Slow Relaxation in a Constrained Ising Spin Chain: a Toy Model for Granular Compaction

We present detailed analytical studies on the zero temperature coarsening dynamics in an Ising spin chain in presence of a dynamically induced field that favors locally the `-' phase compared to the `+' phase. We show that the presence of such a local kinetic bias drives the system into a late time state with average magnetization m=-1. However the magnetization relaxes into this final value extremely slowly in an inverse logarithmic fashion. We further map this spin model exactly onto a simple lattice model of granular compaction that includes the minimal microscopic moves needed for compaction. This toy model then predicts analytically an inverse logarithmic law for the growth of density of granular particles, as seen in recent experiments and thereby provides a new mechanism for the inverse logarithmic relaxation. Our analysis utilizes an independent interval approximation for the particle and the hole clusters and is argued to be exact at late times (supported also by numerical simulations).Comment: 9 pages RevTeX, 1 figures (.eps

Relaxation Properties of Small-World Networks

Recently, Watts and Strogatz introduced the so-called small-world networks in order to describe systems which combine simultaneously properties of regular and of random lattices. In this work we study diffusion processes defined on such structures by considering explicitly the probability for a random walker to be present at the origin. The results are intermediate between the corresponding ones for fractals and for Cayley trees.Comment: 16 pages, 6 figure

Exact Occupation Time Distribution in a Non-Markovian Sequence and Its Relation to Spin Glass Models

We compute exactly the distribution of the occupation time in a discrete {\em non-Markovian} toy sequence which appears in various physical contexts such as the diffusion processes and Ising spin glass chains. The non-Markovian property makes the results nontrivial even for this toy sequence. The distribution is shown to have non-Gaussian tails characterized by a nontrivial large deviation function which is computed explicitly. An exact mapping of this sequence to an Ising spin glass chain via a gauge transformation raises an interesting new question for a generic finite sized spin glass model: at a given temperature, what is the distribution (over disorder) of the thermally averaged number of spins that are aligned to their local fields? We show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the Sherrington-Kirkpatrick model with Gaussian disorder.Comment: 10 pages Revtex (two-column), 1 eps figure (included

Dynamics of systems with isotropic competing interactions in an external field: a Langevin approach

We study the Langevin dynamics of a ferromagnetic Ginzburg-Landau Hamiltonian with a competing long-range repulsive term in the presence of an external magnetic field. The model is analytically solved within the self consistent Hartree approximation for two different initial conditions: disordered or zero field cooled (ZFC), and fully magnetized or field cooled (FC). To test the predictions of the approximation we develop a suitable numerical scheme to ensure the isotropic nature of the interactions. Both the analytical approach and the numerical simulations of two-dimensional finite systems confirm a simple aging scenario at zero temperature and zero field. At zero temperature a critical field $h_c$ is found below which the initial conditions are relevant for the long time dynamics of the system. For $h < h_c$ a logarithmic growth of modulated domains is found in the numerical simulations but this behavior is not captured by the analytical approach which predicts a $t^1/2$ growth law at $T = 0$

Ordering of the lamellar phase under a shear flow

The dynamics of a system quenched into a state with lamellar order and subject to an uniform shear flow is solved in the large-N limit. The description is based on the Brazovskii free-energy and the evolution follows a convection-diffusion equation. Lamellae order preferentially with the normal along the vorticity direction. Typical lengths grow as $\gamma t^{5/4}$ (with logarithmic corrections) in the flow direction and logarithmically in the shear direction. Dynamical scaling holds in the two-dimensional case while it is violated in D=3

Fluctuation-dissipation relations in the non-equilibrium critical dynamics of Ising models

We investigate the relation between two-time, multi-spin, correlation and response functions in the non-equilibrium critical dynamics of Ising models in d=1 and d=2 spatial dimensions. In these non-equilibrium situations, the fluctuation-dissipation theorem (FDT) is not satisfied. We find FDT `violations' qualitatively similar to those reported in various glassy materials, but quantitatively dependent on the chosen observable, in contrast to the results obtained in infinite-range glass models. Nevertheless, all FDT violations can be understood by considering separately the contributions from large wavevectors, which are at quasi-equilibrium and obey FDT, and from small wavevectors where a generalized FDT holds with a non-trivial limit fluctuation-dissipation ratio X. In d=1, we get X = 1/2 for spin observables, which measure the orientation of domains, while X = 0 for observables that are sensitive to the domain-wall motion. Numerical simulations in d=2 reveal a unique X = 0.34 for all observables. Measurement protocols for X are discussed in detail. Our results suggest that the definition of an effective temperature Teff = T / X for large length scales is generically possible in non-equilibrium critical dynamics.Comment: 26 pages, 10 figure

The nature of slow dynamics in a minimal model of frustration-limited domains

We present simulation results for the dynamics of a schematic model based on the frustration-limited domain picture of glass-forming liquids. These results are compared with approximate theoretical predictions analogous to those commonly used for supercooled liquid dynamics. Although model relaxation times increase by several orders of magnitude in a non-Arrhenius manner as a microphase separation transition is approached, the slow relaxation is in many ways dissimilar to that of a liquid. In particular, structural relaxation is nearly exponential in time at each wave vector, indicating that the mode coupling effects dominating liquid relaxation are comparatively weak within this model. Relaxation properties of the model are instead well reproduced by the simplest dynamical extension of a static Hartree approximation. This approach is qualitatively accurate even for temperatures at which the mode coupling approximation predicts loss of ergodicity. These results suggest that the thermodynamically disordered phase of such a minimal model poorly caricatures the slow dynamics of a liquid near its glass transition

Active Brownian Particles. From Individual to Collective Stochastic Dynamics

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte

Longitudinal lung function assessment of patients hospitalised with COVID-19 using 1H and 129Xe lung MRI

BACKGROUND: Microvascular abnormalities and impaired gas transfer have been observed in patients with COVID-19. The progression of pulmonary changes in these patients remains unclear. RESEARCH QUESTION: Do patients hospitalised due to COVID-19 without evidence of architectural distortion on structural imaging show longitudinal improvements in lung function measured using 1H and 129Xe magnetic resonance imaging between 6-52 weeks after hospitalisation? STUDY DESIGN AND METHODS: Patients who were hospitalised due to COVID-19 pneumonia underwent a pulmonary 1H and 129Xe MRI protocol at 6, 12, 25 and 51 weeks after hospital admission in a prospective cohort study between 11/2020 and 02/2022. Imaging protocol: 1H ultra-short echo time, contrast enhanced lung perfusion, 129Xe ventilation, 129Xe diffusion weighted and 129Xe spectroscopic imaging of gas exchange. RESULTS: 9 patients were recruited (57±14 [median±interquartile range] years, 6/9 male). Patients underwent MRI at 6 (N=9), 12 (N=9), 25 (N=6) and 51 (N=8) weeks after hospital admission. Patients with signs of interstitial lung damage were excluded. At 6 weeks, patients demonstrated impaired 129Xe gas transfer (red blood cell to membrane fraction) but lung microstructure was not increased (apparent diffusion coefficient and mean acinar airway dimensions). Minor ventilation abnormalities present in four patients were largely resolved in the 6-25 week period. At 12 weeks, all patients with lung perfusion data (N=6) showed an increase in both pulmonary blood volume and flow when compared to 6 weeks, though this was not statistically significant. At 12 weeks, significant improvements in 129Xe gas transfer were observed compared to 6-week examinations, however 129Xe gas transfer remained abnormally low at weeks 12, 25 and 51. INTERPRETATION: 129Xe gas transfer was impaired up to one year after hospitalisation in patients who were hospitalised due to COVID-19 pneumonia, without evidence of architectural distortion on structural imaging, whereas lung ventilation wa normal at 52 weeks