125 research outputs found
Analytic Lyapunov exponents in a classical nonlinear field equation
It is shown that the nonlinear wave equation , 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.Comment: 4 pages, 1 figur
On the origin of Phase Transitions in the absence of Symmetry-Breaking
In this paper we investigate the Hamiltonian dynamics of a lattice gauge
model in three spatial dimension. Our model Hamiltonian is defined on the basis
of a continuum version of a duality transformation of a three dimensional Ising
model. The system so obtained undergoes a thermodynamic phase transition in the
absence of symmetry-breaking. Besides the well known use of quantities like the
Wilson loop we show how else the phase transition in such a kind of models can
be detected. It is found that the first order phase transition undergone by
this model is characterised according to an Ehrenfest-like classification of
phase transitions applied to the configurational entropy. On the basis of the
topological theory of phase transitions, it is discussed why the seemingly
divergent behaviour of the third derivative of configurational entropy can be
considered as the "shadow" of some suitable topological transition of certain
submanifolds of configuration space.Comment: 31 pages, 9 figure
On the apparent failure of the topological theory of phase transitions
The topological theory of phase transitions has its strong point in two
theorems proving that, for a wide class of physical systems, phase transitions
necessarily stem from topological changes of some submanifolds of configuration
space. It has been recently argued that the lattice -model
provides a counterexample that falsifies this theory. It is here shown that
this is not the case: the phase transition of this model stems from an
asymptotic () change of topology of the energy level sets, in spite
of the absence of critical points of the potential in correspondence of the
transition energy.Comment: 5 pages, 4 figure
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.Comment: 6 pages, 1 figur
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.Comment: REVTEX file, 10 pages, 2 figure
Collective behavior of oscillating electric dipoles
The present work reports about the dynamics of a collection of randomly
distributed, and randomly oriented, oscillators in 3D space, coupled by an
interaction potential falling as , where r stands for the inter-particle
distance. This model schematically represents a collection of identical
biomolecules, coherently vibrating at some common frequency, coupled with a
potential stemming from the electrodynamic interaction between
oscillating dipoles. The oscillating dipole moment of each molecule being a
direct consequence of its coherent (collective) vibration. By changing the
average distance among the molecules, neat and substantial changes in the power
spectrum of the time variation of a collective observable are found. As the
average intermolecular distance can be varied by changing the concentration of
the solvated molecules, and as the collective variable investigated is
proportional to the projection of the total dipole moment of the model
biomolecules on a coordinate plane, we have found a prospective experimental
strategy of spectroscopic kind to check whether the mentioned intermolecular
electrodynamic interactions can be strong enough to be detectable, and thus to
be of possible relevance to biology.Comment: 20 pages, 4 figure
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