Turbulence noise

We show that the large-eddy motions in turbulent fluid flow obey a modified hydrodynamic equation with a stochastic turbulent stress whose distribution is a causal functional of the large-scale velocity field itself. We do so by means of an exact procedure of statistical filtering of the Navier-Stokes equations, which formally solves the closure problem, and we discuss the relation of our analysis with the decimation theory of Kraichnan. We show that the statistical filtering procedure can be formulated using field-theoretic path-integral methods within the Martin-Siggia-Rose (MSR) formalism for classical statistical dynamics. We also establish within the MSR formalism a least-effective-action principle for mean turbulent velocity profiles, which generalizes Onsager's principle of least dissipation. This minimum principle is a consequence of a simple realizability inequality and therefore holds also in any realizable closure. Symanzik's theorem in field theory—which characterizes the static effective action as the minimum expected value of the quantum Hamiltonian over all state vectors with prescribed expectations of fields—is extended to MSR theory with non-Hermitian Hamiltonian. This allows stationary mean velocity profiles and other turbulence statistics to be calculated variationally by a Rayleigh-Ritz procedure. Finally, we develop approximations of the exact Langevin equations for large eddies, e.g., a random-coupling DIA model, which yield new stochastic LES models. These are compared with stochastic subgrid modeling schemes proposed by Rose, Chasnov, Leith, and others, and various applications are discussed.     

Determining Liquid Structure from the Tail of the Direct Correlation Function

In important early work, Stell showed that one can determine the pair correlation function h(r) of the hard-sphere fluid for all distances r by specifying only the tail of the direct correlation function c(r) at separations greater than the hard-core diameter. We extend this idea in a very natural way to potentials with a soft repulsive core of finite extent and a weaker and longer ranged tail. We introduce a new continuous function T(r) which reduces exactly to the tail of c(r) outside the (soft) core region and show that both h(r) and c(r) depend only on the out projection of T(r): i.e., the product of the Boltzmann factor of the repulsive core potential times T(r). Standard integral equation closures can thus be reinterpreted and assessed in terms of their predictions for the tail of c(r) and simple approximations for its form suggest new closures. A new and very efficient variational method is proposed for solving the Ornstein–Zernike equation given an approximation for the tail of c. Initial applications of these ideas to the Lennard-Jones and the hard-core Yukawa fluid are discussed.     

Stability of general relativistic gaseous masses and variational principles

The Einstein field equations for a self-gravitating fluid that obeys an equation of state of the formp=p(w),p the pressure andw the energy density may be derived from a variational principle. The perturbations of the metric tensor and the fluid dynamic variables satisfy equations which may be derived from a related variational principle, namely the principle associated with the second variation problem. It is shown that the variational principle given by Chandrasekhar from which a sufficient criterion may be obtained for deciding when a self gravitating spherical gaseous mass is unstable against spherically symmetric perturbations is that given by the second variation problem. It is further shown that this criterion is equivalent to requiring that the integral entering into the second variation be negative. The latter form of the criterion may be used in general situations.This work was supported in part by the United States Atomic Energy Commission under contract number AT (11-1)-34, Project Agreement Number 125.     

Local observables and particle statistics I

We consider the family of those states which become asymptotically indistinguishable from the vacuum for observations in far away regions of space. The pure states of this family may be subdivided into superselection sectors labelled by generalized charge quantum numbers. The principle of locality implies that within this family one may define a natural product composition (leading for instance from single particle states ton-particle states). Intrinsically associated with then-fold product of states of one sector there is a unitary representation ofP ⁽ⁿ⁾, the permutation group ofn elements, analogous in its role to that arising in wave mechanics from the permutations of the arguments of ann-particle wave function. We show that each sector possesses a statistics parameter which determines the nature of the representation ofP ⁽ⁿ⁾ for alln and whose possible values are 0, ±d ^–1 (d a positive integer). A sector with 0 has a unique charge conjugate (antiparticle states); if =d ^–1 the states of the sector obey para-Bose statistics of orderd, if =–d ^–1 they obey para-Fermi statistics of orderd. Some conditions which restrict to ± 1 (ordinary Bose or Fermi statistics) are given.     

Most Probable Histories for Nonlinear Dynamics: Tracking Climate Transitions

The effective action provides an appropriate cost function to determine most probable (or optimal) histories for nonlinear dynamics with strong noise. In such strong-coupling problems, a nonperturbative technique is required to calculate the effective action. We have proposed a Rayleigh–Ritz variational approximation, which employs simple moment-closures or intuitive guesses of the statistics to calculate the effective action. We consider here an application to climate dynamics, within a simple bimodal Langevin model similar to that proposed by C. Nicolis and G. Nicolis [Tellus 33:225 (1981)]. Capturing climate state transitions even in this simple model is known to present a serious problem for standard methods of data assimilation. In contrast, it is shown that the effective action for the climate history is already well-approximated by a one-moment closure and that the optimal, minimizing history robustly tracks climate change, even with large observation errors. Furthermore, the Hessian of the effective action provides the ensemble variance as a realistic measure of confidence level in the predicted optimal history.     

The lattice deformation and the energy of antiphase boundaries in Fe-Si alloys

The lattice deformation across the antiphase boundary and the energy of both types (a/2111 anda(100) of antiphase boundaries lying in {110} plane are calculated using a series of three interatomic potentials fitted to experimental data. It is shown that the relaxation of atomic planes in the vicinity of antiphase boundary is important for thea/2111 antiphase boundary and is negligible for a 100 antiphase boundary in the DO₃ structure.The author is grateful to Dr. F.Kroupa and to Dr. A.Gemperle for valuable discussions.     

Transfer Matrix Spectrum and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We obtain new properties of general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high-temperature region (1). Each model is characterized by a single-site a priori spin distribution, taken to be even. We state our results in terms of the parameter =s ⁴–3s ²², where s ^k denotes the kth moment of the a priori distribution. Associated with the model is a lattice quantum field theory which is known to contain particles. We show that for >0, small, there exists a bound state with mass below the two-particle threshold. The existence of the bound state has implications for the decay of correlations, i.e., the 4-point functions decay at a slower rate than twice that of the 2-point function. These results are obtained using a lattice version of the Bethe–Salpeter equation. The existence results generalize to N-component models with rotationally invariant a priori spin distributions.     

Intermittency in the Navier-Stokes dynamics

Intermittency effects in high Reynolds number turbulence (Re10³–10⁶) are calculated from the Navier-Stokes equation in Fourier-Weierstrass approximation. First, the probability density functions (PDF) of scale resolved turbulent signals is found to be Gaussian for large scales, whereas for smaller scales the PDF changes (in agreement with experiment) to a more and more stretched exponential type. This is due to intermittent small scale fluctuations which are caused by the competition between turbulent energy transfer downscale and viscous energy loss. Second, we calculate the moments of ther-averaged energy dissipation rate _r (x) and theirr-scaling exponents (m/3). Our results well agree with experiment and numerical simulations of the full Navier-Stokes equations ((2)=0.29±0.02). We analytically show that the common identification between the (m/3) and the corrections (m) to classical scaling of the velocity structure functions (Kolmogorov's refined similarity hypothesis) is doubtful, because even Gaussian ₁ u ₁-PDFs (characterizing non intermittent flow) lead to (m/3)0.     

New Aspects in Physics on Gel'fand Triplets

New observations in Gel'fand triplets are studied. An interesting one is vortex phenomena that stem from zero energy solutions of two-dimensional Schrödinger equations with central potentials V() ^r (² = x ² + y ² and r –2), which are eigenstates of conjugate spaces of Gel'fand triplets. The zero energy solutions for all the potentials V() are shown to have the same structure with infinite degeneracy by making use of the conformal transformation = z with z = x + iy. The infinite degeneracy is observed as variety of vortex patterns in real physical phenomena. Some simple vortex patterns such as vortex lines and vortex lattices are presented. Such a new freedom on the Gel'fand triplets can be treated in a statistical mechanics. In the theory a new entropy being different from the so-called Boltzmann entropy appears. Transitions between the two entropies occur in thermal nonequilibrium phenomena, where energy emissions are observed.     

Photon counting statistics of the superposition of coherent and chaotic light of arbitrary spectrum passed through the turbulent atmosphere or a Gaussian medium

We derive the approximate photon counting distribution and its factorial moments of arbitrary order for the superposition of coherent and chaotic light of arbitrary spectrum and for arbitrary counting time intervals passing through the turbulent atmosphere or a Gaussian scattering medium; the frequency of coherent light and the mean frequency of chaotic light need not coincide. It is shown by discussion of numerical results for various levels of turbulence and various values of the parameter, which is the halfwidth of the Lorentzian spectrum of chaotic light multiplied by the detection time, that peaks of the photon counting distributions are decreasing (the distributions are broadening) with increasing level of turbulence, they are shifted to lower number of counts and values of the normalized factorial moments are rapidly increasing; these values tend to a limit given by the level of turbulence for.Leninova 26, Olomouc, Czechoslovakia.The authors thank Dr. Z. Braunerová of the Computer Centre of the Palacký University for performing the numerical calculations.     

Transfer matrix analysis of the two-dimensional ANNNI model

The two-dimensional axial next-nearest neighbour Ising (ANNNI) model is studied for the first time by a transfer matrix method for semi-infinite strips (N×;N13). One obtains the thermodynamic properties and, more important, the correlation functions, which display directly the various magnetic structures. In one part of the phase diagram where a disorder line is believed to follow closely the ferromagnetic transition line the critical exponents appear to deviate from the expected ones. ForN13 a variational procedure is proposed.     

N-fold joint photocounting distribution for modulated laser radiation: Transmission through the turbulent atmosphere

In this paper a generalised result for theN-fold joint photoelectron counting distribution for independently modulated radiation is given. We extend the recent results of Diament and Teich, for the one-fold photoelectron counting distribution for light propagated through an atmosphere characterised by log-normal irradiance fluctuations, to theN-fold joint photoelectron counting distribution. An approximate solution for thisN-fold distribution is obtained, for detection intervals {T_i} « _a where _a is the characteristic time of the atmospheric turbulence. We present specifically the two-fold joint photocounting distribution for amplitude-stabilised laser radiation passing through such an atmosphere for several levels of turbulence and degrees of correlation. Cases including additive, independent, non-interfering Poisson noise are considered. Computer generated plots of the photocounting distribution are presented. For noise-free detection, the otherwise narrow-peaked photocounting distribution is seen to broaden markedly and shift its peak to lower counts as the turbulence level increases. Furthermore, a non-singular counting distribution is obtained for fully correlated detection. In the presence of additive noise and varying only the signal-to-noise ratio, the probability surface is intermediate between that of the Poisson and that of the noise-free log-normal fading counting distribution. The peak, however, is observed to decrease and then again increase in magnitude as , for correlated detection only. These results are expected to be of use in the study of atmospheric turbulence, as well as in the evaluation of certain stochastic functionals that occur in optical communication theory for the turbulent atmospheric channel.This work was supported in part by the US National Science Foundation under Grant Number NSF-GK-16649.     

Relativistic mass distribution in event-anti-event system and “realistic” equation of state for hot hadronic matter

We find the equation of state p, T ⁶,which gives the value of the sound velocity c ²7 = 0.20,in agreement with the realistic equation of state for hot hadronic matter suggested by Shuryak, in the framework of a covariant relativistic statistical mechanics of an event-anti-event system with small chemical and mass potentials. The relativistic mass distribution for such a system is obtained and shown to be a good candidate for fitting hadronic resonances, in agreement with the phenomenological models of Hagedorn, Shuryak, et al. This distribution provides a correction to the value of specific heat 3/2,of the order of 5.5%,at low temperatures.     

Analogy between the Lorenz strange attractor and a bistable stochastic oscillator

The relation between the aperiodic solution of the Lorenz model and that of a stochastic anharmonic oscillator is explored. The stochastic oscillator is constructed by replacing (t) in the Lorenz model by a stochastic variable(t) of specified statistics. The resulting system is of course not isomorphic to the Lorenz model, but does share with it a number of statistical properties. Thus, within the confines of these measures the two systems are physically very similar.     

Generalized classical theory of magnetism

We consider an Ising model with Kac potential ^dK(¦x¦) which may have arbitrary sign, and show, following Gates and Penrose, that the free energy in the classical limit0+ can be obtained from a variational principle. When the Fourier transform of the potential has its maximum atp=0 one recovers the usual mean-field theory of magnetism. When the maximum occurs forp ₀0, however, one obtains an oscillatory or helicoidal phase in which the magnetization near the critical point oscillates with period 2/¦p ₀¦. An example with a potential possessing parameter-dependent oscillations is shown to exhibit crossover phenomena and a multicritical Lifshitz point in the classical limit.     

Variational theory for correlated lattice fermions in high dimensions

A diagrammatic approach to the evaluation of correlated variational wave functions for strongly interacting fermions is presented. Diagrammatic rules for the calculation of the one-particle density matrix and the Hubbard interaction are derived which are valid for arbitraryd-dimensional lattices. An exact evaluation of expectation values is performed in the limitd=. The wellknown Gutzwiller approximation is seen to become the exact result for the expectation value of the Hubbard Hamiltonian in terms of the Gutzwiller wave function ind=. An efficient procedure to correct the Gutzwiller approximation in finite dimensions is developed. A detailed discussion of expectation values ind= in terms of explicit antiferromagnetic wave functions is given. Thereby an approximate result for the ground state energy of the Hubbard model, obtained recently within a slave-boson approach, is recovered.     

Dissipations and derivations

We show a usefulness of the notion of dissipative operators in the study of derivations ofC*-algebras and prove that the closure of a normal *-derivation of UHF algebra satisfying a special condition is a generator of a one-parameter group of *-automorphisms.     

On learning and energy-entropy dependence in recurrent and nonrecurrent signed networks

Learning of patterns by neural networks obeying general rules of sensory transduction and of converting membrane potentials to spiking frequencies is considered. Any finite number of cellsA can sample a pattern playing on any finite number of cells without causing irrevocable sampling bias ifA = orA =. Total energy transfer from inputs ofA to outputs of depends on the entropy of the input distribution. Pattern completion on recall trials can occur without destroying perfect memory even ifA = by choosing the signal thresholds sufficiently large. The mathematical results are global limit and oscillation theorems for a class of nonlinear functional-differential systems.The preparation of this work was supported in part by the National Science Foundation (GP 9003), the Office of Naval Research (N00014-67-A-024-OQ16), and the A.P. Sloan Foundation.     

F-Information, a Unitless Variant of Fisher Information

A new information matrix [F] with elements F _mn = (y _m - a _m )(y _n - a n) ( ln p(y | a)/a _m ) ( ln p(y | a)/a _n ) is analyzed. The PDF p(y | a) is the usual likelihood law. [F] differs from the Fisher information matrix by the presence of the first two factors in the given expectation. These factors make F _mn unitless, in contrast with the Fisher information. This lack of units allows F _mn values from entirely different phenomena to be compared as, for example, Shannon information values can be compared. Each element F _mn defines an error inequality analogous to the Cramer-Rao inequality. In the scalar case F _mn F, for a normal p(y|a) law F = 3, while for an exponential law F = 9. A variational principle F = min (called FMIN) allows an unknown PDF p(x) to be estimated in the presence of weak information. Under certain conditions F obeys a Boltzmann F-theorem F/t 0, indicating that F is a physical entropy. Finally, the trace of [F] may be used as the scalar information quantity in an information-based principle for deriving distribution laws p of physics.     

Mach's principle and the reciprocal symmetry of nature

This article develops the postulate that spacetime-charge inversion invariance reflects a fundamental reciprocal symmetry in nature between the two long range forces, from which the derivation of Mach's principle (i.e., the principle that the fundamental parameters of the electromagnetically elementary charged particle are related to those describing the electromagnetically observable universe) follows quite easily. Interpreting this result, it is argued that relativity and quantum mechanics can be made conceptually compatible and mathematically consistent by this reciprocal symmetry if one realizes that relativity isboth a macroscopic, semiclassical theory (i.e., the global half of relativity, described by Eq. (1.1), including special and general relativity) and a microscopic theory (i.e., the local half of relativity, described by Eq. (2.1), including relativistic quantum mechanics and field theory). The reciprocal symmetry of nature, then, promises unique (differential and/or integral) relationships between the coordinate variables of the observers of these tworeciprocally related theories, which implies unique, consistent numerical values for the scalar curvatureR, the massM, and the critical density for closure, _c, of the observable universe [derived from the elementary particle parameters (i.e., the electron mass and Coulomb radius)]. With this symmetry we also postulate a plausible mechanism for spontaneous generation of matter from the ubiquitous (zero-mass ether) nothingness of the Dirac sea of filled negative energy states, and can consistently interpret both the positive and negative-energy state solutions of Dirac's equation for massive, spin-1/2 (i.e., fermion) particles and both the advanced and retarded potential solutions of electromagnetic field equations. It is pointed out that, with this interpretation of the advanced potential solutions from electromagnetic field theory, one can actuallyderive causality from electromagnetic theory.