Nonanalyticities of the entropy induced by saddle points of the potential energy landscape

The relation between saddle points of the potential of a classical many-particle system and the analyticity properties of its Boltzmann entropy is studied. For finite systems, each saddle point is found to cause a nonanalyticity in the Boltzmann entropy, and the functional form of this nonanalytic term is derived for the generic case of potentials having the Morse property. With increasing system size the order of the nonanalytic term grows unboundedly, leading to an increasing differentiability of the entropy. Nonetheless, a distribution of an unboundedly growing number of saddle points may cause a phase transition in the thermodynamic limit. Analysing the contribution of the saddle points to the density of states in the thermodynamic limit, conditions on the distribution of saddle points and their curvatures are derived which are necessary for a phase transition to occur. With these results, the puzzling absence of topological signatures in the spherical model is elucidated. As further applications, the phase transitions of the mean-field XY model and the mean-field k-trigonometric model are shown to be induced by saddle points of vanishing curvature.

[1]  M. Pettini.  Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics , 2007, 0711.1484.

[2]  Michael Kastner,et al.  Phase transitions induced by saddle points of vanishing curvature. , 2007, Physical review letters.

[3]  M. Kastner.  Phase transitions and configuration space topology , 2007, cond-mat/0703401.

[4]  M. Kastner,et al.  Phase transitions from saddles of the potential energy landscape. , 2007, Physical review letters.

[5]  M. Kastner,et al.  Partial equivalence of statistical ensembles and kinetic energy , 2007, cond-mat/0701339.

[6]  D. Stariolo,et al.  Topology and Phase Transitions: The Case of the Short Range Spherical Model , 2006 .

[7]  M. Kastner,et al.  Nonanalyticities of entropy functions of finite and infinite systems. , 2006, Physical review letters.

[8]  S. Hilbert,et al.  Nonanalytic microscopic phase transitions and temperature oscillations in the microcanonical ensemble: an exactly solvable one-dimensional model for evaporation. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  S. Hilbert,et al.  Phase transitions in small systems: Microcanonical vs. canonical ensembles , 2005, cond-mat/0511501.

[10]  Daniel W. Hook,et al.  Quantum phase transitions without thermodynamic limits , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[11]  D. Stariolo,et al.  Topology, phase transitions, and the spherical model. , 2005, Physical review letters.

[12]  M. Kastner,et al.  The mean-field model Phi4: entropy, analyticity, and configuration space topology. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  H. Behringer.  Critical properties of the spherical model in the microcanonical formalism , 2005, cond-mat/0509486.

[14]  Firenze,et al.  Topology and phase transitions II. Theorem on a necessary relation , 2005, math-ph/0505057.

[15]  M. Kastner,et al.  On the Mean-Field Spherical Model , 2005, cond-mat/0503046.

[16]  F. Zamponi,et al.  Topology and phase transitions: from an exactly solvable model to a relation between topology and thermodynamics. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  R. Franzosi,et al.  Theorem on the origin of phase transitions. , 2003, Physical review letters.

[18]  Marco Pettini,et al.  Topological signature of first-order phase transitions in a mean-field model , 2003 .

[19]  E. Cohen,et al.  Phase Transitions and Topology Changes in Configuration Space , 2003, cond-mat/0303200.

[20]  G.Ruocco,et al.  Topological Signature of First Order Phase Transitions , 2002, cond-mat/0205483.

[21]  Tony Crilly,et al.  Bifurcations and Catastrophes , 2000 .

[22]  Michel Demazure,et al.  Bifurcations and Catastrophes: Geometry Of Solutions To Nonlinear Problems , 2000 .

[23]  D. Ruelle.  Statistical Mechanics: Rigorous Results , 1999 .

[24]  T. Keyes,et al.  Unstable modes in liquids density of states, potential energy, and heat capacity , 1993 .

[25]  S. Rutkevich.  Instantons in spherical model thermodynamics , 1992 .

[26]  H. W. Lewis,et al.  Spherical Model of a Ferromagnet , 1952 .

[27]  Mark Kac,et al.  The Spherical Model of a Ferromagnet , 1952 .

[28]  Roberto Franzosi,et al.  0 30 50 32 v 1 1 5 M ay 2 00 3 Topology and Phase Transitions : Theorem on a necessary relation , 2005 .

[29]  D. Wales.  Energy Landscapes by David Wales , 2004 .

[30]  松本 幸夫.  An introduction to Morse theory , 2002 .

[31]  B. Dundas,et al.  DIFFERENTIAL TOPOLOGY , 2002 .