640 research outputs found
Kinetic hierarchy and propagation of chaos in biological swarm models
We consider two models of biological swarm behavior. In these models, pairs
of particles interact to adjust their velocities one to each other. In the
first process, called 'BDG', they join their average velocity up to some noise.
In the second process, called 'CL', one of the two particles tries to join the
other one's velocity. This paper establishes the master equations and BBGKY
hierarchies of these two processes. It investigates the infinite particle limit
of the hierarchies at large time-scale. It shows that the resulting kinetic
hierarchy for the CL process does not satisfy propagation of chaos. Numerical
simulations indicate that the BDG process has similar behavior to the CL
process
On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials
The paper concerns - convergence to equilibrium for weak solutions of
the spatially homogeneous Boltzmann Equation for soft potentials (-4\le
\gm<0), with and without angular cutoff. We prove the time-averaged
-convergence to equilibrium for all weak solutions whose initial data have
finite entropy and finite moments up to order greater than 2+|\gm|. For the
usual -convergence we prove that the convergence rate can be controlled
from below by the initial energy tails, and hence, for initial data with long
energy tails, the convergence can be arbitrarily slow. We also show that under
the integrable angular cutoff on the collision kernel with -1\le \gm<0, there
are algebraic upper and lower bounds on the rate of -convergence to
equilibrium. Our methods of proof are based on entropy inequalities and moment
estimates.Comment: This version contains a strengthened theorem 3, on rate of
convergence, considerably relaxing the hypotheses on the initial data, and
introducing a new method for avoiding use of poitwise lower bounds in
applications of entropy production to convergence problem
Celebrating Cercignani's conjecture for the Boltzmann equation
Cercignani's conjecture assumes a linear inequality between the entropy and
entropy production functionals for Boltzmann's nonlinear integral operator in
rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities
and spectral gap inequalities, this issue has been at the core of the renewal
of the mathematical theory of convergence to thermodynamical equilibrium for
rarefied gases over the past decade. In this review paper, we survey the
various positive and negative results which were obtained since the conjecture
was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani,
powerful mind and great scientist, one of the founders of the modern theory
of the Boltzmann equation. 24 pages. V2: correction of some typos and one
ref. adde
Heat Kernel Bounds for the Laplacian on Metric Graphs of Polygonal Tilings
We obtain an upper heat kernel bound for the Laplacian on metric graphs
arising as one skeletons of certain polygonal tilings of the plane, which
reflects the one dimensional as well as the two dimensional nature of these
graphs.Comment: 8 page
Complete characterization of convergence to equilibrium for an inelastic Kac model
Pulvirenti and Toscani introduced an equation which extends the Kac
caricature of a Maxwellian gas to inelastic particles. We show that the
probability distribution, solution of the relative Cauchy problem, converges
weakly to a probability distribution if and only if the symmetrized initial
distribution belongs to the standard domain of attraction of a symmetric stable
law, whose index is determined by the so-called degree of
inelasticity, , of the particles: . This result is
then used: (1) To state that the class of all stationary solutions coincides
with that of all symmetric stable laws with index . (2) To determine
the solution of a well-known stochastic functional equation in the absence of
extra-conditions usually adopted
A Stronger Subadditivity of Entropy
The strong subadditivity of entropy plays a key role in several areas of
physics and mathematics. It states that the entropy S[\rho]= - Tr (\rho \ln
\rho) of a density matrix \rho_{123} on the product of three Hilbert spaces
satisfies S[\rho_{123}] - S[\rho_{23}] \leq S[\rho_{12}]- S[\rho_2]. We
strengthen this to S[\rho_{123}] - S[\rho_{12}] \leq \sum_\alpha n^\alpha
(S[\rho_{23}^\alpha ] - S[\rho_2^\alpha ]), where the n^\alpha are weights and
the \rho_{23}^\alpha are partitions of \rho_{23}. Correspondingly, there is a
strengthening of the theorem that the map A -> Tr \exp[L + \ln A] is concave.
As applications we prove some monotonicity and convexity properties of the
Wehrl entropy and entropy inequalities for quantum gases.Comment: LaTeX2e, 24 page
Molecular dynamics simulation of nanocolloidal amorphous silica particles: Part II
Explicit molecular dynamics simulations were applied to a pair of amorphous
silica nanoparticles of diameter 3.2 nm immersed in a background electrolyte.
Mean forces acting between the pair of silica nanoparticles were extracted at
four different background electrolyte concentrations. Dependence of the
inter-particle potential of mean force on the separation and the silicon to
sodium ratio, as well as on the background electrolyte concentration, are
demonstrated. The pH was indirectly accounted for via the ratio of silicon to
sodium used in the simulations. The nature of the interaction of the
counter-ions with charged silica surface sites (deprotonated silanols) was also
investigated. The effect of the sodium double layer on the water ordering was
investigated for three Si:Na+ ratios. The number of water molecules trapped
inside the nanoparticles was investigated as the Si:Na+ ratio was varied.
Differences in this number between the two nanoparticles in the simulations are
attributed to differences in the calculated electric dipole moment. The
implications of the form of the potentials for aggregation are also discussed.Comment: v1. 33 pages, 7 figures (screen-quality PDF), submitted to J. Chem.
Phys v2. 15 pages, 4 tables, 6 figures. Content, author list and title
changed; single space
Comment on "Why quantum mechanics cannot be formulated as a Markov process"
In the paper with the above title, D. T. Gillespie [Phys. Rev. A 49, 1607,
(1994)] claims that the theory of Markov stochastic processes cannot provide an
adequate mathematical framework for quantum mechanics. In conjunction with the
specific quantum dynamics considered there, we give a general analysis of the
associated dichotomic jump processes. If we assume that Gillespie's
"measurement probabilities" \it are \rm the transition probabilities of a
stochastic process, then the process must have an invariant (time independent)
probability measure. Alternatively, if we demand the probability measure of the
process to follow the quantally implemented (via the Born statistical
postulate) evolution, then we arrive at the jump process which \it can \rm be
interpreted as a Markov process if restricted to a suitable duration time.
However, there is no corresponding Markov process consistent with the
event space assumption, if we require its existence for all times .Comment: Latex file, resubm. to Phys. Rev.
A negative mass theorem for surfaces of positive genus
We define the "sum of squares of the wavelengths" of a Riemannian surface
(M,g) to be the regularized trace of the inverse of the Laplacian. We normalize
by scaling and adding a constant, to obtain a "mass", which is scale invariant
and vanishes at the round sphere. This is an anlaog for closed surfaces of the
ADM mass from general relativity. We show that if M has positive genus then on
each conformal class, the mass attains a negative minimum. For the minimizing
metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a
Moser-Trudinger-Onofri type inequality.Comment: 8 page
Froth-like minimizers of a non local free energy functional with competing interactions
We investigate the ground and low energy states of a one dimensional non
local free energy functional describing at a mean field level a spin system
with both ferromagnetic and antiferromagnetic interactions. In particular, the
antiferromagnetic interaction is assumed to have a range much larger than the
ferromagnetic one. The competition between these two effects is expected to
lead to the spontaneous emergence of a regular alternation of long intervals on
which the spin profile is magnetized either up or down, with an oscillation
scale intermediate between the range of the ferromagnetic and that of the
antiferromagnetic interaction. In this sense, the optimal or quasi-optimal
profiles are "froth-like": if seen on the scale of the antiferromagnetic
potential they look neutral, but if seen at the microscope they actually
consist of big bubbles of two different phases alternating among each other. In
this paper we prove the validity of this picture, we compute the oscillation
scale of the quasi-optimal profiles and we quantify their distance in norm from
a reference periodic profile. The proof consists of two main steps: we first
coarse grain the system on a scale intermediate between the range of the
ferromagnetic potential and the expected optimal oscillation scale; in this way
we reduce the original functional to an effective "sharp interface" one. Next,
we study the latter by reflection positivity methods, which require as a key
ingredient the exact locality of the short range term. Our proof has the
conceptual interest of combining coarse graining with reflection positivity
methods, an idea that is presumably useful in much more general contexts than
the one studied here.Comment: 38 pages, 2 figure
- …