19 research outputs found
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
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 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