arXiv subjects

Lapo Casetti

Publications and source records attributed to Lapo Casetti.

At least 55 records · Page 3Linked to original sources

arXiv2009

Energy landscape and phase transitions in the self-gravitating ring model

We apply a recently proposed criterion for the existence of phase transitions, which is based on the properties of the saddles of the energy landscape, to a simplified model of a system with gravitational interactions, referred to as the self-gravitating ring model. We show analytically that the criterion correctly singles out the phase transition between a homogeneous and a clustered phase and also suggests the presence of another phase transition, not previously known. On the basis of the properties of the energy landscape we conjecture on the nature of the latter transition.

cond-mat.stat-mech↗

arXiv2009

A graph theoretical analysis of the energy landscape of model polymers

In systems characterized by a rough potential energy landscape, local energetic minima and saddles define a network of metastable states whose topology strongly influences the dynamics. Changes in temperature, causing the merging and splitting of metastable states, have non trivial effects on such networks and must be taken into account. We do this by means of a recently proposed renormalization procedure. This method is applied to analyze the topology of the network of metastable states for different polypeptidic sequences in a minimalistic polymer model. A smaller spectral dimension emerges as a hallmark of stability of the global energy minimum and highlights a non-obvious link between dynamic and thermodynamic properties.

cond-mat.soft↗

arXiv2009

Kinetic energy and microcanonical nonanalyticities in finite and infinite systems

In contrast to the canonical case, microcanonical thermodynamic functions can show nonanalyticities also for finite systems. In this paper we contribute to the understanding of these nonanalyticities by working out the relation between nonanalyticities of the microcanonical entropy and its configurational counterpart. If the configurational microcanonical entropy $ω_N^c(v)$ has a nonanalyticity at $v=v_c$, then the microcanonical entropy $ω_N(ε)$ has a nonanalyticity at the same value $ε=v_c$ of its argument for any finite value of the number of degrees of freedom $N$. The presence of the kinetic energy weakens the nonanalyticities such that, if the configurational entropy is $p$ times differentiable, the entropy is $p+\lfloor N/2 \rfloor$-times differentiable. In the thermodynamic limit, however, the behaviour is very different: The nonanalyticities do not longer occur at the same values of the arguments, but the nonanalyticity of the microcanonical entropy is shifted to a larger energy. These results give a general explanation of the peculiar behaviour previously observed for the mean-field spherical model. With the hypercubic model we provide a further example illustrating our results.

cond-mat.stat-mech↗

arXiv2009

Vibrational entropy and the structural organization of proteins

In this paper we analyze the vibrational spectra of a large ensemble of non-homologous protein structures by means of a novel tool, that we coin the Hierarchical Network Model (HNM). Our coarse-grained scheme accounts for the intrinsic heterogeneity of force constants displayed by protein arrangements and also incorporates side-chain degrees of freedom. Our analysis shows that vibrational entropy per unit residue correlates with the content of secondary structure. Furthermore, we assess the individual contribution to vibrational entropy of the novel features of our scheme as compared with the predictions of state-of-the-art network models. This analysis highlights the importance of properly accounting for the intrinsic hierarchy in force strengths typical of the different atomic bonds that build up and stabilize protein scaffolds. Finally, we discuss possible implications of our findings in the context of protein aggregation phenomena.

q-bio.BM↗

arXiv2008

Geometry of the energy landscape and folding transition in a simple model of a protein

A geometric analysis of the global properties of the energy landscape of a minimalistic model of a polypeptide is presented, which is based on the relation between dynamical trajectories and geodesics of a suitable manifold, whose metric is completely determined by the potential energy. We consider different sequences, some with a definite protein-like behavior, a unique native state and a folding transition, and the others undergoing a hydrophobic collapse with no tendency to a unique native state. The global geometry of the energy landscape appears to contain relevant information on the behavior of the various sequences: in particular, the fluctuations of the curvature of the energy landscape, measured by means of numerical simulations, clearly mark the folding transition and allow to distinguish the protein-like sequences from the others.

cond-mat.stat-mech↗

arXiv2007

Partial equivalence of statistical ensembles and kinetic energy

The phenomenon of partial equivalence of statistical ensembles is illustrated by discussing two examples, the mean-field XY and the mean-field spherical model. The configurational parts of these systems exhibit partial equivalence of the microcanonical and the canonical ensemble. Furthermore, the configurational microcanonical entropy is a smooth function, whereas a nonanalytic point of the configurational free energy indicates the presence of a phase transition in the canonical ensemble. In the presence of a standard kinetic energy contribution, partial equivalence is removed and a nonanalyticity arises also microcanonically. Hence in contrast to the common belief, kinetic energy, even though a quadratic form in the momenta, has a non-trivial effect on the thermodynamic behaviour. As a by-product we present the microcanonical solution of the mean-field spherical model with kinetic energy for finite and infinite system sizes.

cond-mat.stat-mech↗

arXiv2006

Curvature of the energy landscape and folding of model proteins

We study the geometric properties of the energy landscape of coarse-grained, off-lattice models of polymers by endowing the configuration space with a suitable metric, depending on the potential energy function, such that the dynamical trajectories are the geodesics of the metric. Using numerical simulations, we show that the fluctuations of the curvature clearly mark the folding transition, and that this quantity allows to distinguish between polymers having a protein-like behavior (i.e., that fold to a unique configuration) and polymers which undergo a hydrophobic collapse but do not have a folding transition. These geometrical properties are defined by the potential energy without requiring any prior knowledge of the native configuration.

cond-mat.stat-mech↗

arXiv2006

Nonanalyticities of entropy functions of finite and infinite systems

In contrast to the canonical ensemble where thermodynamic functions are smooth for all finite system sizes, the microcanonical entropy can show nonanalytic points also for finite systems, even if the Hamiltonian is smooth. The relation between finite and infinite system nonanalyticities is illustrated by means of a simple classical spin-like model which is exactly solvable for both, finite and infinite system sizes, showing a phase transition in the latter case. The microcanonical entropy is found to have exactly one nonanalytic point in the interior of its domain. For all finite system sizes, this point is located at the same fixed energy value $ε_{c}^{finite}$, jumping discontinuously to a different value $ε_{c}^{infinite}$ in the thermodynamic limit. Remarkably, $ε_{c}^{finite}$ equals the average potential energy of the infinite system at the phase transition point. The result, supplemented with results on nonanalyticities of the microcanonical entropy for other models, indicates that care is required when trying to infer infinite system properties from finite system nonanalyticities.

cond-mat.stat-mech↗

arXiv2005

Topological conditions for discrete symmetry breaking and phase transitions

In the framework of a recently proposed topological approach to phase transitions, some sufficient conditions ensuring the presence of the spontaneous breaking of a Z_2 symmetry and of a symmetry-breaking phase transition are introduced and discussed. A very simple model, which we refer to as the hypercubic model, is introduced and solved. The main purpose of this model is that of illustrating the content of the sufficient conditions, but it is interesting also in itself due to its simplicity. Then some mean-field models already known in the literature are discussed in the light of the sufficient conditions introduced here.

cond-mat.stat-mech↗

arXiv2004

Weak and strong chaos in FPU models and beyond

We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. A first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: i) a Stochasticity Threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. ii) a Strong Stochasticity Threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weakly and strongly chaotic regimes. It is stable with N. A second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. The starting of this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N, or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, a third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions.

cond-mat.stat-mech↗

arXiv2004

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

The elsewhere surmised topological origin of phase transitions is given here new important evidence through the analytic study of an exactly solvable model for which both topology 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 completely new 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 new 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 behaviour. 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.

cond-mat.stat-mech↗

arXiv2003

Phase transitions and topology changes in configuration space

The relation between thermodynamic phase transitions in classical systems and topological changes in their configuration space is discussed for two physical models and contains the first exact analytic computation of a topologic invariant (the Euler characteristic) of certain submanifolds in the configuration space of two physical models. The models are the mean-field XY model and the one-dimensional XY model with nearest-neighbor interactions. The former model undergoes a second-order phase transition at a finite critical temperature while the latter has no phase transitions. The computation of this topologic invariant is performed within the framework of Morse theory. In both models topology changes in configuration space are present as the potential energy is varied; however, in the mean-field model there is a particularly "strong" topology change, corresponding to a big jump in the Euler characteristic, connected with the phase transition, which is absent in the one-dimensional model with no phase transition. The comparison between the two models has two major consequences: i) it lends new and strong support to a recently proposed topological approach to the study of phase transitions; ii) it allows us to conjecture which particular topology changes could entail a phase transition in general. We also discuss a simplified illustrative model of the topology changes connected to phase transitions using of two-dimensional surfaces, and a possible direct connection between topological invariants and thermodynamic quantities.

cond-mat.stat-mech↗

arXiv2002

Modeling hydration water and its role in polymer folding

The hydrophobic effect is the dominant force which drives a protein towards its native state, but its physics has not been thoroughly understood yet. We introduce an exactly solvable model of the solvation of non-polar molecules in water, which shows that the reduced number of allowed configurations of water molecules when the solute is present is enough to give rise to hydrophobic behaviour. We apply our model to a non-polar homopolymer in aqueous solution, obtaining a clear evidence of both ``cold'' and ``warm'' collapse transitions that recall those of proteins. Finally we show how the model can be adapted to describe the solvation of aromatic and polar molecules.

cond-mat.stat-mech↗

arXiv2002

Model for the hydration of non-polar compounds and polymers

We introduce an exactly solvable statistical-mechanical model of the hydration of non-polar compounds, based on grouping water molecules in clusters where hydrogen bonds and isotropic interactions occur; interactions between clusters are neglected. Analytical results show that an effective strengthening of hydrogen bonds in the presence of the solute, together with a geometric reorganization of water molecules, are enough to yield hydrophobic behavior. We extend our model to describe a non-polar homopolymer in aqueous solution, obtaining a clear evidence of both ``cold'' and ``warm'' swelling transitions. This suggests that our model could be relevant to describe some features of protein folding.

cond-mat.stat-mech↗

arXiv2001

Phase transitions as topology changes in configuration space: an exact result

The phase transition in the mean-field XY model is shown analytically to be related to a topological change in its configuration space. Such a topology change is completely described by means of Morse theory allowing a computation of the Euler characteristic--of suitable submanifolds of configuration space--which shows a sharp discontinuity at the phase transition point, also at finite N. The present analytic result provides, with previous work, a new key to a possible connection of topological changes in configuration space as the origin of phase transitions in a variety of systems.

cond-mat.stat-mech↗

arXiv1999

Geometric approach to Hamiltonian dynamics and statistical mechanics

This paper is a review of results which have been recently obtained by applying mathematical concepts drawn, in particular, from differential geometry and topology, to the physics of Hamiltonian dynamical systems with many degrees of freedom of interest for statistical mechanics. The first part of the paper concerns the applications of methods used in classical differential geometry to study the chaotic dynamics of Hamiltonian systems. Starting from the identity between the trajectories of a dynamical system and the geodesics in its configuration space, a geometric theory of chaotic dynamics can be developed, which sheds new light on the origin of chaos in Hamiltonian systems. In fact, it appears that chaos can be induced not only by negative curvatures, as was originally surmised, but also by positive curvatures, provided the curvatures are fluctuating along the geodesics. In the case of a system with a large number of degrees of freedom it is possible to give an analytical estimate of the largest Lyapunov exponent by means of a geometric model independent of the dynamics. In the second part of the paper the phenomenon of phase transitions is addressed and it is here that topology comes into play. In fact, when a system undergoes a phase transition, the fluctuations of the configuration-space curvature exhibit a singular behavior at the phase transition point, which can be qualitatively reproduced using geometric models. In these models the origin of the singular behavior of the curvature fluctuations appears to be caused by a topological transition in configuration space. This leads us to put forward a Topological Hypothesis (TH). The content of the TH is that phase transitions would be related at a deeper level to a change in the topology of the configuration space of the system.

cond-mat.stat-mech↗

arXiv1999

Lattice model for cold and warm swelling of polymers in water

We define a lattice model for the interaction of a polymer with water. We solve the model in a suitable approximation. In the case of a non-polar homopolymer, for reasonable values of the parameters, the polymer is found in a non-compact conformation at low temperature; as the temperature grows, there is a sharp transition towards a compact state, then, at higher temperatures, the polymer swells again. This behaviour closely reminds that of proteins, that are unfolded at both low and high temperatures.

cond-mat.stat-mech↗

arXiv1999

Topological origin of the phase transition in a mean-field model

We argue that the phase transition in the mean-field XY model is related to a particular change in the topology of its configuration space. The nature of this topological transition can be discussed on the basis of elementary Morse theory using the potential energy per particle V as a Morse function. The value of V where such a topological transition occurs equals the thermodynamic value of V at the phase transition and the number of (Morse) critical points grows very fast with the number of particles N. Furthermore, as in statistical mechanics, also in topology the way the thermodynamic limit is taken is crucial.

cond-mat.stat-mech↗