408 research outputs found
Scaling laws for the largest Lyapunov exponent in long-range systems: A random matrix approach
We investigate the laws that rule the behavior of the largest Lyapunov
exponent (LLE) in many particle systems with long range interactions. We
consider as a representative system the so-called Hamiltonian alpha-XY model
where the adjustable parameter alpha controls the range of the interactions of
N ferromagnetic spins in a lattice of dimension d. In previous work the
dependence of the LLE with the system size N, for sufficiently high energies,
was established through numerical simulations. In the thermodynamic limit, the
LLE becomes constant for alpha greater than d whereas it decays as an inverse
power law of N for alpha smaller than d. A recent theoretical calculation based
on Pettini's geometrization of the dynamics is consistent with these numerical
results (M.-C. Firpo and S. Ruffo, cond-mat/0108158). Here we show that the
scaling behavior can also be explained by a random matrix approach, in which
the tangent mappings that define the Lyapunov exponents are modeled by random
simplectic matrices drawn from a suitable ensemble.Comment: 5 pages, no figure
Hamiltonian Dynamics and the Phase Transition of the XY Model
A Hamiltonian dynamics is defined for the XY model by adding a kinetic energy
term. Thermodynamical properties (total energy, magnetization, vorticity)
derived from microcanonical simulations of this model are found to be in
agreement with canonical Monte-Carlo results in the explored temperature
region. The behavior of the magnetization and the energy as functions of the
temperature are thoroughly investigated, taking into account finite size
effects. By representing the spin field as a superposition of random phased
waves, we derive a nonlinear dispersion relation whose solutions allow the
computation of thermodynamical quantities, which agree quantitatively with
those obtained in numerical experiments, up to temperatures close to the
transition. At low temperatures the propagation of phonons is the dominant
phenomenon, while above the phase transition the system splits into ordered
domains separated by interfaces populated by topological defects. In the high
temperature phase, spins rotate, and an analogy with an Ising-like system can
be established, leading to a theoretical prediction of the critical temperature
.Comment: 10 figures, Revte
Riemannian theory of Hamiltonian chaos and Lyapunov exponents
This paper deals with the problem of analytically computing the largest
Lyapunov exponent for many degrees of freedom Hamiltonian systems. This aim is
succesfully reached within a theoretical framework that makes use of a
geometrization of newtonian dynamics in the language of Riemannian geometry. A
new point of view about the origin of chaos in these systems is obtained
independently of homoclinic intersections. Chaos is here related to curvature
fluctuations of the manifolds whose geodesics are natural motions and is
described by means of Jacobi equation for geodesic spread. Under general
conditions ane effective stability equation is derived; an analytic formula for
the growth-rate of its solutions is worked out and applied to the
Fermi-Pasta-Ulam beta model and to a chain of coupled rotators. An excellent
agreement is found the theoretical prediction and the values of the Lyapunov
exponent obtained by numerical simulations for both models.Comment: RevTex, 40 pages, 8 PostScript figures, to be published in Phys. Rev.
E (scheduled for November 1996
An equilibrium model for RFP plasmas in the presence of resonant tearing modes
The equilibrium of a finite-beta RFP plasma in the presence of
saturated-amplitude tearing modes is investigated. The singularities of the MHD
force balance equation JXB=grad(p) at the modes rational surfaces are resolved
through a proper regularization of the zeroth-order (equilibrium) profiles, by
setting to zero there the gradient of the pressure and parallel current
density. An equilibrium model, which satisfies the regularization rule at the
various rational surfaces, is developed. The comparison with the experimental
data from the Reversed Field eXperiment (RFX) gives encouraging results. The
model provides an easy tool for magnetic analysis: many aspects of the
perturbations can be analyzed and reconstructed.Comment: Final accepted version. 36 page
Statistical features of edge turbulence in RFX-mod from Gas Puffing Imaging
Plasma density fluctuations in the edge plasma of the RFX-mod device are
measured through the Gas Puffing Imaging Diagnostics. Statistical features of
the signal are quantified in terms of the Probability Distribution Function
(PDF), and computed for several kinds of discharges. The PDFs from discharges
without particular control methods are found to be adequately described by a
Gamma function, consistently with the recent results by Graves et al [J.P.
Graves, et al, Plasma Phys. Control. Fusion 47, L1 (2005)]. On the other hand,
pulses with external methods for plasma control feature modified PDFs. A first
empirical analysis suggests that they may be interpolated through a linear
combination of simple functions. An inspection of the literature shows that
this kind of PDFs is common to other devices as well, and has been suggested to
be due to the simultaneous presence of different mechanisms driving
respectively coherent bursts and gaussian background turbulence. An attempt is
made to relate differences in the PDFs to plasma conditions such as the local
shift of the plasma column. A simple phenomenological model to interpret the
nature of the PDF and assign a meaning to its parameters is also developed.Comment: 27 pages. Published in PPC
Landau model for uniaxial systems with complex order parameter
We study the Landau model for uniaxial incommensurate-commensurate systems of
the I class by keeping Umklapp terms of third and fourth order in the expansion
of the free energy. It applies to systems in which the soft mode minimum lies
between the corresponding commensurate wave numbers. The minimization of the
Landau functional leads to the sine-Gordon equation with two nonlinear terms,
equivalent to the equation of motion for the well-known classical mechanical
problem of two mixing resonances. We calculate the average free energies for
periodic, quasiperiodic and chaotic solutions of this equation, and show that
in the regime of finite strengths of Umklapp terms only periodic solutions are
absolute minima of the free energy, so that the phase diagram contains only
commensurate configurations. The phase transitions between neighboring
configurations are of the first order, and the wave number of ordering goes
through harmless staircase with a finite number of steps. These results are the
basis for the interpretation of phase diagrams for some materials from the I
class of incommensurate-commensurate systems, in particular of those for
ABX and BCCD compounds. Also, we argue that chaotic barriers which
separate metastable periodic solutions represent an intrinsic mechanism for
observed memory effects and thermal hystereses.Comment: 12 pages, 14 figures, LaTeX, to be published in Phys. Rev.
Universal diffusion near the golden chaos border
We study local diffusion rate in Chirikov standard map near the critical
golden curve. Numerical simulations confirm the predicted exponent
for the power law decay of as approaching the golden curve via principal
resonances with period (). The universal
self-similar structure of diffusion between principal resonances is
demonstrated and it is shown that resonances of other type play also an
important role.Comment: 4 pages Latex, revtex, 3 uuencoded postscript figure
Phase transition in the collisionless regime for wave-particle interaction
Gibbs statistical mechanics is derived for the Hamiltonian system coupling
self-consistently a wave to N particles. This identifies Landau damping with a
regime where a second order phase transition occurs. For nonequilibrium initial
data with warm particles, a critical initial wave intensity is found: above it,
thermodynamics predicts a finite wave amplitude in the limit of infinite N;
below it, the equilibrium amplitude vanishes. Simulations support these
predictions providing new insight on the long-time nonlinear fate of the wave
due to Landau damping in plasmas.Comment: 12 pages (RevTeX), 2 figures (PostScript
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