arXiv subjects

Marco Pettini

Publications and source records attributed to Marco Pettini.

At least 55 records · Page 3Linked to original sources

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↗

arXiv2004

Controlling chaos in area-preserving maps

We describe a method of control of chaos that occurs in area-preserving maps. This method is based on small modifications of the original map by addition of a small control term. We apply this control technique to the standard map and to the tokamap.

nlin.CD↗

arXiv2004

Tailoring Phase Space : A Way to Control Hamiltonian Transport

We present a method to control transport in Hamiltonian systems. We provide an algorithm - based on a perturbation of the original Hamiltonian localized in phase space - to design small control terms that are able to create isolated barriers of transport without modifying other parts of phase space. We apply this method of localized control to a forced pendulum model and to a system describing the motion of charged particles in a model of turbulent electric field.

nlin.CD↗

arXiv2004

Controlling chaotic transport in a Hamiltonian model of interest to magnetized plasmas

We present a technique to control chaos in Hamiltonian systems which are close to integrable. By adding a small and simple control term to the perturbation, the system becomes more regular than the original one. We apply this technique to a model that reproduces turbulent ExB drift and show numerically that the control is able to drastically reduce chaotic transport.

nlin.CD↗

arXiv2003

A Theorem on the origin of Phase Transitions

For physical systems described by smooth, finite-range and confining microscopic interaction potentials V with continuously varying coordinates, we announce and outline the proof of a theorem that establishes that unless the equipotential hypersurfaces of configuration space Σ_v ={(q_1,...,q_N)\in R^N | V(q_1,...,q_N) = v}, v \in R, change topology at some v_c in a given interval [v_0, v_1] of values v of V, the Helmoltz free energy must be at least twice differentiable in the corresponding interval of inverse temperature (β(v_0), β(v_1)) also in the N -> \infty$ limit. Thus the occurrence of a phase transition at some β_c =β(v_c) is necessarily the consequence of the loss of diffeomorphicity among the {Σ_v}_{v < v_c}$ and the {Σ_v}_{v > v_c}, which is the consequence of the existence of critical points of V on Σ_{v=v_c}, that is points where \nabla V=0.

cond-mat.stat-mech↗

arXiv2003

Control of chaotic transport in Hamiltonian systems

It is shown that a relevant control of Hamiltonian chaos is possible through suitable small perturbations whose form can be explicitly computed. In particular, it is possible to control (reduce) the chaotic diffusion in the phase space of a Hamiltonian system with 1.5 degrees of freedom which models the diffusion of charged test particles in a ``turbulent'' electric field across the confining magnetic field in controlled thermonuclear fusion devices. Though still far from practical applications, this result suggests that some strategy to control turbulent transport in magnetized plasmas, in particular tokamaks, is conceivable.

nlin.CD↗

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↗

arXiv2001

On the clustering phase transition in self-gravitating N-body systems

The thermodynamic behaviour of self-gravitating $N$-body systems has been worked out by borrowing a standard method from Molecular Dynamics: the time averages of suitable quantities are numerically computed along the dynamical trajectories to yield thermodynamic observables. The link between dynamics and thermodynamics is made in the microcanonical ensemble of statistical mechanics. The dynamics of self-gravitating $N$-body systems has been computed using two different kinds of regularization of the newtonian interaction: the usual softening and a truncation of the Fourier expansion series of the two-body potential. $N$ particles of equal masses are constrained in a finite three dimensional volume. Through the computation of basic thermodynamic observables and of the equation of state in the $P - V$ plane, new evidence is given of the existence of a second order phase transition from a homogeneous phase to a clustered phase. This corresponds to a crossover from a polytrope of index $n=3$, i.e. $p=K V^{-4/3}$, to a perfect gas law $p=K V^{-1}$, as is shown by the isoenergetic curves on the $P - V$ plane. The dynamical-microcanonical averages are compared to their corresponding canonical ensemble averages, obtained through standard Monte Carlo computations. A major disagreement is found, because the canonical ensemble seems to have completely lost any information about the phase transition. The microcanonical ensemble appears as the only reliable statistical framework to tackle self-gravitating systems. Finally, our results -- obtained in a ``microscopic'' framework -- are compared with some existing theoretical predictions -- obtained in a ``macroscopic'' (thermodynamic) framework: qualitative and quantitative agreement is found, with an interesting exception.

astro-ph↗

arXiv2001

A geometric interpretation of integrable motions

Integrability, one of the classic issues in galactic dynamics and in general in celestial mechanics, is here revisited in a Riemannian geometric framework, where newtonian motions are seen as geodesics of suitable ``mechanical'' manifolds. The existence of constants of motion that entail integrability is associated with the existence of Killing tensor fields on the mechanical manifolds. Such tensor fields correspond to hidden symmetries of non-Noetherian kind. Explicit expressions for Killing tensor fields are given for the N=2 Toda model, and for a modified Henon-Heiles model, recovering the already known analytic expressions of the second conserved quantity besides energy for each model respectively.

astro-ph↗

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↗

arXiv2001

Topology and Phase Transitions: towards a proper mathematical definition of finite N transitions

A new point of view about the deep origin of thermodynamic phase transitions is sketched. The main idea is to link the appearance of phase transitions to some major topology change of suitable submanifolds of phase space instead of linking them to non-analyticity, as is usual in the Yang-Lee and in the Dobrushin-Ruelle-Sinai theories. In the new framework a new possibility appears to properly define a mathematical counterpart of phase transitions also at finite number of degrees of freedom. This is of prospective interest to the study of those systems that challenge the conventional approaches, as is the case of phase transitions in nuclear clusters.

cond-mat.stat-mech↗

arXiv2001

Strong Chaos in the N-body problem and Microcanonical Thermodynamics of collisionless self gravitating systems

The dynamical justifications which lie at the basis of an effective Statistical Mechanics for self gravitating systems are formulated, analyzing some among the well known obstacles thought to prevent a rigorous Statistical treatment. It is shown that N-body gravitational systems satisfy a strong chaos criterion, so supporting the assumption of an increasingly uniform spreading of orbits over the constant energy surface, i.e., the asymptotic evolution towards a microcanonical distribution. We then focus on the necessary conditions for the equivalence of statistical ensembles and remark that this equivalence is broken for any N-body system whose interaction has a range comparable with its spatial extension. Once realized that the obstacles originate from the long range nature of the Newtonian interaction, leading to the non-extensivity of canonical and grand-canonical thermodynamic potentials, we show that instead a suitably generalized microcanonical ensemble constitutes an orthode, i.e., a reliable framework for a correct Thermodynamics. Within this setting we use the thermodynamic relations and find consistent definitions for Entropy (which turns out to be extensive, despite the non stable nature of the interaction), Temperature and Heat Capacity. Then, a Second Law-like criterion is used to select the hierarchy of secular equilibria describing, for any finite time, the macroscopic behaviour of self gravitating systems.

astro-ph↗

arXiv2000

Analytic Lyapunov exponents in a classical nonlinear field equation

It is shown that the nonlinear wave equation $\partial_t^2ϕ- \partial^2_x ϕ-μ_0\partial_x(\partial_xϕ)^3 =0$, which is the continuum limit of the Fermi-Pasta-Ulam (FPU) beta model, has a positive Lyapunov exponent lambda_1, whose analytic energy dependence is given. The result (a first example for field equations) is achieved by evaluating the lattice-spacing dependence of lambda_1 for the FPU model within the framework of a Riemannian description of Hamiltonian chaos. We also discuss a difficulty of the statistical mechanical treatment of this classical field system, which is absent in the dynamical description.

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

Topology and phase transitions: a paradigmatic evidence

We report upon the numerical computation of the Euler characteristic χ(a topologic invariant) of the equipotential hypersurfaces Σ_v of the configuration space of the two-dimensional lattice $ϕ^4$ model. The pattern χ(Σ_v) vs. v (potential energy) reveals that a major topology change in the family {Σ_v}_{v\in R} is at the origin of the phase transition in the model considered. The direct evidence given here - of the relevance of topology for phase transitions - is obtained through a general method that can be applied to any other model.

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↗

arXiv1998

Geometric approach to chaos in the classical dynamics of abelian lattice gauge theory

A Riemannian geometrization of dynamics is used to study chaoticity in the classical Hamiltonian dynamics of a U(1) lattice gauge theory. This approach allows one to obtain analytical estimates of the largest Lyapunov exponent in terms of time averages of geometric quantities. These estimates are compared with the results of numerical simulations, and turn out to be very close to the values extrapolated for very large lattice sizes even when the geometric quantities are computed using small lattices. The scaling of the Lyapunov exponent with the energy density is found to be well described by a quadratic power law.

chao-dyn↗

arXiv1998

Lyapunov exponents from geodesic spread in configuration space

The exact form of the Jacobi -- Levi-Civita (JLC) equation for geodesic spread is here explicitly worked out at arbitrary dimension for the configuration space manifold M_E = {q in R^N | V(q) < E} of a standard Hamiltonian system, equipped with the Jacobi (or kinetic energy) metric g_J. As the Hamiltonian flow corresponds to a geodesic flow on (M_E,g_J), the JLC equation can be used to study the degree of instability of the Hamiltonian flow. It is found that the solutions of the JLC equation are closely resembling the solutions of the standard tangent dynamics equation which is used to compute Lyapunov exponents. Therefore the instability exponents obtained through the JLC equation are in perfect quantitative agreement with usual Lyapunov exponents. This work completes a previous investigation that was limited only to two-degrees of freedom systems.

chao-dyn↗