# Grant Lythe: Research Interests

Differential equations 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 equations is being replaced by a methodology that profits from a judicious mixture of numerical generation of paths, bifurcation theory and asymptotic analysis.

Solving a stochastic differential equation 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. In dynamical systems, possible behaviours of solutions of differential equations are classified according to qualitative properties such as whether, for long times, the trajectory is close to a fixed point or limit cycle. Numerics are used when necessary and the improvement of numerical methods is an essential part of my research program.

Stochastic Partial Differential equations have emerged as a powerful new tool in the arsenal of the applied mathematician. They describe spatially extended 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 theory of such equations is still in its infancy, but the rapid increase in the computing power available to scientists has permitted their increasing use. The focus of my research in stochastic partial differential equations is on the statistics and dynamics of coherent structures that maintain their identity as they move and are buffeted by local fluctuations. In their simplest manifestation they are known as kinks and a systematic study is possible, which will serve as a stepping stone towards the study of more complicated structures such as vortices.

## The exponential timestepping method for stochastic differential equations

Recently, with Kalvis Jansons (UCL), a new type of timestepping method for stochastic differential equations was presented, that does not require Gaussian random variables to be generated. Time is incremented in steps that are exponentially-distributed random variables; boundaries can be explicitly accounted for at each timestep. The method was illustrated by numerical solution of a system of diffusing particles. Journal of Statistical Physics abstract

### Efficient numerical solution of stochastic differential equations using exponential timestepping

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

## Kink dynamics

Many extended systems described by stochastic partial differential equations 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 Ø4 field theory at finite tempearture are being combined with new theoretical results.

Equilibrium properties of finite 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.

### Dynamics of kinks: nucleation, diffusion and annihilation.

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

### Controlling One-Dimensional Langevin Dynamics on the Lattice

Luis M. A. Bettencourt, Salman Habib and Grant Lythe.       Physical Review D 60 105039 (1999)

# More on kinks

## Diffusion-limited reaction

Reaction rates controlled by collisions between diffusing particles depend on the distribution of distances between particles as well as on the density of particles. In particular, as Noyes stated in 1961

Any rigorous treatment of chemical kinetics in solution must consider concentration gradients that are established by the existence of the reaction itself
We study the dynamics of point particles in one dimension, nucleated at random positions and times then diffusing until colliding and annihilating with another particle. Competition between nucleation and annihilation produces a statistically steady state with a well-defined mean density of particles and distribution of distances between particles.

The dynamics is as follows:

1. Particles are nucleated in pairs with initial separation $b$;
2. Nucleation occurs at random times and positions with rate G;
3. Once born, all particles diffuse independently with diffusivity $D$;
4. Particles annihilate on collision.

For unpaired nucleation (i) and (ii) are replaced by

4.'  Particles are nucleated at random times and positions with rate $Q$.
A striking difference between paired and unpaired nucleation is the scaling of the steady state density of particles, $r$0, with the nucleation rate:
$r$0   µ   G 1/2 (paired)
versus
$r$0   µ   Q1/3 (unpaired).
We exhibit the crossover between these two cases in terms of the following dimensionless quantity:
e = (2G/D)1/3 b. 

### Diffusion-limited reaction in one dimension: Paired and unpaired nucleation

Salman Habib, Katja Lindenberg, and Grant Lythe and Carmen Molina-París.       Journal of Chemical Physics   115    73

## Noise and nonlinear dielectrics

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 equations 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.

### Dielectric nonlinearity and stochastic effects in strontium titanate

Alp Findikoglu, Roberto Camassa, Grant Lythe and Q.X. Jia       Applied Physics Letters      (2002)

### New potential applications of nonlinear dielectrics

Alp Findikoglu et al       Integrated Ferroelectrics  22  259-268

## Stochastic differential equations

Differential equations 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 equations 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 equations, stochastic partial differential equations and differential delay equations.

### Stochastic Stokes' drift

Kalvis Jansons and Grant Lythe       Physical Review Letters   81  3136-3139

### Stochastic calculus: application to dynamic bifurcations and threshold crossings

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

## Stochastic partial differential equations

(.ps file)

Stochastic partial differential equations 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 the rate of change of the parameter and the amplitude of the noise.

### Domain formation in transitions with noise and a time-dependent bifurcation parameter

Grant Lythe   Physical Review E 53   R4271-4274 (1996)

### Dynamics of defect formation

Esteban Moro and Grant Lythe   Physical Review E   59    R1303-1306 (1999)

### Stochastic PDEs: convergence to the continuum?

Salman Habib and Grant Lythe.       Computer Physics Communications   142    29 (2001)

## Semiconductor laser dynamics

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.

### Slowly passing through resonance strongly depends on noise

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)

## Noise-controlled dynamics

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 equations 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.

### Noise and slow-fast dynamics in a three-wave resonance problem

Grant Lythe and MRE Proctor Physical Review E 47  3122-3127 (1993)

### Predictability of noise-controlled dynamics

Grant Lythe and MRE Proctor Physica D   (1999)
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