Mathematics Department - Mathematical Physics Seminar - Spring 2015

Mathematical Physics Seminar - Spring 2015



Organizer(s)

Joel Lebowitz, Michael Kiessling

Archive

Website

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



Upcoming Talks


Thursday, April 23rd

Andrea Liu , University of Pennsylvania

"A tale of two rigidities"

Time: 12:00 PM
Location: Hill 705
Abstract: When we first learn the physics of solids, we are taught the theory of perfect crystals. Only later do we learn that in the real world, all solids are imperfect. The perfect crystal is invaluable because we can describe real solids by perturbing around this extreme limit by adding defects. But such an approach fails to describe a glass, another ubiquitous form of rigid matter. I will argue that the jammed solid is an extreme limit that is the anticrystal--an opposite pole to perfect order. Like the perfect crystal, it is an abstraction that can be understood in depth and used as a starting point for understanding the mechanical properties of solids with surprisingly high amounts of order. Unlike the crystal, it is also remarkably adaptable so that mechanical properties such as the Poisson ratio can be tuned over the whole of the allowed range.





Past Talks


Thursday, April 16th

John Barton , Massachusetts Institute of Technology

"Viewing HIV evolution through the lens of statistical physics"

Time: 2:00 PM
Location: Hill 705
Abstract: Human immunodeficiency virus (HIV) evolves rapidly within infected individuals, accumulating mutations that allow the virus to escape targeting by the host immune system and thereby preventing effective immune control of infection. In this talk I will show how we can use techniques from statistical physics to infer information about the fitness landscapes of HIV proteins from sequence data, providing insight into HIV evolution and mutational escape from immune control. I will discuss experimental tests of these proposed fitness landscapes and applications of this technique in understanding viral evolution within infected individuals. Additionally, I will show some analogies between sets of collectively coupled mutations in HIV and Hopfield neural networks.


Thursday, April 16th

Chris Jarzynski , University of Maryland

"Quantal and classical shortcuts to adiabaticity"

Time: 12:00 PM
Location: Hill 705
Abstract: Adiabatic invariants play an important role in classical mechanics, quantum mechanics and thermodynamics. Within the field of quantum control, the term "shortcuts to adiabaticity" refers to strategies for gaining the benefits of quantum adiabatic evolution without paying the price of slow driving. Specifically, the goal is to design protocols that cause a system to evolve from the n'th eigenstate of an initial Hamiltonian to the n'th eigenstate of a final Hamiltonian, over a finite -- perhaps short -- interval of time. In other words, we wish to preserve the adiabatic invariant during a rapid (non-adiabatic) process. This topic has emerged as an active field of study in the past few years. I will discuss a number of quantal and classical shortcuts that have recently been discovered and I will illustrate them using model systems. The derivations of these shortcuts are often surprisingly simple, involving basic analyses of the Schrodinger equation or Hamiltonian dynamics.

BROWN BAG LUNCH FROM 1-2PM


Thursday, April 9th

Roman Kotecky , The University of Warwick

"Long range order for random colourings on planar graphs "

Time: 2:00 PM
Location: Hill 705
Abstract: We establish a phase transition for the uniform random proper 3-colourings on a class of planar quasi-transitive graphs. This is a case of purely entropicaly driven long range order and the proof is based on an enhanced Peierls argument (which is of independent interest even for the Ising model for which it extends the range of temperatures with proven long range order) combined with an additional percolation argument.

Based on a joint work with Alan Sokal and Jan Swart.


Thursday, April 9th

Gregory Falkovich, Weizmann Institute of Science

"Flight-crash events in turbulence"

Time: 12:00 PM
Location: Hill 705
Abstract: The statistical properties of turbulence differ in an essential way from those of systems in or near thermal equilibrium because of the flux of energy between vastly different scales at which energy is supplied and at which it is dissipated. We elucidate this difference by studying experimentally and numerically the fluctuations of the energy of a small fluid particle moving in a turbulent fluid. We demonstrate how the fundamental property of detailed balance is broken, so that the probabilities of forward and backward transitions are not equal for turbulence. In physical terms, we found that in a large set of flow configurations, fluid elements decelerate faster than accelerate, a feature known all too well from driving in dense traffic. The statistical signature of rare “flight–crash” events, associated with fast particle deceleration, provides a way to quantify irreversibility in a turbulent flow. Namely, we find that the third moment of the power fluctuations along a trajectory, nondimensionalized by the energy flux, displays a remarkable power law as a function of the Reynolds number, both in two and in three spatial dimensions. This establishes a relation between the irreversibility of the system and the range of active scales. We speculate that the breakdown of the detailed balance characterized here is a general feature of other systems very far from equilibrium, displaying a wide range of spatial scales.

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


Wednesday, April 8th

Special Mathematical Physics Seminar

Amit Einav, Cambridge University

"On Entropic Equivalence of Ensembles on Kac's Sphere"

Time: 3:30 PM
Location: Hill 705
Abstract: It is an interesting well known fact that the relative entropy with respect to the Gaussian measure on N dimensional space satisfies a simple subadditivity property. Surprisingly enough, when one tries to achieve a similar result on the sphere in N dimensional space, a factor of 2 appears in the right hand side of the inequality (a result due to Carlen, Lieb and Loss), and the constant is sharp. Besides a deviation from the simple equivalence of ensembles principle in equilibrium Statistical Mechanics, this entropic inequality on the sphere has interesting ramifications in other fields - for instance in the study of the so-called Cercignani's Conjecture. In this talk we will present conditions on the density function under which we can get an almost subaditivity property; i.e. the factor 2 can be replaced with a factor of 1+epsilon_N, with epsilon_N given explicitly and going to zero. We will discuss what this may mean for the investigation of Cercignani's conjecture and, time permitting, we will give an example to many families of functions that satisfy these conditions.


Thursday, April 2nd

Almut Burchard , University of Toronto

"Geometric Stability results for the Coulomb Energy"

Time: 2:00 PM
Location: Hill 705


Thursday, April 2nd

Subhro Ghosh, Princeton University

"Large deviations and random polynomials"

Time: 12:00 PM
Location: Hill 705
Abstract: We obtain a large deviations principle (in the space of probability measures on $C$) for the empirical measure of zeroes of random polynomials with i.i.d. exponential coefficients. One of the key challenges here is the fact that the coefficients are a.s. all positive, which enforces a growing number of non-linear constraints on the locations of the zeroes. En route, we will discuss a recent characterization theorem of Bergweiler and Eremenko, and its application in the proof of our main theorem.

Based on joint work with Ofer Zeitouni.

BROWN BAG LUNCH FROM 1-2PM


Thursday, March 26th

Gael Raoul , Ecole Polytechnique

"Dynamics of a species structured by a space variable and a phenotypic trait"

Time: 2:00 PM
Location: Hill 705
Abstract: In many current ecological problems, both the spacial structure and the genetic diversity of species have to be taken into account. Combining those two aspects leads to challenging mathematical questions. We will present two such problems:

- Evolutionary epidemiology. Microbial population have typically a high mutation rate, and a large population size. As a consequence, evolutionary effects can affect the spacial dynamics of the population.

- Effect of climate change. We consider a population of trees that reproduces sexually, and we wonder whether the high dispersion of pollen will help the population to survive global warming.


Thursday, March 26th

Natan Andrei , Rutgers University

"Quench evolution of quantum integrable many body systems"

Time: 12:00 PM
Location: Hill 705
Abstract: Recently, with the appearance of experimental systems ranging from nano-devices to optically trapped cold atom gases, much progress was in our understanding of many fundamental aspects of nonequilibrium processes in isolated quantum systems. Quench evolutions, where a Hamiltonian is suddenly applied to a system and its evolution is followed in time, provide a means of studying the dynamics of these systems and to reveal their intrinsic relaxation mechanisms and time scales, whether the system thermalizes, the dynamic spectrum of states driving the evolution, the energy exchange among the various modes and the phase transitions in time it may cross as it evolves, among many other. In this talk I will describe the quench dynamics of isolated interacting systems in 1-d, governed by integrable Hamiltonians. I shall study the time evolution of a gas of interacting bosons moving on the continuous infinite line and interacting via a short range potential - the Lieb-Liniger model. For a system with a finite number of bosons we find that independently of the initial state the system asymptotes towards a strongly repulsive gas for any value of repulsive coupling, while for any value of attractive coupling the system is dominated by the maximal bound state. I shall then discuss the boson system in the thermodynamic limit and consider several circumstances: quench from a Mott insulator, quench in a box, quench from a domain wall. I will discuss the appearance of a GGE (generalized Gibbs ensemble) in the long time limit and why it usually fails. If time permits I shall discuss also the quench dynamics of the XXZ Heisenberg chain in different parameter regimes.

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


Thursday, March 5th

Senya Shlosman , University of Marseille, Luminy

"How the Ising Crystal Grows, and Where to Look for Airy Diffusion"

Time: 12:00 PM
Location: Hill 705
Abstract: I will describe the process of the crystal growth, as the crystal acquires mode and more atoms. I will also explain where one should look in order to see the (fashionable nowadays) N^{1/3} exponent, where N is the linear size of the crystal.


Thursday, February 26th

David Huse , Princeton University

"Many-body localization"

Time: 2:00 PM
Location: Hill 705
Abstract: I will discuss some aspects of our present understanding of the physics of the many-body localized phase, and of the quantum phase transition between many-body localization and quantum thermalization. The many-body localized phase can, in certain cases, be understood as a new type of integrable system, where the emergent conserved quantities are localized operators (H., Nandkishore, Oganesyan, PRB 2014). The low frequency dynamics in this phase differs in important ways from that of noninteracting Anderson localization due to rare many-body "resonances" (Gopalakrishnan, H., et al, in progress).

The delocalization phase transition is due to the proliferation of these resonances (Vosk, Altman, H., arXiv:1412.3117).


Thursday, February 26th

Michael Kiessling , Rutgers University

"A novel quantum-mechanical interpretation of Dirac's equation"

Time: 12:00 PM
Location: Hill 705
Abstract: A novel interpretation is given of Dirac's ``wave equation for the relativistic electron'' as a quantum-mechanical one-particle equation in which electron and positron are merely the two different ``topological spin'' states of a single more fundamental particle, not distinct particles in their own right.

This is joint work with A.Shadi Tahvildar-Zadeh

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


Thursday, February 19th

Dan Pirjol , National Institute for Physics and Nuclear Engineering, Romania

"Phase transition in a stochastic growth process with multiplicative noise"

Time: 2:00 PM
Location: Hill 705
Abstract: The talk will discuss the random linear recursion x(i+1) = a(i)*x(i)+b(i) where a(i) > 1 are stochastic multipliers related to the exponential of a standard Brownian motion, and b(i) are positive uncorrelated noise. This is a growth process, which is motivated by problems in mathematical finance related to interest rate modeling and numerical simulation of stochastic volatility models. Under certain conditions x(i) develops heavy tailed distributions, which are manifested as numerical explosions of the positive integer moments <(x(i))^q>, q=1,2,.... This phenomenon can be studied by mapping the problem to a one-dimensional lattice gas with linear attractive potentials, which can be solved exactly. The moment explosions can be related to a phase transition in the equivalent lattice gas.


Thursday, February 19th

Luca Peliti , Istituto Nazionale di Fisica Nucleare

"Beneficial mutations in a range-expansion wave"

Time: 12:00 PM
Location: Hill 705
Abstract: Many theoretical and experimental studies suggest that range expansions can have severe consequences for the gene pool of the expanding population. Due to strongly enhanced genetic drift at the advancing frontier, neutral and weakly deleterious mutations can reach large frequencies in the newly colonized regions, as if they were surfing the front of the range expansion. These findings raise the question of how frequently beneficial mutations successfully surf at shifting range margins, thereby promoting adaptation towards a range-expansion phenotype. We studied this problem by means of individual-based simulations, as a function of two strongly antagonistic factors, the probability of surfing given the spatial location of a novel mutation and the rate of occurrence of mutations at that location. We find that small amounts of genetic drift increase the fixation rate of beneficial mutations at the advancing front, and thus could be important for adaptation during species invasions.

Joint work with R. Lehe (Paris) and O. Hallatschek (now at Berkeley).

BROWN BAG LUNCH FROM 1-2PM!


Thursday, February 12th

Peter Nandori , New York University

"Local thermal equilibrium for certain stochastic models of heat transport "

Time: 2:00 PM
Location: Hill 705
Abstract: This talk is about nonequilibrium steady states (NESS) of a class of stochastic models in which particles exchange energy with their 'local environments' rather than directly with one another. The physical domain of the system can be a bounded region of R^d for any dimension d. We assume that the temperature at the boundary of the domain is prescribed and is nonconstant, so that the system is forced out of equilibrium. Our main result is local thermal equilibrium in the infinite volume limit. We also prove that the mean energy profile of NESS satisfies Laplace's equation for the prescribed boundary condition. Our method of proof is duality: by reversing the sample paths of particle movements, we convert the problem of studying local marginal energy distributions at x to that of joint hitting distributions of certain random walks starting from x.

This is a joint work with Yao Li and Lai-Sang Young.


Thursday, February 12th

Roger Nussbaum, Rutgers University

"Perron-Frobenius Operators, Positive C^m Eigenvectors and the Computation of Hausdorff Dimension"

Time: 12:00 PM
Location: Hill 705
Abstract: We shall discuss a class of linear Perron-Frobenius operators L which, under added assumptions, arise in the computation of Hausdorff dimension for invariant sets of iterated function systems or graph directed iterated function systems. We shall describe theorems which insure the existence of a strictly positive, C^m eigenvector of L. In important cases it is possible to obtain explicit bounds on second order (and higher) partial derivatives of v. We shall indicate how (joint work with Richard Falk) information about partial derivatives of v can be used to obtain rigorous estimates of Hausdorff dimension (at least three to four decimal point accuracy) for some previously intractable examples like the set of complex continued fractions.

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


Thursday, February 5th

Alessandro Giuliani , University of Roma Tre

"Height fluctuations in interacing dimers"

Time: 2:00 PM
Location: Hill 705
Abstract: Perfect matchings of Z^2 (also known as non-interacting dimers on the square lattice) are an exactly solvable 2D statistical mechanics model. It is known that the associated height function behaves at large distances like a massless gaussian field, with the variance of height gradients growing logarithmically with the distance. As soon as dimers mutually interact, via e.g. a local energy function favoring the alignment among neighboring dimers, the model is not solvable anymore and the dimer-dimer correlation functions decay polynomially at infinity with a non-universal (interaction-dependent) critical exponent. We prove that, nevertheless, the height fluctuations remain gaussian even in the presence of interactions, in the sense that all their moments converge to the gaussian ones at large distances. The proof is based on a combination of multiscale methods with the path-independence properties of the height function.

Joint work with V. Mastropietro and F. Toninelli.


Thursday, February 5th

Haim Brezis, Rutgers University and Technion

"From the characterization of constant functions to isoperimetric inequalities"

Time: 12:00 PM
Location: Hill 705
Abstract: I will present a "common roof" to various, seemingly unrelated, known statements asserting that integer-valued functions satisfying some kind of mild regularity are constant. For this purpose I will introduce a new function space B which is so large that it contains many classical spaces, such as BV (=functions of bounded variation) , BMO (=John-Nirenberg space of functions of bounded mean oscillation) and some fractional Sobolev spaces. I will then define a fundamental closed subspace B_0 of B contaning in particular W^{1,1}, VMO--- and thus continuous functions---- H^{1/2} etc. A remarkable fact is that integer-valued functions belonging to B_0 are necessarily constant. I will also discuss connections of the B-norm to geometric concepts, such as the perimeter of sets.

This is joint work with L. Ambrosio, J. Bourgain, A. Figalli and P. Mironescu.

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


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