New results for phase transitions from catastrophe theory.

Catastrophe theory predicts that in certain limits universal relations should exist between barrier heights, curvatures and the positions of local maxima and minima on a potential or free energy surface. In the present work we investigate these relations for both first- and second-order phase transitions, revealing that the ideal ratios often hold quite well over a wide range of conditions. This elementary catastrophe theory is illustrated for the melting transition of an atomic cluster, the isotropic-to-nematic transition in a liquid crystal, and the ferromagnetic-to-paramagnetic phase transition in the two-dimensional Ising model.

[1]  V. Lobaskin,et al.  Critical exponents in the single-parameter highest catastrophe theory , 1993 .

[2]  Richard Jones Physical and Mechanistic Organic Chemistry , 1979 .

[3]  Chen Ning Yang,et al.  The Spontaneous Magnetization of a Two-Dimensional Ising Model , 1952 .

[4]  P. Mukherjee,et al.  Simple Landau model of the smectic- A-isotropic phase transition , 2001 .

[5]  Christophe Chipot,et al.  Free Energy Calculations. The Long and Winding Gilded Road , 2002 .

[6]  THE CRITICAL POINTS OF A FUNCTION OF n VARIABLES. , 1930, Proceedings of the National Academy of Sciences of the United States of America.

[7]  A. Berezin Application of catastrophe theory to phase transitions of trapped particles , 1991 .

[8]  D. Ronis,et al.  Unified model of the smectic-A, nematic, and isotropic phases for bulk, interfaces, and thin films: Bulk , 1980 .

[9]  K. Baldridge,et al.  Structure/energy correlation of bowl depth and inversion barrier in corannulene derivatives: combined experimental and quantum mechanical analysis. , 2001, Journal of the American Chemical Society.

[10]  E. Ising Beitrag zur Theorie des Ferromagnetismus , 1925 .

[11]  U. D. Priyakumar,et al.  Heterobuckybowls: a theoretical study on the structure, bowl-to-bowl inversion barrier, bond length alternation, structure-inversion barrier relationship, stability, and synthetic feasibility. , 2001, The Journal of organic chemistry.

[12]  Pierre Papon,et al.  The physics of phase transitions : concepts and applications , 2002 .

[13]  Jonathan P. K. Doye,et al.  Calculation of thermodynamic properties of small Lennard‐Jones clusters incorporating anharmonicity , 1995 .

[14]  C. Thompson Classical Equilibrium Statistical Mechanics , 1988 .

[15]  David J. Wales,et al.  Free energy barriers to melting in atomic clusters , 1994 .

[16]  D. Ronis,et al.  Unified model of the smectic-A, nematic, and isotropic phases for bulk, interfaces, and thin films. II. Interfaces and thin films , 1981 .

[17]  M. Magnuson,et al.  On the temperature dependence of the order parameter of liquid crystals over a wide nematic range , 1995 .

[18]  T. W. Barrett,et al.  Catastrophe Theory, Selected Papers 1972-1977 , 1978, IEEE Transactions on Systems, Man, and Cybernetics.

[19]  David J. Wales,et al.  Coexistence in small inert gas clusters , 1993 .

[20]  H. Bürgi What we can learn about fast chemical processes from slow diffraction experiments. , 2003, Faraday discussions.

[21]  J. Mather,et al.  Stability of $C^\infty $ mappings, III. Finitely determined map-germs , 1968 .

[22]  O. Castaños,et al.  Shapes and stability within the interacting boson model: effective Hamiltonians , 1998 .

[23]  Beale Exact distribution of energies in the two-dimensional ising model. , 1996, Physical review letters.

[24]  R. Thom Topological models in biology , 1969 .

[25]  J. M. Arias,et al.  Phase transitions and critical points in the rare-earth region , 2003, nucl-th/0304008.

[26]  The Antonov problem for rotating systems , 2002, cond-mat/0208230.

[27]  Wolff,et al.  Collective Monte Carlo updating for spin systems. , 1989, Physical review letters.

[28]  R. Baxter,et al.  399th solution of the Ising model , 1978 .

[29]  L. Schulman Tricritical Points and Type-Three Phase Transitions , 1973 .

[30]  R. Moldovan Density Dependence in the Landau‐de Gennes Theory for the Nematic‐Isotropic Phase Transition , 2000 .

[31]  P. Gennes Phenomenology of short-range-order effects in the isotropic phase of nematic materials , 1969 .

[32]  D. Landau,et al.  Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  L. Onsager Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[34]  J. Margalef-Roig,et al.  Analysis of a three-component model phase diagram by Catastrophe Theory , 1997, cond-mat/9807301.

[35]  D. Peregoudov Effective potentials and Bogoliubov’s quasi-averages , 1997 .

[36]  Timothy A. Davis,et al.  Finite Element Analysis of the Landau--de Gennes Minimization Problem for Liquid Crystals , 1998 .

[37]  Evidence of exactness of the mean-field theory in the nonextensive regime of long-range classical spin models , 1999, cond-mat/9906340.

[38]  K. Okada Successive phase transition of a cubic system , 1992 .

[39]  D. Wales A Microscopic Basis for the Global Appearance of Energy Landscapes , 2001, Science.

[40]  L. Hammett,et al.  Linear free energy relationships in rate and equilibrium phenomena , 1938 .

[41]  J. Ilja Siepmann,et al.  Monte carlo methods in chemical physics , 1999 .

[42]  Eberhard R. Hilf,et al.  The structure of small clusters: Multiple normal-modes model , 1993 .

[43]  B. Chakrabarti,et al.  Mean-field and Monte Carlo studies of the magnetization-reversal transition in the Ising model , 2000, cond-mat/0003294.

[44]  P. G. de Gennes,et al.  Short Range Order Effects in the Isotropic Phase of Nematics and Cholesterics , 1971 .

[45]  T. Lubensky,et al.  Mean-field theory of the nematic-smectic-Aphase change in liquid crystals , 1976 .

[46]  Daan Frenkel,et al.  Monte Carlo study of the isotropic-nematic transition in a fluid of thin hard disk , 1982 .

[47]  L. Onsager THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES , 1949 .

[48]  D. Wales Locating stationary points for clusters in cartesian coordinates , 1993 .

[49]  T. D. Lee,et al.  Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model , 1952 .

[50]  K. Keller Modelling critical behaviour in terms of catastrophe theory and fractal lattices , 1981 .

[51]  Mark Kac,et al.  On the van der Waals Theory of the Vapor‐Liquid Equilibrium. I. Discussion of a One‐Dimensional Model , 1963 .

[52]  P. Gennes,et al.  The physics of liquid crystals , 1974 .

[53]  Lars Onsager,et al.  Crystal Statistics. III. Short-Range Order in a Binary Ising Lattice , 1949 .

[54]  René Thom,et al.  Structural stability and morphogenesis , 1977, Pattern Recognit..

[55]  William P. Jencks,et al.  A primer for the Bema Hapothle. An empirical approach to the characterization of changing transition-state structures , 1985 .

[56]  P. Flory,et al.  Phase equilibria in solutions of rod-like particles , 1956, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[57]  Robert Gilmore,et al.  Catastrophe Theory for Scientists and Engineers , 1981 .

[58]  P. Saunders An Introduction to Catastrophe Theory , 1980 .

[59]  D. Landau,et al.  Efficient, multiple-range random walk algorithm to calculate the density of states. , 2000, Physical review letters.

[60]  P. Morse Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels , 1929 .