Directed waves in random media

Dynamic criticality far from equilibrium: One-loop flow of Burgers-Kardar-Parisi-Zhang systems with broken Galilean invariance

Burgers-Kardar-Parisi-Zhang (KPZ) scaling has recently (re-) surfaced in a variety of physical contexts, ranging from anharmonic chains to quantum systems such as open superfluids, in which a variety of random forces may be encountered and/or engineered. Motivated by these developments, we here provide a generalization of the KPZ universality class to situations with long-ranged temporal correlations in the noise, which purposefully break the Galilean invariance that is central to the conventional KPZ solution. We compute the phase diagram and critical exponents of the KPZ equation with 1/f noise (KPZ1/f) in spatial dimensions 1≤d<4 using the dynamic renormalization group with a frequency cutoff technique in a one-loop truncation. Distinct features of KPZ1/f are (i) a generically scale-invariant, rough phase at high noise levels that violates fluctuation-dissipation relations and exhibits hyperthermal statistics even in d=1, (ii) a fine-tuned roughening transition at which the flow fulfills an emergent thermal-like fluctuation-dissipation relation, that separates the rough phase from (iii) a massive phase in 1<d<4 (in d=1 the interface is always rough). We point out potential connections to nonlinear hydrodynamics with a reduced set of conservation laws and noisy quantum liquids.

Two-frequency mutual coherence function and pulse propagation in random media

In this work an analysis of transient wave propagation in forward scattering random media is presented. The analysis is based on evaluation of the two-frequency mutual coherence function, which is an important quantity in itself since it provides a measure of the coherence bandwidth. The coherence function is calculated by using the path integral technique; specifically, by resorting to a cumulant expansion of the path integral. In contrast to the formulas available in the literature, the solution obtained is not limited by the strength of disorder and applies equally well to both dispersive and nondispersive media, with arbitrary spectra of inhomogeneities. For the regime of weak scattering (or relatively short propagation distances) the first cumulant gives an excellent approximation coinciding with the results obtained earlier in a particular case of the Kolmogorov turbulence by solving the corresponding differential equation numerically. In the regime of strong scattering (long distances), which to our knowledge has not been covered previously, our solution demonstrates a different type of scaling dependence. It is shown that, even for power spectra with fractal behavior in a wide range of spatial frequencies, the coherence function is very sensitive to fine details of the spectrum at both small and large spatial scales. Using the cumulant expansion, the temporal moments of the pulsed wave propagating in a random medium are also considered. It is found that the temporal moments of the pulse are determined exactly by accounting for a corresponding number of the cumulants. In particular, the average time delay of the pulse is determined by the first cumulant, and the pulse width is obtained by accounting for the first two cumulants. Although the consideration of the problem is based on the model of a continuous medium, the results are also applicable to wave propagation in media containing discrete particles scattering predominantly in the forward direction.

Dynamics and critical behavior of the q model

The q model, a random walk model rich in behavior and applications, is investigated. We introduce and motivate the q model via its application proposed by Coppersmith et al. to the flow of stress through granular matter at rest. For a special value of its parameters the q model has a critical point that we analyze. To characterize the critical point we imagine that a uniform load has been applied to the top of the granular medium and we study the evolution with depth of fluctuations in the distribution of load. Close to the critical point explicit calculation reveals that the evolution of load exhibits scaling behavior analogous to thermodynamic critical phenomena. The critical behavior is remarkably tractable: the harvest of analytic results includes scaling functions that describe the evolution of the variance of the load distribution close to the critical point and of the entire load distribution right at the critical point, values of the associated critical exponents, and determination of the upper critical dimension. These results are of intrinsic interest as a tractable example of a random critical point. Of the many applications of the q model, the critical behavior is particularly relevant to network models of river basins, as we briefly discuss. Finally we discuss circumstances under which quantum network models that describe the surface electronic states of a quantum Hall multilayer can be mapped onto the classical q model. For mesoscopic multilayers of finite circumference the mapping fails; instead a mapping to a ferromagnetic supersymmetric spin chain has proved fruitful. We discuss aspects of the superspin mapping and give an elementary derivation of it making use of operator rather than functional methods.

Effect of noise on a quantum bound state

The evolution of a quantum wave function on a lattice with time-dependent random disorder is studied. It is found that the bound state wave function of a quantum system exhibits intrinsic instability with respect to noisy disorder in the statistical sense. Under the general scheme of time coarse graining, we show nonperturbatively that the stationary attractive potential is completely washed out at large time scales by the existence of the noisy disorder to any order of density correlations.