Passive scalar turbulence in high dimensions

Exploiting a Lagrangian strategy we present a numerical study for both perturbative and nonperturbative regions of the Kraichnan advection model. The major result is the numerical assessment of the first-order $1/d$-expansion by M. Chertkov, G. Falkovich, I. Kolokolov and V. Lebedev ({\it Phys. Rev. E}, {\bf 52}, 4924 (1995)) for the fourth-order scalar structure function in the limit of high dimensions $d$'s. %Two values of the velocity scaling exponent $\xi$ have been considered: %$\xi=0.8$ and $\xi=0.6$. In the first case, the perturbative regime %takes place at $d\sim 30$, while in the second at $d\sim 25$, %in agreement with the fact that the relevant small parameter %of the theory is $\propto 1/(d (2-\xi))$. In addition to the perturbative results, the behavior of the anomaly for the sixth-order structure functions {\it vs} the velocity scaling exponent, $\xi$, is investigated and the resulting behavior discussed.Comment: 4 pages, Latex, 4 figure

A non-perturbative renormalization group study of the stochastic Navier--Stokes equation

We study the renormalization group flow of the average action of the stochastic Navier--Stokes equation with power-law forcing. Using Galilean invariance we introduce a non-perturbative approximation adapted to the zero frequency sector of the theory in the parametric range of the H\"older exponent $4-2\,\varepsilon$ of the forcing where real-space local interactions are relevant. In any spatial dimension $d$, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov's -5/3 law is, thus, recovered for $\varepsilon=2$ as also predicted by perturbative renormalization. At variance with the perturbative prediction, the -5/3 law emerges in the presence of a \emph{saturation} in the $\varepsilon$-dependence of the scaling dimension of the eddy diffusivity at $\varepsilon=3/2$ when, according to perturbative renormalization, the velocity field becomes infra-red relevant.Comment: RevTeX, 18 pages, 5 figures. Minor changes and new discussion

Pressure and intermittency in passive vector turbulence

We investigate the scaling properties a model of passive vector turbulence with pressure and in the presence of a large-scale anisotropy. The leading scaling exponents of the structure functions are proven to be anomalous. The anisotropic exponents are organized in hierarchical families growing without bound with the degree of anisotropy. Nonlocality produces poles in the inertial-range dynamics corresponding to the dimensional scaling solution. The increase with the P\'{e}clet number of hyperskewness and higher odd-dimensional ratios signals the persistence of anisotropy effects also in the inertial range.Comment: 4 pages, 1 figur

Shell Model for Time-correlated Random Advection of Passive Scalars

We study a minimal shell model for the advection of a passive scalar by a Gaussian time correlated velocity field. The anomalous scaling properties of the white noise limit are studied analytically. The effect of the time correlations are investigated using perturbation theory around the white noise limit and non-perturbatively by numerical integration. The time correlation of the velocity field is seen to enhance the intermittency of the passive scalar.Comment: Replaced with final version + updated figure

Scaling, renormalization and statistical conservation laws in the Kraichnan model of turbulent advection

We present a systematic way to compute the scaling exponents of the structure functions of the Kraichnan model of turbulent advection in a series of powers of $\xi$, adimensional coupling constant measuring the degree of roughness of the advecting velocity field. We also investigate the relation between standard and renormalization group improved perturbation theory. The aim is to shed light on the relation between renormalization group methods and the statistical conservation laws of the Kraichnan model, also known as zero modes.Comment: Latex (11pt) 43 pages, 22 figures (Feynman diagrams). The reader interested in the technical details of the calculations presented in the paper may want to visit: http://www.math.helsinki.fi/mathphys/paolo_files/passive_scalar/passcal.htm

Heat release by controlled continuous-time Markov jump processes

We derive the equations governing the protocols minimizing the heat released by a continuous-time Markov jump process on a one-dimensional countable state space during a transition between assigned initial and final probability distributions in a finite time horizon. In particular, we identify the hypotheses on the transition rates under which the optimal control strategy and the probability distribution of the Markov jump problem obey a system of differential equations of Hamilton-Bellman-Jacobi-type. As the state-space mesh tends to zero, these equations converge to those satisfied by the diffusion process minimizing the heat released in the Langevin formulation of the same problem. We also show that in full analogy with the continuum case, heat minimization is equivalent to entropy production minimization. Thus, our results may be interpreted as a refined version of the second law of thermodynamics.Comment: final version, section 2.1 revised, 26 pages, 3 figure

Perturbations of Noise: The origins of Isothermal Flows

We make a detailed analysis of both phenomenological and analytic background for the "Brownian recoil principle" hypothesis (Phys. Rev. A 46, (1992), 4634). A corresponding theory of the isothermal Brownian motion of particle ensembles (Smoluchowski diffusion process approximation), gives account of the environmental recoil effects due to locally induced tiny heat flows. By means of local expectation values we elevate the individually negligible phenomena to a non-negligible (accumulated) recoil effect on the ensemble average. The main technical input is a consequent exploitation of the Hamilton-Jacobi equation as a natural substitute for the local momentum conservation law. Together with the continuity equation (alternatively, Fokker-Planck), it forms a closed system of partial differential equations which uniquely determines an associated Markovian diffusion process. The third Newton law in the mean is utilised to generate diffusion-type processes which are either anomalous (enhanced), or generically non-dispersive.Comment: Latex fil