Particle systems and statistical physics
For more than a hundred years, physicists have put great effort into
trying to understand the connection between the microscopic and
macroscopic laws of nature. How can the seemingly ordered structure of
matter on the macroscopic level be consistent with the highly
disordered structure on the atomic and molecular level? This
subdiscipline of physics is known as statistical mechanics.
During the last few decades, probability theory has played an
important role in statistical mechanics, through the construction
and analysis of large (sometimes infinite) spatial stochastic particle
system models that capture various dichotomies between the microscopic and
macroscopic behaviors observed in nature.
Much of our work is concerned with such models. A famous example is
the Ising model for spontaneous magnetization. Each integer point in 2-
or 3-dimensional space is assigned the value +1 or -1 randomly,
according to rules which favor agreement between nearest neighbors.
How strongly favored, depends on the interaction parameter J. When
J=0, different spins are independent, while as J increases, the
tendency to take the same values becomes stronger and stronger.
It turns out that there is a critical value for J, below which
the macroscopic behavior is disordered and above which it is ordered;
this is an example of an important phenomenon known as phase
transition, which in this case explains the occurrence of spontaneous
The Ising model and many of its relatives are so called Markov random
fields, meaning that they exhibit a spatial variant of the property
that characterizes Markov chains (see the
Markov theory group). Such models are
used extensively in many fields outside of physics, such as
spatial statistics and image analysis.
We also study so called percolation models, which are models for the
connectivity behavior of random media. These have a variety of
applications, ranging from the spread of liquid through porous media
and electrical conductivity, to forest fires and epidemics. An example
is given in the figure below: disks of unit radius are spread out
randomly in 2-dimensional space (according to a so called Poisson
process) with an average of L disks per unit area. Again we see a
phase transition phenomenon: there is a critical value for L, below which
the set of disks exhibits short-range connectivity only, and above
which there are long- (in fact, infinite-) range connections. The
figure shows realizations below, near, and above the critical value.
A poetic name for this model is the lilypad model. Imagine a pond with
lilypads, and a snail trying to cross the pond but being limited to
going on the lilypads. How large must the "lilypad density" L be for
this to be possible?
We also study clumping phenomena in such models. This is relevant e.g.
for understanding strength of inhomogeneous materials (see
the material fatigue group) and can also be
viewed as an extreme value problem.
Rigorous mathmematical solutions to a large number
of difficult problems in this
area have been obtained in recent years, by us and others. However,
many of the problems appear to be, at present, too
difficult to admit such solutions. When that happens, it is common practice
to revert to simulation studies, in which case the so
called Markov chain Monte Carlo algorithms (developed e.g. in the
Markov theory group) are of great use.
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by Olle Häggström