*Barry M McCoy*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199556632
- eISBN:
- 9780191723278
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556632.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This book begins where elementary books and courses leave off and covers the advances made in statistical mechanics in the past fifty years. The book is divided into three parts. The first part is on ...
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This book begins where elementary books and courses leave off and covers the advances made in statistical mechanics in the past fifty years. The book is divided into three parts. The first part is on general theory which includes a summary of the basic principles of statistical mechanics; a presentation of the physical phenomena covered and the models used to discuss them; theorems on the existence and uniqueness of partition functions; theorems on order; and critical phenomena and scaling theory. The second part is on series and numerical methods which includes derivations of the Mayer and Ree–Hoover expansions of the low density virial equation of state; Groeneveld's theorems; the application to hard spheres and discs; a summary of numerical studies of systems at high density; and the use of high temperature series expansions to estimate critical exponents for magnets. The third part covers exactly solvable models which includes a detailed presentation of the Pfaffian methods of computing the Ising partition function, magnetization, correlation functions, and susceptibility; the star-triangle (Yang–Baxter equation); functional equations and the free energy for the eight-vertex model; and the hard hexagon and chiral Potts models. All needed mathematics is developed in detail and many open questions are discussed. The goal is to guide the reader to the current forefront of research.Less

This book begins where elementary books and courses leave off and covers the advances made in statistical mechanics in the past fifty years. The book is divided into three parts. The first part is on general theory which includes a summary of the basic principles of statistical mechanics; a presentation of the physical phenomena covered and the models used to discuss them; theorems on the existence and uniqueness of partition functions; theorems on order; and critical phenomena and scaling theory. The second part is on series and numerical methods which includes derivations of the Mayer and Ree–Hoover expansions of the low density virial equation of state; Groeneveld's theorems; the application to hard spheres and discs; a summary of numerical studies of systems at high density; and the use of high temperature series expansions to estimate critical exponents for magnets. The third part covers exactly solvable models which includes a detailed presentation of the Pfaffian methods of computing the Ising partition function, magnetization, correlation functions, and susceptibility; the star-triangle (Yang–Baxter equation); functional equations and the free energy for the eight-vertex model; and the hard hexagon and chiral Potts models. All needed mathematics is developed in detail and many open questions are discussed. The goal is to guide the reader to the current forefront of research.

*Barry M. McCoy*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199556632
- eISBN:
- 9780191723278
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556632.003.0015
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter presents the exact results for the grand partition function of the hard hexagon model in both low and high density regions are presented. In the low density region, these results are ...
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This chapter presents the exact results for the grand partition function of the hard hexagon model in both low and high density regions are presented. In the low density region, these results are used to derive the first 25 virial coefficients of the virial expansion. The analyticity of the pressure in the density plane in both the low and high density regions are then presented. The general theory of the chiral Potts model as a two-dimensional statistical model is presented and the eigenvalues of the three-component superintegrable case are computed in detail. The order parameter is discussed and the phase diagram of the general three-component integrable chiral Potts spin chain is given. Open questions are discussed about Q operators, eight-vertex model degeneracies, and conjectures for correlations functions of the superintegrable chiral Potts model.Less

This chapter presents the exact results for the grand partition function of the hard hexagon model in both low and high density regions are presented. In the low density region, these results are used to derive the first 25 virial coefficients of the virial expansion. The analyticity of the pressure in the density plane in both the low and high density regions are then presented. The general theory of the chiral Potts model as a two-dimensional statistical model is presented and the eigenvalues of the three-component superintegrable case are computed in detail. The order parameter is discussed and the phase diagram of the general three-component integrable chiral Potts spin chain is given. Open questions are discussed about *Q* operators, eight-vertex model degeneracies, and conjectures for correlations functions of the superintegrable chiral Potts model.

*Barry M. McCoy*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199556632
- eISBN:
- 9780191723278
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556632.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) ...
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This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) equation is introduced for vertex, spin, and face models and used to demonstrate the commutation of the transfer matrices. The star-triangle equation is solved for the six-vertex, eight-vertex, SOS, RSOS, hard hexagon, and chiral Potts models. The commutation of the transfer matrix with the related quantum spin chain is derived.Less

This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) equation is introduced for vertex, spin, and face models and used to demonstrate the commutation of the transfer matrices. The star-triangle equation is solved for the six-vertex, eight-vertex, SOS, RSOS, hard hexagon, and chiral Potts models. The commutation of the transfer matrix with the related quantum spin chain is derived.

*Barry M. McCoy*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199556632
- eISBN:
- 9780191723278
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556632.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter derives the modification of the Mayer expansion made by Ree and Hoover. Analytic expressions for the virial coefficients B2,B3, and B4 are given and Monte–Carlo results for Bn for 5 ≤ n ...
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This chapter derives the modification of the Mayer expansion made by Ree and Hoover. Analytic expressions for the virial coefficients B2,B3, and B4 are given and Monte–Carlo results for Bn for 5 ≤ n ≤ 10 in dimensions 1 ≤ D ≤ 10 are presented. Various approximate equations of state used to ‘fit’ these coefficients are summarized. Low order virial coefficients for hard squares, cubes and hexagons are given. Open questions relating to the signs of the virial coefficients for hard spheres and discs and to the relation of virial expansions to freezing are discussed.Less

This chapter derives the modification of the Mayer expansion made by Ree and Hoover. Analytic expressions for the virial coefficients *B2*,*B3*, and *B4* are given and Monte–Carlo results for *Bn* for 5 *≤ n ≤* 10 in dimensions 1 ≤ D ≤ 10 are presented. Various approximate equations of state used to ‘fit’ these coefficients are summarized. Low order virial coefficients for hard squares, cubes and hexagons are given. Open questions relating to the signs of the virial coefficients for hard spheres and discs and to the relation of virial expansions to freezing are discussed.