Mathematics Department - Mathematical Physics Seminar - Fall 2016

# Mathematical Physics Seminar - Fall 2016

### Organizer(s)

Joel Lebowitz, Michael Kiessling

### Website

http://www.sas.rutgers.edu/cms/math/news-events-cmsr/mathematical-physics-seminar/range.listevents/-

## Thursday, October 27th

### "DETAILED BALANCE IN NONEQUILIBRIUM STAT MECH"

Time: 12:00 PM
Location: Hill 705
Abstract: Detailed balance relates the transition probabilities J -> K and K -> J between two states J,K of a system M in equilibrium with a bath at a certain temperature. This relation is based on time-reversal invariance of physical laws. It can give very specific answers to certain questions. We discuss detailed balance in the case of an “active” bath (containing chemical with which our system M can react). This situation has been considered by J. England, and remains intriguing and interesting.

## Thursday, October 20th

### "Non-equilibrium Dynamics of Quantum Integrable Systems"

Time: 2:00 PM
Location: Hill 705
Abstract: The study of non-equilibrium dynamics of interacting many body systems is currently one of the main challenges of modern condensed matter physics, driven by the spectacular progress in the ability to create experimental systems - trapped cold atomic gases are a prime example - that can be isolated from their environment and be highly controlled. Many of the system so studied are integrable. In this talk I will describe nonequilibrium quench and Floquet dynamics in some integrable quantum systems. I'll discuss the time evolution of the Lieb-Liniger system, a gas of interacting bosons moving on the continuous infinite line and interacting via a short range potential. Considering a finite number of bosons on the line we find that for any value of repulsive coupling the system asymptotes towards a strongly repulsive gas for any initial state, while for an attractive coupling, the system forms a maximal bound state that dominates at longer times. In the thermodynamic limit -with the number of bosons and the system size sent to infinity at a constant density and the long time limit taken subsequently- I'll show that the density and density-density correlation functions for strong but finite positive coupling are described by GGE for translationally invariant initial states with short range correlations. As examples I’ll discuss quenches from a Mott insulator initial state or a Newton’s Cradle. Then I will show that if the initial state is strongly non translational invariant, e.g. a domain wall configuration, the system does not equilibrate but evolves into a nonequilibrium steady state (NESS). A related NESS arises when the quench consists of coupling a quantum dot to two leads held at different chemical potential, leading in the long time limit to a steady state current. I will also present some results on Floquet dynamics for interacting bosons. Time permitting I will discuss the quench dynamics of the XXZ Heisenberg chain.

## Thursday, October 20th

### "Steady States in the non local Luttinger Model"

Time: 12:00 PM
Location: Hill 705
Abstract: We study transport in a one-dimensional system of interacting fermions described by the Luttinger model with finite range interactions, with a domain wall initial state with different densities or temperatures on its left and right sides. Asymptotically in time the system approaches a translation invariant steady state carrying a non vanishing current showing universality properties. The non-locality of interaction, acting as an ultraviolet cut-off, breaks Lorentz and scale invariance and leads to dispersive effects manifested by the shape of the fronts changing with time.

Based on joint work with E.Langmann, J.Lebowitz, P. Moosavi

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

## Thursday, October 13th

### "Universality of Transport Coefficients in the Haldane-Hubbard Model"

Time: 2:00 PM
Location: Hill 705
Abstract: In this talk I will review some selected aspects of the theory of interacting electrons on the honeycomb lattice, with special emphasis on the Haldane-Hubbard model: this is a model for interacting electrons on the hexagonal lattice, in the presence of nearest and next-to-nearest neighbor hopping, as well as of a transverse dipolar magnetic field. I will discuss the key properties of its phase diagram, most notably the phase transition from a standard insulating phase to a Chern insulator, across a critical line, where the system exhibits semi-metallic behavior. I will also review the universality of its transport coefficients, including the quantization of the transverse conductivity within the gapped phases, and that of the longitudinal conductivity on the critical line. The methods of proof combine constructive Renormalization Group methods with the use of Ward Identities and the Schwinger-Dyson equation.

Based on joint works with Vieri Mastropietro, Marcello Porta, Ian Jauslin.

## Thursday, October 13th

### "Ergodicity: an early paper by Boltzmann and its relevance"

Time: 12:00 PM
Location: Hill 705
Abstract: A little known 1868 paper by Boltzmann illustrates the ergodic hypotesis in the sense of Boltzmann and Maxwell, after having a little earlier derived the microcanonical distribution. The property should be illustrated by a simple dynamical system: his results ends up connecting to modern KAM theory and to the ancient theory of conics. They seem to indicate the need for a revisitation of the properties of the Hamiltonian system considered.

BROWN BAG LUNCH FROM 1-2PM

## Thursday, October 6th

### "Relativistic zero-range interactions for multi-time wave functions"

Time: 2:00 PM
Location: Hill 705
Abstract: Multi-time wave functions, a concept introduced by Dirac in 1932, are quantum-mechanical wave functions with N space-time arguments for N particles. In this way, they naturally extend the non-relativistic (single-time) Schrödinger picture to the relativistic domain. However, because of the many time coordinates, the nature of time evolution changes. Setting up an interacting multi-time theory has proven a major challenge; a recent no-go theorem by Petrat and Tumulka e.g. excludes interaction by potentials. In this talk, I will present a multi-time model which achieves interaction in a different way, namley by zero-range (or delta) interactions. After briefly introducing the general multi-time formalism, I will give an overview of the main results and an idea of the novel techniques required to implement zero-range interactions in this setting.

## Thursday, October 6th

### "The truncated moment problem"

Time: 12:00 PM
Location: Hill 705
Abstract: Let K be a subset of the real numbers. The (one-dimensional) truncated moment problem on K is to find, for given numbers m_1,...,m_n, a random variable X which takes values in K and whose moments are given by the m_k: E[X^k]=m_k. More accurately, one wants to find necessary and sufficient conditions, in term of the m_k, for the existence of such a random variable. The multi-dimensional version of this problem, in which K is a subset of a Euclidean space of higher dimension, is surprisingly hard and is far from being resolved; we give a short introduction to the problem and to the state of the art.

Finally, we describe a recent result concerning the truncated moment problem for a discrete set in one dimension; this is work in collaboration with M. Infusino, J. Lebowitz, and E. Speer.

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

## Thursday, September 29th

### "Almost surely recurrent motions in the Euclidean space"

Time: 2:00 PM
Location: Hill 705
Abstract: We will show that measure-preserving transformations of$R^n$ are recurrent if they satisfy a certain growth condition depending on the dimension $n$. Moreover, it is also shown that this condition is sharp. Examples will include non-autonomous Hamiltonian systems $dot{z}=Jnabla_z H(t, z)$ of one degree of freedom and $T$-periodic in $t$, for which our result will imply the existence of a periodic solution, provided that $nabla_z H(t, z) ={cal O}(|z|^{-alpha})$ as $|z|toinfty$ for some $alpha>1$ uniformly in $t$.

This is joint work with Rafael Ortega (Granada).

## Thursday, September 29th

### "Macroscopic temperature profiles in non-equilibrium stationary states"

Time: 12:00 PM
Location: Hill 705
Abstract: Systems that have more than one conserved quantity (i.e. energy plus momentum, density etc.), can exhibit quite interesting temperature profiles. I will present some numerical experiment and mathematical result.

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

## Thursday, September 22nd

### "Emergence of a nematic phase in a system of hard plates in three dimensions with discrete orientations"

Time: 2:00 PM
Location: Hill 705
Abstract: We consider a system of hard parallelepipedes, which we call plates, of size 1 by k^a by k in which a is larger than 5/6 and no larger than 1. Each plate is in one of six orthogonal allowed orientations. We prove that, when the density of plates is sufficiently larger than k^(2-5a) and sufficiently smaller than k^(3-a), the rotational symmetry of the system is broken, but its translational invariance is not. In other words, the system is in a nematic phase. The argument is based on a two-scale cluster expansion, and uses ideas from the Pirogov-Sinai construction.

## Thursday, September 22nd

### "A Gaussian Gibbs variational principle and geometric inequalities"

Time: 12:00 PM
Location: Hill 705
Abstract: The Gibbs variational principle has been a cornerstone of statistical mechanics since at least J. W. Gibbs' seminal 1902 treatise. It has also proved to be remarkably useful in other areas of mathematics, such as in the study of geometric inequalities of Brunn-Minkowski and Brascamp-Lieb type. This fundamental connection, pioneered by C. Borell, is however not sufficiently powerful to obtain the sharp isoperimetric and Brunn-Minkowski inequalities for Gaussian measures. In this talk, I will describe an unexpected Gaussian refinement of the Gibbs variational principle that makes it possible to recover these sharp inequalities. I will aim to explain how this gives rise to new Gaussian inequalities---in particular, a Gaussian improvement of Barthe's reverse Brascamp-Lieb inequality---and why the apparent duality between the Prekopa-Leindler and Holder inequalities is manifestly absent in the Gaussian setting.

BROWN BAG LUNCH.....1-2:00pm

## Thursday, September 15th

### "Relative Entropy Principles with Complex Measures"

Time: 2:00 PM
Location: Hill 705
Abstract: The notion of relative entropy for probabilitiy measures relative to a given a-priori probability measure is generalized to signed and complex measures relative to a given a-priori signed measure. This generalization is motivated by some problems at the intersection of statistical mechanics and differential geometry. Computer algebra-produced evidence for the utility of this new notion is presented using the example of complex random polynomials.

## Thursday, September 15th

### "Eigenvalues of sums and products of matrices"

Time: 12:00 PM
Location: Hill 705
Abstract: Given two Hermitian real nxn matrices, given their eigenvalues what can one say about the eigenvalues of the sum? Similar, given two unitary nxn matrices what can one say about the eigenvalues of the product?

I will survey some results on this, joint with Agnihotri and Knutson-Tao, and talk about some more recent results by others on generalizing these results to other kinds of matrices.

(Joint with Agnihotri and separately Knutson and Tao)

BROWN BAG LUNCH FROM 1-2PM

## Thursday, September 8th

### "A `liquid-solid' phase transition in a simple model for swarming"

Time: 12:00 PM
Location: Hill 705
Abstract: We consider a non-local shape optimization problem, which is motivated by a simple model for swarming and other self-assembly/aggregation models, and prove the existence of different phases. A technical key ingredient, which we establish, is that a strictly subharmonic function cannot be constant on a set of positive measure.

(With Rupert Frank)

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