Abstract:
We prove that the solution to the Cauchy problem for the
time-dependent Hartree-Fock equation may be approximated in
L2-norm step by step in time by the Gauss wave
packets. It gives a way to obtain the solution the time-dependent
Hartree-Fock equation by some modification of a
classical particle method just like as in a case of the Vlasov equation.
Abstract:
We consider the linear Boltzmann equation (in
the Lorentz gas limit) describing elastic and inelastic
collisions of test particles with the background.
Depending on the interplay between the magnitude of
elastic and inelastic Knudsen numbers and the Mach
number we derive a number of possible "hydrodynamic"
limits and prove the strong convergence in L1 of respective
solutions of Boltzmann equation to them.
Abstract:
The Chapman-Enskog method (CEM) of kinetic theory has long been used
to derive hydrodynamic equations of parabolic type
from kinetic equations of the Boltzmann type. The complexity of these
equations has perhaps hindered the realization
that the CEM is a powerful singular perturbation
method that can be used advantageously as a viable
alternative to multiple scales or normal form
calculations in different contexts. I will illustrate how to use a
modification
of CEM to obtain reduced equations in the high-field limit of the
Vlasov-Poisson-Fokker-Planck
system, the small inertia limit of a model of Josephson junction arrays and
an $O(2)$-symmetric Takens-Bogdanov bifurcation for the Kuramoto model of
oscillator synchronization with a bimodal distribution of natural
frequencies. This latter example shows that the CEM is a convenient
alternative to normal form calculations.
Abstract:
I will talk about one dimensional granular media. First of all
will give some reuslts on the inelastic particle system which is probably
the simplest model ogf granular media. In particular i will give a
characterization of the phenomenum of the inelastic collapse.
Then I will consider some kinetic description for granular media in
particular whne the particle system is in a thermal bath.
In this case it is possible consider, starting from the kinetic
description,
the hydrodynamical limit and the diffusive limit that obtaining
interesting equations that I will discuss.
Abstract:
The basic aim of kinetic theory is to study a system made of an extremely
large number of discrete entities. This leads immediately to the remark
that the dynamics is impossible to follow in detail. The way out is to
study the dynamics of a statistical description, as Boltzmann first
indicated. Previous authors, including Maxwell, in fact, had envisaged
statistical tools of growing accuracy, but not the study of the time
evolution of probability.
The discrete entities were originally molecules, but later authors started
to consider electrons(classical and quantum), ions, and more sizable
"particles": colloidal particles, grains, cars, members of a population,...
We should remark that the qualitative behavior follows from the general
structure of the governing kinetic equation, but a more basic understanding
of the underlying dynamics is required in order to obtain accurate results.
The mathematical problems that arise are of various nature:
1) Validation of the kinetic equation from the underlying dynamics (here
the problem of irreversibility arrising from a reversible model shows up).
This problem is reasonably well understood at a formal level, but still
incomplete at a rigorous level for nonlinear equations(a celebrated result
by Lanford is local in time). The same problem is essentially well
understood at the rigorous level in the linear case.
2) Existence and uniqueness results. Great results were obtained in the
last 15 years (including some very recent within the European TMR). Great
problems are, however, still open. There are many subfields:
a) The space
homogeneous case, almost complete (thanks also to work done within the
TMR);
b) 1d problems (better understood);
c) role of boundaries;
d) discrete models.
3) Models for complex situations
(polyatomic gases, granular materials).
4) Asymptotics to study limiting behaviors and simpler
models (Example:High fields in semiconductors)
5) Numerical methods (Monte Carlo and deterministic)
not only for applications but also to gain insight into the structure
of solutions (Examples: instabilities, strange attractors).
Abstract:
Protein channels conduct ions (Na+, K+, Ca++, and Cl-) through a
narrow tunnel of fixed charge ('doping') thereby acting as gatekeepers
for cells and cell compartments. Hundreds of types of channels are
studied everyday in thousands of laboratories because of their
biological and medical importance: a substantial fraction of all drugs
used by physicians act directly or indirectly on channels. The charges
of channels can be manipulated one at a time with the powerful
techniques of molecular biology and in favorable cases the location of
every atom can be determined within a few tenths of an Ĺ.
Computing the movement of spheres through a 'hole in the wall' should
be easier than computing most other biological functions, yet it is
nearly as important as any from a medical and technological point of
view. Ionic objects are ideal objects for mathematical and
computational investigation.
The function of open channels can be described if the electric field
and current flow are computed by the Poisson-Drift-Diffusion (called
PNP, for Poisson Nernst Planck, in biology) equations and the channel
protein is described as an invariant arrangement of fixed charges-not
as an invariant potential of mean force or set of rate constants, as
is done in the chemical and biological tradition. The
Poisson-Drift-Diffusion equations describe the flux of individual ions
(each moving randomly in the Langevin trajectories of Brownian motion)
in the mean electric field. They are nearly identical to the drift
diffusion equations of semiconductor physics used there to describe
the diffusion and migration of quasi-particles, holes and
electrons. They are closely related to the Vlasov equations of plasma
physics.
PNP fits a wide range of current voltage (I-V ) relations-whether
sublinear, linear or superlinear-from 7 types of channels, over ±180
mV of membrane potential, in symmetrical and asymmetrical solutions of
20 mM to 2 M salt. The I-V relation of the gramicidin channel can be
predicted directly from its structure, known from NMR, using an
independently measured diffusion coefficient and no adjustable
parameters. Porin channels with known structure have been studied, and
parameter estimates (in mutations also of known structure) are
surprisingly close to those predicted (i.e., within 7%). Selectivity
has been studied extensively in the calcium release channel of cardiac
muscle: I-V relations in Li+, K+, Na+, Rb+, and Cs+ and their mixtures
can be explained with a few invariant parameters (of reasonable value)
over the full range of concentrations and potentials. Complex
selectivity properties of channels are easily explained: the anomalous
mole fraction effect in K+ and L-type calcium channels arise naturally
as a consequence of binding. Indeed, the selectivity of the L-type
calcium channel can be predicted quantitatively if permeating ions are
treated as finite objects with the entropy and electrostatic energy of
spheres. The L-type Ca channels is of particular clinical importance
because it controls the heart beat and is the target of calcium
channel blockers, drugs taken by a substantial fraction of the
population.
Taken together, these results suggest that open ionic channels are
natural nanotubes, dominated by the enormous fixed charge lining their
walls (~5 M, arising from 1 charge in 7´10Ĺ). Physical chemists (e.g.,
Henderson, Blum, and Lebowitz, 1979, J. Electronal. Chem. 102: 315)
have shown that highly charged systems are dominated by their mean
electric field and the changes in the shape of the electric
field. Atomic detail is unexpectedly unimportant because correlation
effects are small. Biologists and biochemists have traditionally
focussed on correlation effects and more or less ignored the electric
field. Thus, the success of the Poisson-Drift-Diffusion equations and
the predominant role of the electric field has been a surprise to
biologists, although the role of the electric field is hardly a
surprise to physicists who have long understood the importance of the
electric field in semiconductors and ionized gases, i.e., plasmas.
Ionic channels form a biological system of great clinical significance
and potential technological importance that can be immediately studied
by the techniques of computational physics. Many of those techniques
have not yet been used to analyze other biological systems. Perhaps
they should be: the application of the even the lowest resolution
techniques involving the Poisson-Drift-Diffusion equation has
revolutionized the study of channels. It is likely that application of
higher resolution methods, like self-consistent Monte Carlo/Molecular
dynamics, would have a comparable effect on other areas of
computational biology. Self-consistent/nonequilibrium models should be
used to describe protein structure, protein folding, nucleic acids
(i.e., DNA), and drug binding, because the systems are highly charged
and nearly always involve flux.
Protein structure, protein folding, nucleic acids (i.e., DNA), and
drug binding are systems of the greatest importance to biological
research and computational biology, and are the recipients of
considerable funding, yet traditional analysis more or less ignores
the electric field. Traditional analysis never treats the electric
field self consistently, as a result of charge, in the presence of
flux.
An opportunity exists to apply the well established methods of
computational physics to the central problems of computational
biology. The plasmas of biology need to be analyzed like the plasmas
of physics. The mathematics of semiconductors and ionized gases should
be the starting point for the mathematics of ions and proteins, for
the analysis of protein structure, protein folding, nucleic acids
(i.e., DNA), and the binding of drugs to proteins and nucleic acids.
Abstract:
Ecological interactions such as predation, resource competition,
parasitism, epidemic transmission, and reproduction often occur at
spatial scales much smaller than that of the whole population. For
organisms with limited mobility, it is extremely important to
explicitly consider the spatial pattern of individuals when predicting
population dynamics. Lattice or cellular automata models are useful
for modeling spatially structured population dynamics. Most analyses
of lattice models have relied on computer simulations of spatial
stochastic processes. Here we will describe the pair-approximation
(PA) method as an alternative to numerical computations. The PA method
yields a system of ODEs that govern the temporal dynamics of average
densities and local densities, with the latter describing the
correlation of states of nearest-neighbor cells. PAs are capable of
predicting the behavior of lattice models even when mean-field
approximations fail. In this talk, I will use the PA to investigate
the spatial dynamics of a population where individuals may interact
altruistically with their nearest neighbors. >From there I will show
how to use the PA to define a measure of fitness when there are
several classes of interacting individuals that differ in their
respective degrees of altruism. This fitness measure allows one to
predict how the altruism degree evolves in the long run by means of a
Darwinian mutation-selection process.
Abstract:
Granular materials are `infamous' for their numerous `unusual'
properties, part of which still
resist theoretical understanding and/or description. While the
basic reason for all problems
encountered in this field is the dissipative nature of the grain
interactions (friction, inelasticity),
it is convenient to list a number of properties which follow from this
fundamental fact and which are directly
responsible for the difficulties encountered in attempts to produce
theoretical descriptions of granular
matter.
It seems that one can identify several sources of the challenges
these materials pose to the
theoretician (as well as to experimentalists and industry). A major
reason for the difficulties
encountered in attempts to understand granular matter is their
mesoscopic nature. The latter has
two facets. The first is `trivial': typical granular assemblies
contain a relative small number of
particles (e.g. compared to Avogadro's number) and thus many of the
`thermodynamic limit' considerations
may not apply to these systems. This practical side of the
difficulties is not nearly as important as
the more fundamental property of granular materials, namely that they
are of mesoscopic nature even when
impractically large assemblies are considered. For instance one can
show that in fluidized
granular systems the mean free time (a microscopic scale) is
comparable with the macroscopic time
scales (e.g. shear rate) and that mean free paths can be
macroscopic. Dense and even static granular
matter exhibits arches, stress chains and other `microstructures' that
can span the entire system and thus
they too are mesoscopic. Other, related properties, are the scale
dependence of stress fields and
the observability of `microscopic fluctuations'. In addition nearly
all states of granular matter are
metastable and multistable, a fact which may `explain' the
irreproducibility of some experiments in this
field.
Some problems concerning granular materials exist in the realm of
molecular systems as well except that
there they are considered to be relatively esoteric as in
e.g. the case of very strongly
sheared gases (the `continuum transition regime') which requires the
study of beyond-Navier-Stokes
hydrodynamics, a problem known for its difficulty (but in the realm of granular
matter practically every shear rate is `large'). Static and
quasistatic granular matter is usually
disordered; the description of these states of granular matter
encounters difficulties similar to
those that are e.g. faced in the theory of glasses. In particular,
finding a `sufficient' characterization
for the state of such a system (so as to be able to `close' the
equation of motion) is highly
non-trivial. Interestingly, even the construction of theories for
some of the simplest states
of granular matter (`the elastic regime') requires the rethinking and
perhaps the redefining
of some basic entities (stress, strain).
The above comments notwithstanding, some theories are quite successful
in this field. This includes
certain kinetic descriptions, numerous phenomenological equations and
explanations of some physical
phenomena (e.g. collapse, clustering). Moreover, some of the theories
seem to work beyond their
expected range of validity, a fact which needs to be understood as well.
Numerous other states/phenomena (e.g. oscillons) and some
engineering problems (pressure distributions in silos) still await
proper theories.
Abstract:
Hydrodynamical semiconductor models have been derived from the
Boltzmann equation by various authors using a variety of moment
closure assumptions. These hydrodynamical models can describe
behaviour in semiconductor devices which the traditional
drift-diffusion model does not account for. Adaptive numerical
methods are used here to generate approximate solutions to
hydrodynamical models for semiconductor devices in which
time-dependent effects are important. The aim of this work is
two-fold: to assess how well the hydrodynamical models describe
physical semiconductor behaviour, and to develop numerical
techniques that could contribute to the industrial design of
semiconductor devices.
Abstract:
In this talk we will be concerned with a kinetic model for
diluted solutions of polymeric liquid and its viscoelastic fluid
limit when the elastic character of the fluid is small. Differential
relationships between stress and velocity are derived by the use
of Chapman-Enskog expansions. These viscoelastic fluid models
are derived in the case of unsteady, nonhomogeneous and nonpotential flows.
Furthermore,
the limiting model
recovers both the linear (as Oldroyd-B models) and
the nonlinear (as FENE models) elastic character of the fluid.
Abstract:
We present some existence results for some models of
Non Newtonian fluids (Oldroyd B) as well as for
some models coming from polymer kinetic theory.
Abstract:
In this talk, we rigorously derive a non-linear diffusion model
from the Boltzmann equation in a semiconductor device, when
collision term consists in the (non-linear) Pauli's operator,
without assuming the detailed balance principle (in particular,
equilibrium states are no long the Fermi-Dirac distributions).
The proof of convergence relies on the Hilbert expansion and the
Chapman-Enskog method.
Abstract:
In this talk I will discuss some properties of weak solutions to
the nonlinear kinetic equations of the Povzner and Enskog type.
The main topics are the weak L1 compactness of solutions
(which is in the stationary case obtained by using the entropy
production estimates) and the stability properties of the equations
and the boundary conditions. I will specifically consider the case
of the boundary conditions of the diffuse reflection type. The
above properties of solutions lead to some new existence results
for the stationary kinetic equations.
Abstract:
In a perfect crystal an electron which moves under the action of a constant
electric field, oscillates around its mean position. These oscillations was
predicted by Brillouin and has been experimentally reproduced in
super-lattices. We give a mathematical proof of this paradoxal phenomenon.
Starting from the Schrodinger equation, homogeneization together with a
semiclassical limit end up with a semi-classical transport equation which
explains Brillouin oscillations. This result is a consequence of joint works
with P. Bechouche (Monatsh. Math 129 (2000)) and with P. Bechouche and N.
Mauser.
Abstract:
An efficient difference scheme for solution of non-steady state
Boltzmann transport equation in diffusion approximation with account
of phonons, as well as ions and electron-electron scattering
mechanisms, and the Pauli exclusion principle is proposed.
Semiconductor drift-diffusion and hydrodynamic models are based on
diffusion approximation for momentum distribution function. Both
models are well suited for device simulation purposes since they
describe efficiently the main physical effects. However, the
diffusion approximation itself is a more universal approach than
drift-diffusion and hydrodynamic models traditionally used for
semiconductor device optimization.
A mathematical formulation of the diffusion approximation for momentum
distribution function is based on expression of the momentum
distribution function as sum of the first and second Legendre
polynomials in the momentum space. As a result a coupled set of
differential equations for symmetrical and asymmetrical momentum
distribution functions has been obtained. In a general form the
Boltzmann equation which includes quantum exclusion effects is well
known. We have included scattering mechanisms on acoustic, polar
optical, and piezoelectric phonons, as well as ions and the
electron-electron scattering. Collision integrals are reduced, where
possible, to a form containing the first and second derivatives of
the distribution function as in the Fokker-Planck kinetic equation. In
the case of optical phonons collision integrals are expressed, in
general, in terms of finite differences of the distribution
function. Electron-electron collisions are represented in a form
containing the first and the second derivatives and integral operators
for the distribution function. The energy dispersion is approximated
by a parabolic relation.
A difference scheme for time dependent, non-linear system of
differential equations for symmetrical and asymmetrical momentum
distribution functions is proposed. The difference scheme for the
system of non-linear differential equations is built up by a special
method of exponential type substitution. By using this substitution,
a conservative and absolutely stable exponential type difference
scheme has been elaborated to solve the system of equations for
symmetrical and asymmetrical momentum distribution functions. It has
been shown by numerical calculations that the difference scheme,
developed for the symmetrical distribution function, is monotonous.
A physical situation is of interest where the distribution of electrons is
spatially homogeneous and a nonzero external field exists up to a certain time
moment. Then the field is switched off and we consider the relaxation
of the momentum distribution function to the stationary state.
Results of numerical calculations are presented for kinetics of the
momentum distribution function in GaAs.
Abstract:
We derive the linear Boltzmann equation with the hard-sphere cross
section for a Lorentz gas in the plane where the scatterers have random
positions on a square lattice.The scatterers are identical disks of diameter
e, which is also the size of the side of a
cell and the probability, for a given cell,
to be occupied by a scatterer, and the light particle moves freely between
them, interacting with them through elastic collisions.
The same result can be proved for other periodic configurations of obstacles.
Abstract:
A special form of the Boltzmann collision operator for the
Variable Hard Spheres model of interaction is considered.
The possibilities of the fast numerical computation of the
collision operator based on the Fast Fourier Transform and
numerical quadratures are discussed. A new deterministic
method based on the smoothing of singularities of the
collision kernel in the Fourier representation is presented.
Abstract:
(for a text version with references, see
textversion )
A consistent hydrodynamical model for electron transport in
silicon semiconductors,
free of any fitting parameter,
has been formulated on the basis of the maximum entropy
principle, by considering the energy band described by the Kane dispersion
relation. Explicit constitutive functions for fluxes and production terms
in the macroscopic balance equations of density, crystal momentum, energy
and energy-flux have been obtained.
Scatterings of electrons with non polar optical phonons (both for
intervalley and
intravalley interactions), acoustic phonons and impurities have been
taken into account.
Here we show the link with other macroscopic models
describing the motion of charge carriers. In particular, under suitable
scaling assumptions, an energy transport model is recovered.
An analysis of the
formal properties is given by showing that the evolution equations
form a hyperbolic system in the physically relevant region of the space of
the dependent variables. Al last, by using a numerical method,
simulations for bulk
silicon and n+-n-n+ silicon diode are
performed. The obtained results are in good
agreement with Monte Carlo data.
Abstract:
In this talk I propose a derivation of Boltzmann-type operators
which is not based as usual on the convergence of some N-body
model, but rather on abstract physical requirements of the model
(e. g. preservation of positiveness of the solution of the
corresponding equation, Galilean invariance, etc...).
This is part of a joint work with Laurent Desvillettes.
Abstract:
Many models in the theory of natural selection depend on the
frequencies of interacting populations and can be
profitably understood in terms of game theory. This talk describes some
of them, with particular emphasis on the often striking effects of
introducing spatial structure.
Abstract:
We present and discuss one-dimensional kinetic models of the Boltzmann equation
with dissipative collisions. It is shown that in the quasi-elastic limit the
resulting Fokker-Planck type equation is a nonlinear friction, in which the
nonlinearity depends on the details of the collision kernel in the Boltzmann
operator. Finite time cooling of the solution is discussed. In the second part
of the talk we discuss properties of some models. The Maxwellian case is of
particular interest. In this case in fact, steady states different from
concentration exist. Connection of these models with the central limit theorem
for stable laws is discussed.
Abstract:
Starting from Smoluchowski's coagulation equation, we discuss the relationship
between interacting stochastic particle systems and deterministic equations
occuring in the limit of large particle numbers. Then we consider a more
general type of coagulation-fragmentation equations, including both cases of
discrete and continuous size parameters. For a certain class of unbounded
coagulation kernels and fragmentation rates, relative compactness of the
stochastic system is established and weak accumulation points are
characterized in terms of solutions. These probabilistic limit theorems imply
new existence results for the deterministic equations.
Abstract:
In this talk, I shall consider the problem of trend to equilibrium for a
system which evolves by diffusion, drift and nonlinear friction
of granular type. In this context I shall present generalizations of
the usual logarithmic Sobolev inequalities and in particular analogs
of the Bakry-Emery criterion when the confinement energy is replaced
by an interaction energy. This is part of a joint work with
J.Carrillo and R.J.McCann.
Abstract:
Many solid propellant rocket motors (such as Ariane 5 boosters) use
aluminized propellants to improve their performances. The aluminium
combustion process produces a condensed phase and therefore a two-phase
flow
in the rocket chamber which has to be precisely modelled to predict
the motor performance in terms of acoustic stability, slag
accumulation,
nozle erosion and so on ....
This contribution will be devoted to the presentation of a kinetic
model
for droplet collision. This model takes into account the effects of
colasecence on the size and velocity on the droplets and the influence
of
the surrounding gas on the collisional cross section.
A stochastic particle method has been used for its numerical
discretization.
Numerical results concerning the application of this model to the
prediction
of the slag accumulation in Ariane 5 boosters will be shown at the
conference.
Abstract:
A kinetic model for vehicular traffic is presented and
investigated in detail. For this model the stationary distributions
can be determined explicitly. A derivation of associated
macroscopic traffic flow equations from the kinetic equation is given.
The coefficients appearing in these equations are identified
from the solutions of the underlying stationary kinetic equation.
Moreover, numerical experiments and comparisons between different
macroscopic models are presented. The results were worked out in
cooperation with Prof. A. Klar (Darmstadt), Dr. Marco Guenther
(Kaiserslautern) and Thorsten Materne (Darmstadt).