F. Leyvraz

Connectivity of Growing Random Networks

A solution for the time- and age-dependent connectivity distribution of a growing random network is presented. The network is built by adding sites that link to earlier sites with a probability A(k) which depends on the number of preexisting links k to that site. For homogeneous connection kernels, A(k) approximately k(gamma), different behaviors arise for gamma<1, gamma>1, and gamma = 1. For gamma<1, the number of sites with k links, N(k), varies as a stretched exponential. For gamma>1, a single site connects to nearly all other sites. In the borderline case A(k) approximately k, the power law N(k) approximately k(-nu) is found, where the exponent nu can be tuned to any value in the range 2

Singularities in the kinetics of coagulation processes

The authors consider a system of substances Ai reacting according to the following scheme: Ak+Al-RKl to k+l, (Rkl=Rlk>or=0). (The reaction is taken as irreversible.) They discuss the existence of global solutions of the kinetic equations derived for the concentrations. It is shown that one cannot expect the total number of monomers to remain constant. Rather, it can decrease as the result of the formation of infinite clusters (gelation). With this restriction, they obtain that a physically reasonable global solution exists if Rkl<or=rkrl and rk=o(k). It is further conjectured that no gelation will take place if rk=o( square root k). The case Rjk=(Aj+B)(Ak+B) is also solved explicitly and shown to exhibit gelation at t=1/A(A+B).

Identifying States of a Financial Market

The understanding of complex systems has become a central issue because such systems exist in a wide range of scientific disciplines. We here focus on financial markets as an example of a complex system. In particular we analyze financial data from the S&P 500 stocks in the 19-year period 1992–2010. We propose a definition of state for a financial market and use it to identify points of drastic change in the correlation structure. These points are mapped to occurrences of financial crises. We find that a wide variety of characteristic correlation structure patterns exist in the observation time window, and that these characteristic correlation structure patterns can be classified into several typical “market states”. Using this classification we recognize transitions between different market states. A similarity measure we develop thus affords means of understanding changes in states and of recognizing developments not previously seen.

Coupled normal heat and matter transport in a simple model system.

We introduce the first simple mechanical system that shows fully realistic transport behavior while still being exactly solvable at the level of equilibrium statistical mechanics. The system is a Lorentz gas with fixed freely rotating circular scatterers which scatter point particles via perfectly rough collisions. Upon imposing either a temperature gradient and/or a chemical potential gradient, a stationary state is attained for which local thermal equilibrium holds. Transport in this system is normal in the sense that the transport coefficients which characterize the flow of heat and matter are finite in the thermodynamic limit. Moreover, the two flows are nontrivially coupled, satisfying Onsagers reciprocity relations.

Transport Properties of a Modified Lorentz Gas

We present a detailed study of the first simple mechanical system that shows fully realistic transport behavior while still being exactly solvable at the level of equilibrium statistical mechanics. The system under consideration is a Lorentz gas with fixed freely-rotating circular scatterers interacting with point particles via perfectly rough collisions. Upon imposing a temperature and/or a chemical potential gradient, a stationary state is attained for which local thermal equilibrium holds for low values of the imposed gradients. Transport in this system is normal, in the sense that the transport coefficients which characterize the flow of heat and matter are finite in the thermodynamic limit. Moreover, the two flows are non-trivially coupled, satisfying Onsagers reciprocity relations to within numerical accuracy as well as the Green–Kubo relations. We further show numerically that an applied electric field causes the same currents as the corresponding chemical potential gradient in first order of the applied field. Puzzling discrepancies in higher order effects (Joule heating) are also observed. Finally, the role of entropy production in this purely Hamiltonian system is shortly discussed.

Scaling theory for migration-driven aggregate growth.

We give a comprehensive description for the irreversible growth of aggregates by migration from small to large aggregates. For a homogeneous rate K(i;j) at which monomers migrate from aggregates of size i to those of size j, that is, K(ai;aj)similar to a(lambda)K(i;j), the mean aggregate size grows with time as t(1/(2-lambda)) for lambda<2. The aggregate size distribution exhibits distinct regimes of behavior that are controlled by the scaling properties of the migration rate from the smallest to the largest aggregates. Our theory applies to diverse phenomena such as the distribution of city populations, late stage coarsening of nonsymmetric binary systems, and models for wealth exchange.

Spatial organization in two-species annihilation

The spatial structure of reactants in the two-species annihilation reaction A+B→0 is described. In one dimension, we investigate the distribution of domain sizes and the distributions of nearest-neighbor distances between particles of the same and of opposite species. The latter two quantities are characterized by a new length scale which is intermediate to the domain size t1/2 and the typical interparticle spacing t1/4. A scaling argument suggests that the typical distance between particles of opposite species, or equivalently the gaps between domains, grows astζ, with ζ = 3/8 and 1/3, respectively, in spatial dimensiond=1 and 2. The average density profile of a single domain is spatially nonuniform, with the density decaying to zero linearly as the domain edge is approached. This behavior permits a determination of the distribution of nearest-neighbor distances of same-species reactants. The corresponding moments of this distribution exhibit multiscaling which involves geometric averages of different powers of the domain size and the interparticle spacing.

THE LEVEL SPLITTING DISTRIBUTION IN CHAOS-ASSISTED TUNNELLING

A compound tunneling mechanism from one integrable region to another mediated by a delocalized state in an intermediate chaotic region of phase space was recently introduced to explain peculiar features of tunneling in certain two-dimensional systems. This mechanism is known as chaos-assisted tunneling. We study its consequences for the distribution of the level splittings and obtain a general analytical form for this distribution under the assumption that chaos assisted tunneling is the only operative mechanism. We have checked that the analytical form we obtain agrees with splitting distributions calculated numerically for a model system in which chaos-assisted tunneling is known to be the dominant mechanism. The distribution depends on two parameters: The first gives the scale of the splittings and is related to the magnitude of the classically forbidden processes, the second gives a measure of the efficiency of possible barriers to classical transportwhich may exist in the chaotic region. If these are weak, this latter parameter is irrelevant; otherwise it sets an energy scale at which the splitting distribution crosses over from one type of behavior to another. The detailed form of the crossover is also obtained and found to be in good agreement with numerical results for models for chaos-assisted tunneling.

Isochronous and partially isochronous Hamiltonian systems are not rare

A technique is provided that allows to associate to a Hamiltonian another, ω-modified, Hamiltonian, which reduces to the original one when the parameter ω vanishes, and for ω>0 features an open, hence fully dimensional, region in its phase space where all its solutions are isochronous, i.e., completely periodic with the same period. The class of Hamiltonians to which this technique is applicable is large: it includes for instance the Hamiltonian characterizing the classical many-body problem with potentials that are translation-invariant but otherwise completely arbitrary, which is largely used in this paper to illustrate these findings. We also discuss variants of this technique that yield partially isochronous Hamiltonians, which also feature a region in their phase space where all solutions are isochronous, that region having however a bit less than full dimensionality (for instance codimension one or two) in phase space.

Rigorous Results in the Scaling Theory of Irreversible Aggregation Kinetics

Abstract The kinetic equations describing irreversible aggregation and the scaling approach developed to describe them in the limit of large times and large sizes are tersely reviewed. Next, a system is considered in which aggregates can only react with aggregates of their own size. The existence of a scaling solution of the kinetic equations can then be shown rigorously in the case in which the total mass of the system is conserved. A large number of detailed properties of the solution, previously predicted by qualitative arguments, can be shown rigorously as well in this system. In the case in which gelation occurs, some sketchy rigorous results are shown, and numerical evidence for the existence of a scaling solution is presented. These are the first explicit examples of typical scaling behaviour for systems exhibiting gelation.