Solution of the Smoluchowski equation using Sparse Grids
Dr Jochen Garcke
Dr Markus Hegland
Description of Project
The Smoluchowski (or Fokker-Plank) equation is a partial
differential equation of the form
where ρ is a probability density function ρ(x,t) with real
state vector x and time t, F(x) is a force field, and
∆ x and ∇ x denote the Laplacian and the gradient,
respectively. The Smoluchowski equation describes stochastic
processes in physics, chemistry and biology. In many applications,
the state vectors x are high-dimensional. In Brownian dynamics the
forces F are of the form
where the potential Φ is given by
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Φ(X) = |
∑
i ≠ j
|
|
1
||xi − xj||12
|
. |
|
In this project you will study numerical techniques to solve the
Smoluchowski equation and compare these techniques with stochastic
simulation methods. The aim is to approximately compute the evolving
probability distribution ρ(x,t) of a particle system, and
compare it to current methods which use algorithms for molecular
dynamic simulations to generate a large number of random sample
paths X(t).
The challenge for the numerical treatment of this partial
differential equation is two-fold. First there is the number of
dimensions, as standard finite element methods cannot be used in
(much) more than 3 dimensions due to computational complexity.
Sparse Grid methods allows to overcome this limitation and will be
used in this project as the underlying approximation scheme for the
solution of the Smoluchowski equation. The second challenge relates
to the singularities of the potentials Φ(x). Thus it is
suggested that this project proceeds in two steps: First the
numerical solution is developed and analysed for a smooth potential
and after this is completed the singular potential is considered.
Particular examples will be studied in close collaboration with
members of the Research School of Chemistry. This includes the
problem of crystallisation kinetics and of the geometry of polymers.
First, relatively small systems will be considered (with around 5
atoms in 2 dimensions) and then methods for more complex systems
will be discussed.
Requirements
The student should have completed Math2320 Analysis I.
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