Mathematics Department - Mathematical Physics Seminar - Fall 2015

Mathematical Physics Seminar - Fall 2015

Organizer(s)

Joel Lebowitz, Michael Kiessling

Website

http://www.math.rutgers.edu/~lebowitz/Fall2014seminars.html

Thursday, November 19th

"Mixed order phase transitions: from DNA denaturation to jamming processes"

Time: 12:00 PM
Location: Hill 705
Abstract: Phase transitions of mixed nature, which on the one hand exhibit a diverging correlation length as in second order transitions and on the other hand display a discontinuous order parameter as in first order transitions have been observed in a diverse classes of physical systems. Examples include DNA denaturation, models of wetting, glass and jamming transitions, rewiring networks and some one-dimensional models with long-range interactions. An exactly soluble Ising model which provides a link between some of these rather distinct classes of systems is introduced. Renormalization group analysis which provides a common framework for studying some of these systems, elucidating the relation between them will be discussed.

BROWN BAG LUNCH 1:00 - 2:00PM

SPECIAL COLLOQUIUM TALK FOLLOWS AT 2PM

Thursday, October 29th

"Nonlinear waves in granular crystals: Theory, computation, experiments and applications"

Time: 2:00 PM
Location: Hill 705
Abstract: In this talk, we will provide an overview of results in the setting of granular crystals, consisting of beads interacting through Hertzian contacts. We will start from the simplest setting of one-dimensional, monoatomic chains where highly localized traveling waves exist and we will also examine states in the form of (dark) discrete breathers and shock waves therein. Wherever possible, we will corroborate these considerations with recent experimental results. We will then extend our considerations to the case of diatomic chains and examine how the properties of traveling waves and also of discrete breathers are modified in the latter setting. More highly heterogeneous chains will be briefly examined as well. In addition to considering the purely Hamiltonian case, select examples of the damped-driven variant of the system and its rich phenomenology, including chaotic response and bistability/ hysteresis will also be shown. In the last part of the talk, time-permitting, a number of recent aspects will be touched upon including:

Thursday, October 29th

"Ising model from the Random Current perspective"

Time: 12:00 PM
Location: Hill 705
Abstract: The random current representation yields stochastic geometric insight on the structure of correlation functions and on the phase transition in the Ising model. Specifically mentioned will be: A) general results on the Ising phase transition on transitive graphs [joint works with D. Barsky, R. Fernandez, H. Duminil-Copin, and V. Sidoravicious], B) results specific to high dimensions (d>4), C) results specific to the low dimension d=2. The latter include some insight on the phenomenon of emergent planarity at the critical point of non-planar 2D models [work in progress with H. D-C].

BROWN BAG LUNCH FROM 1-2PM

Thursday, October 22nd

"On the time-asymptotic evolution of Gaussian Vlasov fluctuation fields"

Time: 2:00 PM
Location: Hill 705
Abstract: After reviewing some basic results about the mean-field limit for systems of interacting particles, I will discuss a recent result about the long-time evolution of Gaussian fluctuations around an infinitely extended, spatially uniform Vlasov equilibrium. I will also highlight the significance of the time-asymptotic fluctuating force field for a rigorous justification (and possible generalizations) of the Balescu-Guernsey-Lenard kinetic equation.

Thursday, October 22nd

"How a Dirichlet character pretends to be the mobius function"

Time: 12:00 PM
Location: Hill 705
Abstract: THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM

Thursday, October 15th

"Norm lower bounds, correlations and the area law"

Time: 2:00 PM
Location: Hill 705
Abstract: Consider quantum lattice system with nearest neighbor interactions. Let A be a subset of the lattice. For a typical pure state of the system, the von Neumann entropy of the reduced density matrix of the pure state on A has an entropy that is proportional to the number of sites in A. The area law conjecture is that for ground states, the entropy is bounded by the number of sites on the boundary of A. Hastings has proved this for certain one dimensional systems. We present a partially alternative and more constructive proof for one dimensional systems.

This is joint work with Elliott Lieb.

Thursday, October 15th

" Non-equilibrium statistical mechanics of turbulence"

Time: 12:00 PM
Location: Hill 705
Abstract: The recent proposal by D. Ruelle for a theory of the corrections to the OK theory, iintermittency corrections, is to take into account that the Kolmogorov scale itsef should be regarded as a fluctuating variable. Some quantitative aspects of the theory can be quite easily studied also via computer and will be presented (comments elaborated in collaboration with P. Garrido)

THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM

Thursday, October 8th

"Twist Maps and Very Fast Ping Pong"

Time: 2:00 PM
Location: Hill 705
Abstract: Twist maps are very important in dynamical systems, and in particular in the realm of K.A.M. (Kolmogorov-Arnold-Moser) theory. First it is explained how such maps arise and why they are a natural object to study. Afterwards we consider the Fermi-Ulam ping pong system (not to be confused with the Fermi-Pasta-Ulam chain) and try to determine how likely it is that the ball can reach infinite speed. Depending on the forcing function, this will lead to maps which cannot be treated by means of KAM theory or Aubry-Mather theory.

This is joint work with R. Ortega, Granada.)

Thursday, October 8th

"Statistical mechanics of random walks, with application to molecular evolution"

Time: 12:00 PM
Location: Hill 705
Abstract: Understanding properties of transport in networks and other complex media is crucial for quantitative analysis of numerous biological, physical, social and technological systems, from protein-protein interactions in biology to the World Wide Web. While transport on networks with unweighted edges can be described using scaling techniques, few analytical results are available for networks with weighted and directed edges, which appear naturally in evolutionary dynamics, chemical reactions, and protein folding. In this talk I will describe an efficient recursive approach for studying continuous-time random walks on weighted networks and fitness or energy landscapes of arbitrary complexity. Using simple examples, I will discuss the importance of non-exponential waiting time distributions, memory, and course-graining in first-passage processes. Finally, I will apply our approach to the problem of protein adaptation. Specifically, I will investigate how structural coupling between protein folding and binding (the fact that most proteins can only function when folded) gives rise to evolutionary coupling between the traits of folding stability and binding strength, facilitating the emergence of evolutionary "spandrels" (features that appear through adaptation even though the feature itself does not contribute to the organism's fitness).

THERE WILL BE A BROWN BAG LUNCH FROM 1-2:00PM

Thursday, October 1st

"Yet another criterion for global existence in the 3D relativistic Vlasov-Maxwell system"

Time: 2:00 PM
Location: Hill 705
Abstract: The relativistic Vlasov-Maxwell system describes the time evolution of a classical plasma in position-momentum phase space. Since the particles can move at relativistic speeds, the motion of a single particle is governed by the Lorentz force, whereas the self-generated electric and magnetic fields are described by Maxwell's equations. The theory of existence of smooth solution was initiated by Glassey and Strauss in the eighties, but an unrestricted result on global existence of solutions is still not available. We will review the state of the art and present a new criterion for global existence, which in particular applies if a moment bound of order larger than three is available.

Thursday, October 1st

"Holder continuity of solutions to Vlasov-Fokker-Planck type equations with rough coefficients"

Time: 12:00 PM
Location: Hill 705
Abstract: THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM

Thursday, September 24th

"Asymptotics independent of structure"

Time: 2:00 PM
Location: Hill 705
Abstract: We'll be interested in situations in which enumerative information for a large discrete system is nearly determined by relatively simple parameters of the system; thus even very different systems will in some respects behave similarly if the parameters in question agree. (One may view such behavior as a reflection of weak long range interactions.) We'll discuss a few examples of this type and try to mention a few more that are conjecturally so.

Thursday, September 17th

"Derivation of Mean-Field Dynamics for Fermions"

Time: 2:00 PM
Location: Hill 705
Abstract: I present recent results about the derivation of the time-dependent Hartree(-Fock) equations from the Schroedinger equation for fermions in a mean-field limit. I introduce and discuss different scaling limits that were proposed for such a derivation. Several new methods were recently developed for tackling this kind of problems and I will focus on introducing the one I used myself in recent works. This method is based on controlling a well-chosen functional by estimating certain transition amplitudes in the Fermi gas. The functional is directly related to the trace norm difference of reduced density matrices and the estimates lead to explicit rates of convergence.

Thursday, September 17th

"Asymptotics of spacing distributions in RMT"

Time: 12:00 PM
Location: Hill 705
Abstract: Spacing distributions between neighboring eigenvalues are a primary observable quantity in random matrix theory. More than 50 years ago F.J. Dyson introduced a log-gas heuristic which predicted their asymptotic forms. I'll discuss subsequent developments on this topic, including my recent joint work with J.L. Lebowitz relating on local limit theorems.

THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM.

Thursday, September 10th

"Entropy and Entanglement Bounds for Reduced Density Matrices of Fermionic states"

Time: 2:00 PM
Location: Hill 705
Abstract: One of the important aspects of many-body quantum mechanics of electrons is the analysis of two-body density matrices. While the characterization of one-body density matrices is well known and simple to state, that of two-body matrices is far from simple -- indeed, it is not fully known. In this talk I will present joint work with Eric Carlen in which we study the possible entropy of such matrices. We find, inter alia, that minimum entropy is achieved for Slater determinant N-body parent functions. Thus, from the entropic point of view, Slater determinants play the same role as condensates play for bosons.

Thursday, September 10th

"Many-particle dynamical systems"

Time: 12:00 PM
Location: Hill 705
Abstract: In the summer I finished two books about large dynamical systems. The (temporary) title of the first book is Many-particle dynamical systems, intrinsic entropy and the Second Law". The (temporary) title of the second book is Realistic ergodic theorems and large dynamical systems". I will give a short preview of the two books.

THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM!

Thursday, September 3rd

" Non-perturbative renormalization group in a hierarchical Kondo model"

Time: 12:00 PM
Location: Hill 705
Abstract: In this seminar, a hierarchical version of the Kondo model is introduced and shown to be exactly solvable via renormalization group techniques. The qualitative behavior of the magnetic susceptibility of the system is shown to be governed by a non-trivial fixed point in the renormalization group flow. The hierarchical Kondo model therefore provides an example of a system that can be studied exactly using renormalization group methods without being asymptotically free.

The work presented here was carried out in collaboration with Giuseppe Benfatto and Giovanni Gallavotti.

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