On the Existence of Invariant Measure for Lagrangian Velocity in Compressible Environments

We study transport of a passive tracer particle in a time dependent turbulent flow in the medium with positive molecular diffusivity. We show that there exists then a probability measure equivalent to the underlying physical probability, corresponding to the Eulerian velocity field, under which the particle Lagrangian velocity observations are stationary. As an application we derive the existence of the Stokes drift and the effective diffusivity—the characteristics of the long time behavior of the particle motion.     

Enhanced diffusion due to active swimmers at a solid surface

We consider two systems of active swimmers moving close to a solid surface, one being a living population of wild-type E.?coli and the other being an assembly of self-propelled Au-Pt rods. In both situations, we have identified two different types of motion at the surface and evaluated the fraction of the population that displayed ballistic trajectories (active swimmers) with respect to those showing randomlike behavior. We studied the effect of this complex swimming activity on the diffusivity of passive tracers also present at the surface. We found that the tracer diffusivity is enhanced with respect to standard Brownian motion and increases linearly with the activity of the fluid, defined as the product of the fraction of active swimmers and their mean velocity. This result can be understood in terms of series of elementary encounters between the active swimmers and the tracers.     

Transport of a passive scalar and Lagrangian chaos in a Hamiltonian hydrodynamic system

An experimental and numerical study is made of the chaotic behavior of Lagrangian trajectories and transport of a passive tracer in a quasi-two-dimensional four-vortex flow with a periodic time dependence of the Euler velocity field. Quantitative measurements are made of tracer transport between isolated vortices in physical space and in “action” variable space. The theory of adiabatic chaos is used to interpret the measurements. The simplest phenomenological models of liquid particle random walks are proposed to describe the anomalous transport in terms of the action.     

Path Lengths in Turbulence

By tracking tracer particles at high speeds and for long times, we study the geometric statistics of Lagrangian trajectories in an intensely turbulent laboratory flow. In particular, we consider the distinction between the displacement of particles from their initial positions and the total distance they travel. The difference of these two quantities shows power-law scaling in the inertial range. By comparing them with simulations of a chaotic but non-turbulent flow and a Lagrangian Stochastic model, we suggest that our results are a signature of turbulence.     

Mesoscopic spin-dependent reflection experiments on InSb- and InAs-based heterostructures

Spin–orbit interaction in two-dimensional electron systems can lead to a spin-dependent reflection of carriers off a lithographic barrier. Scattering of a spin-unpolarized beam from the barrier leads to the creation of two fully spin-polarized side beams in addition to an unpolarized specularly reflected beam. We experimentally demonstrate a method to create spin-polarized beams of ballistic electrons in mesoscopic samples fabricated on InSb/InAlSb and InAs/AlGaSb heterostructures. We describe two geometries, one open and one closed, in which the spin-dependent reflection and spin-dependent semiclassical trajectories were observed.     

Relative distance between tracers as a measure of diffusivity within moving aggregates

Tracking of particles, be it a passive tracer or an actively moving bacterium in the growing bacterial colony, is a powerful technique to probe the physical properties of the environment of the particles. One of the most common measures of particle motion driven by fluctuations and random forces is its diffusivity, which is routinely obtained by measuring the mean squared displacement of the particles. However, often the tracer particles may be moving in a domain or an aggregate which itself experiences some regular or random motion and thus masks the diffusivity of tracers. Here we provide a method for assessing the diffusivity of tracer particles within mobile aggregates by measuring the so-called mean squared relative distance (MSRD) between two tracers. We provide analytical expressions for both the ensemble and time averaged MSRD allowing for direct identification of diffusivities from experimental data.     

Lagrangian Dispersion in Gaussian Self-Similar Velocity Ensembles

We analyze the Lagrangian flow in a family of simple Gaussian scale-invariant velocity ensembles that exhibit both spatial roughness and temporal correlations. We argue that the behavior of the Lagrangian dispersion of pairs of fluid particles in such models is determined by the scale dependence of the ratio between the correlation time of velocity differences and the eddy turnover time. For a non-trivial scale dependence, the asymptotic regimes of the dispersion at small and large scales are described by the models with either rapidly decorrelating or frozen velocities. In contrast to the decorrelated case, known as the Kraichnan model and exhibiting Lagrangian flows with deterministic or stochastic trajectories, fast separating or trapped together, the frozen model is poorly understood. We examine the pair dispersion behavior in its simplest, one-dimensional version, reinforcing analytic arguments by numerical analysis. The collected information about the pair dispersion statistics in the limiting models allows to partially predict the extent of different phases of the Lagrangian flow in the model with time-correlated velocities.     

Exceptional Point of the Neutral Heavy Higgs System H2–H3

We study the singularity of the surface that represents the masses of the isolated doublet of heavy, neutral Higgs bosons, H ₂–H ₃, in a toy model based on the MSSM with CP violation, in parameter space. These two heavy, neutral Higgs bosons are coherent and, for large values of the masses, nearly degenerate. In this scenario, mixing between the mass eigenstates of the H ₂–H ₃ system could be very large and exact degeneracy is possible. As function of the Lagrangian parameters, the physical mass of the doublet has an algebraic branch point of rank one at the exceptional point where the two masses are equal. The real and imaginary parts of the masses in the doublet have branch cuts that start at the same branch point but extend in opposite directions in parameter space. Associated with this branch point, the propagator of the mixing doublet of neutral heavy Higgs bosons has a double pole in the complex s-plane of the energy squared. We computed the mass surface of the isolated doublet of H ₂–H ₃ bosons as function of the Lagrangian parameters in the neighbourhood of the exceptional point in a toy model of the system H ₂–H ₃. We also computed the trajectories of the poles of the transition matrix for values of the Lagrangian parameters close to the exceptional point and explained the characteristic change of identity seen in these trajectories in the s-plane as a manifestation of the topology of the two-sheeted mass surfaces in the space of Lagrangian parameters.     

Thermal properties of compressed expanded graphite: photothermal measurements

The thermal diffusivity and the thermal conductivity of compressed expanded graphite (CEG) samples were investigated by photothermal measurements in two geometries differing by a place of temperature disturbance detection. This disturbance can be detected on a surface opposite to the one at which the disturbance was generated (rear detection) or on the same surface (front detection). A measurement based on the rear detection allowed us to determine the effective thermal diffusivity of the sample, while the method with front detection gives the possibility of analysis of homogeneity of the sample. It is shown that the thermal diffusivity of CEG strongly depends on its apparent density. Moreover, CEG samples reveal anisotropy of the thermal properties. The thermal diffusivity in the direction parallel to the compacting axis is lower than the one in the direction perpendicular to it. The parallel thermal diffusivity decreases with growing apparent density, while the perpendicular thermal diffusivity significantly grows when the apparent density grows. The perpendicular thermal conductivity exhibits the same behavior as the perpendicular thermal diffusivity. The parallel thermal conductivity slightly grows with growing density and then reaches a plateau. The anisotropy of CEG samples grows with growing apparent density and vanishes for low-density samples. The photothermal measurement with front signal detection revealed that the CEG samples are non-homogeneous in the direction of the compacting axis and can be modeled by a two-layer system.     

Quantum chaotic scattering with a mixed phase space: the three-disk billiard in a magnetic field

We study the classical and semiclassical scattering behavior of electrons in an open three-disk billard in the presence of a homogeneous magnetic field, which is confined to the inner part of the scattering region. As the magnetic field is increased the phase space of the invariant set of the classical scattering trajectories changes from hyperbolic (fully chaotic) to a mixed situation, where KAM tori are present. The "stickiness" of the stable trajectories leads to a much slower decay of the survival probability of trajectories as compared to the hyperbolic case. We show that this effect influences strongly the quantum fluctuations of the scattering amplitude and cross sections.     

Diffusion of asymmetric swimmers

Particles moving along curved trajectories will diffuse if the curvature fluctuates sufficiently in either magnitude or orientation. We consider particles moving at a constant speed with either a fixed or a Gaussian distributed magnitude of curvature. At small speeds the diffusivity is independent of the speed. At larger particle speeds, the diffusivity depends on the speed through a novel exponent. We apply our results to intracellular transport of vesicles. In sharp contrast to thermal diffusion, the effective diffusivity increases with vesicle size and so may provide an effective means of intracellular transport.     

Slow Modes in Passive Advection

The anomalous scaling in the Kraichnan model of advection of the passive scalar by a random velocity field with nonsmooth spatial behavior is traced to the presence of slow resonance-type collective modes of the stochastic evolution of fluid trajectories. We show that the slow modes are organized into infinite multiplets of descendants of the primary conserved modes. Their presence is linked to the nondeterministic behavior of the Lagrangian trajectories at high Reynolds numbers caused by the sensitive dependence on initial conditions within the viscous range where the velocity fields are more regular. Revisiting the Kraichnan model with smooth velocities, we describe the explicit solution for the stationary state of the scalar. The properties of the probability distribution function of the smeared scalar in this state are related to a quantum mechanical problem involving the Calogero–Sutherland Hamiltonian with a potential.Partially supported by NSF Grant DMS-9205296     

Surface bound states and spin currents in noncentrosymmetric superconductors

We investigate the ground state properties of a noncentrosymmetric superconductor near a surface. We determine the spectrum of Andreev bound states due to surface-induced mixing of bands with opposite spin helicities for a Rashba-type spin-orbit coupling. We find that the order parameter suppression qualitatively changes the bound state spectrum. The spin structure of Andreev states leads to a spin supercurrent along the interface, which is strongly enhanced compared to the normal state spin current. Particle and hole coherence amplitudes show Faraday-like rotations of the spin along quasiparticle trajectories.     

A Lagrangian description of transport associated with a front-eddy interaction: Application to data from the North-Western Mediterranean Sea

We discuss the Lagrangian transport in a time-dependent oceanic system involving a Lagrangian barrier associated with a salinity front which interacts intermittently with a set of Lagrangian eddies — ‘leaky’ coherent structures that entrain and detrain fluid as they move. A theoretical framework, rooted in the dynamical systems theory, is developed in order to describe and analyse this situation. We show that such an analysis can be successfully applied to a realistic ocean model. Here, we use the output of the numerical ocean model DieCAST from Dietrich et al. (2004) [17] and Fernández et al. (2005) [18] studied earlier in Mancho et al. (2008) [15] where a Lagrangian barrier associated with the North Balearic Front in the North-Western Mediterranean Sea was identified. The numerical model provides an Eulerian view of the flow and we employ the dynamical systems approach to identify relevant hyperbolic trajectories and their stable and unstable manifolds. These manifolds are used to understand the Lagrangian geometry of the evolving front-eddy system. Transport in this system is effected by the turnstile mechanism whose spatio-temporal geometry reveals intermittent pathways along which transport occurs. Particular attention is paid to the ‘Lagrangian’ interactions between the front and the eddies, and to transport implications associated with the transition between the one-eddy and two-eddy situation. The analysis of this ‘Lagrangian’ transition is aided by a local kinematic model that provides insight into the nature of the change in hyperbolic trajectories and their stable and unstable manifolds associated with the ‘birth’ and ‘death’ of leaky Lagrangian eddies.     

Chaotic behavior of a Galerkin model of a two-dimensional flow

Chaotic behavior of a Galerkin model of the Kolmogorov fluid motion equations is demonstrated. The study focuses on the dynamical behavior of limit trajectories branching off secondary periodic solutions. It is shown that four limit trajectories exist and transform simultaneously from periodic solutions to chaotic attractors through a sequence of bifurcations including a periodic-doubling scenario. Some instability regimes display close similarities to those of a discrete dynamical system generated by an interval map.     

A points-splitting and dimensional renormalization

We consider Feynman amplitudes which are doubly regularized by means of complete points splitting of vertices and continuation in the dimension of space-time. We show how to construct a subtraction operator which leads to polynomial counterterms and to a renormalized amplitude which is finite as the regularizations are removed in either order, and corresponds to the dimensionally renormalized result in the limit of no points splitting.     

Transport of a Passive Tracer by an Irregular Velocity Field

The trajectories of a passive tracer in a turbulent flow satisfy the ordinary differential equation x′(t)=V(t,x(t)), where V(t,x) is a stationary random field, the so-called Eulerian velocity. It is a nontrivial question to define the dynamics of the tracer in the case when the realizations of the Eulerian field are only spatially Hölder regular because the ordinary differential equation in question lacks then uniqueness. The most obvious approach is to regularize the dynamics, either by smoothing the velocity field (the so-called ε-regularization), or by adding a small molecular diffusivity (the so-called κ-regularization) and then pass to the appropriate limit with the respective regularization parameter. The first procedure corresponds to the Prandtl number Pr=∞, while the second to Pr=0. In the present paper we consider a two parameter family of Gaussian, Markovian time correlated fields V(t,x), with the power-law spectrum. Using the infinite dimensional stochastic calculus we show the existence and uniqueness of the law of the trajectory process corresponding to a given field V(t,x) for a certain regime of parameters characterizing the spectrum of the field. Additionally, this law is the limit, in the sense of the weak convergence of measures, of the laws obtained as a result of any of the described regularizations. The so-called Kolmogorov point, that corresponds to the parameters characterizing the relaxation time and energy spectrum of a turbulent, three dimensional flow, belongs to the boundary of the parameter regime considered in the paper.