In this talk, we make quantitative comparisons between two widely-used liquid crystal modelling approaches - the continuum Landau-de Gennes theory and mesoscopic mean-field theories, such as the Maier-Saupe and Onsager theories. We use maximum principle arguments for elliptic partial differential equations to compute explicit bounds for the norm of static equilibria within the Landau-de Gennes framework. These bounds yield an explicit prescription of the temperature regime within which the LdG and the mean-field predictions are consistent, for both spatially homogeneous and inhomogeneous systems. We find that the Landau-de Gennes theory can make physically unrealistic predictions in the low-temperature regime. In my joint work with John Ball, we formulate a new theory that interpolates between mean-field and continuum approaches and remedies the deficiencies of the Landau-de Gennes theory in the low-temperature regime. In particular, we define a new thermotropic potential that blows up whenever the mean-field constraints are violated. The main novelty of this work is the incorporation of spatial inhomogeneities (outside the scope of mean-field theory) along with retention of mean-field level information.
When modelling the motion of a sperm cell in the female reproductive tract, the Reynolds number is found to be very small, thus allowing the nonlinear Navier-Stokes equations to simplify to the linear Stokes equations stating that pressure, viscous and body forces balance each other at any instant in time. A wide range of analytical techniques can be applied to investigate the Stokes flow past a moving body. In this talk, we introduce various Stokes flow singularities and illustrate how they can provide a handy starting point (ansatz) when trying to determine the form of the flow field around certain bodies, from simple translating spheres to beating sperm tails.
In this talk we aim to filter the Majda-McLaughlin-Tabak(MMT) model, which is a one-dimensional prototypical turbulence system. Due to its inherent high dimensionality, we first try to find a low dimensional dynamical system whose statistical property is similar to the original complexity system. This dimensional reduction, called stochastic parametrization, is clearly well-known method but the value of current work lies in the derivation of an analytic closure for the parameters. We then discuss the necessity of the accurate filtering algorithm for this effective dynamics, and introduce the particle filter using the cubature on Wiener space and the recombination skill.
Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that impose a number of computational constraints. In this talk we discuss efficient, scalable techniques for solving fractional-in-space reaction diffusion equations combining the finite element method with robust techniques for computing the fractional power of a matrix times a vector. We shall demonstrate the methods on a number examples which show the qualitative difference in solution profiles between standard and fractional diffusion models.
When modelling biochemical reactions within cells, it is vitally important to take into account the effect of intrinsic noise in the system, due to the small copy numbers of some of the chemical species. Deterministic systems can give vastly different types of behaviour for the same parameter sets of reaction rates as their stochastic analogues, giving us an incorrect view of the bifurcation behaviour.
\newlineThe stochastic description of this problem gives rise to a multi-dimensional Markov jump process, which can be approximated by a system of stochastic differential equations. Long-time behaviour of the process can be better understood by looking at the steady-state solution of the corresponding Fokker-Planck equation.
\newlineIn this talk we consider a new finite element method which uses simulated trajectories of the Markov-jump process to inform the choice of mesh in order to approximate this invariant distribution. The method has been implemented for systems in 3 dimensions, but we shall also consider systems of higher dimension.
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