Peter Sollich

Research Interests
Peter Sollich

King's College London

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 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


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 type-II 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 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 (power-law 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 (stress-strain 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.


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 low-dimensional 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 (cloud-point 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 double-tangent 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 re-entrant melting and solid-solid coexistence; the prediction of nematic ordering at arbitrary low densities in hard rods with fat-tailed length distributions; and the quantitative analysis of polydispersity effects in colloid-polymer 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 low-temperature 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 (one-spin facilitiated Ising models) that map to reaction-diffusion systems. Here we have obtained exact symmetries that affect the critical dimension (with Robert Jack), exact long-time 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.

Fluctuation-dissipation relations and effective temperatures

Kinetically constrained models also provide an interesting scenario in which to look at so-called out-of-equilibrium fluctuation-dissipation theorems and the related issue of effective temperatures. This has so far mainly been done for mean-field systems (effectively corresponding to infinite dimension), so studying the opposite limit of short-range systems with trivial equilibrium properties has provided very useful insights into how generic the properties observed in mean-field 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 non-equilibrium fluctuation-dissipation relations, but the fluctuation-dissipation ratio, which measures by how much the equilibrium fluctuation-dissipation theorem is violated, is then also negative. We have also looked at non-equilibrium fluctuation-dissipation 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 real-valued 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 high-dimensional representations. This requires sophisticated statistical inference techniques. Results so far are encouraging: inference from waveforms achieves better performance in very noisy conditions.

Last updated 16 Apr 2008
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