Uniqueness of the Gibbs state of the BEG model in the disordered region of parameters

Even though the general framework of statistical mechanics is ultimately targeted at the description of macroscopic systems, it is illustrative to apply it first to some simple systems: a harmonic oscillator, a rotor, and a spin in a magnetic field. These applications serve to illustrate how a key function associated with the Gibbs state, the so-called partition function, is calculated in practice, how the entropy function is obtained via a Legendre transformation, and how such systems behave in the limits of high and low temperatures. After discussing these simple systems, this chapter considers a first example where multiple constituents are assembled into a macroscopic system: a basic model of a paramagnetic salt. It also investigates the size of energy fluctuations and how—in the case of the paramagnet—these fluctuations scale with the number of constituents.

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THE CRITICAL BEHAVIOR OF THE 2D SPIN-1 ISING MODEL WITH THE BILINEAR AND POSITIVE BIQUADRATIC NEAREST NEIGHBOR INTERACTIONS ON A CELLULAR AUTOMATON

International Journal of Modern Physics C ◽

10.1142/s012918310400687x ◽

2004 ◽

Vol 15 (10) ◽

pp. 1425-1438 ◽

Cited By ~ 5

Author(s):

A. SOLAK ◽

B. KUTLU

Keyword(s):

Cellular Automaton ◽

Phase Boundary ◽

Critical Behavior ◽

Nearest Neighbor ◽

Finite Size ◽

Square Lattice ◽

Two Dimensional ◽

Finite Size Scaling ◽

Creutz Cellular Automaton ◽

Beg Model

The two-dimensional BEG model with nearest neighbor bilinear and positive biquadratic interaction is simulated on a cellular automaton, which is based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transitions of the model are presented for comparison with those obtained from other calculations. We confirm the existence of the tricritical points over the phase boundary for D/K>0. The values of static critical exponents (α, β, γ and ν) are estimated within the framework of the finite size scaling theory along D/K=-1 and 1 lines. The results are compatible with the universal Ising critical behavior except the points over phase boundary.

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PHASE DIAGRAMS OF THE SPIN-1 BLUME–EMERY–GRIFFITHS MODEL IN A RANDOM TRANSVERSE FIELD

International Journal of Modern Physics B ◽

10.1142/s021797920402775x ◽

2004 ◽

Vol 18 (31n32) ◽

pp. 4129-4142 ◽

Cited By ~ 4

Author(s):

H. EZ-ZAHRAOUY ◽

H. MAHBOUB ◽

A. BENYOUSSEF ◽

M. J. OUAZZANI

Keyword(s):

Field Theory ◽

Phase Diagrams ◽

Effective Field Theory ◽

Low Temperatures ◽

Transverse Field ◽

Effective Field ◽

Long Range Order ◽

Special Value ◽

Tricritical Behavior ◽

Beg Model

The effect of the random quantum transverse field Ω on the tricritical behavior of the spin-1 Blume–Emery–Griffiths (BEG) model is studied using an effective field theory. It is found that the tricritical behavior depends on both the biquadratic interaction K, single-ion anisotropy Δ and the concentration p of the disorder of Ω. Indeed, there exists a special value p1 of the probability p below which the tricritical behavior disappears. In addition, at sufficiently low temperatures, the system exhibits long-range order accompanied by the tricritical behavior below a special value p2 of the probability p.

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Thermodynamics of an alternate σ=1 and Ising chain mapped onto an effective BEG model

Physica A Statistical Mechanics and its Applications ◽

10.1016/s0378-4371(03)00596-x ◽

2003 ◽

Vol 329 (1-2) ◽

pp. 147-160 ◽

Cited By ~ 4

Author(s):

E.C. Fireman ◽

J.C. Cressoni ◽

R.J.V. dos Santos

Keyword(s):

Ising Chain ◽

Beg Model

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Free transport for convex potentials

New Zealand Journal of Mathematics ◽

10.53733/102 ◽

2021 ◽

Vol 52 ◽

pp. 259-359

Author(s):

Yoann Dabrowski ◽

Alice Guionnet ◽

Dima Shlyakhtenko

Keyword(s):

Tensor Product ◽

Stochastic Analysis ◽

Gibbs State ◽

Gibbs States ◽

Convexity Condition ◽

Semicircle Law ◽

Free Transport ◽

Transport Maps

We construct non-commutative analogs of transport maps among free Gibbs state satisfying a certain convexity condition. Unlike previous constructions, our approach is non-perturbative in nature and thus can be used to construct transport maps between free Gibbs states associated to potentials which are far from quadratic, i.e., states which are far from the semicircle law. An essential technical ingredient in our approach is the extension of free stochastic analysis to non-commutative spaces of functions based on the Haagerup tensor product.

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