Table of contents

We consider the problem of the definition of an effective temperature via the long-time limit of the fluctuation-dissipation ratio after a quench from the disordered state to the critical point of an O (N) model with dissipative dynamics. The scaling forms of the response and correlation functions of a generic observable are derived from the solutions of the corresponding renormalization group equations. We show that within the Gaussian approximation all the local observables have the same , allowing for a definition of a unique effective temperature. This is no longer the case when fluctuations are taken into account beyond that approximation, as shown by a computation up to the first order in the -expansion for two quadratic observables. This implies that, contrarily to what is often conjectured, a unique effective temperature cannot be defined for this class of models.

An external magnetic field induces large relative changes in the entropy of one-dimensional quantum spin systems at finite temperatures. This leads to a magnetocaloric effect, i.e. a change in temperature during an adiabatic (de)magnetization process. Several examples of one-dimensional spin-1/2 models are studied by employing the Jordan–Wigner transformation and exact diagonalization. During an adiabatic (de)magnetization process, the temperature drops in the vicinity of a field-induced zero-temperature quantum phase transition. Comparing different levels of frustration, we find that more frustrated systems cool down to lower temperatures. For geometrically frustrated spin models a finite entropy survives down to zero temperature at certain magnetic fields. This property suggests frustrated quantum spin systems as promising alternative refrigerant materials for low-temperature magnetic refrigeration.

We investigate a model of epidemic spreading with partial immunization which is controlled by two probabilities, namely, for first infections, p₀, and for reinfections, p. When the two probabilities are equal, the model reduces to directed percolation, while for perfect immunization one obtains the general epidemic process belonging to the universality class of dynamical percolation. We focus on the critical behaviour in the vicinity of the directed percolation point, especially for high number of dimensions d>2. It is argued that the clusters of immune sites are compact for . This observation implies that a recently introduced scaling argument, suggesting a stretched exponential decay of the survival probability for p = p_c, in one spatial dimension, where p_c denotes the critical threshold for directed percolation, should apply for any number of dimensions and perhaps for d = 4 as well. Moreover, we show that the phase transition line, connecting the critical points of directed percolation and of dynamical percolation, terminates at the critical point of directed percolation with vanishing slope for d<4 and with finite slope for . Furthermore, an exponent is identified for the temporal correlation length for the case of p = p_c and p₀ = p_c−, , which is different from the exponent of directed percolation. We also improve numerical estimates of several critical parameters and exponents, especially for dynamical percolation for d = 4,5.

We investigate the properties of an autoassociative network of threshold-linear units whose synaptic connectivity is spatially structured and asymmetric. Since the methods of equilibrium statistical mechanics cannot be applied to such a network due to the lack of a Hamiltonian, we approach the problem through a signal-to-noise analysis, that we adapt to spatially organized networks. The conditions are analysed for the appearance of stable, spatially non-uniform profiles of activity with large overlaps with one of the stored patterns. It is also shown, with simulations and analytic results, that the storage capacity does not decrease much when the connectivity of the network becomes short range. In addition, the method used here enables us to calculate exactly the storage capacity of a randomly connected network with arbitrary degree of dilution.

Detrended fluctuation analysis is used to investigate the power law relationship between the monthly averages of the maximum daily temperatures for different locations in the western US. On the map created by the power law exponents, we can distinguish different geographical regions with different power law exponents. When the power law exponents obtained from the detrended fluctuation analysis are plotted versus the standard deviation of the temperature fluctuations, we observe different data points belonging to the different climates, hence indicating that by observing the long-time trends in the fluctuations of temperature we may be able to distinguish between different climates.

We investigate the performance of flat-histogram methods based on a multicanonical ensemble and the Wang–Landau algorithm for the three-dimensional ± J spin glass by measuring round-trip times in the energy range between the zero-temperature ground state and the state of highest energy. Strong sample-to-sample variations are found for fixed system size and the distribution of round-trip times follows a fat-tailed Fréchet extremal value distribution. Rare events in the fat tails of these distributions corresponding to extremely slowly equilibrating spin glass realizations dominate the calculations of statistical averages. While the typical round-trip times scale exponentially as expected for this NP-hard problem, we find that the average round-trip time is no longer well defined for systems with spins. We relate the round-trip times for multicanonical sampling to intrinsic properties of the energy landscape and compare with the numerical effort needed by the genetic cluster-exact approximation to calculate the exact ground-state energies. For systems with spins the simulation of these rare events becomes increasingly hard. For there are samples where the Wang–Landau algorithm fails to find the true ground state within reasonable simulation times. We expect similar behaviour for other algorithms based on multicanonical sampling.

A four-state mutation–selection model for the evolution of populations of DNA sequences is investigated with particular interest taken in the phenomenon of error thresholds. The mutation model considered is the Kimura 3ST mutation scheme; the fitness functions, which determine the selection process, come from the permutation invariant class. Error thresholds can be found for various fitness functions; the phase diagrams are more interesting than for equivalent two-state models. Results for (small) finite sequence lengths are compared with those for infinite sequence length, obtained via a maximum principle that is equivalent to the principle of minimal free energy in physics.

The Jarzynski equality relates the free energy difference between two equilibrium states of a system to the average of the work over all irreversible phase space trajectories for going from one state to the other. We claim that the derivation of this equality is flawed, introducing an ad hoc and unjustified weighting factor which handles improperly the heat exchange with a heat bath. Therefore the experiment of Liphardt et al cannot be viewed as a confirmation of this equality, although the numerical deviations between the two are small. However, the Jarzynski equality may well be a useful approximation, e.g. in measurements on single molecules in solution.

We investigate critical properties of a class of number-conserving cellular automata (CAs) which can be interpreted as deterministic models of traffic flow with anticipatory driving. These rules are among the only known CA rules for which the shape of the fundamental diagram has been rigorously derived. In addition, their fundamental diagrams contain nonlinear segments, as opposed to the majority of number-conserving CAs which exhibit piecewise-linear diagrams. We found that the nature of singularities in the fundamental diagram of these rules is the same as for rules with piecewise-linear diagrams. The current converges toward its equilibrium value as t^−1/2, and the critical exponent β is equal to unity. This supports the conjecture of universal behaviour at singularities in number-conserving rules. We discuss properties of phase transitions occurring at singularities as well as properties of the intermediate phase.

We propose a new exactly solvable pairing model for bosons corresponding to the attractive pairing interaction. Using the electrostatic analogy, the solution of this model in the thermodynamic limit is found. The transition from the superfluid phase, with a Bose condensate and a Bogoliubov-type spectrum of excitations in the weak coupling regime, to the incompressible phase, with a gap in the excitation spectrum in the strong coupling regime, is observed.

We examine the distribution of the extent of criminal activity by individuals in two widely cited data bases. The Cambridge Study in Delinquent Development records criminal convictions amongst a group of working class youths in the UK over a 14 year period. The Pittsburgh Youth Study measures self-reported criminal acts over intervals of six months or a year in three groups of boys in the public school system in Pittsburgh, PA.

The range of the data is very substantially different between these two measures of criminal activity, one of which is convictions and the other self-reported acts. However, there are many similarities between the characteristics of the data sets.

A power law relationship between the frequency and rank of the number of criminal acts describes the data well in both cases, and fits the data better than an exponential relationship. Power law distributions of macroscopic observables are ubiquitous in both the natural and social sciences. They are indicative of correlated, cooperative phenomena between groups of interacting agents at the microscopic level.

However, there is evidence of a bimodal distribution, again in each case. Excluding the frequency with which zero crimes are committed or reported reduces the absolute size of the estimated exponent in the power law relationship. The exponent is virtually identical in both cases. A better fit is obtained for the tail of the distribution.

In other words, there appears to be a subtle deviation from straightforward power law behaviour. The description of the data when the number of boys committing or reporting zero crimes are excluded is different from that when they are included. The crucial step in the criminal progress of an individual appears to be committing the first act. Once this happens, the number of criminal acts committed by an individual can take place on all scales.

We show how the symmetries of the Ising field theory on a pseudosphere can be exploited to derive the form factors of the spin fields as well as the non-linear differential equations satisfied by the corresponding two-point correlation functions. The latter are studied in detail and, in particular, we present a solution to the so-called connection problem relating two of the singular points of the associated Painlevé VI equation. A brief discussion of the thermodynamic properties is also presented.

We apply the concept of reflection–transmission (RT) algebra, originally developed in the context of integrable systems in 1+1 space–time dimensions, to the study of finite temperature quantum field theory with impurities in higher dimensions. We consider a scalar field in (s+1)+1 space–time dimensions, interacting with impurities localized on s-dimensional hyperplanes, but without self-interaction. We discuss first the case s = 0 and extend afterwards all results to s>0. Constructing the Gibbs state over an appropriate RT algebra, we derive the energy density at finite temperature and establish the correction to the Stefan–Boltzmann law generated by the impurity. The contribution of the impurity bound states is taken into account. The charge density profiles for various impurities are also investigated.