A Comment on the Letter by Keiko M. Aoki and Tetsuo Akiyama, Phys. Rev. Lett. 77, 4166 (1996). The authors of the Letter offer a Reply.

A Reply to the Comment by C. Bizon et al.

The Hamiltonian dynamics of the classical planar Heisenberg model is numerically investigated in two and three dimensions. In three dimensions peculiar behaviors are found in the temperature dependence of the largest Lyapunov exponent and of other observables related to the geometrization of the dynamics. On the basis of a heuristic argument it is conjectured that the phase transition might correspond to a change in the topology of the manifolds whose geodesics are the motions of the system.

For many-body Fermi systems we determine the dependence of the Breit-Wigner width $\gamma$ and inverse participation ratio $\xi$ on interaction strength $U\ge U_c$ and energy excitation $\delta E\ge \delta E_{\mathrm{ch}}$ when a crossover from Poisson to Wigner-Dyson $P\left(s\right)$ statistics takes place. At $U\ge U_c$ the eigenstates are composed of a large number of noninteracting states and even for $U<\mathrm{}{U}_c$ there is a regime where $P\left(s\right)$ is close to the Poisson distribution but $\xi \gg 1$.

We present results of experiments on the transport of light through thin random media. Total transmission and time-resolved propagation measurements were performed using strongly scattering samples of varying thickness $L$. For $L/\ell <8\mathrm{}$, where $\ell$ is the transport mean free path, the observed decay times from the long-time exponential behavior exhibit strong deviations from diffusion theory and radiative transport theory, whereas the total transmission measurements do not. For the thinnest sample $\left(L/\ell \simeq 2\right)$ a reduction of the diffusion coefficient with a factor of 2 was observed.

The cross-stream dimension of a horizontal, thermally stratified intrusion flow, that travels underneath an insulating wall into a heavier, isothermal fluid, is investigated experimentally. The observed development of a regular temperature variation along that third dimension is linked to a Rayleigh-Bénard-like instability, which is caused by the fact that the upper part of the intrusion flow is effectively heated from below through internal heat advection. These results can be understood by linear stability calculations and contradict the assumption of all previous investigations that the flow is essentially two dimensional.

The plane Couette flow displays a globally subcritical transition to turbulence. When forced by a thin wire introduced in the central plane, the basic flow bifurcates towards a new state of streamwise vortices which break down through a complex spatiotemporal regime. When a bead is introduced in the central plane, only a few vortices are generated. Their destabilization occurs via a new type of temporal intermittency hard to understand within the framework of dynamical systems.

A surfactant driven instability is observed in the air-liquid interface of an aqueous surfactant solution moving inside a prewetted glass Hele-Shaw cell with a gap width of $300\mu \mathrm{m}$ when the viscous surfactant solution is pushing on the less viscous air. This is the reverse direction of the well-known Saffman-Taylor instability. The fluctuations produced by the instability will develop into a steady state cellular pattern with the sharp fingers pointing in the opposite direction of the motion of the interface. This instability occurs only when the velocity of the interface exceeds a critical value which depends on the geometry of the cell and the wetting layer in the air side of the interface.

Modeling transport in a disordered medium with broadly distributed heterogeneities is often hampered by the fact that a very large set of equations has to be solved if the model contains detailed information about the medium's morphology. We propose a novel method, based on the application of the wavelets, that drastically reduces the number of equations to be solved without sacrificing any important information about the medium's morphology. The method is applicable to any disordered medium with any types of heterogeneities.

Strange nonchaotic attractors (SNA) arise in quasiperiodically driven systems in the neighborhood of a saddle-node bifurcation whereby a strange attractor is replaced by a periodic (torus) attractor. This transition is accompanied by Type-I intermittency. The largest nontrivial Lyapunov exponent $\Lambda$ is a good order parameter for this route from chaos to SNA to periodic motion: the signature is distinctive and unlike that for other routes to SNA. In particular, $\Lambda$ changes sharply at the SNA to torus transition, as does the distribution of finite-time or $N$-step Lyapunov exponents, $P\left({\Lambda }_N\right)$.

We use a recent result to show that the rate of loss of coherence of a quantum system increases with increasing system phase-space structure and that a chaotic quantal system in the semiclassical limit decoheres exponentially with rate $2{\lambda }_2$, where ${\lambda }_2$ is a generalized Lyapunov exponent. As a result, for example, the dephasing time for classically chaotic systems goes to infinity logarithmically with the temperature, in accord with recent experimental results.

The peculiar spatiotemporal profile of an optical realization of the nonspreading axisymmetric $X$ wave has been recorded. For treatment of the experiment a simple representation of $X$-type waves and their correlation with plane waves has been introduced, which covers also a broader class of incoherent wave fields.

We study experimentally the role played by cavity boundary conditions in the onset of modulational instability in nonlinear and dispersive waves. This study is performed by means of an externally driven optical cavity made of a single-mode silica fiber. Spectral measurements of the intracavity field allow us to identify modulational instability in the presence of normal dispersion. This finding demonstrates that cavity boundary conditions fundamentally alter the physical mechanisms which are at the origin of modulational instability.

We predict phase transitions in the quantum noise characteristics of systems described by the quantum nonlinear Schrödinger equation, showing them to be related to the solitonic field transition at half the fundamental soliton amplitude. These phase transitions are robust with respect to Raman noise and scattering losses. We also describe the rich internal quantum noise structure of the solitonic fields in the vicinity of the phase transition. For optical coherent quantum solitons, this leads to the prediction that eliminating the peak sideband noise due to the electronic nonlinearity of silica fiber by spectral filtering leads to the optimal photon-number noise reduction of a fundamental soliton.

We present quantitative measurements of the interaction between a guided optical wave and a two-dimensional photonic crystal using spontaneous emission of the material as an internal point source. This is the first analysis at near-infrared wavelengths where transmission, reflection, and in-plane diffraction are quantified at the same time. Low transmission coincides with high reflection or in-plane diffraction, indicating that the light remains guided upon interaction. Also, good qualitative agreement is found with a two-dimensional simulation based on the transfer matrix method.

We detected thermal radiation forces produced by heat flow through solid slabs suspended in nonisothermal liquids. Lateral effects are avoided, allowing a unidimensional approach. The results confirm expectations. Numerically simulated gravity-dependent perturbations are smaller than the measured forces. More precise measurements will be attempted on a European Space Agency suborbital flight in 1998.

A reduced description of disturbances on marginally stable shear flows is presented. The system is the Boussinesq equation coupled to a critical-layer, vorticity equation.

We show that the tails of the single-point velocity probability distribution function (PDF) are generally non-Gaussian in developed turbulence. By using instanton formalism for the randomly forced Navier-Stokes equation, we establish the relation between the PDF tails of the velocity and those of the external forcing. In particular, we show that a Gaussian random force having correlation scale $L$ and correlation time τ produces velocity PDF tails $\ln P\left(v\right)\propto {-v}^4$ at $v\gg v_{\mathrm{rms}},L/\tau$. For a short-correlated forcing when $\tau \ll {L/v}_{\mathrm{rms}}$ there is an intermediate asymptotics $\ln P\left(v\right)\propto {-v}^3$ at $L/\tau \gg v\gg v_{\mathrm{rms}}$.

We report the first clear experimental evidence of a stationary two-dimensional inverse energy cascade. The experiments are performed in electromagnetically driven flows, using thin, stably stratified layers; measurements of the instantaneous velocity fields show that the fluctuations are homogeneous and isotropic in the inertial range. The energy spectrum displays the expected $k^{-5\mathrm{}/3}$ law with a Kolmogorov constant lying in the range 5.5–7, which is consistent with the current numerical estimates.

We present results from direct numerical simulations of the Kraichnan model for passive scalar advection by a rapidly varying random scaling velocity field for intermediate values of the velocity scaling exponent. These results are compared with the scaling exponents predicted for this model by Kraichnan. Further, we test the recently proposed fusion rules which govern the scaling properties of multipoint correlations, and present results on the linearity of the conditional statistics of the Laplacian operator on the scalar field.