H. J. Hilhorst

A note on q-Gaussians and non-Gaussians in statistical mechanics

The sum of N sufficiently strongly correlated random variables will not in general be Gaussian distributed in the limit . We revisit examples of sums x that have recently been put forward as instances of variables obeying a q-Gaussian law, that is, one of type cst × [1−(1−q)x2]1/(1−q). We show by explicit calculation that the probability distributions in the examples are actually analytically different from q-Gaussians, in spite of numerically resembling them very closely. Although q-Gaussians exhibit many interesting properties, the examples investigated do not support the idea that they play a special role as limit distributions of correlated sums.

WILSON RENORMALIZATION OF A REACTION-DIFFUSION PROCESS

Healthy and sick individuals (A and B particles) diffuse independently with diffusion constants DA and DB. Sick individuals upon encounter infect healthy ones (at rate k), but may also spontaneously recover (at rate 1/τ). The propagation of the epidemic therefore couples to the fluctuations in the total population density. Global extinction occurs below a critical value ρc of the spatially averaged total density. The epidemic evolves as the diffusion–reaction–decay process A+B→2B,B→A, for which we write down the field theory. The stationary-state properties of this theory when DA=DB were obtained by Kree et al. The critical behavior for DA DB remains unsolved.

Triangular SOS models and cubic-crystal shapes

A solid-on-solid (SOS) model in a field h conjugate to the orientation of the surface is exactly solved with the aid of Pfaffians. The free energy (h) directly gives the equilibrium shape of a finite crystal. The phase diagram exhibits rough and smooth phases, corresponding to rounded and flat portions of the crystal surface. The solid-on-solid model undergoes transitions of the Pokrovsky-Talapov type (1979) characterised by a specific heat exponent alpha =1/2. One special point of the phase diagram corresponds to the appearance of a facet via an alpha =0 transition. Height-height correlations are derived along a special line in the phase diagram. With the aid of the known equivalence of this SOS model with an Ising model, several exponents can be translated from one model to the other. This enables one to derive the topology of the phase diagram of the antiferromagnetic triangular Ising model with first- and second-neighbour couplings in a field.

Lévy-flight spreading of epidemic processes leading to percolating clusters

Abstract:We consider two stochastic processes, the Gribov process and the general epidemic process, that describe the spreading of an infectious disease. In contrast to the usually assumed case of short-range infections that lead, at the critical point, to directed and isotropic percolation respectively, we consider long-range infections with a probability distribution decaying in d dimensions with the distance as . By means of Wilsons momentum shell renormalization-group recursion relations, the critical exponents characterizing the growing fractal clusters are calculated to first order in an -expansion. It is shown that the long-range critical behavior changes continuously to its short-range counterpart for a decay exponent of the infection .

Pinning of a rough interface by an external potential

An analysis is given of the behavior of an interface between two phases in the presence of an external pinning potential in the solid-on-solid limit of the two-dimensional Ising model. It is found that the potential turns a rough interface into a smooth one, except in the case of a boundary potential, where a minimum potential strength is required. The connection with the roughening transition found by Abraham is discussed. The interface width is calculated as a function of the potential parameters in the limit of a weak pining potential.

Note on a q-modified central limit theorem

A q-modified version of the central limit theorem due to Umarov et alaffirms that q-Gaussians are attractors under addition and rescaling of certain classes of strongly correlated random variables. The proof of this theorem rests on a nonlinear q-modified Fourier transform. By exhibiting an invariance property we show that this Fourier transform does not have an inverse. As a consequence, the theorem falls short of achieving its stated goal.

Covering of a finite lattice by a random walk

Abstract We study the covering process by a simple random walk of a d -dimensional periodic hypercubic lattice of N sites. In d = 1, the probability L N ( X ) for site x to be the last site visited in this covering process does not depend on x , as long as x is not the starting point of the walk. We argue that in dimensions d > 2, the probability L N ( X ) approaches a constant value according to a Coulomb law: L N (x)⋍ 1 2 1− const |x| d−2 valid for ∥ x ∥ small on the scale N 1 d , whereas it behaves logarithmically in d = 2. Also, there is a dimension-dependent characteristic time scale on which the last site is visited. The structure of the set of sites not yet visited on this characteristic time scale is fractal-like in d = 2. In d ⩾ 3, on the other hand, this set is essentially distributed randomly through the lattice.

Molecular dynamics of 16000 Lennard-Jones particles

Abstract We report a molecular-dynamics simulation of very large two-dimensional Lennard-Jones systems (up to 16383 particles). The simulation was carried out on a special-purpose processor built by Bakker. The processors principal features and possibilities are described. Preliminary results of measurements along the isochore ϱ ∗ = 0.94 are presented.

Central limit theorems for correlated variables: some critical remarks

In this talk I first review at an elementary level a selection of central limit theorems, including some lesser known cases, for sums and maxima of uncorrelated and correlated random variables. I recall why several of them appear in physics. Next, I show that there is room for new versions of central limit theorems applicable to specific classes of problems. Finally, I argue that we have insufficient evidence that, as a consequence of such a theorem, q-Gaussians occupy a special place in statistical physics.

Special purpose computers in physics

This talk describes a new approach for large-scale computational problems which is particularly effective when a relatively simple algorithm is used. We demonstrate that it is possible to design and construct, at modest cost, special purpose computers for various classes of problems. By exploiting the principles of pipelining and parallel processing, and by adapting the hardware design to the specific structure of a particular algorithm, one can obtain a device which is as fast as or faster than general-purpose commercial supercomputers. The user of a such a processor has the double advantage of its speed and of its continuous availability for the particular problem for which it was constructed. In statistical mechanics special purpose computers have been built recently (i) for Monte Carlo simulation of the Ising model, and (ii) for the molecular dynamics of classical many-particle systems with short-range interactions. The design and performance of these machines are discussed and compared to those of commercial computers.