (paramagnetic) phases at subcritical temperatures. As a result, there is no latent heat r/T) and disorder parameter associated with the ferromagnetic transition. The condition dh/dT = 0 must be added to h=0. The known CXC-dependence of the lattice-gas chemical potential ... 

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