Warwick Statistical Mechanics Seminar
in the Department of Mathematics
 |
This term, all seminars take place Thursdays at 2pm, room MS.04 (Zeeman Building), unless indicated otherwise. |
 Charles-Edouard Pfister, 21.02.2008 |
| 29.04.2010 | Volker Betz (University of Warwick) Effective density of states for a quantum oscillator coupled to a photon field. When a quantum particle is coupled to a photon field, higher energy eigenstates dissolve into resonances. Physically, this means that a system state prepared at the energy of a former eigenvalue is metastable, and takes a long time to decay to the ground state. Mathematically, the most successful method of describing resonances so far is by using Hunziker's complex dilation method. In this talk, I will present an alternative point of view, based on quantum statistical mechanics. Here, the quantum particle is regarded as a closed system that feels the effective influence of the field. I will introduce the effective density of states, and show that in the important special case where the particle is a harmonic oscillator, it displays the behaviour that we expect from the natural line spectrum of a charged particle (energy level shift and line broadening), even quantitatively. On the other hand, not much is known about the effective density of states beyond the case I present. I will highlight the most important open questions. |
| 06.05.2010 | Freddy Bouchet (ENS Lyon) Building invariant measures of the 2D Euler and Vlasov equations, non-ergodicity, and proof of the irreversible behavior of these reversible equations In any dynamical systems, the knowledge of invariant measures is extremely useful. For instance in a turbulence problem, it gives a solution to the hierarchy of equations that describes the moments of the velocity field. I will explain how it is possible to explicitly built sets of invariant measures that generalize statistical equilibrium measures for the two dimensional Euler and the Vlasov equations. For the two-dimensional Navier-Stokes equations with weak stochastic forcing and dissipation, the existence of an invariant measure has been mathematically proved recently, together with mixing and ergodic properties. I will sketch how to use the invariant measures of the two dimensional Euler equations to describe self-consistently the invariant measures for the two dimensional Navier-Stokes equations. In a second part of the talk, I will describe new mathematical results about the asymptotic stability and explicit predictions of the large time asymptotics for the two dimensional Euler equation and the Vlasov equations. These results are one of the few examples in statistical mechanics where the apparent paradox between microscopic reversibility and macroscopic irreversibility can be understood and analyzed thoroughly mathematically. Contact: Stefan Adams and Oleg Zaboronski |
| 18.05.2010 | Karel Netočný (Czech Academy of Sciences) Nonequilibrium variational principles from dynamical fluctuations As well known since the pioneering work of I. Prigogine, the steady state of thermodynamically open system not too far from equilibrium approximately minimizes a certain functional which derives from entropy changes. Within the stochastic framework, this amounts to a variational principle for the leading correction term when expanding the stationary distribution around equilibrium. We argue that this principle is just a consequence of the structure of dynamical fluctuations in the sense of the Donsker-Varadhan theory; somewhat analogously to the Gibbs variational principle being intimately related with equilibrium fluctuations. A more general large deviation approach including current fluctuations further reveals connection between the far-from-equilibium breakdown of the entropy production principles and the appearance of time- symmetric-antisymmetric correlations. (Jointly with C. Maes and B. Wynants.) Special date and time: Tuesday 11am in MS.04 Contact: Roman Kotecký |
| 20.05.2010 | Nathanaël Berestycki (University of Cambridge) Emergence of giant cycles in random transpositions: a new proof of a result of Oded Schramm Consider the permutation-valued process obtained by performing random transpositions at rate 1, started from the identity permutation. In 2005 Oded Schramm proved a beautiful result, conjectured by David Aldous, that the distribution of large cycle sizes has Poisson-Dirichlet asymptotics after (1+eps)n/2 steps. In particular this is also the time for emergence of giant cycles. I will present a very short and non-computational proof of this fact, which can also be generalised to some other models. I will discuss related conjectures and ideas, and (time-permitting) explain briefly how this is connected to a problem on mixing times of random walks (this last bit being joint work with Oded Schramm and Ofer Zeitouni). Contact: Daniel Ueltschi |
| 27.05.2010 | |
| 03.06.2010 | No seminars (too many people away) |
| 10.06.2010 | Francesco Caravenna (University of Padova) The Weak Coupling Limit of Disordered Copolymer Models A copolymer is a chain of repetitive units (monomers) that are almost identical, but they differ in their degree of affinity for certain solvents. This difference leads to striking phenomena when the polymer fluctuates in a non-homogeneous medium, for example made of two solvents separated by an interface. One may observe, for instance, the localization of the polymer at the interface between the two solvents. A discrete model of such system, based on the simple symmetric random walk, has been investigated in [Bolthausen and den Hollander, AP 1997], notably in the weak polymer-solvent coupling limit, where the convergence of the discrete model toward a continuum model, based on Brownian motion, has been established. This result is remarkable because it strongly suggests a universal feature of copolymer models. In this talk we prove that this is indeed the case. More precisely, we determine the weak coupling limit for a general class of discrete copolymer models based on renewal processes, obtaining as limits a one-parameter family of continuum models, based on stable regenerative sets. (Joint work with G. Giacomin) Contact: Nikos Zygouras |
| 17.06.2010 | Douglas Abraham (University of Oxford) Some Recent Results for the Thermal Casimir Interaction Contact: Daniel Ueltschi |
| 24.06.2010 | Oliver Johnson (University of Bristol) Monotonicity, thinning and discrete versions of the Entropy Power Inequality We consider the entropy of sums of discrete random variables, in analogy with Shannon's Entropy Power Inequality, where equality holds for normals. In our case, infinite divisibility suggests that equality should hold for Poisson variables. Some natural analogues of the Entropy Power Inequality fail, but we propose an alternative formulation which does hold. In the spirit of the monotonicity results established by Artstein, Ball, Barthe and Naor, we prove a stronger version of concavity of entropy, which implies a strengthened form of our discrete Entropy Power Inequality. (Joint work with Y.Yu) Contact: Stefan Grosskinsky |
| 01.07.2010 | |