Rescaling Of Space Research Articles

Asymptotic behaviour of a tagged particle in an inhomogeneous zero-range process

We consider an infinite particle system living on a torus; from the microscopic point of view, the particles move in a lattice and their evolution is described by a spatially inhomogeneous zero-range process: each particle jumps following a symmetric random walk, with rate depending on the number of particles belonging to the same site, and weakly on the position of the site itself. In particular, we study the asymptotic behaviour of a tagged particle: under a diffusive rescaling of space and time, we prove that the process described by the tagged particle converges in law to a driftless diffusion. The proof is based on a local ergodic property and martingale convergence arguments.

Hydrodynamical limit for a nongradient system: The generalized symmetric exclusion process

We consider a symmetric simple exclusion process where at most two particles per site are permitted. This model turns out to be nongradient. We prove that the particles' densities, under a diffusive rescaling of space and time, converge to the solution of a diffusion equation. We give a variational characterization of the diffusion coefficent. We also prove, for the generator of the process in finite volume, a lower bound on the spectral gap uniform in the volume. © 1994 John Wiley & Sons, Inc.

Scaling of decay lengths for surface states in the gaps of one-dimensional semi-infinite superlattices

We look at some one-dimensional semi-infinite superlattices with an underlying Hamiltonian that is of the nearest neighbour, tight binding type. A real space rescaling procedure which is exact in one dimension is applied to obtain the location of the subbands. It has been found that these subbands never overlap in 1D, and we interpret this as a ‘band repulsion’ effect. Relevance in the case of a disordered system where this band repulsion crosses over to the well-known level repulsion is discussed. Then with a proper matching at the boundary we solve for the sets of denumerably infinite number of decaying solutions (the ‘surface states’) in the gaps. These types of states have been proposed quite some time ago. We look at detail theirexact analytical solutions in 1D and find that their decay lengths near the band edges diverge as ζ∼|E−Eb|−v, wherev=1/2 andEb is the nearest band edge. The decay lengths and their divergence exponent match extremely well with those obtained from transfer matrix method. Some recent experiments on quantum well structures seem to have observed such states.

On the universal form of energy spectra in fully developed turbulence

Several experimentally measured energy spectra of fully developed incompressible turbulent flows with Taylor microscale Reynolds numbers ranging from 130–13 000 have been reexamined. It is shown that all spectra collapse to a universal curve after a simple rescaling of space and time. This rescaling does not involve any multifractal-type transformation, and the rescaled wave numbers scatter around the Kolmogorov dissipation wave numbers derived from experimentally measured quantities. An empirical formula for the resulting universal form is proposed, indicating a nontrivial form of the energy spectrum near the viscous dissipation cutoff.

Limit Theorems for the Frontier of a One-Dimensional Branching Diffusion

Let $R_t$ be the position of the rightmost particle at time $t$ in a time-homogeneous one-dimensional branching diffusion process. Let $\gamma(\alpha,t)$ be the $\alpha$th quantile of $R_t$ under $P^0$, where $P^x$ denotes the probability measure of the branching diffusion process starting with a single particle at position $x$. We show that $\gamma(\alpha,t)$ is a limiting quantile of $R_t$ under $P^x$ in the sense that $\lim_{t \rightarrow\infty}P^x\{R_t \leq \gamma(\alpha,t)\}$ exists for all $\alpha \in (0,1)$ and all $x \in \mathbb{R}$. If the underlying diffusion is recurrent, we show that, after an appropriate rescaling of space, the $P^x$ distribution of $R_t - t$ converges weakly to a nontrivial limiting distribution $w_x$.

Monte Carlo renormalization-group study of the dynamics of an unstable state.

The kinetics of an order-disorder transition is studied by Monte Carlo renormalization-group methods. The block-spin transformation acts to renormalize both the growing domains and the moving interfaces separating them. After several iterations, the growth is found to be self-similar under a simultaneous rescaling of space and time. The growth law for the average size of domains is then determined by a matching condition. Good agreement with the results of previous studies is obtained.

Nonlinear time-dependent anharmonic oscillator: Asymptotic behavior and connected invariants

The motion of a particle in a potential decreasing with time as ‖X‖n is considered. Different time and space rescaling are considered in order to obtain the asymptotic solutions. The validity of adiabatic invariants is discussed. The classical critical case corresponds to the obtainment of self-similar solutions for the quantum problem.

Real space rescaling study of spin glass behaviour in three dimensions

The Edwards and Anderson model (1975) for spin glasses is investigated by a real space rescaling transformation on an S=1/2 Ising Hamiltonian with nearest-neighbour-only interactions in three dimensions. A transition to a spin glass ordered phase is found at about half the mean transition temperature, in contrast to earlier work on a two-dimensional model where no transition was found. The results suggest that the lower critical dimensionality, dc, for spin glass order is slightly greater than 2, although possibly dc=2 exactly. The specific heat is calculated and found to exhibit a broad maximum above Tc, with no discernable singularity at the transition since the exponent alpha is large and negative.