 Andrea Collevecchio, 15.10.2009 |
Department of Mathematics Warwick Statistical Mechanics Seminar |
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| 14.01.2010 | Jani Lukkarinen (Helsinki) Kinetic Equations for a Homogeneous Bose Fluid with Condensate We consider the kinetics of a three-dimensional fluid of weakly interacting bosons with supercritical densities. More precisely, we consider the postulated nonlinear Boltzmann-Nordheim equations for this system, in a spatially homogeneous state which has an isotropic momentum distribution. The resulting evolution equations have a surprisingly rich mathematical structure, where proper definitions play an important role. In fact, it is not even clear if the standard kinetic scheme for derivation of the Boltzmann-Nordheim equations can be trusted in the presence of a condensate. Elaborating on the work of Herbert Spohn [arXiv:0809.4551], we propose a definition of the coupled equations for which the thermal equilibrium states are stationary. To test the validity of the equations, we have studied global existence and uniqueness of solutions, as a problem about return to equilibrium from a perturbation of a thermal state with a condensate. I will discuss here preliminary results from this joint work with Jogia Bandyopadhyay and Antti Kupiainen. |
| 21.01.2010 | Michalis Loulakis (Crete) Zero-range condensation at criticality Zero-range processes with decreasing jump rates exhibit a continuous condensation transition, where a finite fraction of all particles condenses on a single lattice site when the total density exceeds a critical value. We study the onset of condensation, i.e. the behaviour of the maximum occupation number after adding a subextensive excess mass of particles at the critical density. We establish a law of large numbers for the excess mass fraction in the maximum, which turns out to jump from 0 to a positive value at a critical scale. Our results also include distributional limits for the fluctuations of the maximum in both regimes, which change from standard extreme value statistics to Gaussian. We identify the detailed behaviour at the critical scale including sub-leading terms, providing a full understanding of the crossover between the two regimes. |
| 28.01.2010 | Paul Chleboun (University of Warwick) Finite size effects in a stochastic condensation model We study finite size effects on the condensation transition in the zero-range process. The transition is already well understood in the thermodynamic limit, but finite size effects can be significantly large and counter intuitive. We observe a high current overshoot above the limiting critical value and an abrupt change between a putative fluid and condensed phase. This is reminiscent of a first order transition, although condensation is known to be continuous in the thermodynamic limit. Close to the abrupt transition we also observe metastable switching between the two phases. We can define an effective free energy landscape and thus predict the scaling of the lifetime of the two phases. We formulate approximations for the fluid and condensed phases, and use these to derive the leading order finite size effects. |
| 04.02.2010 | Thomas Østergaard Sørensen (Imperial College) Regularity properties of Coulombic wavefunctions and their one-electron densities We review recent results on the regularity and structure of wavefunctions psi of the non-relativistic Schrödinger operator describing atoms and molecules (that is, with Coulomb interactions). We also discuss the regularity of the associated one-electron densities rho. In particular, we characterize the structure of psi around 'two-particle coalescence points'. The method of proof of the latter extends to the study of the structure of the solutions to the multiconfiguration equations, and their densities, at the positions of the nuclei. This is joint work with S. Fournais and M. and T. Hoffmann-Ostenhof. |
| 11.02.2010 | Amin Coja-Oghlan (University of Warwick) A statistical mechanics perspective on hard computational problems A large variety of computational problems can be classified as "computationally hard". In recent years researchers from statistical mechanics have investigated such problems via methods from the theory of spin glasses. The aim of this talk is to give a brief overview of this work, and of the extent to which these considerations can be turned into rigorous mathematics. |
| 18.02.2010 | Christina Goldschmidt (University of Warwick) The scaling limit of critical random graphs Consider the Erdos-Renyi random graph G(n,p): we have n vertices, every pair of which is joined by an edge independently with probability p. As is well known, this model undergoes a phase transition as p passes through 1/n. Below 1/n, there are only "small" components; above, there is one so-called giant component, which contains a positive proportion of the vertices. We are interested in the "critical window", where p = n^{-1} + \lambda n^{-4/3} for some real \lambda. In this regime, the largest components are of size n^{2/3} and have finite surpluses (where the surplus of a component is the number of edges more than a tree that it has). Using a bijective correspondence between graphs and certain "marked random walks", we are able to give a (surprisingly simple) metric space description of the scaling limit of the ordered sequence of components, where edges in the original graph are re-scaled by n^{-1/3}. Along the way, I will talk about the notion of a continuum random tree; such trees form an important part of the description of a limiting component in the random graph. Our convergence holds in a sufficiently strong sense that we are able to deduce the convergence in distribution of the diameter of G(n,p), rescaled by n^{-1/3}, to a non-degenerate random variable, for p in the critical window. This is joint work with Louigi Addario-Berry (Universite de Montreal) and Nicolas Broutin (INRIA Rocquencourt). |
| 25.02.2010 | Marek Biskup (UCLA) tba |
| 03.03.2010 | Oliver Riordan (Oxford) tba Joint seminar with DIMAP, takes place Wednesday in tba at tba. |
| 11.03.2010 | |
| 18.03.2010 | Hillel Raz (Cardiff) |