# Forum Archive to Summer 2005

## Fall Term, 2001

The seminar is held at Room MI2 (Mathematics Institute) at 3 pm on specified Wednesdays.
1. 31.10.2001.

### Jean-Sebastien Caux (Oxford).

"The Josephson Current in Luttinger Liquid-Superconductor Junctions".
2. 21.11.2001.

### Shura Nersesyan (ICTP,Trieste).

"Applications of Majorana Fermions to quantum spin chains and ladders".
3. 05.12.2001

### John Wheater (Oxford).

" Young-Mills Matrix Integrals".

## Spring Term, 2002

The seminar is held at Room 100 (Mathematics Institute) at 3 pm on specified Wednesdays.
1. ## 06.02.2002.

### Robin C. Ball (Warwick).

"DIFFUSION CONTROLLED GROWTH."

### Abstract.

1. The simple scaling phenomenology of Diffusion Limited Aggregation and its Harmonic measure wil be introduced.

2. Recent numerical developments will be presented showing the evidence that hitherto notable departures from scaling are just transients.

3. The choice of spatial cut-off proceedure will be argued to be quite crucial; only within the wider class of Dielectric Breakdown Models is it possible to propose equivalences.

4. Assuming the proposed equivalences, the reduction of Diffusion Controlled Growth models in 2+1 dimensions to non-linear equations in 1 spatial + 1 time dimension will be discussed. Numerical evidence suggests these are well behaved and do not req uire explicit stochastic terms. Theoretical approaches based on mode- mode coupling theory will be discussed.

Recent collaborators in this work include: E Somfai, NE Bowler and LM Sander.

RCB 16.1.02.

2. ## 20.02.2002.

### John Cardy (Oxford).

"SCALING FUNCTIONS FOR VESICLES: WILSON LOOPS, SUPERSYMMETRY AND AN EXACT SOLUTION".

### Abstract.

The thermodynamic properties of a system near a second-order phase transition are characterized by simple power laws. For example, near the Curie point of a ferromagnet, we get different exponents if we vary the temperature in zero external magnetic field, or switch on a magnetic field at the critical temperature. These critical exponents are very well understood. However, if we vary both parameters simultaneously, the thermodynamics depends on a particular combination of them through a scaling function. These functions are believed to be just as universal as exponents, yet very little precise information is known about them. In the first half of the seminar I introduce a recently found exactly solvable example in a simple model for the collapse of biological vesicles under pressure, and show that its mathematical properties are far from simple. The analysis uses concepts of theoretical physics as diverse as lattice gauge theories and supersymmetry. In the second half I can discuss some of the mathematical details, especially of the supersymmetry.
3. ## 06.03.2002.

### Laszlo Erdos (Georgia Tech).

"ZERO MODES OF THE 3D PAULI OPERATOR."

### Abstract.

The Dirac operator, ${\cal D} = \sigma\cdot (-i\nabla + A)$, describes an electron in a classical magnetic field $B= \mbox{curl} A$. The zero energy localized eigenstates are called zero modes. In $d=2$ dimensions the Index Theorem guarantees an abundance of zero modes, but their existence is somewhat unexpected in $d=3$ dimensions. A particular example was miraculously found by Loss and Yau 15 years ago and it has been used to prove a theoretical upper bound on the fine structure constant. The number of zero modes remained poorly understood since then; neither good lower nor upper bounds are known. In the first part of the talk I present a systematic way to construct several new families of zero modes. In the second part I show a robust upper bound on their densities and explain its connection with semiclassical formulas for magnetic eigenvalues.

## Fall Term, 2002

The seminar is held at Room 100 (Mathematics Institute) at 3 pm on specified Wednesdays.
1. ## 02.10.2002. (Cancelled!)

### Misha Chertkov (Los-Alamos National Laboratory, Theoretical Division).

"Acceleration of chemical reaction by chaotic mixing."

### Abstract.

Comprehensive theory of binary chemical reaction, ${\cal A +{\cal B \to\emptyset$, in a fluid at large Schmidt number $\mbox{Sc$ and large Damk\"ohler number $\mbox{Da$ is developed. We consider a case of chaotic flow in a finite volume (chemical reactor). The major question addressed is: what is the law of temporal decay of the overall amount of chemicals, $N_{a,b (t)$? Four subsequent stages of the decay are identified: (i) $N_{a,b$ remain practically the same during the major part of the stage, which is characterized by formation of a stripe-like distribution of chemicals. Exponentially fast (with a decrement of the order of the Lyapunov exponent $\lambda$) decay of chemical concentration starts closer to the end of the $\ln(\mbox{Sc )/\lambda$-long stage in the bulk part of the flow, which becomes almost empty of chemicals. (ii) The empty region starts to propagate towards the boundary. Chemicals are remained mainly in a $L/\sqrt{\lambda t$-wide vicinity of the boundary, where $L$ is the system size. Total amount of chemicals decreases according to, $N_{a,b \propto 1/\sqrt{t$. (iii) Chemicals are mainly left in a narrow, diffusion controlled, boundary layer. The decay law is exponential, $N_{a,b \propto \exp(-\gamma t)$, where $\gamma\sim \lambda/\sqrt{Sc$. (iv) Neither advection no diffusion are essential during the final spatially uniform stage. This is a joint work with V. Lebedev (Landau Inst., Moscow).
2. ## 16.10.2002.

### Abstract.

Directed percolation describes the critical behaviour of a large number of reaction-diffusion systems. In this talk, the super-critical properties of directed percolation will be discussed. A mapping of d-dimensional directed percolation to (d-2)-dimensional Kardar-Parisi-Zhang equation for interface growth will be established.
3. ## 30.10.2002.

### Ian Kogan (Th. Phys., Oxford) (DELAYED UNTILL NEXT TERM!) .

"MATTER-GHOST MIXING, LOGARITHMIC CFT AND C=0 STRINGS."

### Abstract.

In this talk an attempt will be given to discuss several topics. The first one is the so-called Logarithmic conformal field theory (LCFT) which is a special class of Conformal field theories (CFT) with logarithmic singularities in correlation functions. These theories have operators with degenerate scaling dimensions and Virasoro (as well as other chiral operators) have the form of Jordan cell. Special emphasis will be given to the theories with central charge c=0 which have a very unusual structure of extended Virasoro algebra. There are several important areas of physics in which c=0 LCFT play important role: one is the theory of planar quenched disorder another is critical string theory. The second topic will be the ghost-matter mixing in string theory in a presence of a new type of BRST nontrivial brane-like vertex operators which naturally leads to c=0 LCFT on world sheet. The final goal will be to show that String/M theory in dynamical background is described by the world-sheet c=0 LCFT with matter-ghost mixing.

5. ## 27.11.2002.

### Abstract.

Consider a non-equilibrium state of non-interacting Fermi gas obtained by application of an arbitrary time-dependent potential to the Fermi gas in the ground state. We develop a general method that yields closed analytical expressions for quantum statistics of any single-particle operator (total energy, total momentum, etc...) through a solution of an auxiliary matrix Riemann-Hilbert problem. As an application the statistics of the charge transfered during a finite time interval through a parametric quantum pump is discussed
6. ## 03.12.2002.

### Abstract.

A wide range of classical particle models can be described, surprisingly, by quantum field theories. The mapping takes the classical Fokker-Planck equation to a quantum mechanical description via a second-quantized representation, and finally uses the coherent state representation to take the field theory limit. This field theory is well-suited to the study of reaction-diffusion processes, which, in analogy with equilibrium critical phenomena, have an upper critical dimension d_c, above which mean-field theory applies. In spatial dimensions d below the upper critical dimension, renormalization group methods applied to the field theory demonstrate explicitly universal quantities, and provide a perturbative (but non-analytic) calculation of these quantities in (fractional) powers of d_c - d.

## Spring and Summer Terms, 2003

The seminar is held at Room 100 (Mathematics Institute) at 3 pm on specified Wednesdays.

2. ## 05.02.2003.

### Abstract.

In this talk an attempt will be given to discuss several topics. The first one is the so-called Logarithmic conformal field theory (LCFT) which is a special class of Conformal field theories (CFT) with logarithmic singularities in correlation functions. These theories have operators with degenerate scaling dimensions and Virasoro (as well as other chiral operators) have the form of Jordan cell. Special emphasis will be given to the theories with central charge c=0 which have a very unusual structure of extended Virasoro algebra. There are several important areas of physics in which c=0 LCFT play important role: one is the theory of planar quenched disorder another is critical string theory. The second topic will be the ghost-matter mixing in string theory in a presence of a new type of BRST nontrivial brane-like vertex operators which naturally leads to c=0 LCFT on world sheet. The final goal will be to show that String/M theory in dynamical background is described by the world-sheet c=0 LCFT with matter-ghost mixing.

4. ## 07.05.2003.

### Abstract.

In this talk, I will discuss the physical principles lying behind the acquisition of accurate positional information in bacteria. A good application of these ideas is to the rod-shaped bacterium E. coli which divides precisely at its cellular midplane. This positioning is controlled by the Min system of proteins: MinC, MinD, and MinE. These proteins coherently oscillate from end to end of the bacterium. I will present a reaction-diffusion model that describes the diffusion of the Min proteins, and their binding/unbinding from the cell membrane. The system possesses a Turing-like instability that spontaneously generates the Min oscillations, which then control the accurate placement of the midcell division site. I will then discuss the role of fluctuations in protein dynamics, and examine the extent to which fluctuations can set optimal protein concentration levels. Possible theoretical methods of handling fluctuation effects in systems with few particles will also be examined.

M. Howard, A. Rutenberg, S. de Vet: Phys. Rev. Lett. 87 278102 (2001)

M. Howard, A. Rutenberg: Phys. Rev. Lett. 90 128102 (2003)

5. ## 28.05.2003.

### Abstract.

We shall generalize the Harish-Chandra-Itzykson-Zuber and certain other integrals {Gross-Witten integral, integrals over complex matrices, integrals over rectangle matrices) using a notion of a tau function of matrix argument. The answers are presented in the form of determinants. In this case one can reduce multi-matrix integrals to integrals over eigenvalues, which in turn are certain tau functions having determinant representation. We also consider a generalization of the Kontsevich integral.

## Autumn Term, 2003

The seminar is held at Room 123 (Mathematics Institute) at 3 pm on specified Tuesdays.
1. ## 07.10.2003.

### Abstract.

The talk is dedicated to the analysis of strong fluctuation effects in the problem of cluster-cluster aggregation in the arbitrary number of dimensions. We will start by reformulating the problem in terms of recently derived stochastic Smoluchowski equation (SSE), which correctly accounts for fluctuations of local mass distribution in the original particle system. This will allow us to review the celebrated Doi-Zel'dovich-Ovchinnikov formalism of the theory of reaction-diffusion systems in rigorous terms.

We will find that the average mass distribution is correctly described by Kolmogorov's theory of turbulence. This result will be obtained by applying Zakharov transformation to the properly renormalized SSE.

We will also find that high order moments of the average mass distribution reflect the intermittent nature of stochastic aggregation and fail to be correctly predicted by Kolmogorov theory. We use the formalism of perturbative renormalization group to compute the the (non-linear) correction to Kolmogorov scaling.

Finally, we will discuss the implications of our work for the understanding of the directed abelian sandpile model.

The reported results have been obtained in collaboration with Colm Connaughton (Laboratoire de Physique Statistique de l'ENS, Paris), R. Rajesh (Brendais University, Boston) and Roger Tribe (Warwick).

2. ## 21.10.2003.

### Abstract.

There are now pretty accurate numerical estimates of the fractional quantum Hall effect gaps. However, the gaps extracted from analysing the temperature dependence of the dissipative transport at the center of a quantum Hall plateau are always smaller, sometimes significantly so, than predicted values. The finger seems to point at the effects of disorder and the question is what are the experiments actually measuring. I will present two possible scenarios, one old (variable range hopping) and one new, both of which certainly apply if the conditions are right. I will then discuss in which regime the experiments are working.

4. ## 18.11.2003.

### Abstract.

For each s in S let X_s be a finite set and q^s the transition rate matrix for a continuous-time communicating Markov process on X_s, so it has a unique stationary measure p^s and it attracts exponentially. Now introduce coupling eps between the Markov processes, to make one big Markov process on the product of the X_s, e.g. make q^s_{ij depend weakly on the states of the neighbouring units in S. When S is finite there is a unique stationary measure p(eps), but how close is it to p(0), the product of the p^s ? And what about infinite S? An example of application is models of nicotinic acetylcholine receptors for ion channels.
5. ## 02.12.2003.

### Abstract.

Phase transition in a system at phase coexistence is discussed in canonical ensemble. It is shown that a novel phase transition at mesoscopic scale is responsible for an abrupt creation of equilibrium droplets. Rigorous results are formulated and explained in the case of Ising model.

2. ## 27.01.2004.

### Abstract.

I consider spin correlations in the 1D repulsive Hubbard model at half-filling. For very large values of the on-site repulsion $U$, the spin correlations are dominated by virtual hopping processes of electrons and are described in terms of a spin-1/2 Heisenberg chain. As U is decreased, real hopping processes of electrons become important and eventually dominate the spin response. I discuss the evolution of the dynamical structure factor as a function of U and comment on the relevance of these results to inelastic Neutron scattering experiments on the quasi-1D cuprates.

4. ## 24.02.2004.

### Abstract.

When a magnetic moment (a spin) is coupled to a sea of electrons, the "Kondo effect" occurs: the electrons form a collective singlet with the moment as the temperature is decreased, until the spin is completely screened. A typical - if expensive - experimental realisation would be an iron atom in a block of solid gold. Within the last five years or so, the Kondo effect has been observed in a new situation: quantum dots. The essential difference in the quantum dot case is that the spin is connected to _two_ seas of electrons rather than one, and the resulting possibility of a current's flowing from one sea to the other makes this an out-of-equilibrium problem. In the first half of my presentation, I shall review the Kondo effect in its traditional guise, and in the quantum dot case close to equilibrium. I shall also give a brief survey of the (largely dubious, and often mutually contradictory) theoretical attempts to address the out-of-equilibrium problem. In the second half, I shall present my and others' recent work aimed at giving a more controlled analysis. This is based on treating the model perturbatively using Schwinger-Keldysh non-equilibrium field theory. I shall exhibit a few of the interesting tricks required, highlight a couple of surprising results, and indicate the likely future and limitations of this perturbative programme.
5. ## 09.03.2004.

### Abstract.

There is a growing understanding that transport properties of complex oxides and individual molecules are dominated by polaron physics. In superconducting oxides the long-range Froehlich and short-range Jahn-Teller electron-phonon interactions bind carriers into real space bosons - small bipolarons with surprisingly low mass but sufficient binding energy, while the long-range Coulomb repulsion keeps bipolarons apart preventing their clustering. The bipolaron theory numerically explains high Tc values without any fitting parameters and describes other key features of the cuprates. The same approach provides a new insight into the theory of transport through molecular nanowires and quantum dots (MQD). Attractive polaron-polaron correlations lead to a "switching" phenomenon in the current-voltage characteristics of MQD. The degenerate MQD with strong electron-vibron coupling has two stable current states (a volatile memory), which might be useful in molecular electronic.

## Spring Term, 2005

The seminar is held at Room B3.03 (Mathematics Institute) at 4pm on specified Wednesdays or Thursdays.
1. ### 12.01.2005, 16:00-17:00, Rm B3.03.

Stefan Adams (Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Germany)

"On probabilistic approaches to the Gross-Pitaevskii theory for dilute systems of Bosons."

Abstract.In this talk we first review some basic facts about the Gross-Pitaveskii theory for dilute systems of interacting Bosons in inhomogenous magnetic traps. Then we present a probabilistic approach and interpretation for the Gross-Pitaveskii variational formula. Starting point are N interacting Bosons moving in an external field which keeps them in a bounded region of space. Our probabilistic ansatz starts with a transformed path measure for N Brownian particles given by a Feynman-Kac formula. The aim is then to study two limits, one of vanishing temperature (infinite time for the the probabilistic model) and one for diverging number of Bosons (Brownian particles). In particular we introduce two probabilistic models for N interacting Brownian motions moving in a trap under mutually repellent forces. The first pair interaction exhibits a particle repellency, while the second one imposes a path repellency. We analyse both models in the limit of diverging time with fixed number N of Brownian motions. In particular, we prove large deviations principles for the normalised occupation measures. The minimisers of the rate functions are related to a certain associated operator, the Hamilton operator for a system of N interacting trapped particles. More precisely, in the particle-repellency model, the minimiser is its ground state, and in the path-repellency model, the minimisers are its ground product-states. In the case of path- repellency, we also discuss the case of a Dirac-type interaction, which is rigorously defined in terms of Brownian intersection local times. We prove a large-deviation result for a discrete variant of the model. This study is a contribution to the search for a mathematical formulation of the quantum system of N trapped interacting bosons as a model for Bose-Einstein condensation, motivated by the success of the famous 1995 experiments. Recently, Lieb et al. described the large-N behaviour of the ground state in terms of the well-known Gross-Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-N behaviour of the ground product-states is also described by the Gross-Pitaevskii formula, however with the scattering length of the pair potential replaced by its integral.

2. ### 19.01.2005, 16:00-17:00, Rm B3.03.

Yurii Suhov (Cambridge University)

"Anderson localisation for multi-particle systems."

AbstractAnderson localisation is an important phenomenon describing a transition between insulation and conductivity. The problem is to analyse the spectrum of a Schroedinger operator with a random potential in the Euclidean space or on a lattice. We say that the system exhibits (exponential) localisation if with probability one the spectrum is pure point and the corresponding eigen-functions decay exponentially fast. So far in the literature one considered a single-particle model where the potential at different sites is IID or has a controlled decay of correlations. The present talk aims at $N$-particle systems (bosons or fermions) where the potential sums over different sites, and the traditional approach needs serious modifications. The main result is that if the `randomness' is strong enough, the $N$-particle system exhibits localisation. The proof exploits the muli-scale analysis scheme going back to Froehlich, Martinelli, Scoppola and Spencer and refined by von Drefus and Klein. No preliminary knowledge of the related material will be assumed from the audience, apart from basic facts. This is a joint work with V. Chulaevsky (University of Reims, France).

3. ### 08.02.2005, 16:00-17:00, Rm B3.03.

Alan Sokal (New York University)

Chromatic polynomials, Tutte polynomials, Potts models, and all that.

AbstractChromatic polynomials, Tutte polynomials, Potts models, and all that The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information about the graph indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte polynomial --- and therefore also its one-variable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial --- but is considerably more flexible. I begin by giving an introduction to all these problems, stressing the advantages of working with the multivariate version. I then discuss some questions concerning the complex zeros of the multivariate Tutte polynomial, along with their physical interpretations in statistical mechanics (in connection with the Yang--Lee approach to phase transitions) and electrical circuit theory. Along the way I mention numerous open problems. This survey is intended to be understandable to mathematicians with no prior knowledge of physics.

4. ### 01.03.2005, 15:00-17:00, Rm B3.03.

Colm Connaughton (ENS and CNLS LANL)

Constant Flux Relations in Non-Equilibrium Statistical Mechanics.

AbstractThe presence of constant flux of a conserved quantity in the steady state of a turbulent system often allows an exact determination of the scaling exponent of a particular correlation function, namely the one measuring average flux. Such universal scaling laws generalize the well known 4/5 law of Navier-Stokes turbulence and are not the result of any mean field approximations. They must be respected by any effective description of non-equilibrium statistical systems. In this talk we shall introduce the general philosophy which leads to such constant flux relations and discuss the conditions under which the theory is applicable. We shall then study in more detail the application of these ideas to turbulence-like cascades in cluster-cluster aggregation and wave turbulence.

 Forum - Stats A1.01 (upstairs by front door of building) 15:00, Tue 10 Jan '06 Ben Simons, Cavendish Labs Pattern Formation in Exciton Systems near Quantum Degeneracy (provisional title) [RCB] Fri, 9 Sep 2005
 Forum - B3.02 Maths 15:00, Tue 28 Feb '06 Neil Johnson, University of Oxford Two's Company, Three's a Crowd: from Traders to Terrorists Why do people decide to join or leave groups? And what is the effect of the resulting group dynamics on the macroscopic behavior of the system in question? From financial markets through to insurgent warfare, the importance of understanding how the ecology of a given population evolves and acts over time, is of great importance. Depending on the situation, it may pay to seek strength in numbers -- on the other hand, such aggregation may lead to over-use of some limited resource. In this talk I will analyze several real-world situations, comparing theoretical models to empirical data. The goal is to understand when crowds might be more likely to emerge based on the characteristics of the individuals concerned, and whether these crowds then represent a potential threat to the system's stability. In addition to such endogenous crowd formation, I will look at the effect of exogenous 'kicks' from outside influences such as news, rumors, memes or antigens. The results have possible relevance to a wide variety of social, economic and biological phenomena -- in addition, they yield as a by-product an intriguing generalization of conventional many-body theory in Physics. Thu, 16 Feb 2006