A numerical comparison of theories of violent relaxation

Using N-body simulations with a large set of massless test particles we compare the predictions of two theories of violent relaxation, the well known Lynden-Bell theory and the more recent theory by Nakamura. We derive ``weaken'' versions of both theories in which we use the whole equilibrium coarse-grained distribution function as a constraint instead of the total energy constraint. We use these weaken theories to construct expressions for the conditional probability $K_i(\tau)$ that a test particle initially at the phase-space coordinate $\tau$ would end-up in the $i$'th macro-cell at equilibrium. We show that the logarithm of the ratio $R_{ij}(\tau) \equiv K_i(\tau)/K_j(\tau)$ is directly proportional to the initial phase-space density $f_0(\tau)$ for the Lynden-Bell theory and inversely proportional to $f_0(\tau)$ for the Nakamura theory. We then measure $R_{ij}(\tau)$ using a set of N-body simulations of a system undergoing a gravitational collapse to check the validity of the two theories of violent relaxation. We find that both theories are at odds with the numerical results, qualitatively and quantitatively.Comment: Replaced with a revised version, which is now accepted to MNRAS. LaTeX, 12 pages, 6 figure

Statistics of pressure and of pressure-velocity correlations in isotropic turbulence

Some pressure and pressure-velocity correlation in a direct numerical simulations of a three-dimensional turbulent flow at moderate Reynolds numbers have been analyzed. We have identified a set of pressure-velocity correlations which posseses a good scaling behaviour. Such a class of pressure-velocity correlations are determined by looking at the energy-balance across any sub-volume of the flow. According to our analysis, pressure scaling is determined by the dimensional assumption that pressure behaves as a ``velocity squared'', unless finite-Reynolds effects are overwhelming. The SO(3) decompositions of pressure structure functions has also been applied in order to investigate anisotropic effects on the pressure scaling.Comment: 21 pages, 8 figur

Molecular adaptation to calsequestrin 2 (CASQ2) point mutations leading to catecholaminergic polymorphic ventricular tachycardia (CPVT): comparative analysis of R33Q and D307H mutants

Homozygous calsequestrin 2 (CASQ2) point mutations leads to catecholaminergic polymorphic ventricular tachycardia: a common pathogenetic feature appears to be the drastic reduction of mutant CASQ2 in spite of normal transcription. Comparative biochemical analysis of R33Q and D307H knock in mutant mice identifies different pathogenetic mechanisms for CASQ2 degradation and different molecular adaptive mechanisms. In particular, each CASQ2 point mutation evokes specific adaptive cellular and molecular processes in each of the four adaptive pathways investigated. Thus, similar clinical phenotypes and identical cellular mechanism for cardiac arrhythmia might imply different molecular adaptive mechanisms

Manifestation of anisotropy persistence in the hierarchies of MHD scaling exponents

The first example of a turbulent system where the failure of the hypothesis of small-scale isotropy restoration is detectable both in the `flattening' of the inertial-range scaling exponent hierarchy, and in the behavior of odd-order dimensionless ratios, e.g., skewness and hyperskewness, is presented. Specifically, within the kinematic approximation in magnetohydrodynamical turbulence, we show that for compressible flows, the isotropic contribution to the scaling of magnetic correlation functions and the first anisotropic ones may become practically indistinguishable. Moreover, skewness factor now diverges as the P\'eclet number goes to infinity, a further indication of small-scale anisotropy.Comment: 4 pages Latex, 1 figur

Derivative moments in turbulent shear flows

We propose a generalized perspective on the behavior of high-order derivative moments in turbulent shear flows by taking account of the roles of small-scale intermittency and mean shear, in addition to the Reynolds number. Two asymptotic regimes are discussed with respect to shear effects. By these means, some existing disagreements on the Reynolds number dependence of derivative moments can be explained. That odd-order moments of transverse velocity derivatives tend not vanish as expected from elementary scaling considerations does not necessarily imply that small-scale anisotropy persists at all Reynolds numbers.Comment: 11 pages, 7 Postscript figure

Active and Passive Fields in Turbulent Transport: the Role of Statistically Preserved Structures

We have recently proposed that the statistics of active fields (which affect the velocity field itself) in well-developed turbulence are also dominated by the Statistically Preserved Structures of auxiliary passive fields which are advected by the same velocity field. The Statistically Preserved Structures are eigenmodes of eigenvalue 1 of an appropriate propagator of the decaying (unforced) passive field, or equivalently, the zero modes of a related operator. In this paper we investigate further this surprising finding via two examples, one akin to turbulent convection in which the temperature is the active scalar, and the other akin to magneto-hydrodynamics in which the magnetic field is the active vector. In the first example, all the even correlation functions of the active and passive fields exhibit identical scaling behavior. The second example appears at first sight to be a counter-example: the statistical objects of the active and passive fields have entirely different scaling exponents. We demonstrate nevertheless that the Statistically Preserved Structures of the passive vector dominate again the statistics of the active field, except that due to a dynamical conservation law the amplitude of the leading zero mode cancels exactly. The active vector is then dominated by the sub-leading zero mode of the passive vector. Our work thus suggests that the statistical properties of active fields in turbulence can be understood with the same generality as those of passive fields.Comment: 13 pages, 13 figures, submitted to Phys. Rev.

Universality and saturation of intermittency in passive scalar turbulence

The statistical properties of a scalar field advected by the non-intermittent Navier-Stokes flow arising from a two-dimensional inverse energy cascade are investigated. The universality properties of the scalar field are directly probed by comparing the results obtained with two different types of injection mechanisms. Scaling properties are shown to be universal, even though anisotropies injected at large scales persist down to the smallest scales and local isotropy is not fully restored. Scalar statistics is strongly intermittent and scaling exponents saturate to a constant for sufficiently high orders. This is observed also for the advection by a velocity field rapidly changing in time, pointing to the genericity of the phenomenon. The persistence of anisotropies and the saturation are both statistical signatures of the ramp-and-cliff structures observed in the scalar field.Comment: 4 pages, 8 figure

Completeness of classical spin models and universal quantum computation

We study mappings between distinct classical spin systems that leave the partition function invariant. As recently shown in [Phys. Rev. Lett. 100, 110501 (2008)], the partition function of the 2D square lattice Ising model in the presence of an inhomogeneous magnetic field, can specialize to the partition function of any Ising system on an arbitrary graph. In this sense the 2D Ising model is said to be "complete". However, in order to obtain the above result, the coupling strengths on the 2D lattice must assume complex values, and thus do not allow for a physical interpretation. Here we show how a complete model with real -and, hence, "physical"- couplings can be obtained if the 3D Ising model is considered. We furthermore show how to map general q-state systems with possibly many-body interactions to the 2D Ising model with complex parameters, and give completeness results for these models with real parameters. We also demonstrate that the computational overhead in these constructions is in all relevant cases polynomial. These results are proved by invoking a recently found cross-connection between statistical mechanics and quantum information theory, where partition functions are expressed as quantum mechanical amplitudes. Within this framework, there exists a natural correspondence between many-body quantum states that allow universal quantum computation via local measurements only, and complete classical spin systems.Comment: 43 pages, 28 figure

Stochastic attractors for shell phenomenological models of turbulence

Recently, it has been proposed that the Navier-Stokes equations and a relevant linear advection model have the same long-time statistical properties, in particular, they have the same scaling exponents of their structure functions. This assertion has been investigate rigorously in the context of certain nonlinear deterministic phenomenological shell model, the Sabra shell model, of turbulence and its corresponding linear advection counterpart model. This relationship has been established through a "homotopy-like" coefficient $\lambda$ which bridges continuously between the two systems. That is, for $\lambda=1$ one obtains the full nonlinear model, and the corresponding linear advection model is achieved for $\lambda=0$. In this paper, we investigate the validity of this assertion for certain stochastic phenomenological shell models of turbulence driven by an additive noise. We prove the continuous dependence of the solutions with respect to the parameter $\lambda$. Moreover, we show the existence of a finite-dimensional random attractor for each value of $\lambda$ and establish the upper semicontinuity property of this random attractors, with respect to the parameter $\lambda$. This property is proved by a pathwise argument. Our study aims toward the development of basic results and techniques that may contribute to the understanding of the relation between the long-time statistical properties of the nonlinear and linear models

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