Quasifree Quantum Statistical Models for Tunnelling Junction
by N. Angelescu, M. Bundaru and R. Bundaru


When several reservoirs, initially in different equilibrium states, are brought into
contact (directly or, possibly, mediated by a finite quantum system), it is to be
expected that the state of the composite system will eventually converge to a
nonequilibrium stationary state (NESS), characterized by constant energy and particle
currents between reservoirs. The scattering approach, initiated by D. Ruelle, to modeling
this physical setup in the operator algebra framework of quantum statistical mechanics is
briefly described. In the body of the paper, a class of solvable microscopic quantum
models is considered, in which the reservoirs are (Fermi or) Bose gases at thermal
equilibrium and for which the dynamics, both of the isolated reservoirs and of the
coupled system, are quasifree automorphism groups of a suitable canonical
(anti)commutation relation algebra. Within this class, the existence problem of the NESS
is reduced to the scattering and spectral problems for the interacting and noninteracting
"oneparticle" automorphisms. Though this simplifying assumption restricts the contacts
only to particle tunnelling between reservoirs, it allows to prove the existence, and
derive analytically many properties, of stationary states in various physical instances.
Recent results on the transport properties of
quasifree Fermions substantiating the LandauerBüttiker formula for conductance and linear response theory (GreenKubo formulas, Onsager
relations, etc.) are reviewed. The case of Bose particles is more complex due to the
phenomenon of BoseEinstein condensation, implying nonuniqueness of the equilibrium
state of the reservoirs. We derive for this case expressions of the currents in the NESS,
including the ''Josephson supercurrent'' exhibiting a characteristic sindependence
on the phase difference between condensates.

An Introduction to
Monotonicity Methods for Nonlinear Kinetic Equations
by Cecil Pompiliu Grünfeld


We survey the recent progress on the application of monotonicity
methods (with respect to order) to the existence theory for
various Boltzmannlike, nonlinear kinetic equations. To motivate
the topic, we first provide several examples of Boltzmann models
for complex systems, with similar monotonicity properties, which
present interest in applications. These are Smoluchowski's
coagulation equation, Povznerlike models with dissipative
collisions and reactive collisions, respectively, a Boltzmann
model for several chemical species (with reactions), and a von
NeumannBoltzmann quantum model. The common properties of the
above models can be abstracted into a very general setting. One
obtains a class of nonlinear evolution equations, formulated into
an abstract Lebesgue space, for which one can state general
criteria for the existence, uniqueness and positivity of global
(in time) solutions. The proofs extend techniques that were
initially developed in the more particular context of the
spacehomogeneous version of the classical Boltzmann equation.
Finally we show how the abstract results can be applied to our
examples of Boltzmannlike models.

Estimating the number of negative eigenvalues of a
relativistic Hamiltonian
with regular magnetic field
by Viorel Iftimie, Marius Măntoiu and Radu Purice


We prove the analog of the CwickelLiebRosenblum estimation for
the number of negative eigenvalues of a relativistic Hamiltonian
with magnetic field $B\inC^\infty_{\rm{pol}}(\mathbb
R^d)$ and an electric potential $V\in L^1_{\rm{loc}}(\mathbb
R^d)$, $V_\in L^d(\mathbb R^d)\cap L^{d/2}(\mathbb R^d)$.
Compared to the nonrelativistic case, this estimation involves both norms of $V_$ in $L^{d/2}(\mathbb R^d)$ and in $L^{d}(\mathbb R^d)$. A
direct consequence is a LiebThirring inequality for the sum of
powers of the absolute values of the negative eigenvalues.

Approximate inertial manifolds, induced trajectories, and approximate
solutions for semilinear parabolic equations, based upon these;
applications to flow and diffusion problems
by Anca Veronica Ion


The lecture presents the notion
of approximate inertial manifold of a semidynamical system
generated by a nonlinear evolution PDE (more precisely, a
semilinear parabolic equation), as it appeared in the literature
of the last twenty years. The localization of the attractors in
the space of phases was a first interesting application field of
the a.i.m.s. Besides, a.i.m.s found very interesting applications
in the construction of some approximate solutions (and
consequently in the numerical integration) of the nonlinear
evolution problems. These are contained in the socalled nonlinear
Galerkin and postprocessed Galerkin methods.

Diffusion processes. Physical models
and numerical approximation
by Stelian Ion


The paper deals with the numerical approximation of a class of nonlinear
diffusion processes that includes the unsaturated water flow
through porous media and the fast diffusion. The approximation
method consists in the discretization of space derivative
operators using the finite volume scheme and keeping the continuum
time differentiation. Consequently, the solution of the partial
differential equations is approximated by the solution of a system
of ordinary differential equations. A scheme to approximate the
diffusion and convective term such that one can obtain a
quasimonotone ODE system is defined. Further, it is proved that
there exists a discrete comparison principle, the solutions of the
discrete model are bounded and the upper and lower bounds are
independent of the mesh size of triangulation. To perform the time
numerical integration a class of implicit backward differentiation
formulae with adaptive time step is used. Since the implicit
schemes require a nonlinear solver a method that mixes Broyden
method and an inexact Newton method is constructed. The
performances of the new method are illustrated by some numerical
results concerning the fast diffusion equation and water
infiltration through a layered soil.

On a convergent numerical method for
nonlinear Boltzmanntype models
by Dorin Marinescu


The paper deals with the extensions of approximation techniques of
Nambu, Babovsky and Illner for the solutions of the classical
Boltzmann equation to a nonlinear generalized Boltzmanntype
system of equations solving nontrivial transport flows in dilute
gas mixtures. First, one proves the global existence and
uniqueness of solutions. Then a weak timediscretized version of
equations for positive measures is provided. To obtain an
algorithm, with small numerical effort (of order N\log N)
stochastic methods are introduced. Finally a numerical
approximation scheme, converging almost surely, in some sense, to
the solutions of exact equations is provided.

Mathematical models
of diffusion in nonhomogeneous porous media
by Gabriela Marinoschi


The first part of the paper introduces diffusive models of water flow in
saturatedunsaturated media, characterized by a space variation of
the porosity. Then the analysis focuses on a model with mixed
boundary conditions involving a flux on a part of the boundary and
a nonhomogeneous Dirichlet condition corresponding to a singular
situation (i.e., the blowing up diffusion coefficient) on the
other part of the domain boundary. From the mathematical point of
view, the problem resides in the study of a degenerate nonlinear
variational inequality which can be reduced to a multivalued
inclusion by an appropriate change of the unknown function.
Finally, existence, uniqueness and other properties of the
solution are established.
