The theory of fully developed turbulence is usually considered in an idealized homogeneous and isotropic state. Real turbulent flows exhibit the effects of anisotropic forcing. The analysis of correlation functions and structure functions in isotropic and anisotropic situations is facilitated and made rational when performed in terms of the irreducible representations of the relevant symmetry group which is the group of all rotations SO(3). In this paper we firstly consider the needed general theory and explain why we expect different (universal) scaling exponents in the different sectors of the symmetry group. We exemplify the theory context of isotropic turbulence (for third order tensorial structure functions) and in weakly anisotropic turbulence (for the second order structure function). The utility of the resulting expressions for the analysis of experimental data is demonstrated in the context of high Reynolds number measurements of turbulence in the atmosphere.

We analyze turbulent velocity signals in the atmospheric surface layer, obtained by pairs of probes separated by inertial-range distances parallel to the ground and (nominally) orthogonal to the mean wind. The Taylor microscale Reynolds number ranges up to 20 000. Choosing a suitable coordinate system with respect to the mean wind, we derive theoretical forms for second order structure functions and fit them to experimental data. The effect of flow anisotropy is small for the longitudinal component but significant for the transverse component. The data provide an estimate for a universal exponent from among a hierarchy that governs the decay of flow anisotropy with the scale size.[S0031-9007(98)07959-9] PACS numbers: 47.27. Gs, 05.40. + j, 47.27.Jv, 94.10.Jd Experimental studies of turbulent flows at very high Reynolds numbers are usually limited in the sense that one measures the velocity field at a single spatial point as a function of time [1], and uses Taylor's hypothesis to identify velocity increments at different times with those across spatial length scales, R. The standard outputs of such measurements are the longitudinal twopoint differences of the Eulerian velocity field and their moments, termed structure functions:where ͗?͘ denotes averaging over time. In homogeneous and isotropic turbulence, these structure functions are observed to vary as a power law in R, S n ͑R͒ ϳ R z n , with apparently universal scaling exponents z n [2]. [4] begins to offer information about the tensorial nature of structure functions. Ideally, one would like to measure the tensorial nth order structure functions defined as S a 1 ···a n n ͑R͒ ϵ ͓͗u a 1 ͑r 1 R͒ 2 u a 1 ͑r͔͒ 3 ͓u a 2 ͑r 1 R͒ 2 u a 2 ͑r͔͒ · · · ͓u a n ͑r 1 R͒ 2 u a n ͑r͔͒͘ , Recent progress in measurements [3] and in simulationswhere the superscript a i indicates the velocity component in the direction i. Such information should be useful in studying the anisotropic effects induced by all practical means of forcing.In analyzing experimental data the model of "homogeneous and isotropic small scale" is universally adopted, but it is important to examine the relevance of this model for realistic flows. One of the points of this Letter is that keeping the tensorial information helps significantly in disentangling different scaling contributions to structure functions [5]. Especially when anisotropy might lead to different scaling exponents for different tensorial components, a careful study of the various contributions is needed. We will show below that atmospheric measurements contain important anisotropic contributions to one type of transverse structure functions.We analyze measurements in atmospheric turbulence at heights of 6 and 35 m above the ground (data sets I and II). Set I was acquired over a flat desert with a long fetch, and the Taylor microscale Reynolds number was about 10 000. Set II was acquired over a rough terrain with illdefined fetch, and the microscale Reynolds number was 20 000 [6]. The data were acquired simultaneously from two single hot-...

Recovering an unknown Hamiltonian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here, we show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling linearly with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian for a long time, allowing to recover a large family of time-dependent Hamiltonians. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.

We address scaling in inhomogeneous and anisotropic turbulent flows by decomposing structure functions into their irreducible representation of the SO(3) symmetry group which are designated by j, m indices. Employing simulations of channel flows with Re l ഠ 70 we demonstrate that different components characterized by different j display different scaling exponents, but for a given j these remain the same at different distances from the wall. The j 0 exponent agrees extremely well with high Re measurements of the scaling exponents, demonstrating the vitality of the SO (3) decomposition. [S0031-9007(99)09384-9] PACS numbers: 47.27.EqMost of the available data analysis and theoretical thinking about the universal statistics of the small scale structure of turbulence assume the existence of an idealized model of homogeneous and isotropic flows. In fact most realistic flows are neither homogeneous nor isotropic. Accordingly, one can analyze the data pertaining to such flows in two ways. The traditional one has been to disregard the inhomogeneity and anisotropy and proceed with the data analysis assuming that the results pertain to the homogeneous and isotropic flows. The second, which is advocated in this Letter, is to take the anisotropy explicitly into account, to carefully decompose the relevant statistical objects into their isotropic and anisotropic contributions, and to assess the degree of universality of each component separately. We analyze here direct numerical simulations (DNS) of a channel flow with Re l ഠ 70 [1-3]. The main conclusion of this Letter is that this procedure is unavoidable; in particular, it highlights the universality of the scaling exponents of the isotropic sector which are presumably those governing the universal small scale statistics at very high Reynolds numbers. In agreement with recent studies of this subject [4,5] we report that different irreducible representations of the symmetry group (characterized by indices j, m) exhibit scalar functions that scale with apparently universal exponents that differ for different j. The exponents found at low values of the Reynolds number for the j 0 (isotropic) sector are in excellent agreement with high Re results; these exponents are invariant to the position in the inhomogeneous flow, leading to reinterpretation of recent findings of position dependence as resulting from the intervention of the anisotropic sectors. The latter have nonuniversal weights that depend on the position in the flow.We consider here channel flow simulations on a grid of 256 points in the streamwise directionx, and ͑128 3 128͒ in the other two directions,ŷ,ẑ. We denote bŷ z the direction perpendicular to the walls and byŷ the spanwise direction in planes parallel to the walls. We employ periodic boundary conditions in the spanwise and streamwise directions and no-slip boundary conditions on the walls. The Reynolds number based on the Taylor scale is Re l ഠ 70 in the center of the channel ͑z 64͒. The simulation is fully symmetric with respect to the central plane. The f...

Abstract:One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n O(log n) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time isÕ(nM(n)), where M(n) is the time required to multiply two n × n matrices.