scholarly journals INTEGRABLE MODELS IN STATISTICAL MECHANICS: THE HIDDEN FIELD WITH UNSOLVED PROBLEMS

1999 ◽  
Vol 14 (25) ◽  
pp. 3921-3933 ◽  
Author(s):  
BARRY M. McCOY
Keyword(s):  
Ising Model ◽  
Potts Model ◽  
Nonlinear Differential Equations ◽  
Chiral Potts Model ◽  
Level Crossing ◽  
The Past ◽  
Fermi Statistics ◽  
Mechanical Models ◽  
Statistical Mechanical ◽  
Model Correlation

In the past 30 years there have been extensive discoveries in the theory of integrable statistical mechanical models including the discovery of nonlinear differential equations for Ising model correlation functions, the theory of random impurities, level crossing transitions in the chiral Potts model and the use of Rogers–Ramanujan identities to generalize our concepts of Bose/Fermi statistics. Each of these advances has led to the further discovery of major unsolved problems of great mathematical and physical interest. I will here discuss the mathematical advances, the physical insights and extraordinary lack of visibility of this field of physics.

scholarly journals Ferroic Multipolar Order and Disorder in Cyanoelpasolite Molecular Perovskites

2018 ◽  
Author(s):  
Chloe Coates ◽  
Harry Gray ◽  
Johnathan Bulled ◽  
Hanna Boström ◽  
Arkadiy Simonov ◽  
...  
Keyword(s):  
Potts Model ◽  
Degrees Of Freedom ◽  
Variable Temperature ◽  
First Order ◽  
Order And Disorder ◽  
Simple Cubic ◽  
Different Types ◽  
Mechanical Models ◽  
Statistical Mechanical ◽  
First Order Transition

<div>We use a combination of variable-temperature high-resolution synchrotron X-ray powder diffraction measurements and Monte Carlo simulations to characterise the evolution of two different types of ferroic multipolar order in a series of cyano elpasolite molecular perovskites. We show that ferroquadrupolar order in [C3N2H5]2Rb[Co(CN)6] is a first-order process that is well described by a 4-state Potts model on the simple cubic lattice. Likewise, ferrooctupolar order in [NMe4]2B[Co(CN)6] (B = K, Rb, Cs) also emerges via a first-order transition that now corresponds to a 6-state Potts model. Hence, for these particular cases, the dominant symmetry breaking mechanisms are well understood in terms of simple statistical mechanical models. By varying composition, we find that the effective coupling between multipolar degrees of freedom—and hence the temperature at which ferromultipolar order emerges—can be tuned in a chemically sensible manner.</div><div><br></div>

scholarly journals Ferroic multipolar order and disorder in cyanoelpasolite molecular perovskites

2019 ◽  
Vol 377 (2149) ◽  
pp. 20180219 ◽  
Author(s):  
C. S. Coates ◽  
H. J. Gray ◽  
J. M. Bulled ◽  
H. L. B. Boström ◽  
A. Simonov ◽  
...  
Keyword(s):  
Potts Model ◽  
Degrees Of Freedom ◽  
Variable Temperature ◽  
First Order ◽  
Order And Disorder ◽  
Simple Cubic ◽  
Different Types ◽  
Mechanical Models ◽  
Statistical Mechanical ◽  
First Order Transition

We use a combination of variable-temperature high-resolution synchrotron X-ray powder diffraction measurements and Monte Carlo simulations to characterize the evolution of two different types of ferroic multipolar order in a series of cyanoelpasolite molecular perovskites. We show that ferroquadrupolar order in [C 3 N 2 H 5 ] 2 Rb[Co(CN) 6 ] is a first-order process that is well described by a four-state Potts model on the simple cubic lattice. Likewise, ferrooctupolar order in [NMe 4 ] 2 B[Co(CN) 6 ] (B = K, Rb, Cs) also emerges via a first-order transition that now corresponds to a six-state Potts model. Hence, for these particular cases, the dominant symmetry breaking mechanisms are well understood in terms of simple statistical mechanical models. By varying composition, we find that the effective coupling between multipolar degrees of freedom—and hence the temperature at which ferromultipolar order emerges—can be tuned in a chemically sensible manner. This article is part of the theme issue ‘Mineralomimesis: natural and synthetic frameworks in science and technology’.

scholarly journals Ferroic Multipolar Order and Disorder in Cyanoelpasolite Molecular Perovskites

2018 ◽  
Author(s):  
Chloe Coates ◽  
Harry Gray ◽  
Johnathan Bulled ◽  
Hanna Boström ◽  
Arkadiy Simonov ◽  
...  
Keyword(s):  
Potts Model ◽  
Degrees Of Freedom ◽  
Variable Temperature ◽  
First Order ◽  
Order And Disorder ◽  
Simple Cubic ◽  
Different Types ◽  
Mechanical Models ◽  
Statistical Mechanical ◽  
First Order Transition

<div>We use a combination of variable-temperature high-resolution synchrotron X-ray powder diffraction measurements and Monte Carlo simulations to characterise the evolution of two different types of ferroic multipolar order in a series of cyano elpasolite molecular perovskites. We show that ferroquadrupolar order in [C3N2H5]2Rb[Co(CN)6] is a first-order process that is well described by a 4-state Potts model on the simple cubic lattice. Likewise, ferrooctupolar order in [NMe4]2B[Co(CN)6] (B = K, Rb, Cs) also emerges via a first-order transition that now corresponds to a 6-state Potts model. Hence, for these particular cases, the dominant symmetry breaking mechanisms are well understood in terms of simple statistical mechanical models. By varying composition, we find that the effective coupling between multipolar degrees of freedom—and hence the temperature at which ferromultipolar order emerges—can be tuned in a chemically sensible manner.</div><div><br></div>

Fluctuations and Critical Behavior

2005 ◽  
Author(s):  
Eldred H. Chimowitz
Keyword(s):  
Ising Model ◽  
Critical Behavior ◽  
Exact Analytical Solution ◽  
Mean Field ◽  
Field Analysis ◽  
Mechanical Models ◽  
Statistical Mechanical ◽  
Field Approaches ◽  
Insight Into

An important historical result with the Ising model was the achievement of an exact analytical solution by Onsager for the two-dimensional case. It is one of the few statistical-mechanical models for which an exact solution is available, and yields insight into the role played by thermodynamic fluctuations in determining critical behavior. We know that mean-field approaches, for example, which ignore the effects of fluctuations, do not predict the correct scaling exponents in the critical region. Also, we showed in the previous chapter that a mean-field analysis of the Ising model showed its specific heat CH to have a discontinuity at Tc remaining finite on either side of Tc , namely for T > Tc and T < Tc .

scholarly journals PROOF OF THE DETERMINANTAL FORM OF THE SPONTANEOUS MAGNETIZATION OF THE SUPERINTEGRABLE CHIRAL POTTS MODEL

2010 ◽  
Vol 51 (3) ◽  
pp. 309-316 ◽  
Author(s):  
R. J. BAXTER
Keyword(s):  
Ising Model ◽  
Potts Model ◽  
General Model ◽  
Spontaneous Magnetization ◽  
Order Parameters ◽  
Chiral Potts Model ◽  
Algebraic Calculation ◽  
Algebraic Properties ◽  
Determinantal Form

AbstractThe superintegrable chiral Potts model has many resemblances to the Ising model, so it is natural to look for algebraic properties similar to those found for the Ising model by Onsager, Kaufman and Yang. The spontaneous magnetization ℳr can be written in terms of a sum over the elements of a matrix Sr. The author conjectured the form of the elements, and this conjecture has been verified by Iorgov et al. The author also conjectured in 2008 that this sum could be expressed as a determinant, and has recently evaluated the determinant to obtain the known result for ℳr. Here we prove that the sum and the determinant are indeed identical expressions. Since the order parameters of the superintegrable chiral Potts model are also those of the more general solvable chiral Potts model, this completes the algebraic calculation of ℳr for the general model.

Chiral segregation in three microscopic statistical-mechanical models

2009 ◽  
Vol 80 (1) ◽  
Author(s):  
Igor Medved’ ◽  
Anton Trník ◽  
Dale A. Huckaby
Keyword(s):  
Mechanical Models ◽  
Statistical Mechanical
Sign in / Sign up

Export Citation Format

Share Document