Dynamics and geometric properties of thek-trigonometric model

Role of saddles in topologically driven phase transitions: the case of the d -dimensional spherical model

Analyzing the d -dimensional spherical model, we show that underlying saddles, defined through a map in the configuration space, play a central role in driving the phase transition. At the phase transition point the underlying saddle energy reaches its lowest value, corresponding to the trivial boundary topological singularity.

Understanding fragility in supercooled Lennard-Jones mixtures. II. Potential energy surface

The connection between isobaric fragility and the properties of high-order stationary points of the potential energy surface in different supercooled Lennard-Jones mixtures was investigated. The increase of effective activation energies upon supercooling appears to be driven by the increase of average potential energy barriers measured by the energy dependence of the fraction of unstable modes. Such an increase is sharper, the more fragile the mixture. Correlations between fragility and other properties of high-order stationary points, including the vibrational density of states and the localization features of unstable modes, are also discussed.

Phase transitions and topology in 2+k XY mean-field models

The thermodynamics and topology of mean-field models with 2+k body interaction terms (generalizing XY model) are derived. Focusing on two particular cases (2+4 and 2+6 body interaction terms), a comparison between thermodynamic (phase transition energy, thermodynamically forbidden energy regions) and topological (singularity and curvature of saddle entropy) properties is performed. We find that (i) a topological change is present at the phase transition energy; however, (ii) only one topological change occurs, also for those models exhibiting two phase transitions; (iii) the order of a phase transition is not completely signaled by the curvature of topological quantities.

Geometrical properties of the potential energy of the soft-sphere binary mixture

We report a detailed study of the stationary points (zero-force points) of the potential energy surface (PES) of a model structural glassformer. We compare stationary points found with two different algorithms (eigenvector following and square gradient minimization), and show that the mapping between instantaneous configuration and stationary points defined by those algorithms is as different as to strongly influence the instability index K versus temperature plot, which relevance in analyzing the liquid dynamics is thus questioned. On the other hand, the plot of K versus energy is much less sensitive to the algorithm employed, showing that the energy is the good variable to discuss geometric properties of the PES. We find new evidence of a geometric transition between a minima-dominated phase and a saddle-point-dominated one. We analyze the distances between instantaneous configurations and stationary points, and find that above the glass transition, the system is closer to saddle points than to minima.

Relationship between phase transitions and topological changes in one-dimensional models

We address the question of the quantitative relationship between thermodynamic phase transitions and topological changes in the potential energy manifold analyzing two classes of one dimensional models, the Burkhardt solid-on-solid model and the Peyrard-Bishop model for DNA thermal denaturation, both in the confining and nonconfining version. These models, apparently, do not fit [M. Kastner, Phys. Rev. Lett. 93, 150601 (2004)] in the general idea that the phase transition is signaled by a topological discontinuity. We show that in both models the phase transition energy v(c) is actually noncoincident with, and always higher than, the energy v(theta) at which a topological change appears. However, applying a procedure already successfully employed in other cases as the mean field phi4 model, i.e., introducing a map M:v-->v(s) from levels of the energy hypersurface V to the level of the stationary points "visited" at temperature T, we find that M (v(c))=v(theta). This result enhances the relevance of the underlying stationary points in determining the thermodynamics of a system, and extends the validity of the topological approach to the study of phase transition to the elusive one-dimensional systems considered here.

Topology and phase transitions: from an exactly solvable model to a relation between topology and thermodynamics

The elsewhere surmized topological origin of phase transitions is given here important evidence through the analytic study of an exactly solvable model for which both topology of submanifolds of configuration space and thermodynamics are worked out. The model is a mean-field one with a k-body interaction. It undergoes a second-order phase transition for k=2 and a first-order one for k >2 . This opens a perspective for the understanding of the deep origin of first and second-order phase transitions, respectively. In particular, a remarkable theoretical result consists of a mathematical characterization of first-order transitions. Moreover, we show that a "reduced" configuration space can be defined in terms of collective variables, such that the correspondence between phase transitions and topology changes becomes one-to-one, for this model. Finally, an unusual relationship is worked out between the microscopic description of a classical N -body system and its macroscopic thermodynamic behavior. This consists of a functional dependence of thermodynamic entropy upon the Morse indexes of the critical points (saddles) of the constant energy hypersurfaces of the microscopic 2N-dimensional phase space. Thus phase space (and configuration space) topology is directly related to thermodynamics.

Topological properties of the mean-field phi4 model

We study the thermodynamics and the properties of the stationary points (saddles and minima) of the potential energy for a phi4 mean-field model. We compare the critical energy vc [i.e., the potential energy vT evaluated at the phase transition temperature Tc ] with the energy vtheta at which the saddle energy distribution show a discontinuity in its derivative. We find that, in this model, vc >> vtheta, at variance to what has been found in different mean-field and short ranged systems, where the thermodynamic phase transitions take place at vc=vtheta [Phys. Rep. 337, 237 (2000)]]. By direct calculation of the energy vs T of the "inherent saddles," i.e., the saddles visited by the equilibrated system at temperature T , we find that vsTc approximately vtheta. Thus, we argue that the thermodynamic phase transition is related to a change in the properties of the inherent saddles rather than to a change of the topology of the potential energy surface at T= Tc. Finally, we discuss the approximation involved in our analysis and the generality of our method.

Saddle index properties, singular topology, and its relation to thermodynamic singularities for a phi4 mean-field model

We investigate the potential energy surface of a phi(4) model with infinite range interactions. All stationary points can be uniquely characterized by three real numbers alpha(+) , alpha(0) , alpha(-) with alpha(+) + alpha(0) + alpha(-) =1 , provided that the interaction strength mu is smaller than a critical value. The saddle index n(s) is equal to alpha(0) and its distribution function has a maximum at n(max)(s) =1/3 . The density p (e) of stationary points with energy per particle e , as well as the Euler characteristic chi (e) , are singular at a critical energy e(c) (mu) , if the external field H is zero. However, e(c) (mu) not equal upsilon(c) (mu) , where upsilon(c) (mu) is the mean potential energy per particle at the thermodynamic phase transition point T(c) . This proves that previous claims that the topological and thermodynamic transition points coincide is not valid, in general. Both types of singularities disappear for H not equal 0 . The average saddle index n (s) as function of e decreases monotonically with e and vanishes at the ground state energy, only. In contrast, the saddle index n(s) as function of the average energy e ( n(s) ) is given by n(s) ( e ) =1+4 e (for H=0 ) that vanishes at e =-1/4> upsilon(0) , the ground state energy.