64 research outputs found
Global validity of the Master kinetic equation for hard-sphere systems
Following the recent establishment of an exact kinetic theory realized by the Master kinetic equation
which describes the statistical behavior of the Boltzmann-Sinai Classical Dynamical System (CDS), in
this paper the problem is posed of the construction of the related global existence and regularity theorems.
For this purpose, based on the global prescription of the same CDS for arbitrary single- and multiplecollision
events, first global existence is extablished for the N-body Liouville equation which is written
in Lagrangian differential and integral forms. This permits to reach the proof of global existence both
of generic N-body probability density functions (PDF) as well as of particular solutions which maximize
the statistical Boltzmann-Shannon entropy and are factorized in terms of the corresponding 1-body PDF.
The latter PDF is shown to be uniquely defined and to satisfy the Master kinetic equation globally in the
extended 1-body phase space. Implications concerning the global validity of the asymptotic Boltzmann
equation and Boltzmann H-theorem are discussed
A dynamical classification of the range of pair interactions
We formalize a classification of pair interactions based on the convergence
properties of the {\it forces} acting on particles as a function of system
size. We do so by considering the behavior of the probability distribution
function (PDF) P(F) of the force field F in a particle distribution in the
limit that the size of the system is taken to infinity at constant particle
density, i.e., in the "usual" thermodynamic limit. For a pair interaction
potential V(r) with V(r) \rightarrow \infty) \sim 1/r^a defining a {\it
bounded} pair force, we show that P(F) converges continuously to a well-defined
and rapidly decreasing PDF if and only if the {\it pair force} is absolutely
integrable, i.e., for a > d-1, where d is the spatial dimension. We refer to
this case as {\it dynamically short-range}, because the dominant contribution
to the force on a typical particle in this limit arises from particles in a
finite neighborhood around it. For the {\it dynamically long-range} case, i.e.,
a \leq d-1, on the other hand, the dominant contribution to the force comes
from the mean field due to the bulk, which becomes undefined in this limit. We
discuss also how, for a \leq d-1 (and notably, for the case of gravity, a=d-2)
P(F) may, in some cases, be defined in a weaker sense. This involves a
regularization of the force summation which is generalization of the procedure
employed to define gravitational forces in an infinite static homogeneous
universe. We explain that the relevant classification in this context is,
however, that which divides pair forces with a > d-2 (or a < d-2), for which
the PDF of the {\it difference in forces} is defined (or not defined) in the
infinite system limit, without any regularization. In the former case dynamics
can, as for the (marginal) case of gravity, be defined consistently in an
infinite uniform system.Comment: 12 pages, 1 figure; significantly shortened and focussed, additional
references, version to appear in J. Stat. Phy
Multifractality of the Feigenbaum attractor and fractional derivatives
It is shown that fractional derivatives of the (integrated) invariant measure
of the Feigenbaum map at the onset of chaos have power-law tails in their
cumulative distributions, whose exponents can be related to the spectrum of
singularities . This is a new way of characterizing multifractality
in dynamical systems, so far applied only to multifractal random functions
(Frisch and Matsumoto (J. Stat. Phys. 108:1181, 2002)). The relation between
the thermodynamic approach (Vul, Sinai and Khanin (Russian Math. Surveys 39:1,
1984)) and that based on singularities of the invariant measures is also
examined. The theory for fractional derivatives is developed from a heuristic
point view and tested by very accurate simulations.Comment: 20 pages, 5 figures, J.Stat.Phys. in pres
Translation-invariance of two-dimensional Gibbsian point processes
The conservation of translation as a symmetry in two-dimensional systems with
interaction is a classical subject of statistical mechanics. Here we establish
such a result for Gibbsian particle systems with two-body interaction, where
the interesting cases of singular, hard-core and discontinuous interaction are
included. We start with the special case of pure hard core repulsion in order
to show how to treat hard cores in general.Comment: 44 pages, 6 figure
An approximate renormalization-group transformation for Hamiltonian systems with three degrees of freedom
We construct an approximate renormalization transformation that combines
Kolmogorov-Arnold-Moser (KAM)and renormalization-group techniques, to analyze
instabilities in Hamiltonian systems with three degrees of freedom. This scheme
is implemented both for isoenergetically nondegenerate and for degenerate
Hamiltonians. For the spiral mean frequency vector, we find numerically that
the iterations of the transformation on nondegenerate Hamiltonians tend to
degenerate ones on the critical surface. As a consequence, isoenergetically
degenerate and nondegenerate Hamiltonians belong to the same universality
class, and thus the corresponding critical invariant tori have the same type of
scaling properties. We numerically investigate the structure of the attracting
set on the critical surface and find that it is a strange nonchaotic attractor.
We compute exponents that characterize its universality class.Comment: 10 pages typeset using REVTeX, 7 PS figure
Probability distributions consistent with a mixed state
A density matrix may be represented in many different ways as a
mixture of pure states, \rho = \sum_i p_i |\psi_i\ra \la \psi_i|. This paper
characterizes the class of probability distributions that may appear in
such a decomposition, for a fixed density matrix . Several illustrative
applications of this result to quantum mechanics and quantum information theory
are given.Comment: 6 pages, submitted to Physical Review
Probabilistic ballistic annihilation with continuous velocity distributions
We investigate the problem of ballistically controlled reactions where
particles either annihilate upon collision with probability , or undergo an
elastic shock with probability . Restricting to homogeneous systems, we
provide in the scaling regime that emerges in the long time limit, analytical
expressions for the exponents describing the time decay of the density and the
root-mean-square velocity, as continuous functions of the probability and
of a parameter related to the dissipation of energy. We work at the level of
molecular chaos (non-linear Boltzmann equation), and using a systematic Sonine
polynomials expansion of the velocity distribution, we obtain in arbitrary
dimension the first non-Gaussian correction and the corresponding expressions
for the decay exponents. We implement Monte-Carlo simulations in two
dimensions, that are in excellent agreement with our analytical predictions.
For , numerical simulations lead to conjecture that unlike for pure
annihilation (), the velocity distribution becomes universal, i.e. does
not depend on the initial conditions.Comment: 10 pages, 9 eps figures include
On the dynamical behavior of the ABC model
We consider the ABC dynamics, with equal density of the three species, on the
discrete ring with sites. In this case, the process is reversible with
respect to a Gibbs measure with a mean field interaction that undergoes a
second order phase transition. We analyze the relaxation time of the dynamics
and show that at high temperature it grows at most as while it grows at
least as at low temperature
Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems
We prove Lieb-Robinson bounds for the dynamics of systems with an infinite
dimensional Hilbert space and generated by unbounded Hamiltonians. In
particular, we consider quantum harmonic and certain anharmonic lattice
systems
On Ruelle's construction of the thermodynamic limit for the classical microcanonical entropy
In this note we make a very elementary technical observation to the effect
that Ruelle's construction of the thermodynamic limit of the classical entropy
density defined with a regularized microcanonical measure actually establishes
the thermodynamic limit for the entropy density defined with the proper
microcanonical measure. At this stage a key formula is still derived from the
regularized measures. We also show that with only minor changes in the proof
the regularization of the microcanonical measure is actually not needed at all.Comment: Short communication (7p), accepted for publication in J.Stat.Phy
- …