Mathematics Department - Mathematical Physics Seminar - Spring 2014

Mathematical Physics Seminar - Spring 2014



Organizer(s)

Joel Lebowitz, Michael Kiessling

Archive

Website

http://www.math.rutgers.edu/~lebowitz/Fall2013seminars.html




Past Talks


Thursday, April 17th

Mikko Stenlund, University of Helsinki

"A limit theorem for a quasi-static dynamical system"

Time: 2:00 PM
Location: Hill 705
Abstract: The statistical properties of dynamical systems are traditionally studied in the context of their invariant measures. Motivated by non-equilibrium phenomena in nature, we wish to step out of the above setup. To this end, we introduce a model whose characteristics change slowly with time. Solving a martingale problem in the spirit of Stroock and Varadhan, we show that repeated observations of the state of the system yield a certain stochastic diffusion process.


Thursday, April 17th

Leonid Pastur , Institute for Low Temperatures, Kharkiv, Ukraine

"Qubits Dynamics in Random Matrix Environment"

Time: 12:00 PM
Location: Hill 705
Abstract: We consider two models of dynamics of the system of two qubits whose environment is modeled by random matrix Hamiltonian. In the first model each qubit interacts with its own environment, however the initial condition for qubits is entangled. In the second model only one of two qubits interacts with environment while another qubit is free and the same entangled initial condition is assumed. We find the time dependence of concurrence, negativity and quantum discord, three widely used entanglement characteristics, in the Bogolyubov - van Hove asymptotic regime of the dynamics and we comment on the non-Markov evolution, the entanglement sudden death for concurrence and negativity and the large time asymptotics of quantum discord.

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


Thursday, April 10th

Jeremy Quastel , IAS and University of Toronto

"The KPZ equation and its universality class: exact solutions"

Time: 2:00 PM
Location: Hill 705
Abstract: We will survey recent progress on exact formulas for fluctuations of the KPZ equation and models in the universality class.


Thursday, April 10th

Herbert Spohn , IAS and Technical University Munich

"The KPZ equation and its universality class: replica solutions"

Time: 12:00 PM
Location: Hill 705
Abstract: In 1986 Kardar Parisi and Zhang proposed a stochastic PDE for interface motion. On the one-dimensional case (two-dimensional bulk and one-dimensional interface) there has been much activity recently. In my talk I provide some physical background and discuss replica solutions which are based on the Lieb Liniger delta-Bose gas with attractive interactions.

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


Friday, March 28th

Special Mathematical Physics Seminar

Michael Loss , Georgia Tech (NOTE: NEW DAY AND TIME)

"The Kac model coupled to a thermal bath"

Time: 1:30 PM
Location: Hill 705
Abstract: I present a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature $beta$. The system admits the canonical distribution at inverse temperature $beta$ as the unique equilibrium state. I talk about a number of issues but chief among them is the rate at which the system tends to equilibrium, both, in the sense of the spactral gap and in the sense of relative entropy.

This is joint work with Federico Bonetto and Ranjini Vaidyanathan.


Thursday, March 13th

Thierry Bodineau, Ecole Normale Superieure

"Lyapunov functionals for boundary-driven nonlinear drift-diffusions"

Time: 2:00 PM
Location: Hill 705
Abstract: We will describe a large class of Lyapunov functionals for nonlinear drift-diffusion equations with non-homogeneous Dirichlet boundary conditions. These are generalizations of large deviation functionals for underlying stochastic many-particle systems, the zero range process and the Ginzburg-Landau dynamics. More generally, we will discuss the connection between Lyapunov functionals and large deviation functionals of particle systems.


Thursday, March 13th

Peter Pickl, LMU

"Dynamics of Sound Waves in an Interacting Bose Gas"

Time: 12:00 PM
Location: Hill 705
Abstract: We consider a non-relativistic quantum gas of $N$ bosonic atoms confined to a box of volume $Lambda$ in physical space. The atoms interact with each other through a pair potential whose strength is inversely proportional to the density, $rho=frac{N}{Lambda}$, of the gas. We study the time evolution of coherent excitations above the ground state of the gas in a regime of large volume $Lambda$ and small $frac{Lambda}{rho}$. The initial state of the gas is assumed to be close to a textit{product state} of one-particle wave functions that are approximately constant throughout the box. The initial one-particle wave function of an excitation is assumed to have a compact support independent of $Lambda$. We derive an effective non-linear equation for the time evolution of the one-particle wave function of an excitation and establish an explicit error bound tracking the accuracy of the effective non-linear dynamics in terms of the fraction $frac{Lambda}{rho}$. We conclude with a discussion of the dispersion law of low-energy excitations, recovering Bogolyubov's well-known formula for the speed of sound in the gas, and a dynamical instability for attractive two-body potentials.

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


Thursday, March 6th

Pierre Germain , Courant Institute

"An extension of the Derrida-Lebowitz-Speer-Spohn equation"

Time: 2:00 PM
Location: Hill 705
Abstract: The Derrida-Lebowitz-Speer-Spohn (DLSS) equation is a PDE which was introduced as the continuum limit for the invariant measure in a spin model related to random interfaces. It recently drew a lot of attention as a nonlinear diffusion model. By revisiting the original derivation, we were able to find a higher order correction to the DLSS equation, in the so-called biased case. The resulting PDE could exhibit a trend to equilibrium, which makes it relevant from a probabilistic point of view.

This is joint work with Charles Bordenave.


Thursday, March 6th

Jozsef Beck, Rutgers University

"Uniform Distribution and the Second Law"

Time: 12:00 PM
Location: Hill 705
Abstract: We study the typical time evolution of deterministic many-particle systems in motion in a closed environment, e.g., point billiards in a box or on a torus or on a sphere, also oscillating systems where the particles move on closed curves or on infinite curves wrapped up on a bounded surface (quasi-periodic motions). We focus on the following 4 questions: (1) what is (spatial) equilibrium; (2) how long does it take to reach (spatial) equilibrium; (3) how to define an intrinsic (spatial) entropy; (4) how to prove a Second Law.

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


Thursday, February 20th

Cedric Bernardin , Université de Nice Sophia-Antipolis

"Expansion of the Green-Kubo formula in the weak coupling limit"

Time: 2:00 PM
Location: Hill 705
Abstract: We consider an infinite system of cells coupled by a smooth nearest neighbor potential aV with a<<1 (weak coupling) . The uncoupled (a=0) system (cells) evolve according to Hamiltonian dynamics perturbed stochastically with an energy conserving noise. We study the expansion in a of the Green-Kubo (GK) formula for the heat conductivity of this system. We are in particular interested in the behavior of the expansion when the strength of the noise goes to 0.


Thursday, February 20th

Haim Brezis, Rutgers University

"New approximations of the total variation and filters in Image Processing"

Time: 12:00 PM
Location: Hill 705
Abstract: I will present new results concerning the approximation of the BV- norm by nonlocal, nonconvex, functionals. The mode of convergence introduces mysterious novelties and numerous problems remain open. The original motivation comes from Image Processing.

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


Thursday, February 13th - POSTPONED

Jeremy Quastel, IAS and University of Toronto

"The KPZ equation and its universality class: exact solutions"

Time: 2:00 PM
Location: Hill 705
Abstract: We will survey recent progress on exact formulas for fluctuations of the KPZ equation and models in the universality class.


Thursday, February 13th - POSTPONED

Herbert Spohn , IAS Princeton and Technical University Munich

"The KPZ equation and its universality class: replica solutions"

Time: 12:00 PM
Location: Hill 705
Abstract: In 1986 Kardar Parisi and Zhang proposed a stochastic PDE for interface motion. On the one-dimensional case (two-dimensional bulk and one-dimensional interface) there has been much activity recently. In my talk I provide some physical background and discuss replica solutions which are based on the Lieb Liniger delta-Bose gas with attractive interactions.

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


Thursday, February 6th

David A. Huse , Princeton University

"Eigenstate statistical mechanics: thermalization and localization, II "

Time: 2:00 PM
Location: Hill 705
Abstract: Progress in physics and quantum information science motivates much recent study of the behavior of strongly-interacting many-body quantum systems fully isolated from their environment, and thus undergoing unitary time evolution. What does it mean for such a system to go to thermal equilibrium? I will explain the Eigenstate Thermalization Hypothesis (ETH), which says that each individual exact eigenstate of the system's Hamiltonian is at thermal equilibrium, and which appears to be true for most (but not all) quantum many-body systems. Prominent among the systems that do not obey this hypothesis are quantum systems that are many-body Anderson localized and thus do not constitute a reservoir that can thermalize itself. When the ETH is true, one can do standard statistical mechanics using the `single-eigenstate ensembles', which are the limit of the microcanonical ensemble where the `energy window' contains only a single many-body quantum state. These eigenstate ensembles are more powerful than the traditional ensembles, in that they can also `see' the quantum phase transition in to the localized phase, as well as a rich new world of phases and phase transitions within the localized phase. Ref.: Huse, et al., Phys. Rev. B 88, 014206 (2013).


Thursday, February 6th

David A. Huse , Princeton University

"Eigenstate statistical mechanics: thermalization and localization, I"

Time: 12:00 PM
Location: Hill 705
Abstract: Progress in physics and quantum information science motivates much recent study of the behavior of strongly-interacting many-body quantum systems fully isolated from their environment, and thus undergoing unitary time evolution. What does it mean for such a system to go to thermal equilibrium? I will explain the Eigenstate Thermalization Hypothesis (ETH), which says that each individual exact eigenstate of the system's Hamiltonian is at thermal equilibrium, and which appears to be true for most (but not all) quantum many-body systems. Prominent among the systems that do not obey this hypothesis are quantum systems that are many-body Anderson localized and thus do not constitute a reservoir that can thermalize itself. When the ETH is true, one can do standard statistical mechanics using the `single-eigenstate ensembles', which are the limit of the microcanonical ensemble where the `energy window' contains only a single many-body quantum state. These eigenstate ensembles are more powerful than the traditional ensembles, in that they can also `see' the quantum phase transition in to the localized phase, as well as a rich new world of phases and phase transitions within the localized phase. Ref.: Huse, et al., Phys. Rev. B 88, 014206 (2013).

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


Thursday, January 30th

Abhishek Dhar , Indian Institute of Science

"Heat conduction in the asymmetric Fermi-Pasta-Ulam chain"

Time: 2:00 PM
Location: Hill 705
Abstract: Recent simulation results on heat conduction in one-dimensional systems with asymmetric inter-particle potentials suggest the possibility of normal heat transport in these systems at low temperatures. This is contrary to the general belief that heat conduction in one-dimensional momentum conserving systems is anomalous. I will discuss some of our recent simulation results on this problem. Both non-equilibrium and equilibrium Green-Kubo results will be presented. Some results on decay of equilibrium fluctuations in this system will also be discussed.


Thursday, January 30th

Herbert Spohn , IAS Princeton and Technical University Munich

"Equilibrium time correlations of anharmonic chains"

Time: 12:00 PM
Location: Hill 705
Abstract: We discuss Fermi-Pasta-Ulam type chains. Their equilibrium measures are of product form. But the dynamical correlations of the conserved fields have an interesting structure, which we will try to capture through a nonlinear version of fluctuating hydrodynamics.

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


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