Andreas Nußbaumer

Football fever: goal distributions and non-Gaussian statistics

Analyzing football score data with statistical techniques, we investigate how the not purely random, but highly co-operative nature of the game is reflected in averaged properties such as the probability distributions of scored goals for the home and away teams. As it turns out, especially the tails of the distributions are not well described by the Poissonian or binomial model resulting from the assumption of uncorrelated random events. Instead, a good effective description of the data is provided by less basic distributions such as the negative binomial one or the probability densities of extreme value statistics. To understand this behavior from a microscopical point of view, however, no waiting time problem or extremal process need be invoked. Instead, modifying the Bernoulli random process underlying the Poissonian model to include a simple component of self-affirmation seems to describe the data surprisingly well and allows to understand the observed deviation from Gaussian statistics. The phenomenological distributions used before can be understood as special cases within this framework. We analyzed historical football score data from many leagues in Europe as well as from international tournaments, including data from all past tournaments of the “FIFA World Cup” series, and found the proposed models to be applicable rather universally. In particular, here we analyze the results of the German women’s premier football league and consider the two separate German men’s premier leagues in the East and West during the cold war times as well as the unified league after 1990 to see how scoring in football and the component of self-affirmation depend on cultural and political circumstances.

Self-affirmation model for football goal distributions

Analyzing football score data with statistical techniques, we investigate how the highly co-operative nature of the game is reflected in averaged properties such as the distributions of scored goals for the home and away teams. It turns out that in particular the tails of the distributions are not well described by independent Bernoulli trials, but rather well modeled by negative binomial or generalized extreme value distributions. To understand this behavior from first principles, we suggest to modify the Bernoulli random process to include a simple component of self-affirmation which seems to describe the data surprisingly well and allows to interpret the observed deviation from Gaussian statistics. The phenomenological distributions used before can be understood as special cases within this framework. We analyzed historical football score data from many leagues in Europe as well as from international tournaments and found the proposed models to be applicable rather universally. In particular, here we compare mens and womens leagues and the separate German leagues during the cold war times and find some remarkable differences.

Free-Energy Barrier at Droplet Condensation

We discuss several aspects of a Monte Carlo computer simulation study of the condensation of macroscopic droplets emerging in the two-dimensional Ising lattice-gas model. By varying the particle density at fixed temperature we monitor the droplet formation in detail and compare our results with recent analytical predictions in the infinite-volume limit. Three different lattice discretizations are considered which are found to yield very similar results when presented in properly scaled variables. Particular emphasis is placed on the free-energy barrier associated with droplet formation and its implication for multimagnetical simulations.

Interface tension of the square lattice Ising model with next-nearest-neighbour interactions

In a recent letter, Zandvliet (Europhys. Lett., 73 (2006) 747) presented a simple derivation of an analytical expression for the interface free energy in the (10) direction of the Ising model on a square lattice with nearest- and next-nearest-neighbour couplings, reproducing the famous exact Onsager formula in the special case of only nearest-neighbour interactions. By comparing the resulting transition temperatures, determined as the point where the interface tension vanishes, with previous numerical results in the literature, support for the validity of the new analytical formula in the general case was claimed. Guided by the fact that Zandvliets simple, but rather heuristic derivation neglects overhang configurations and bubble excitations completely, we show that his approach is equivalent to the classic solid-on-solid (SOS) approximation which is known to reproduce accidentally the exact interface tension along one of the two main axes in the case of only nearest-neighbour interactions. In the limiting situation where only next-nearest-neighbour interactions are considered, we prove analytically that such a coincidence no longer holds. To assess the accuracy of Zandvliets formula for the general model we have performed a careful computer simulation study using multicanonical and cluster Monte Carlo techniques combined with finite-size scaling analyses. Our results for the hitherto unknown interface tension and the transition temperatures show that the analytical formula yields fairly good approximations but, in general, is not exact.

Anisotropy of the interface tension of the three-dimensional Ising model

Abstract We determine the interface tension for the 100, 110 and 111 interface of the simple cubic Ising model with nearest-neighbour interaction using novel simulation methods. To overcome the droplet/strip transition and the droplet nucleation barrier we use a newly developed combination of the multimagnetic algorithm with the parallel tempering method. We investigate a large range of inverse temperatures to study the anisotropy of the interface tension in detail.

Parallel Tempering at Second‐Order Phase Transitions

The parallel tempering algorithm has proved to be very powerful in accelerating Monte Carlo simulations of disordered systems with rugged free energy landscapes. The most prominent examples are spin glasses and structural glasses. One of the merits of this algorithm is its simplicity, permitting applications also to models for which cluster, multigrid or other non‐local update algorithms have not yet been discovered. Having such situations in mind, we investigated whether parallel tempering can also be a useful tool at a standard second‐order phase transition. As a very simple test case we considered the 2D Ising model (where, of course, cluster algorithms would be more efficient) and studied the dependence of the integrated autocorrelation time τint,M of the magnetization on the acceptance rate between replicas, on the linear size of the system, and on the number of replicas used. Using a novel FFT implementation for the estimation of the autocorrelation time, it was found that τint,M does depend only we...