Mikhail V. Ivanchenko

q -breathers in Fermi-Pasta-Ulam chains: Existence, localization, and stability

The Fermi-Pasta-Ulam (FPU) problem consists of the nonequipartition of energy among normal modes of a weakly anharmonic atomic chain model. In the harmonic limit each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q. We continue normal modes from the harmonic limit into the FPU parameter regime and obtain persistence of these periodic orbits, termed here q-Breathers (QB). They are characterized by time periodicity, exponential localization in the q-space of normal modes and linear stability up to a size-dependent threshold amplitude. Trajectories computed in the original FPU setting are perturbations around these exact QB solutions. The QB concept is applicable to other nonlinear lattices as well.

Nonlinear lattice waves in heterogeneous media

We review recent progress in the dynamics of nonlinear lattice waves in heterogeneous media, which enforce complete wave localization in the linear wave equation limit, especially Anderson localization for random potentials, and Aubry-Andre localization for quasiperiodic potentials. Additional nonlinear terms in the wave equations can either preserve the phase-coherent localization of waves, or destroy it through nonintegrability and deterministic chaos. Spreading wave packets are observed to show universal features in their dynamics which are related to properties of nonlinear diffusion equations.

Anderson Localization or Nonlinear Waves: A Matter of Probability

In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems (localization versus propagation) is under intense theoretical debate and experimental study. We resolve this dispute showing that, unlike in the common hypotheses, the answer is probabilistic rather than exclusive. At any small but finite nonlinearity (energy) value there is a finite probability for Anderson localization to break up and propagating nonlinear waves to take over. It increases with nonlinearity (energy) and reaches unity at a certain threshold, determined by the initial wave packet size. Moreover, the spreading probability stays finite also in the limit of infinite packet size at fixed total energy. These results generalize to higher dimensions as well.

Correlated metallic two-particle bound states in quasiperiodic chains

Single-particle states in a chain with quasiperiodic potential show a metal-insulator transition upon the change of the potential strength. We consider two particles with local interaction in the single-particle insulating regime. The two-particle states change from being localized to delocalized upon an increase of the interaction strength to a nonperturbative finite value. At even larger interaction strength the states become localized again. This transition of two-particle bound states into correlated metallic ones is due to a resonant mixing of the noninteracting two-particle eigenstates. In the discovered correlated metal states two particles move coherently together through the whole chain.

Assessing T Cell Clonal Size Distribution: A Non-Parametric Approach

Clonal structure of the human peripheral T-cell repertoire is shaped by a number of homeostatic mechanisms, including antigen presentation, cytokine and cell regulation. Its accurate tuning leads to a remarkable ability to combat pathogens in all their variety, while systemic failures may lead to severe consequences like autoimmune diseases. Here we develop and make use of a non-parametric statistical approach to assess T cell clonal size distributions from recent next generation sequencing data. For 41 healthy individuals and a patient with ankylosing spondylitis, who undergone treatment, we invariably find power law scaling over several decades and for the first time calculate quantitatively meaningful values of decay exponent. It has proved to be much the same among healthy donors, significantly different for an autoimmune patient before the therapy, and converging towards a typical value afterwards. We discuss implications of the findings for theoretical understanding and mathematical modeling of adaptive immunity.

q-breathers in FPU-lattices - scaling and properties for large systems

Recently q-breathers - time-periodic solutions which localize in the space of normal modes and maximize the energy density for some mode vector q0 - were obtained for finite nonlinear lattices. We scale these solutions to arbitrarily large lattices in various lattice dimensions. We study the scaling consequence for previously obtained analytical estimates of the localization length of q-breathers for β-FPU and α-FPU lattices. The first finding is that the degree of localization depends only on intensive quantities and is size independent. Secondly, a critical wave vector km is identified, which depends on one effective nonlinearity parameter, q-breathers minimize the localization length at k0 = km and completely delocalize in the limit k0 → 0, π.

Distributed Classifier Based on Genetically Engineered Bacterial Cell Cultures

We describe a conceptual design of a distributed classifier formed by a population of genetically engineered microbial cells. The central idea is to create a complex classifier from a population of weak or simple classifiers. We create a master population of cells with randomized synthetic biosensor circuits that have a broad range of sensitivities toward chemical signals of interest that form the input vectors subject to classification. The randomized sensitivities are achieved by constructing a library of synthetic gene circuits with randomized control sequences (e.g., ribosome-binding sites) in the front element. The training procedure consists in reshaping of the master population in such a way that it collectively responds to the “positive” patterns of input signals by producing above-threshold output (e.g., fluorescent signal), and below-threshold output in case of the “negative” patterns. The population reshaping is achieved by presenting sequential examples and pruning the population using either graded selection/counterselection or by fluorescence-activated cell sorting (FACS). We demonstrate the feasibility of experimental implementation of such system computationally using a realistic model of the synthetic sensing gene circuits.

Localization in Open Quantum Systems

In an isolated single-particle quantum system, a spatial disorder can induce Anderson localization. Being a result of interference, this phenomenon is expected to be fragile in the face of dissipation. Here we show that a proper dissipation can drive a disordered system into a steady state with tunable localization properties. This can be achieved with a set of identical dissipative operators, each one acting nontrivially on a pair of sites. Operators are parametrized by a uniform phase, which controls the selection of Anderson modes contributing to the state. On the microscopic level, quantum trajectories of a system in the asymptotic regime exhibit intermittent dynamics consisting of long-time sticking events near selected modes interrupted by intermode jumps.

Quantum chaotic subdiffusion in random potentials

Two interacting particles (TIP) in a disordered chain propagate beyond the single particle localization length

Periodic orbits, localization in normal mode space, and the Fermi-Pasta-Ulam problem

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