Ballistic aggregation for one-sided Brownian initial velocity

We study the one-dimensional ballistic aggregation process in the continuum limit for one-sided Brownian initial velocity (i.e. particles merge when they collide and move freely between collisions, and in the continuum limit the initial velocity on the right side is a Brownian motion that starts from the origin $x=0$). We consider the cases where the left side is either at rest or empty at $t=0$. We derive explicit expressions for the velocity distribution and the mean density and current profiles built by this out-of-equilibrium system. We find that on the right side the mean density remains constant whereas the mean current is uniform and grows linearly with time. All quantities show an exponential decay on the far left. We also obtain the properties of the leftmost cluster that travels towards the left. We find that in both cases relevant lengths and masses scale as $t^2$ and the evolution is self-similar.Comment: 18 pages, published in Physica

The global picture of self-similar and not self-similar decay in Burgers Turbulence

This paper continue earlier investigations on the decay of Burgers turbulence in one dimension from Gaussian random initial conditions of the power-law spectral type $E_0(k)\sim|k|^n$. Depending on the power $n$, different characteristic regions are distinguished. The main focus of this paper is to delineate the regions in wave-number $k$ and time $t$ in which self-similarity can (and cannot) be observed, taking into account small-$k$ and large-$k$ cutoffs. The evolution of the spectrum can be inferred using physical arguments describing the competition between the initial spectrum and the new frequencies generated by the dynamics. For large wavenumbers, we always have $k^{-2}$ region, associated to the shocks. When $n$ is less than one, the large-scale part of the spectrum is preserved in time and the global evolution is self-similar, so that scaling arguments perfectly predict the behavior in time of the energy and of the integral scale. If $n$ is larger than two, the spectrum tends for long times to a universal scaling form independent of the initial conditions, with universal behavior $k^2$ at small wavenumbers. In the interval $2<n$ the leading behaviour is self-similar, independent of $n$ and with universal behavior $k^2$ at small wavenumber. When $1<n<2$, the spectrum has three scaling regions : first, a $|k|^n$ region at very small $k$\ms1 with a time-independent constant, second, a $k^2$ region at intermediate wavenumbers, finally, the usual $k^{-2}$ region. In the remaining interval, $n<-3$ the small-$k$ cutoff dominates, and $n$ also plays no role. We find also (numerically) the subleading term $\sim k^2$ in the evolution of the spectrum in the interval $-3<n<1$. High-resolution numerical simulations have been performed confirming both scaling predictions and analytical asymptotic theory.Comment: 14 pages, 19 figure

On the dynamics of a self-gravitating medium with random and non-random initial conditions

The dynamics of a one-dimensional self-gravitating medium, with initial density almost uniform is studied. Numerical experiments are performed with ordered and with Gaussian random initial conditions. The phase space portraits are shown to be qualitatively similar to shock waves, in particular with initial conditions of Brownian type. The PDF of the mass distribution is investigated.Comment: Latex, figures in eps, 23 pages, 11 figures. Revised versio

On the decay of Burgers turbulence

This work is devoted to the decay ofrandom solutions of the unforced Burgers equation in one dimension in the limit of vanishing viscosity. The initial velocity is homogeneous and Gaussian with a spectrum proportional to $k^n$ at small wavenumbers $k$ and falling off quickly at large wavenumbers. In physical space, at sufficiently large distances, there is an ``outer region'', where the velocity correlation function preserves exactly its initial form (a power law) when $n$ is not an even integer. When $1<n<2$ the spectrum, at long times, has three scaling regions : first, a $|k|^n$ region at very small $k$\ms1 with a time-independent constant, stemming from this outer region, in which the initial conditions are essentially frozen; second, a $k^2$ region at intermediate wavenumbers, related to a self-similarly evolving ``inner region'' in physical space and, finally, the usual $k^{-2}$ region, associated to the shocks. The switching from the $|k|^n$ to the $k^2$ region occurs around a wave number $k_s(t) \propto t^{-1/[2(2-n)]}$, while the switching from $k^2$ to $k^{-2}$ occurs around $k_L(t)\propto t^{-1/2}$ (ignoring logarithmic corrections in both instances). The key element in the derivation of the results is an extension of the Kida (1979) log-corrected $1/t$ law for the energy decay when $n=2$ to the case of arbitrary integer or non-integer $n>1$. A systematic derivation is given in which both the leading term and estimates of higher order corrections can be obtained. High-resolution numerical simulations are presented which support our findings.Comment: In LaTeX with 11 PostScript figures. 56 pages. One figure contributed by Alain Noullez (Observatoire de Nice, France

Instanton Theory of Burgers Shocks and Intermittency

A lagrangian approach to Burgers turbulence is carried out along the lines of the field theoretical Martin-Siggia-Rose formalism of stochastic hydrodynamics. We derive, from an analysis based on the hypothesis of unbroken galilean invariance, the asymptotic form of the probability distribution function of negative velocity-differences. The origin of Burgers intermittency is found to rely on the dynamical coupling between shocks, identified to instantons, and non-coherent background fluctuations, which, then, cannot be discarded in a consistent statistical description of the flow.Comment: 7 pages; LaTe