# Grant Lythe: Projets de recherche

- calcul stochastique
- resolution numérique des
équations différentielles stochastiques
et des équations stochastiques aux derivées
partielles
- dynamique et thermodynamique hors equilibre
- la méthode de l'intégrale de transfert
- dynamique nonlinéaire
- équations différentielles partielles stochastiques

##
La méthode exponentielle pour les
équations différentielles stochastiques

Journal of Statistical Physics
abstract
Kalvis Jansons and G.D. Lythe
*Journal of Statistical Physics*
**100** 1097 (2000)

## La dynamique des kinks

Many extended systems described by * stochastic partial
differential équations * have localized coherent structures that
maintain their identity as they move and interact. The statistical
mechanics of these objects has diverse applications. Until fairly
recently, computer memory and performance restrictions were
sufficiently severe that Langevin evolutions could only be carried out
at fairly low levels of accuracy and resolution. However, present-day
supercomputers have overcome this problem, at least for low
dimensional problems, and one can well contemplate systematically
studying, understanding, and improving the accuracy of stochastic
evolutions.
In collaboration with Salman Habib (Los Alamos National Laboratory),
large-scale simulations of the stochastic PDEs for Ø^{4}
field theory at finite tempearture are being combined with new
theoretical results.

Equilibrium properties of finte temperature systems described by
stochastic PDEs can be calculated using the transfer integral
method. The calculation is exact, although one typically must evaluate
eigenvalues of the resulting Schrodinger equation numerically. In
equilibrium, the probability of a given set of configurations can be
calculated from the static solution of the Fokker-Planck equation
corresponding to the particular spatial discretization and
time-stepping algorithm applied to the SPDE of interest.
Correlation functions and thermodynamic quantities, which can all be
extracted from the transfer integral, explicitly exhibit lattice
dependences.

Salman Habib and Grant Lythe.
*Physical Review Letters*
**84** 1070 (2000)

Luis M. A. Bettencourt, Salman Habib and Grant Lythe.
*Physical Review D*
**60 ** 105039 (1999)
# en savoir plus

## Le bruit et les diélectriques nonlinéaires

Nonlinear dielectrics, such as SrTiO3 (STO), present unique opportunities
to develop practical electrically tunable devices, and also to explore
novel scientific and technological concepts that exploit strong
nonlinearities. Stochastic resonance (SR), a phenomenon in which random
noise enhances a nonlinear system's response to a deterministic signal,
presents one such opportunity.
Experimental device
(.ps file)

Our theoretical modelling of the coplanar waveguide system uses a
set of coupled partial differential équations in a nonlinear medium with
two boundaries. The nonlinearity depends on the applied bias and is used to
determine the input-output curve of the system, which allows us to
calculate the conditions for the appearance of stochastic resonance.

Alp Findikoglu et al
*Integrated Ferroelectrics*
**22** 259-268

## le calcul stochastique

Differential équations have long been used to model the dynamics of
physical systems. With the availability of computers, the tendency to
focus only on analysis of linear équations is being replaced by a
methodology that profits from a judicious mixture of
numerical generation of paths, bifurcation
theory and asymptotic analysis.
However, when random
perturbations (i.e. * noise*) play an important role, this new spirit is
not so widespread. One reason is that the mathematical tools
appropriate for describing stochastic paths are
not sufficiently well-known.
It is partly as a result of this
that there is a widespread misconception that noise acts only to
smear out deterministic dynamics.
Solving a
stochastic differential equation (SDE)
is akin to solving an ordinary
differential equation: exact analytical solutions
are seldom available, but * paths* can be generated
in a matter of seconds on a computer.
Analysis is based on Ito calculus, that permits
computation of experimentally accessible quantities
inaccessible to traditional methods.
I perform numerics and analysis of
nonlinear stochastic differential équations,
stochastic partial differential équations and
differential delay équations.
Kalvis Jansons and Grant Lythe
*Physical Review Letters
*
**81** 3136-3139

Kalvis Jansons and Grant Lythe
*Journal of Statistical Physics*
**90** 227-251 (1998)

## Les équations
différentielles partielles stochastiques

Stochastic partial differential équations describe
continuum systems with noise.
The noise can be white in space and time,
described solely by an amplitude, or can have space
or time scales of its own.
The numerical and analytical tools for solving SPDEs are
generalisations of the powerful stochastic analysis
used for ODEs with noise. Because they focus on one realisation
at a time, stochastic DEs are natural tools when noise
is an active part of the dynamics, not merely an agent
that rattles the system around in a pre-determined potential.
In a simple example of a non-equilibrium
spatially extended system, microscopic white noise produces a characteristic
macroscopic domain size that is a function
of , where
is the rate of change of the parameter and
the amplitude of the noise.
Grant Lythe
*Physical Review E* **53**
R4271-4274 (1996)
Esteban Moro and Grant Lythe
*Physical Review E*
**59** R1303-1306

## La dynamique des lasers

Semiconductor lasers have a wide range of applications because they
are of relatively small size, they can be massively produced at low
cost, and they are easy to operate. Despite their successful
technology, semiconductor lasers are very sensitive to any external
perturbation. A small amount
of optical feedback resulting from the reflection from an optical disk
or from the end of an optical fiber is sufficient to generate
pulsating instabilities.
Systematic experimental studies of semiconductor lasers, in particular
time series analysis, is not possible because
the timescale of the intensity pulsations is typically in the
picosecond regime. Fourier spectra measurements show a gradual
increase of oscillatory instabilities as parameters are changed but do
not reveal what the bifurcation mechanisms are. Most of the progress
in understanding these bifurcations comes from extensive
numerical studies of simple models and their comparison to the
experimentally obtained Fourier spectra.
J.C. Celet, D. Dangoisse, P. Glorieux,
G. Lythe and T. Erneux

*Physical Review Letters
*
**81** 975-978 (1998)
**
The low pump limit of the bifurcation to periodic intensities in a semiconductor laser subject to external optical feedback**

G.D. Lythe and T. Erneux, A. Gavrielides and V. Kovanis

*Physical Review A* **55** 4443-4448, (1997)

## La dynamique controllé par le bruit

Small amounts of noise can
dramatically change and simplify non-equilibrium dynamics.
Examples include slow-fast dynamics,
when long quiescent phases are occassionally interrupted by bursts of
activity; and spatially extended systems undergoing pattern-forming
transitions such as annealing and the formation of convective
patterns. Macroscopic length- and time-scales are found to emerge
from microscopic fluctuations.
In noise-controlled dynamics, length and time scales
depend on the product of a slow inverse time and the
* logarithm of the noise level*. Examples are described
in ordinary and partial differential équations
modelling laser and fluid systems.
The simplest example of the disproportionate
and * simplifying* effect of noise is a dynamic
bifurcation. In the corresponding situation for a
spatially extended system, microscopic
noise produces a characteristic macroscopic domain size.
Grant Lythe and MRE Proctor
*Physical Review E* **47** 3122-3127 (1993)
Grant Lythe and MRE Proctor
*Physica D* (1999)