Omri Gat

A frictionless microswimmer

We investigate the self-locomotion of an elongated microswimmer by virtue of unidirectional tangential surface treadmilling. We show that the propulsion could be almost frictionless, as the microswimmer is propelled forward with the speed of the backward surface motion, i.e. it moves through an almost quiescent fluid. We investigate this swimming technique using the special spheroidal coordinates and also find an explicit closed-form optimal solution for a two-dimensional treadmiller via complex-variable techniques. Slender-object surface treadmilling is a particularly efficient mode of locomotion because the viscous drag is determined by the smallest length scale of the object rather than by the largest scale, as is the usual case for low Reynolds number flow.

Anomalous scaling in a model of passive scalar advection: Exact results

Kraichnan’s model of passive scalar advection in which the driving velocity field has fast temporal decorrelation is studied as a case model for understanding the appearance of anomalous scaling in turbulent systems. We demonstrate how the techniques of renormalized perturbation theory lead (after exact resummations) to equations for the statistical quantities that reveal also non perturbative effects. It is shown that ultraviolet divergences in the diagrammatic expansion translate into anomalous scaling with the inner length acting as the renormalization scale. In this paper we compute analytically the infinite set of anomalous exponents that stem from the ultraviolet divergences. Notwithstanding, non-perturbative effects furnish a possibility of anomalous scaling based on the outer renormalization scale. The mechanism for this intricate behavior is examined and explained in detail. We show that in the language of L’vov, Procaccia and Fairhall [Phys. Rev. E 50, 4684 (1994)] the problem is “critical” i.e. the anomalous exponent of the scalar primary field � = �c. This is precisely the condition

Critical Behavior of Light in Mode-Locked Lasers

Light is shown to exhibit critical and tricritical behavior in passively mode-locked lasers with externally injected pulses. It is a first and unique example of critical phenomena in a one-dimensional many-body light-mode system. The phase diagrams consist of regimes with continuous wave, driven parapulses, spontaneous pulses via mode condensation, and heterogeneous pulses, separated by phase transition lines that terminate with critical or tricritical points. Enhanced non-Gaussian fluctuations and collective dynamics are present at the critical and tricritical points, showing a mode system analog of the critical opalescence phenomenon. The critical exponents are calculated and shown to comply with the mean field theory, which is rigorous in the light system.

ANOMALOUS SCALING IN PASSIVE SCALAR ADVECTION : MONTE CARLO LAGRANGIAN TRAJECTORIES

We present a numerical method which is used to calculate anomalous scaling exponents of structure functions in the Kraichnan passive scalar advection model (R. H. Kraichnan, Phys. Fluids {\bf11}, 945 (1968)). This Monte-Carlo method, which is applicable in any space dimension, is based on the Lagrangian path interpretation of passive scalar dynamics, and uses the recently discovered equivalence between scaling exponents of structure functions and relaxation rates in the stochastic shape dynamics of groups of Lagrangian particles. We calculate third and fourth order anomalous exponents for several dimensions, comparing with the predictions of perturbative calculations in large dimensions. We find that Kraichnans closure appears to give results in close agreement with the numerics. The third order exponents are compatible with our own previous nonperturbative calculations.

Hydrodynamic Lyapunov Modes in Translation Invariant Systems

We study the implications of translation invariance on the tangent dynamics of extended dynamical systems, within a random matrix approximation. In a model system, we show the existence of hydrodynamic modes in the slowly growing part of the Lyapunov spectrum, which are analogous to the hydrodynamic modes discovered numerically by Dellago, Posch, and Hoover. The hydrodynamic Lyapunov vectors lose the typical random structure and exhibit instead the structure of weakly perturbed coherent long-wavelength waves. We show further that the amplitude of the perturbations vanishes in the thermodynamic limit, and that the associated Lyapunov exponents are universal.

Self-starting of passive mode locking

It has been recently understood that mode locking of lasers has the signification of a thermodynamic phase transition in a system of many interacting light modes subject to noise. In the same framework, self starting of passive mode locking has the thermodynamic significance of a noise-activated escape process across an entropic barrier. Here we present the first dynamical study of the light mode system. While accordant with the predictions of some earlier theories, it is the first to give precise quantitative predictions for the distribution of self-start times, in closed form expressions, resolving the long standing self starting problem. Numerical simulations corroborate these results, which are also in good agreement with experiments.

Experimental study of the stochastic nature of the pulsation self-starting process in passive mode locking

We present an experimental study of the probabilistic nature of pulsation self-starting in passively mode-locked lasers. It is a Poissonian process that results from a noise-activated switching barrier. The switching rate from cw operation to pulsation when the laser pump level is turned on has an exponential dependence that is inversely proportional to the square of the laser power.

Light-mode locking: a new class of solvable statistical physics systems

Passively mode-locked lasers are extended one-dimensional dynamical systems subject to noise, with a nonlinear instability and a global power constraint. We use the recent understanding of the importance of entropy in these systems to study mode locking thermodynamically. We show that this class of problems is solvable by a mean field-like theory, where the nonlinear pulse free energy and entropic continuum free energy compete on the available power, and calculate explicitly the pulse power and mode locking, which occurs when the dimensionless scaled interaction strength γ = 9. A transfer matrix calculation shows that the mean field theory is exact in the thermodynamic limit, where the number of active laser modes tends to infinity.

Semiclassical Analysis and the Magnetization of the Hofstadter Model

The magnetization and the de Haas-van Alphen oscillations of Bloch electrons are calculated near commensurate magnetic fluxes. Two phases that appear in the quantization of mixed systems--the Berry phase and a phase first discovered by Wilkinson--play a key role in the theory.

Symmetry algebras of third‐order ordinary differential equations

The main result of this paper is the complete classification of the third‐order ordinary differential equations according to their symmetries. The same classification was done for second order by Tresse [‘‘Determination des invariants ponctuels de l’equation differentielle ordinaire du second ordre y‘=ω(x,y,y’),’’ Gekronte Preisschrift, Hirzel, Leipzig (1896)], and recently for arbitrary order linear ordinary differential equations. The sections preceding the classification consist of a brief description of the concepts and methods along the lines of Krause and Michel [Lecture Notes Phys. 382, 251 (1991)]. These sections also contain some definitions and a table listing the prolongations of a few vector fields. Finally, two appendices give additional information relevant to equations in real variables and describe how some of the results can be easily generalized to higher orders.