129 research outputs found
Random many-particle systems: applications from biology, and propagation of chaos in abstract models
The paper discusses a family of Markov processes that represent many particle
systems, and their limiting behaviour when the number of particles go to
infinity. The first part concerns model of biological systems: a model for
sympatric speciation, i.e. the process in which a genetically homogeneous
population is split in two or more different species sharing the same habitat,
and models for swarming animals. The second part of the paper deals with
abstract many particle systems, and methods for rigorously deriving mean field
models.Comment: These are notes from a series of lectures given at the 5
Summer School on Methods and Models of Kinetic Theory, Porto Ercole, 2010.
They are submitted for publication in "Rivista di Matematica della
Universit\`a di Parma
Chaotic distributions for relativistic particles
We study a modified Kac model where the classical kinetic energy is replaced
by an arbitrary energy function , . The aim of this
paper is to show that the uniform density with respect to the microcanonical
measure is -chaotic, . The kinetic
energy for relativistic particles is a special case. A generalization to the
case which involves conservation momentum is also formally
discussed
Free path lengths in quasi crystals
The Lorentz gas is a model for a cloud of point particles (electrons) in a
distribution of scatterers in space. The scatterers are often assumed to be
spherical with a fixed diameter , and the point particles move with constant
velocity between the scatterers, and are specularly reflected when hitting a
scatterer. There is no interaction between point particles. An interesting
question concerns the distribution of free path lengths, i.e. the distance a
point particle moves between the scattering events, and how this distribution
scales with scatterer diameter, scatterer density and the distribution of the
scatterers. It is by now well known that in the so-called Boltzmann-Grad limit,
a Poisson distribution of scatters leads to an exponential distribution of free
path lengths, whereas if the scatterer distribution is periodic, the
distribution of free path behaves asymptotically like a Cauchy distribution.
This paper considers the case when the scatters are distributed on a quasi
crystal, i.e. non periodically, but with a long range order. Simulations of a
one-dimensional model are presented, showing that the quasi crystal behaves
very much like a periodic crystal, and in particular, the distribution of free
path lengths is not exponential
The Lorentz Gas with a Nearly Periodic Distribution of Scatterers
We consider the Lorentz gas in a distribution of scatterers which microscopically converges to a periodic distribution, and prove that the Lorentz gas in the low density limit satisfies a linear Boltzmann equation. This is in contrast with the periodic Lorentz gas, which does not satisfy the Boltzmann equation in the limit
Propagation of chaos for the thermostatted Kac master equation
The Kac model is a simplified model of an -particle system in which the
collisions of a real particle system are modeled by random jumps of pairs of
particle velocities. Kac proved propagation of chaos for this model, and hence
provided a rigorous validation of the corresponding Boltzmann equation.
Starting with the same model we consider an -particle system in which the
particles are accelerated between the jumps by a constant uniform force field
which conserves the total energy of the system. We show propagation of chaos
for this model
The BGK equation as the limit of an particle system
The spatially homogeneous BGK equation is obtained as the limit if a model of
a many particle system, similar to Mark Kac's charicature of the spatially
homogeneous Boltzmann equation.Comment: Minor corrections and modifications only. This version is essentially
the same as the published pape
Investigations of a compartmental model for leucine kinetics using nonlinear mixed effects models with ordinary and stochastic differential equations
Nonlinear mixed effects models represent a powerful tool to simultaneously
analyze data from several individuals. In this study a compartmental model of
leucine kinetics is examined and extended with a stochastic differential
equation to model non-steady state concentrations of free leucine in the
plasma. Data obtained from tracer/tracee experiments for a group of healthy
control individuals and a group of individuals suffering from diabetes mellitus
type 2 are analyzed. We find that the interindividual variation of the model
parameters is much smaller for the nonlinear mixed effects models, compared to
traditional estimates obtained from each individual separately. Using the mixed
effects approach, the population parameters are estimated well also when only
half of the data are used for each individual. For a typical individual the
amount of free leucine is predicted to vary with a standard deviation of 8.9%
around a mean value during the experiment. Moreover, leucine degradation and
protein uptake of leucine is smaller, proteolysis larger, and the amount of
free leucine in the body is much larger for the diabetic individuals than the
control individuals. In conclusion nonlinear mixed effects models offers
improved estimates for model parameters in complex models based on
tracer/tracee data and may be a suitable tool to reduce data sampling in
clinical studies
A Boltzmann model for rod alignment and schooling fish
We consider a Boltzmann model introduced by Bertin, Droz and Greegoire as a
binary interaction model of the Vicsek alignment interaction. This model
considers particles lying on the circle. Pairs of particles interact by trying
to reach their mid-point (on the circle) up to some noise. We study the
equilibria of this Boltzmann model and we rigorously show the existence of a
pitchfork bifurcation when a parameter measuring the inverse of the noise
intensity crosses a critical threshold. The analysis is carried over rigorously
when there are only finitely many non-zero Fourier modes of the noise
distribution. In this case, we can show that the critical exponent of the
bifurcation is exactly 1/2. In the case of an infinite number of non-zero
Fourier modes, a similar behavior can be formally obtained thanks to a method
relying on integer partitions first proposed by Ben-Naim and Krapivsky.Comment: 22 pages, 3 figure
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