On the large time asymptotics of decaying Burgers turbulence

The decay of Burgers turbulence with compactly supported Gaussian "white noise" initial conditions is studied in the limit of vanishing viscosity and large time. Probability distribution functions and moments for both velocities and velocity differences are computed exactly, together with the "time-like" structure functions . The analysis of the answers reveals both well known features of Burgers turbulence, such as the presence of dissipative anomaly, the extreme anomalous scaling of the velocity structure functions and self similarity of the statistics of the velocity field, and new features such as the extreme anomalous scaling of the "time-like" structure functions and the non-existence of a global inertial scale due to multiscaling of the Burgers velocity field. We also observe that all the results can be recovered using the one point probability distribution function of the shock strength and discuss the implications of this fact for Burgers turbulence in general.Comment: LATEX, 25 pages, The present paper is an extension of the talk delivered at the workshop on intermittency in turbulent systems, Newton Institute, Cambridge, UK, June 199

Pfaffian formulae for one dimensional coalescing and annihilating systems

The paper considers instantly coalescing, or instantly annihilating, systems of one-dimensional Brownian particles on the real line. Under maximal entrance laws, the distribution of the particles at a fixed time is shown to be Pfaffian point processes closely related to the Pfaffian point process describing one dimensional distribution of real eigenvalues in the real Ginibre ensemble of random matrices. As an application, an exact large time asymptotic for the n-point density function for coalescing particles is derived

Multi-point correlations for two dimensional coalescing random walks

This paper considers an infinite system of instantaneously coalescing rate one simple random walks on $\mathbb{Z}^2$, started from the initial condition with all sites in $\mathbb{Z}^2$ occupied. We show that the correlation functions of the model decay, for any $N \geq 2$, as $$\rho_N (x_1,\ldots,x_N;t) = \frac{c_0(x_1,\ldots,x_N)}{\pi^N} (\log t)^{N-{N \choose 2}} t^{-N} \left(1 + O\left( \frac{1}{\log^{\frac12-\delta}\!t} \right) \right)$$ as $t \to\infty$. This generalises the results for $N=1$ due to Bramson and Griffeath and confirms a prediction in the physics literature for $N>1$. An analogous statement holds for instantaneously annihilating random walks. The key tools are the known asymptotic $\rho_1(t) \sim \log t/\pi t$ due to Bramson and Griffeath, and the non-collision probability $p_{NC}(t)$, that no pair of a finite collection of $N$ two dimensional simple random walks meets by time $t$, whose asymptotic $p_{NC}(t) \sim c_0 (\log t)^{-{N \choose 2}}$ was found by Cox, Merle and Perkins. This paper re-derives the asymptotics both for $\rho_1(t)$ and $p_{NC}(t)$ by proving that these quantities satisfy {\it effective rate equations}, that is approximate differential equations at large times. This approach can be regarded as a generalisation of the Smoluchowski theory of renormalised rate equations to multi-point statistics.Comment: 26 page

Averages of products of characteristic polynomials and the law of real eigenvalues for the real Ginibre ensemble

An elementary derivation of the Borodin-Sinclair-Forrester-Nagao Pfaffian point process, which characterises the law of real eigenvalues for the real Ginibre ensemble in the large matrix size limit, uses the averages of products of characteristic polynomials. This derivation reveals a number of interesting structures associated with the real Ginibre ensemble such as the hidden symplectic symmetry of the statistics of real eigenvalues and an integral representation for the $K$-point correlation function for any $K\in \mathbb{N}$ in terms of an asymptotically exact integral over the symmetric space $U(2K)/USp(2K)$.Comment: 28 pages, 2 figure

One dimensional annihilating particle systems as extended Pfaffian point processes

We prove that the multi-time particle distributions for annihilating Brownian motions, under the maximal entrance law on the real line, are extended Pfaffian point processesComment: An earlier version of this paper falsely stated that the multi-time distributions of coalescing Brownian motions (CBM's) were also an extended Pfaffian point process. This erroneous claim has been removed from the current version. We do not have a simple description of the multi-time distributions for CBM

Constant flux relation for diffusion-limited cluster-cluster aggregation

In a nonequilibrium system, a constant flux relation (CFR) expresses the fact that a constant flux of a conserved quantity exactly determines the scaling of the particular correlation function linked to the flux of that conserved quantity. This is true regardless of whether mean-field theory is applicable or not. We focus on cluster-cluster aggregation and discuss the consequences of mass conservation for the steady state of aggregation models with a monomer source in the diffusion-limited regime. We derive the CFR for the flux-carrying correlation function for binary aggregation with a general scale-invariant kernel and show that this exponent is unique. It is independent of both the dimension and of the details of the spatial transport mechanism, a property which is very atypical in the diffusion-limited regime. We then discuss in detail the "locality criterion" which must be satisfied in order for the CFR scaling to be realizable. Locality may be checked explicitly for the mean-field Smoluchowski equation. We show that if it is satisfied at the mean-field level, it remains true over some finite range as one perturbatively decreases the dimension of the system below the critical dimension, d(c)=2, entering the fluctuation-dominated regime. We turn to numerical simulations to verify locality for a range of systems in one dimension which are, presumably, beyond the perturbative regime. Finally, we illustrate how the CFR scaling may break down as a result of a violation of locality or as a result of finite size effects and discuss the extent to which the results apply to higher order aggregation processes

Stationary mass distribution and nonlocality in models of coalescence and shattering

We study the asymptotic properties of the steady state mass distribution for a class of collision kernels in an aggregation-shattering model in the limit of small shattering probabilities. It is shown that the exponents characterizing the large and small mass asymptotic behavior of the mass distribution depend on whether the collision kernel is local (the aggregation mass flux is essentially generated by collisions between particles of similar masses), or non-local (collision between particles of widely different masses give the main contribution to the mass flux). We show that the non-local regime is further divided into two sub-regimes corresponding to weak and strong non-locality. We also observe that at the boundaries between the local and non-local regimes, the mass distribution acquires logarithmic corrections to scaling and calculate these corrections. Exact solutions for special kernels and numerical simulations are used to validate some non-rigorous steps used in the analysis. Our results show that for local kernels, the scaling solutions carry a constant flux of mass due to aggregation, whereas for the non-local case there is a correction to the constant flux exponent. Our results suggest that for general scale-invariant kernels, the universality classes of mass distributions are labeled by two parameters: the homogeneity degree of the kernel and one further number measuring the degree of the non-locality of the kernel.Comment: Published versio