656 research outputs found
Chaos suppression in the large size limit for long-range systems
We consider the class of long-range Hamiltonian systems first introduced by
Anteneodo and Tsallis and called the alpha-XY model. This involves N classical
rotators on a d-dimensional periodic lattice interacting all to all with an
attractive coupling whose strength decays as r^{-alpha}, r being the distances
between sites. Using a recent geometrical approach, we estimate for any
d-dimensional lattice the scaling of the largest Lyapunov exponent (LLE) with N
as a function of alpha in the large energy regime where rotators behave almost
freely. We find that the LLE vanishes as N^{-kappa}, with kappa=1/3 for alpha/d
between 0 and 1/2 and kappa=2/3(1-alpha/d) for alpha/d between 1/2 and 1. These
analytical results present a nice agreement with numerical results obtained by
Campa et al., including deviations at small N.Comment: 10 pages, 3 eps figure
Statistical mechanics and dynamics of solvable models with long-range interactions
The two-body potential of systems with long-range interactions decays at
large distances as , with , where is the
space dimension. Examples are: gravitational systems, two-dimensional
hydrodynamics, two-dimensional elasticity, charged and dipolar systems.
Although such systems can be made extensive, they are intrinsically non
additive. Moreover, the space of accessible macroscopic thermodynamic
parameters might be non convex. The violation of these two basic properties is
at the origin of ensemble inequivalence, which implies that specific heat can
be negative in the microcanonical ensemble and temperature jumps can appear at
microcanonical first order phase transitions. The lack of convexity implies
that ergodicity may be generically broken. We present here a comprehensive
review of the recent advances on the statistical mechanics and
out-of-equilibrium dynamics of systems with long-range interactions. The core
of the review consists in the detailed presentation of the concept of ensemble
inequivalence, as exemplified by the exact solution, in the microcanonical and
canonical ensembles, of mean-field type models. Relaxation towards
thermodynamic equilibrium can be extremely slow and quasi-stationary states may
be present. The understanding of such unusual relaxation process is obtained by
the introduction of an appropriate kinetic theory based on the Vlasov equation.Comment: 118 pages, review paper, added references, slight change of conten
Time evolution of wave-packets in quasi-1D disordered media
We have investigated numerically the quantum evolution of a wave-packet in a
quenched disordered medium described by a tight-binding Hamiltonian with
long-range hopping (band random matrix approach). We have obtained clean data
for the scaling properties in time and in the bandwidth b of the packet width
and its fluctuations with respect to disorder realizations. We confirm that the
fluctuations of the packet width in the steady-state show an anomalous scaling
and we give a new estimate of the anomalous scaling exponent. This anomalous
behaviour is related to the presence of non-Gaussian tails in the distribution
of the packet width. Finally, we have analysed the steady state probability
profile and we have found finite band corrections of order 1/b with respect to
the theoretical formula derived by Zhirov in the limit of infinite bandwidth.
In a neighbourhood of the origin, however, the corrections are .Comment: 19 pages, 9 Encapsulated Postscript figures; submitted to ``European
Physical Journal B'
Finite times to equipartition in the thermodynamic limit
We study the time scale T to equipartition in a 1D lattice of N masses
coupled by quartic nonlinear (hard) springs (the Fermi-Pasta-Ulam beta model).
We take the initial energy to be either in a single mode gamma or in a package
of low frequency modes centered at gamma and of width delta-gamma, with both
gamma and delta-gamma proportional to N. These initial conditions both give,
for finite energy densities E/N, a scaling in the thermodynamic limit (large
N), of a finite time to equipartition which is inversely proportional to the
central mode frequency times a power of the energy density E/N. A theory of the
scaling with E/N is presented and compared to the numerical results in the
range 0.03 <= E/N <= 0.8.Comment: Plain TeX, 5 `eps' figures, submitted to Phys. Rev.
Non-Gaussian Fluctuations in Biased Resistor Networks: Size Effects versus Universal Behavior
We study the distribution of the resistance fluctuations of biased resistor
networks in nonequilibrium steady states. The stationary conditions arise from
the competition between two stochastic and biased processes of breaking and
recovery of the elementary resistors. The fluctuations of the network
resistance are calculated by Monte Carlo simulations which are performed for
different values of the applied current, for networks of different size and
shape and by considering different levels of intrinsic disorder. The
distribution of the resistance fluctuations generally exhibits relevant
deviations from Gaussianity, in particular when the current approaches the
threshold of electrical breakdown. For two-dimensional systems we have shown
that this non-Gaussianity is in general related to finite size effects, thus it
vanishes in the thermodynamic limit, with the remarkable exception of highly
disordered networks. For these systems, close to the critical point of the
conductor-insulator transition, non-Gaussianity persists in the large size
limit and it is well described by the universal Bramwell-Holdsworth-Pinton
distribution. In particular, here we analyze the role of the shape of the
network on the distribution of the resistance fluctuations. Precisely, we
consider quasi-one-dimensional networks elongated along the direction of the
applied current or trasversal to it. A significant anisotropy is found for the
properties of the distribution. These results apply to conducting thin films or
wires with granular structure stressed by high current densities.Comment: 8 pages, 4 figures. Invited talk at the 18-th International
Conference on Noise and Fluctuations, 19-23 September 2005, Salamanc
Non-Gaussianity of resistance fluctuations near electrical breakdown
We study the resistance fluctuation distribution of a thin film near
electrical breakdown. The film is modeled as a stationary resistor networkunder
biased percolation. Depending on the value of the external current,on the
system sizes and on the level of internal disorder, the fluctuation
distribution can exhibit a non-Gaussian behavior. We analyze this
non-Gaussianity in terms of the generalized Gumbel distribution recently
introduced in the context of highly correlated systems near criticality. We
find that when the average fraction of defects approaches the random
percolation threshold, the resistance fluctuation distribution is well
described by the universal behavior of the Bramwell-Holdsworth-Pinton
distribution.Comment: 3 figures, accepted for publication on Semicond Sci Tec
Inhomogeneous Quasi-stationary States in a Mean-field Model with Repulsive Cosine Interactions
The system of N particles moving on a circle and interacting via a global
repulsive cosine interaction is well known to display spatially inhomogeneous
structures of extraordinary stability starting from certain low energy initial
conditions. The object of this paper is to show in a detailed manner how these
structures arise and to explain their stability. By a convenient canonical
transformation we rewrite the Hamiltonian in such a way that fast and slow
variables are singled out and the canonical coordinates of a collective mode
are naturally introduced. If, initially, enough energy is put in this mode, its
decay can be extremely slow. However, both analytical arguments and numerical
simulations suggest that these structures eventually decay to the spatially
uniform equilibrium state, although this can happen on impressively long time
scales. Finally, we heuristically introduce a one-particle time dependent
Hamiltonian that well reproduces most of the observed phenomenology.Comment: to be published in J. Phys.
Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model
We perform a detailed study of the relaxation towards equilibrium in the
Hamiltonian Mean-Field (HMF) model, a prototype for long-range interactions in
-particle dynamics. In particular, we point out the role played by the
infinity of stationary states of the associated Vlasov dynamics. In this
context, we derive a new general criterion for the stability of any spatially
homogeneous distribution, and compare its analytical predictions with numerical
simulations of the Hamiltonian, finite , dynamics. We then propose and
verify numerically a scenario for the relaxation process, relying on the Vlasov
equation. When starting from a non stationary or a Vlasov unstable stationary
initial state, the system shows initially a rapid convergence towards a stable
stationary state of the Vlasov equation via non stationary states: we
characterize numerically this dynamical instability in the finite system by
introducing appropriate indicators. This first step of the evolution towards
Boltzmann-Gibbs equilibrium is followed by a slow quasi-stationary process,
that proceeds through different stable stationary states of the Vlasov
equation. If the finite system is initialized in a Vlasov stable homogenous
state, it remains trapped in a quasi-stationary state for times that increase
with the nontrivial power law . Single particle momentum distributions
in such a quasi-stationary regime do not have power-law tails, and hence cannot
be fitted by the -exponential distributions derived from Tsallis statistics.Comment: To appear in Physica
Ensemble inequivalence: A formal approach
Ensemble inequivalence has been observed in several systems. In particular it
has been recently shown that negative specific heat can arise in the
microcanonical ensemble in the thermodynamic limit for systems with long-range
interactions. We display a connection between such behaviour and a mean-field
like structure of the partition function. Since short-range models cannot
display this kind of behaviour, this strongly suggests that such systems are
necessarily non-mean field in the sense indicated here. We further show that a
broad class of systems with non-integrable interactions are indeed of
mean-field type in the sense specified, so that they are expected to display
ensemble inequivalence as well as the peculiar behaviour described above in the
microcanonical ensemble.Comment: 4 pages, no figures, given at the NEXT2001 conference on
non-extensive thermodynamic
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