Dynamics of defect formation

A dynamic symmetry-breaking transition with noise and inertia is analyzed. Exact solution of the linearized equation that describes the critical region allows precise calculation (exponent and prefactor) of the number of defects produced as a function of the rate of increase of the critical parameter. The procedure is valid in both the overdamped and underdamped limits. In one space dimension, we perform quantitative comparison with numerical simulations of the nonlinear nonautonomous stochastic partial differential equation and report on signatures of underdamped dynamics.Comment: 4 pages, LaTeX, 4 figures. Submitted to Physical Revie

A new mechanism shapes the naïve CD8+ T cell repertoire: the selection for full diversity

During thymic T cell differentiation, TCR repertoires are shaped by negative, positive and agonist selection. In the thymus and in the periphery, repertoires are also shaped by strong inter-clonal and intra-clonal competition to survive death by neglect. Understanding the impact of these events on the T cell repertoire requires direct evaluation of TCR expression in peripheral naïve T cells. Several studies have evaluated TCR diversity, with contradictory results. Some of these studies had intrinsic technical limitations since they used material obtained from T cell pools, preventing the direct evaluation of clone sizes. Indeed with these approaches, identical TCRs may correspond to different cells expressing the same receptor, or to several amplicons from the same T cell. We here overcame this limitation by evaluating TCRB expression in individual naïve CD8+ T cells. Of the 2269 Tcrb sequences we obtained from 13 mice, 99% were unique. Mathematical analysis of this data showed that the average number of naïve peripheral CD8+ T cells expressing the same TCRB is 1.1 cell. Since TCRA co-expression studies could only increase repertoire diversity, these results reveal that the number of naïve T cells with unique TCRs approaches the number of naïve cells. Since thymocytes undergo multiple rounds of divisions after TCRB rearrangement; and 3–5% of thymocytes survive thymic selection events; the number of cells expressing the same TCRB was expected to be much higher. Thus, these results suggest a new repertoire selection mechanism, which strongly selects for full TCRB diversity

Counting defects with the two-point correlator

We study how topological defects manifest themselves in the equal-time two-point field correlator. We consider a scalar field with Z_2 symmetry in 1, 2 and 3 spatial dimensions, allowing for kinks, domain lines and domain walls, respectively. Using numerical lattice simulations, we find that in any number of dimensions, the correlator in momentum space is to a very good approximation the product of two factors, one describing the spatial distribution of the defects and the other describing the defect shape. When the defects are produced by the Kibble mechanism, the former has a universal form as a function of k/n, which we determine numerically. This signature makes it possible to determine the kink density from the field correlator without having to resort to the Gaussian approximation. This is essential when studying field dynamics with methods relying only on correlators (Schwinger-Dyson, 2PI).Comment: 11 pages, 7 figures

Formation of topological defects in gauge field theories

When a symmetry gets spontaneously broken in a phase transition, topological defects are typically formed. The theoretical picture of how this happens in a breakdown of a global symmetry, the Kibble-Zurek mechanism, is well established and has been tested in various condensed matter experiments. However, from the viewpoint of particle physics and cosmology, gauge field theories are more relevant than global theories. In recent years, there have been significant advances in the theory of defect formation in gauge field theories, which make precise predictions possible, and in experimental techniques that can be used to test these predictions in superconductor experiments. This opens up the possibility of carrying out relatively simple and controlled experiments, in which the non-equilibrium phase transition dynamics of gauge field theories can be studied. This will have a significant impact on our understanding of phase transitions in the early universe and in heavy ion collider experiments. In this paper, I review the current status of the theory and the experiments in which it can be tested.Comment: Review article, 43 pages, 7 figures. Minor changes, some references added. Final version to appear in IJMP

Accurate stationary densities with partitioned numerical methods for stochastic partial differential equations

We consider the numerical solution, by finite differences, of second-order-in-time stochastic partial differential equations (SPDEs) in one space dimension. New timestepping methods are introduced by generalising recently-introduced methods for second-order-in-time stochastic differential equations to multidimensional systems. These stochastic methods, based on leapfrog and Runge–Kutta methods, are designed to give good approximations to the stationary variances and the correlations in the position and velocity variables. In particular, we introduce the reverse leapfrog method and stochastic Runge–Kutta Leapfrog methods, analyse their performance applied to linear SPDEs and perform numerical experiments to examine their accuracy applied to a type of nonlinear SPDE

Critical Dynamics of Symmetry Breaking: Quenches, Dissipation and Cosmology

Symmetry-breaking phase transitions may leave behind topological defects \cite{Kibble} with a density dependent on the quench rate \cite{Zurek}. We investigate the dynamics of such quenches for the one-dimensional, Landau-Ginzburg case and show that the density of kinks, $n$, scales differently with the quench timescale, $\tau_Q$, depending on whether the dynamics in the vicinity of the critical point is overdamped ($n \propto \tau_Q^{-1/4}$) or underdamped ($n \propto \tau_Q^{-1/3}$). Either of these cases may be relevant to the early Universe, and we derive bounds on the initial density of topological defects in cosmological phase transitions.Comment: 5 pages, 4 Postscript figure

Vortex formation in two dimensions: When symmetry breaks, how big are the pieces?

We investigate the dynamics of second order phase transitions in two dimensions, breaking a gauged U(1) symmetry. Using numerical simulations, we show that the density of topological defects formed scales with the quench timescale $\tau_Q$ as $n \sim \tau_Q^{-1/2}$ when the dynamics is overdamped at the instant when the freezeout of thermal fluctuations takes place, and $n \sim \tau_Q^{-2/3}$ in the underdamped case. This is predicted by the scenario proposed by one of us [1].Comment: 5 pages, 4 postscript figures, uses RevTex and epsf.st

A window of opportunity for cooperativity in the T Cell Receptor

The T-cell antigen receptor (TCR) is pre-organised in oligomers, known as nanoclusters. Nanoclusters could provide a framework for inter-TCR cooperativity upon peptide antigen-major histocompatibility complex (pMHC) binding. Here we have used soluble pMHC oligomers in search for cooperativity effects along the plasma membrane plane. We find that initial binding events favour subsequent pMHC binding to additional TCRs, during a narrow temporal window. This behaviour can be explained by a 3-state model of TCR transition from Resting to Active, to a final Inhibited state. By disrupting nanoclusters and hampering the Active conformation, we show that TCR cooperativity is consistent with TCR nanoclusters adopting the Active state in a coordinated manner. Preferential binding of pMHC to the Active TCR at the immunological synapse suggests that there is a transient time frame for signal amplification in the TCR, allowing the T cells to keep track of antigen quantity and binding time

The reproduction number and its probability distribution for stochastic viral dynamics

We consider stochastic models of individual infected cells. The reproduction number, R, is understood as a random variable representing the number of new cells infected by one initial infected cell in an otherwise susceptible (target cell) population. Variability in R results partly from heterogeneity in the viral burst size (the number of viral progeny generated from an infected cell during its lifetime), which depends on the distribution of cellular lifetimes and on the mechanism of virion release. We analyse viral dynamics models with an eclipse phase: the period of time after a cell is infected but before it is capable of releasing virions. The duration of the eclipse, or the subsequent infectious, phase is non-exponential, but composed of stages. We derive the probability distribution of the reproduction number for these viral dynamics models, and show it is a negative binomial distribution in the case of constant viral release from infectious cells, and under the assumption of an excess of target cells. In a deterministic model, the ultimate in-host establishment or extinction of the viral infection depends entirely on whether the mean reproduction number is greater than, or less than, one, respectively. Here, the probability of extinction is determined by the probability distribution of R, not simply its mean value. In particular, we show that in some cases the probability of infection is not an increasing function of the mean reproduction number

Winding up superfluid in a torus via Bose Einstein condensation

Phase transitions are usually treated as equilibrium phenomena, which yields telltale universality classes with scaling behavior of relaxation time and healing length. However, in second-order phase transitions relaxation time diverges near the critical point (“critical slowing down”). Therefore, every such transition traversed at a finite rate is a non-equilibrium process. Kibble-Zurek mechanism (KZM) captures this basic physics, predicting sizes of domains – fragments of broken symmetry – and the density of topological defects, long-lived relics of symmetry breaking that can survive long after the transition. To test KZM we simulate Bose-Einstein condensation in a ring using stochastic Gross-Pitaevskii equation and show that BEC formation can spontaneously generate quantized circulation of the newborn condensate. The magnitude of the resulting winding numbers and the time-lag of BEC density growth – both experimentally measurable – follow scalings predicted by KZM. Our results may also facilitate measuring the dynamical critical exponent for the BEC transition