Areas of research interest
Realizability and Representability:
For random particle systems like liquids, large molecules, lawn made out of grass, etc. quantities of interest can be calculated from the first few correlation functions - often the first two - alone. Given a random distribution in some explicit form, for example as a grand canonical ensemble in thermodynamic limit, one can in principle calculate these correlation functions, although in practice this is often impossible. On the other hand, one may start with certain prescribed correlation functions; these might arise as computable approximations to those of some computationally intractable random distribution, as occurs in the study of equilibrium fluids, or might express some partial information about an as yet unknown random distribution, as in the study of heterogeneous materials. One would like to determine whether or not these given functions could in fact be the correlation functions of some random distribution, i.e., are they realizable?
Phase transition of classical and quantum systems of particles in the continuum with Kac-potentials
In a recent paper Lebowitz, Mazel and Presutti, have introduced a model of identical point particles in the continuum, proving a first order phase transition with the particles density as order parameter. The interaction is given by two and four body, finite range, translational invariant, Kac potentials and the proof of phase transition is based on a Pirogov-Sinai scheme and a local mean-field variation description. As originally argued by Kac, Uhlenbeck and Hemmer, Kac potentials are supposed to model the van der Waals theory of liquid vapor phase transitions, and indeed in the limit of large range and small strength one obtains a phase diagram which agrees with the one proposed by van der Waals (with the Maxwell, equal area law included). For a true proof of the van der Waals theory in a statistical mechanics setting, however, we would need to prove the existence of the phase transition for finite range and strength and indeed this is what was accomplished in LMP. In collaboration with F. Baffioni, I. Merola, E. Presutti a quantum extension of the result was considered. Using pathintegral representation the techniques of LMP can be exploited, the main additional difficulty is to show that the extra correlations arising from the quantumness do not destroy independence in the effective contour model underlying the proof.
Birth, death and jump type dynamics of infinite particle systems and correlation function hierarchies
Stochastic dynamics of infinite particle systems have several interesting and difficult properties. There are three typs of elementary change the system can undergo. A particle can disapear from the system, can diffuse, jump or a new particle can appear in the system. I am in particular interested in such systems at non-zero density. I am studying spectral properties and large time behavior of such systems. I am developing techniques to treat the associated hierarchical equations, the stochastic analogue of BBGKY-hierarchy. To work practically with these hierarchies one often tries to close (truncate and modify) them; I am studying this so-called closure problem which is deeply linked to the realizability problem described above.