
Research Interests
Peter Sollich


Framework: Statistical mechanics of disordered systems
My main research interest is the study of disordered systems
using the methods of statistical mechanics. Such systems can
exhibit a number of interesting features, such as
 slow, `glassy' dynamics
 aging
 metastability
The slow dynamics and the presence of many metastable states mean that
such systems are normally out of equilibrium, which makes
their study both very interesting and very difficult. But even in
equilibrium, disorder poses interesting challenges, as in the phase
behaviour of polydisperse systems (see below). Closely related,
finally, is the subject of statistical inference, where underlying
relations or patterns have to be inferred from noisy and sometimes
incomplete data.
An overview of some areas of my current work can be found below. For
details please see my publications list.
Soft condensed matter
Rheology
There are a number of materials where disorder has long been
recognised as crucial, such as structural glasses (e.g. window glass),
spin glasses (disordered magnets), polymers in random media, vortices
in typeII superconductors etc. One of the aims of my research
is to extend and transfer ideas developed for the study of these
materials and apply them to the Rheology (mechanical and flow
behaviour) of soft condensed matter such as
 foams
 dense emulsions
 colloidal suspensions (in particular colloidal glasses)
 onion and vesicle phases of surfactant systems
 gel bead suspensions
 slurries
The underlying hypothesis is that (structural) disorder and
metastability are crucial to an understanding of the behaviour of
these materials.
This approach led to the "soft glassy rheology" model.
Many striking features of the rheology of soft materials
(powerlaw fluid behaviour, `flat' frequency dependence of linear
moduli) can be explained within this simple mesoscopic model which takes
structural disorder into account. The SGR model
not only establishes interesting connections with previous analyses of
spin glass dynamics, but also allows an exact constitutive quation
(stressstrain relation) to be derived. This has opened the door to
detailed studies of a wide range of nonlinear mechanical
behaviour. The model has also been extended to include full tensorial
descriptions of stress and strain. Ongoing work is aimed at extracting
directly from simulation data measurements of the mesoscopic
quantities that the model starts from.
Polydispersity
Another form of disorder that is very important in soft condensed
matter is polydispersity  variations in the size of colloidal
particles, chemical composition of copolymers etc. This can strongly
affect both the equilibrium phase behaviour and the phase separation
kinetics. The thermodynamic description of such systems in principle
involves a distribution of sizes, chemical compositions etc (a
`functional order parameter', with an infinite number of degrees of
freedom). Together with Prof Michael E Cates at the University of
Edinburgh, I developed a method for `projecting' the free
energies of such systems onto a lowdimensional subspace of `moment
variables' (these are slight generalizations of the ordinary moments of
the density distribution). This can be done in such a way that most of
the information about the thermodynamics is retained; in particular,
from the projected free energy one can find exactly the critical
points, spinodals and the points where phase separation first occurs
(cloudpoint and shadow curves). This is true even though the
projected free energy only depends on a finite number of density
variables, so that its phase behaviour can be extracted by the usual
doubletangent construction.
We have since (with Alessandro Speranza and Moreno Fasolo) applied
this method to a variety of models; key among the results are the
first comprehensive phase diagram of polydisperse hard spheres, which
unified and extended previously disparate predictions of reentrant
melting and solidsolid coexistence; the prediction of nematic
ordering at arbitrary low densities in hard rods with fattailed
length distributions; and the quantitative analysis of polydispersity
effects in colloidpolymer mixtures.
In joint work with Nigel Wilding at the University of Bath I have also
developed advanced simulation techniques for polydisperse systems;
these have been essential for obtaining data in the most
experimentally relevant scenario (where the distribution of particle
sizes across the system is fixed). These simulations have recently
also turned up surprises in how polydisperse liquids wet solid
surfaces: while wetting transitions, where the liquid changes from
forming individual drops to spreading across the surface, conventially
occur only as temperature is varied, in polydisperse systems
variations of density can have the same effect.
Aging and glassy dynamics
Kinetically constrained models
I'm also interested in slow dynamics in systems without
`quenched' disorder (i.e., disorder which is fixed once and for all
before the dynamics begins) disorder. A nice example of this are
kinetically constrained models: their equilibrium behaviour is
normally trivial, but their dynamics can be very intricate because of
constraints on the kinetics which limit the number of `paths'
connecting the various states of the system. Together with Martin
Evans at the University of Edinburgh, I have looked at a model from
this class (the "East model") and shown that it can actually be solved
exactly in the most interesting case, namely the lowtemperature
limit; we found explicitly how the relaxation time diverges, and also
study the unusual behaviour after a quench (rapid cooling down) to low
temperature.
Since then I have looked at several other models, including ones with
less severe constraints (onespin facilitiated Ising models) that map
to reactiondiffusion systems. Here we have obtained exact symmetries
that affect the critical dimension (with Robert Jack), exact longtime
solutions in one dimension (with Peter Mayer) and a lot more. Some of
this work is presented, in a much wider context, in a review written
with Felix Ritort at Barcelona.
Fluctuationdissipation relations and effective temperatures
Kinetically constrained models also provide an interesting scenario in
which to look at socalled outofequilibrium fluctuationdissipation
theorems and the related issue of effective temperatures. This has so
far mainly been done for meanfield systems (effectively corresponding
to infinite dimension), so studying the opposite limit of shortrange
systems with trivial equilibrium properties has provided very useful
insights into how generic the properties observed in meanfield
systems are. One key recent result is that, due to the presence of
thermal activation in many models with glassy dynamics, responses to
external perturbations can be negative. We have found a way to
rationalize this (in work with Ludovic Berthier, Juanpe Garrahan,
Sebastien Leonard and Peter Mayer). Interestingly, response and
correlation functions remain linked via nonequilibrium
fluctuationdissipation relations, but the fluctuationdissipation
ratio, which measures by how much the equilibrium
fluctuationdissipation theorem is violated, is then also negative.
We have also looked at nonequilibrium fluctuationdissipation
relations in other contexts, for example trap models (which are also
at the basis of the work on soft glassy rheology) and more recently
models of phase ordering at the critical temperature (with Alessia
Annibale). Here, the FD ratios are of additional because they are
universal quantities characterizing dynamical universality classes.
Most recently we have extended this to studying the fluctuations of
correlation and response functions, which are linked to dynamical
heterogeneities in such systems.
Statistical inference and neural networks
Disorder also occurs, in more abstract form, in (artificial) neural
networks which are being trained to learn rules from
examples. Training often takes place by gradient descent on a training
error which measures how well the network has learned the
examples. Due to randomness and noise in the examples, this error
`landscape' is often extremely rugged. Local minima act as metastable
states, and learning processes in neural networks can therefore
exhibit `glassy' dynamics. The `disordered' energy landscape also
affects the equilibrium (long training time) behaviour, and
calculations use techniques very similar to those developed for spin
glasses.
I have studied a number of aspects of neural network learning in the
past (see my CV outline). These include as
query learning (asking `intelligent' questions), finite size effects,
online learning and learning with ensembles.
Gaussian processes
More recently, I have looked at learning and inference with Gaussian
processes. These can be thought of as neural networks with an infinite
number of hidden units, in a certain limit. But they are actually
statistical model in their own right, specifying priors over function
spaces which we can then use to perform Bayesian inference. In many
ways, they are easier to handle than neural networks; this is
certainly true for problems where one is trying to learning a
realvalued function (regression); `training' a Gaussian process for
regression is essentially trivial. Finding out how fast these models
can learn is therefore an important question; I have recently derived
some useful approximations to the appropriate learning curves
which get rather closer to the truth than existing bounds. Ongoing
work includes looking at learning of functions that are defined on
random graphs modelling social, economic or biological networks.
Support Vector Machines
Closely related to Gaussian processes are Support Vector Machines
(SVM), which have been used very successfully in many recent
applications (both classification and regression problems). Even
though SVMs were originally derived from a machine learning (PAC)
point of view, it turns out that they can also be interpreted in a
probabilistic, Bayesian framework. This makes it possible to assign
error bars to the outputs of a trained SVM, and also to tune the
parameters of an SVM algorithm by maximizing what is called the
`evidence' (the likelihood of the data given the parameters). In work
with Carl Gold, we have developed numerical methods for estimating the
evidence for SVM and use it for parameter tuning.
Phoneme recognition
An application area which I've become interested in more recently is
recognition of speech from acoustic waveform. In this work, with
Ph.D. students Matthew Ager and Jibran Yousafzai and my colleague
Zoran Cvetkovic in Electronic Engineering, the starting hypothesis is
that waveforms rather than the conventional "compressed"
representations of speech should provide a starting point for
speech recognition that is more robust to noise and variation between
speakers. The downside is that one has to deal with the resulting
highdimensional representations. This requires sophisticated
statistical inference techniques. Results so far are encouraging:
inference from waveforms achieves better performance in very noisy
conditions.