Gustavo Stolovitzky

Feynman's Ratchet and Pawl

While many papers in the last few years have dealt with various equations euphemistically called “ratchets,” the original Feyman two-temperature setup has been left largely unchallenged. We present here a look at the details of how this famous engine actually generates motion from a temperature difference.

Mnemonics for variability: remembering food delay.

Three experiments with White Carneaux pigeons (Columba livia) investigated memory and decision processes under fixed and variable reinforcement intervals. Response rate was measured during the unreinforced trials in the discrete-trial peak procedure in which reinforced trials were mixed with long unreinforced trials. Two decision models differing in assumptions about memory constraints are reviewed. In the complete-memory model (J. Gibbon, R.M. Church, S. Fairhurst, & A. Kacelnik, 1988), all interreinforcement intervals were remembered, whereas in the minimax model (D. Brunner, A. Kacelnik, & J. Gibbon, 1996), only estimates of the shortest and longest possible reinforcement times were remembered. Both models accommodated some features of response rate as a function of trial time, but only the second was compatible with the observed cessation of responding.

Refined similarity hypotheses for passive scalars mixed by turbulence

In analogy with Kolmogorovs refined similarity hypotheses for the velocity field, two hypotheses are stated for passive scalar fields mixed by high-Reynolds-number turbulence. A refined Yaglom equation is derived under the new assumption of local isotropy in pure ensembles, which is stronger than the usual assumption of local isotropy but weaker than the isotropy of the large scale. The new theoretical result is shown to be consistent with the hypotheses of refined similarity for passive scalars. These hypotheses are approximately verified by experimental data on temperature fluctuations obtained (in air) at moderate Reynolds numbers in the wake of a heated cylinder. The fact that the refined similarity hypotheses are stated for high Reynolds (and Peclet) numbers, but verified at moderate Reynolds (and Peclet) numbers suggests that these hypotheses are not sufficiently sensitive tests of universality. It is conjectured that possible departures from universality are hidden by the process of taking conditional expectations.

A simple model of chaotic advection and scattering

In this work, we study a blinking vortex-uniform stream map. This map arises as an idealized, but essential, model of time-dependent convection past concentrated vorticity in a number of fluid systems. The map exhibits a rich variety of phenomena, yet it is simple enough so as to yield to extensive analytical investigation. The maps dynamics is dominated by the chaotic scattering of fluid particles near the vortex core. Studying the paths of fluid particles, it is seen that quantities such as residence time distributions and exit-vs-entry positions scale in self-similar fashions. A bifurcation is identified in which a saddle fixed point is created upstream at infinity. The homoclinic tangle formed by the transversely intersecting stable and unstable manifolds of this saddle is principally responsible for the observed self-similarity. Also, since the model is simple enough, various other properties are quantified analytically in terms of the circulation strength, stream velocity, and blinking period. These properties include: entire hierarchies of fixed points and periodic points, the parameter values at which these points undergo conservative period-doubling bifurcations, the structure of the unstable manifolds of the saddle fixed and periodic points, and the detailed structure of the resonance zones inside the vortex core region. A connection is made between a weakly dissipative version of our map and the Ikeda map from nonlinear optics. Finally, we discuss the essential ingredients that our model contains for studying how chaotic scattering induced by time-dependent flow past vortical structures produces enhanced diffusivities. (c) 1995 American Institute of Physics.

Non-isothermal inertial Brownian motion

Abstract The stationary distribution of a dilute gas of Brownian particles in an inhomogeneous thermal bath and in the presence of a force field is considered. Our aim is to understand the dynamics of the Brownian particles in cases when inertial effects are non-negligible and in conditions of zero mass current. We restrict the analysis to the one-dimensional case. Beyond the requirement that there is no mass transport, the temperature and force fields are arbitrary. The statistical description of this processes is governed by the Kramers equation, whose solution we find as an expansion in powers of the inverse of the friction coefficient. The resulting expressions for the position and velocity distribution, heat flux, etc., are tested against numerical simulations of the corresponding Langevin equation. We show that the appropriate interpretation of the Langevin equations, as inertia becomes less important (overdamped limit), is the Ito interpretation. For sufficiently large friction, the heat flux is linear in the temperature gradient, and the system can be analyzed using the tools of irreversible thermodynamics. We show that the entropy production is never negative, vanishing only at thermodynamic equilibrium.

Independent velocity increments and Kolmogorov’s refined similarity hypotheses

Under the assumption of statistical independence of velocity increments across scales of the order of the Kolmogorov scale, it is shown that a modified version of Kolmogorov’s refined similarity hypotheses follows purely from probabilistic arguments. The connection of this result to three‐dimensional fluid turbulence is discussed briefly.

Multiplicative models for turbulent energy dissipation

We consider models for describing the intermittent distribution of the energy dissipation rate per unit mass, e, in high-Reynolds-number turbulent flows. These models are based on a physical picture in which (in one-dimensional space) an eddy of scale r breaks into b smaller eddies of scale r/b. The energy flux across scales of size r is re r , where e r is the average of e over a linear interval of size r. This energy flux can be written as the product of factors called multipliers. We discuss some properties of the distribution of multipliers. Using measured multiplier distributions obtained from atmospheric surface layer data on e, we show that quasi-deterministic models (multiplicative models) can be developed on a rational basis for multipliers with bases b = 2 and 3 (that is, binary and tertiary breakdown processes). This formalism allows a unified understanding of some apparently unrelated previous work, and its simplicity permits the derivation of explicit analytic expressions for quantities such as the probability density function of re r , which agree very well with measurements. Other related applications of multiplier distributions are presented. The limitations of this approach are discussed when bases larger than three are invoked.