Grant Lythe: Research Interests


theoretical Immuology |  kink stochastics |  diffusion-limited reaction |  numerical methods | 
exponential timestepping |  nonlinear dielectrics |  SDEs |  SPDEs |  lasers |  noise-controlled dynamics | 

Theoretical Immunology

Immunology is one of the most challenging fields in biomedicine. In the last decades its foundations have been revisited and the molecular interactions at the basis of immune responses are far from being accurately known at the microscopic level. The synergetics between microbiology and genetics have been very fruitful in the last three decades. However, there is a new approach coming from the exact sciences that is attracting the attention of some immunologists. Thus, theoretical immunology (in which mathematical and physical tools are combined to build biologically consistent models) has become an active and fruitful field of research.
read ScienceDaily article
The immune system is one of the most complicated multiscale systems imaginable. The adaptive immune system of a vertebrate is a vast army of cells and molecules that cooperate to seek out, mark, bind to and destroy invaders. Stochastic modelling is ideally suited to immunology at many scales. For example,

Kink stochastics

Scientists in many fields face a similar challenge: to understand a large system, driven by many noisy and nonlinear influences, containing persistent identifiable structures. The challenge is being met with a two-pronged approach. Firstly, using the most powerful computers available, perform numerical simulations of the full model on the largest domain and with the highest spatial resolution possible, for as long a time as possible. Secondly, develop theoretical methods that efficiently identify the structures of interest and predict their number, structure, dynamics and interactions. Noise, fluctuations or randomness is part of every real phenomenon. It provides an extra fascinating element for the nonlinear-dynamicist to study. Just as differential equations were invented to describe deterministic dynamics, stochastic differential equations are the tool needed to describe dynamics that have a random, or stochastic, component. In practice, solving a nonlinear stochastic differential equation is like solving an nonlinear ordinary differential equation: exact analytical solutions are seldom available, but paths can be generated in a few seconds on a computer and analysed using tools from pure and applied mathematics. A new area is the study of stochastic partial differential equations, which describe spatially extended systems with randomness, such as vortices in a fluctuating environment. Even though the theory of such equations is still in its infancy, the recent rapid increase in the computing power is starting to make it possible to explore their dynamics. 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.

Kinks are examples of coherent structures: clearly identifiable localized features in a noisy, spatially-extended system that can be followed as they move about under the influence of fluctuations. We study kinks in the Ø4 stochastic partial differential equation, where a steady-state mean density is dynamically maintained: kinks and antikinks are nucleated in pairs, follow Brownian paths and annihilate on meeting. Current computers can attain sufficient resolution to perform direct comparisons with predictions for the steady state, and work at sufficiently low temperatures to unambiguously locate kinks and identify nucleation events. In order to perform precise comparison between numerical and exact results at finite temperature, it is important to use the exact correlation length and not a low-temperature approximation. Numerical convergence of thermodynamic properties, where analytical results are available, makes it possible to proceed with confidence to an exploration of the fascinating stochastic dynamics of kinks.

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.

More on kinks


Many spatially-extended systems in physics, chemistry or biology exhibit two locally-stable states or phases. The two states coexist in the sense that, at any one time, different parts of the system are in different states. Fluctuations cause transitions of part of the system from one state to another, called nucleation events. We consider a string of particles, each coupled to its two neighbours, subject to noise and to the double-well potential:
V(φ) = -1/2φ2 + 1/4φ4.
The equation of motion for the ith particle is

Click on the image below to see the system in real time (java applet)

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 Γ
  3. Once born, all particles diffuse independently with diffusivity D;
  4. Particles annihilate on collision.

For unpaired nucleation 1. and 2. 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, r0, with the nucleation rate:
r0   ∼   Γ1/2 (paired)
r0   ∼   Q1/3 (unpaired).
We exhibit the crossover between these two cases in terms of the following dimensionless quantity:
ε = (2Γ/D)1/3 b.

Diffusion-limited reaction in one dimension

Grant Lythe       Physica D   222  159-163  (2006)

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

Salman Habib, Katja Lindenberg, Grant Lythe and Carmen Molina-París.
Journal of Chemical Physics   115  73  (2001)

Numerical methods for stochastic differential equations

Newton's second law of motion relates force to acceleration. Consequently, second-order differential equations are common in scientific applications, in the guise of ``Langevin'', ``Monte Carlo'', ``molecular'' or ``dissipative particle'' dynamics, and the study of methods for second-order ordinary differential equations is one of the most mature branches of numerical analysis. Numerical methods that preserve geometric properties of a the flow of a differential equation are increasingly useful in deterministic dynamics when it is important that the numerics reproduce the dynamics as accurately as possible over long times. In particular, when solving separable Hamiltonian problems, explicit partitioned Runge-Kutta (PRK) methods can be constructed that are symplectic. In stochastic dynamics, an ensemble of different trajectories originates even from a single initial condition, corresponding to different realizations of the stochastic process. The geometric property of the flow defined by a bundle of neighbouring initial conditions is not relevant. There is, however, an appropriate long-term quantity: the stationary density. We construct explicit PRK methods that approximate the stationary density with high-order accuracy.

Numerical methods for second-order for stochastic differential equations.

Kevin Burrage, Ian Lennane and Grant Lythe
SIAM Journal of Scientific Computing in press (2006)

The exponential timestepping method for stochastic differential equations

Exponential timestepping algorithms are efficient for exit-time problems because a boundary test can be performed at the end of each timestep, giving high-order convergence in numerical evaluation of mean exit times. Successive time increments are independent random variables with an exponential distribution. In the figure on the right, one realization is depicted. Exponential timesteps are jumps from one black point to the next. The values of the process at these times are generated, but not the corresponding times.
Kalvis Jansons and G.D. Lythe:

Efficient numerical solution of stochastic differential equations using exponential timestepping

      Journal of Statistical Physics  100    1097 (2000)

Exponential timestepping with boundary test for stochastic differential equations.

      SIAM Journal of Scientific Computing 24 1809 (2003)

Multidimensional exponential timestepping with boundary test.

      SIAM Journal of Scientific Computing 27 793-808 (2005)

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.

Experimental device

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.

Transmission, reflection and second-harmonic generation in a nonlinear waveguide

Roberto Camassa, Alp T. Findikoglu and Grant Lythe
SIAM Journal of Applied Mathematics 66 1-28 (2005)

Dielectric nonlinearity and stochastic effects in strontium titanate

Alp Findikoglu, Roberto Camassa, Grant Lythe and Q.X. Jia       Applied Physics Letters  80  3391  (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 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. 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.

Coexisting periodic attractors in injection-locked diode lasers

A. Gavrielides, V. Kovanis, P.M. Farangis, T. Erneux and G. Lythe
Quantum and semiclassical optics   9  785   (1997)

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)

Dynamics controlled by additive noise

Grant Lythe   Nuovo Cimento D17  855-861 (1995)

Predictability of noise-controlled dynamics

Grant Lythe and MRE Proctor
Physica D 133  362 (1999)

α = L2|μlogε|/4π2

Stochastic Stokes Drift

Classical Stokes' drift is the small time-averaged drift velocity of suspended non-diffusing particles in a fluid due to the presence of a wave. We consider the effect of adding diffusion to the motion of the particles, and show in particular that a non-zero time-averaged drift velocity exists in general even when the classical Stokes' drift is zero. Our results are obtained from a general procedure for calculating ensemble-averaged Lagrangian mean velocities for motion that is close to Brownian, and are verified by numerical simulations in the case of sinusoidal forcing.

Stochastic Stokes' drift

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

Sensory detection in noisy environments

Many animals manage to detect a weak signal against a noisy background. The key to understanding this may lie in the concept of stochastic resonance, a mechanism by which noise actually assists, rather than hinders, detection of signals. Stochastic resonance has been demonstrated in a wide variety of mathematical models, from simple bistable systems to full-scale models of auditory systems and of neural cell networks. The challenge is to establish whether it is indeed used by living animals. An example that is being actively investigated at Leeds is found in moths, which have tympanic organs sensitive to ultrasound that enable them to detect the echolocation calls of bat predators and take evasive action. Detection involves the identification of a periodic train of bat echolocation calls against a background of biotic and abiotic noise. Resolving the puzzle of how the two-celled moth organ is capable of performing this detection will require integrating experiment and theory from beginning to end, including stochastic modelling and statistical analysis of time series. A systems approach is needed to understand the throughput information from the sensory cells to the behavioural consequences, with input from neurophysiology, signal processing, behavioural ecology and evolutionary modelling.