1,968 research outputs found
Canonical solution of a system of long-range interacting rotators on a lattice
The canonical partition function of a system of rotators (classical X-Y
spins) on a lattice, coupled by terms decaying as the inverse of their distance
to the power alpha, is analytically computed. It is also shown how to compute a
rescaling function that allows to reduce the model, for any d-dimensional
lattice and for any alpha<d, to the mean field (alpha=0) model.Comment: Initially submitted to Physical Review Letters: following referees'
Comments it has been transferred to Phys. Rev. E, because of supposed no
general interest. Divided into sections, corrections in (5) and (20),
reference 5 updated. 8 pages 1 figur
Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems
Classical spin systems with nonadditive long-range interactions are studied
in the microcanonical ensemble. It is expected that the entropy of such a
system is identical to that of the corresponding mean-field model, which is
called "exactness of the mean-field theory". It is found out that this
expectation is not necessarily true if the microcanonical ensemble is not
equivalent to the canonical ensemble in the mean-field model. Moreover,
necessary and sufficient conditions for exactness of the mean-field theory are
obtained. These conditions are investigated for two concrete models, the
\alpha-Potts model with annealed vacancies and the \alpha-Potts model with
invisible states.Comment: 23 pages, to appear in J. Stat. Phy
Dynamical stability criterion for inhomogeneous quasi-stationary states in long-range systems
We derive a necessary and sufficient condition of linear dynamical stability
for inhomogeneous Vlasov stationary states of the Hamiltonian Mean Field (HMF)
model. The condition is expressed by an explicit disequality that has to be
satisfied by the stationary state, and it generalizes the known disequality for
homogeneous stationary states. In addition, we derive analogous disequalities
that express necessary and sufficient conditions of formal stability for the
stationary states. Their usefulness, from the point of view of linear dynamical
stability, is that they are simpler, although they provide only sufficient
criteria of linear stability. We show that for homogeneous stationary states
the relations become equal, and therefore linear dynamical stability and formal
stability become equivalent.Comment: Submitted to Journal of Statistical Mechanics: Theory and Experimen
Relaxation to thermal equilibrium in the self-gravitating sheet model
We revisit the issue of relaxation to thermal equilibrium in the so-called
"sheet model", i.e., particles in one dimension interacting by attractive
forces independent of their separation. We show that this relaxation may be
very clearly detected and characterized by following the evolution of order
parameters defined by appropriately normalized moments of the phase space
distribution which probe its entanglement in space and velocity coordinates.
For a class of quasi-stationary states which result from the violent relaxation
of rectangular waterbag initial conditions, characterized by their virial ratio
R_0, we show that relaxation occurs on a time scale which (i) scales
approximately linearly in the particle number N, and (ii) shows also a strong
dependence on R_0, with quasi-stationary states from colder initial conditions
relaxing much more rapidly. The temporal evolution of the order parameter may
be well described by a stretched exponential function. We study finally the
correlation of the relaxation times with the amplitude of fluctuations in the
relaxing quasi-stationary states, as well as the relation between temporal and
ensemble averages.Comment: 37 pages, 24 figures; some additional discussion of previous
literature and other minor modifications, final published versio
Long-Range Effects in Layered Spin Structures
We study theoretically layered spin systems where long-range dipolar
interactions play a relevant role. By choosing a specific sample shape, we are
able to reduce the complex Hamiltonian of the system to that of a much simpler
coupled rotator model with short-range and mean-field interactions. This latter
model has been studied in the past because of its interesting dynamical and
statistical properties related to exotic features of long-range interactions.
It is suggested that experiments could be conducted such that within a specific
temperature range the presence of long-range interactions crucially affect the
behavior of the system
Canonical Solution of Classical Magnetic Models with Long-Range Couplings
We study the canonical solution of a family of classical spin
models on a generic -dimensional lattice; the couplings between two spins
decay as the inverse of their distance raised to the power , with
. The control of the thermodynamic limit requires the introduction of
a rescaling factor in the potential energy, which makes the model extensive but
not additive. A detailed analysis of the asymptotic spectral properties of the
matrix of couplings was necessary to justify the saddle point method applied to
the integration of functions depending on a diverging number of variables. The
properties of a class of functions related to the modified Bessel functions had
to be investigated. For given , and for any , and lattice
geometry, the solution is equivalent to that of the model, where the
dimensionality and the geometry of the lattice are irrelevant.Comment: Submitted for publication in Journal of Statistical Physic
Bubble propagation in a helicoidal molecular chain
We study the propagation of very large amplitude localized excitations in a
model of DNA that takes explicitly into account the helicoidal structure. These
excitations represent the ``transcription bubble'', where the hydrogen bonds
between complementary bases are disrupted, allowing access to the genetic code.
We propose these kind of excitations in alternative to kinks and breathers. The
model has been introduced by Barbi et al. [Phys. Lett. A 253, 358 (1999)], and
up to now it has been used to study on the one hand low amplitude breather
solutions, and on the other hand the DNA melting transition. We extend the
model to include the case of heterogeneous chains, in order to get closer to a
description of real DNA; in fact, the Morse potential representing the
interaction between complementary bases has two possible depths, one for A-T
and one for G-C base pairs. We first compute the equilibrium configurations of
a chain with a degree of uncoiling, and we find that a static bubble is among
them; then we show, by molecular dynamics simulations, that these bubbles, once
generated, can move along the chain. We find that also in the most unfavourable
case, that of a heterogeneous DNA in the presence of thermal noise, the
excitation can travel for well more 1000 base pairs.Comment: 25 pages, 7 figures. Submitted to Phys. Rev.
Invariant measures of the 2D Euler and Vlasov equations
We discuss invariant measures of partial differential equations such as the
2D Euler or Vlasov equations. For the 2D Euler equations, starting from the
Liouville theorem, valid for N-dimensional approximations of the dynamics, we
define the microcanonical measure as a limit measure where N goes to infinity.
When only the energy and enstrophy invariants are taken into account, we give
an explicit computation to prove the following result: the microcanonical
measure is actually a Young measure corresponding to the maximization of a
mean-field entropy. We explain why this result remains true for more general
microcanonical measures, when all the dynamical invariants are taken into
account. We give an explicit proof that these microcanonical measures are
invariant measures for the dynamics of the 2D Euler equations. We describe a
more general set of invariant measures, and discuss briefly their stability and
their consequence for the ergodicity of the 2D Euler equations. The extension
of these results to the Vlasov equations is also discussed, together with a
proof of the uniqueness of statistical equilibria, for Vlasov equations with
repulsive convex potentials. Even if we consider, in this paper, invariant
measures only for Hamiltonian equations, with no fluxes of conserved
quantities, we think this work is an important step towards the description of
non-equilibrium invariant measures with fluxes.Comment: 40 page
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