Rate equation limit for a combinatorial solution of a stochastic aggregation model.

In a recent series of papers, an exact combinatorial solution was claimed for a variant of the so-called Marcus-Lushnikov model of aggregation. In this model, a finite number of aggregates are initially assumed to be present in the form of monomers. At each time step, two aggregates are chosen according to certain size-dependent probabilities and irreversibly joined to form an aggregate of higher mass. The claimed result given an expression for the full probability distribution over all possible size distributions in terms of the so-called Bell polynomials. In this paper, we develop the asymptotics of this solution in order to check whether the exact solution yields correct expressions for the average cluster size distribution as obtained from the Smoluchowski equations. The answer is surprisingly involved: For the generic case of an arbitrary reaction rate, it is negative, but for the so-called classical rate kernels, constant, additive, and multiplicative, the solutions obtained are indeed exact. On the other hand, for the multiplicative kernel, a discrepancy is found in the full solution between the combinatorial solution and the exact solution. The reasons for this puzzling pattern of agreement and disagreement are unclear. A better understanding of the combinatorial solution's derivation is needed, the better to understand its range of validity.

Exact asymptotic solution of an aggregation model with a bell-shaped distribution.

We present in a detailed manner the scaling theory of irreversible aggregation characterized by the set of reaction rates K(k,l)=1/k+1/l. In this case, it is possible to determine the behavior of large-size aggregates in the limit of large times in a way that allows a highly detailed analysis of the behavior of the system. This is the so-called scaling limit, in which the cluster size distribution collapses to a function of the ratio of the cluster size to a time-dependent typical size. The results confirm the far more general results of earlier work concerning a general scaling theory for so-called reaction rates of Type III, which are characterized by the property that aggregates of very different sizes react faster than comparable aggregates of similar sizes. For these, the cluster size distribution decays rapidly to zero both for sizes much larger and much smaller than the typical size, and is thus often described as being "bell-shaped". For clusters much larger than the typical size, however, an unexpected subleading correction is discovered. Finally, several results going beyond the scope of the scaling limit are obtained: in particular the behavior of concentrations for fixed cluster size in the large-time limit and the large-size behavior for clusters at a fixed time. The latter again shows subleading deviations from the expected behavior.

Systems with negative specific heat in thermal contact: violation of the zeroth law.

Using both numerical simulations and exact expressions for the free energy and microcanonical entropy for a modified Hamiltonian mean-field (HMF) model, we show that when two similar systems with the same intensive parameters but with negative specific heat are weakly coupled, they undergo a process in which the total entropy increases irreversibly. We find that the final equilibrium is such that two phases appear at a temperature (equal in both systems) that is generally different from the initial temperature. We corroborate our results using two different kinds of couplings between the HMF systems. We confirm that our results hold also for the Ising model with long- and short-range interactions, which also has a parameter region with negative specific heat in the microcanonical ensemble. Further, we show that we can couple each system having negative specific heat to a third system that can be used as a thermometer, as long as this thermometer is small enough not to drive the system out of the microcanonical ensemble. Therefore, we show an instance of violation of the zeroth law of thermodynamics.

Analytical approach to the time-dependent probability density function in tilted periodic potentials.

In this work we introduce a scheme for the calculation of an approximate closed expression for the time-dependent probability density function for overdamped particles in tilted periodic potentials. Our derivation is based on an ansatz for the solution of the corresponding Fokker-Planck equation and on a self-consistent cumulant calculation. The high accuracy of our expression for the time-dependent probability density function is exhibited by comparisons with Langevin dynamics simulations and exact analytic results for the drift and diffusion coefficients. Good agreement is found both, for large and intermediate times.

Wigner-Dyson statistics for a class of integrable models.

We construct an ensemble of second-quantized Hamiltonians with two bosonic degrees of freedom, whose members display with probability one Gaussian orthogonal ensemble (GOE) or Gaussian unitary ensemble (GUE) statistics. Nevertheless, these Hamiltonians have a second integral of motion, namely, the boson number, and thus are integrable. To construct this ensemble we use some "reverse engineering" starting from the fact that n bosons in a two-level system with random interactions have an integrable classical limit by the old Heisenberg association of boson operators to actions and angles. By choosing an n-body random interaction and degenerate levels we end up with GOE or GUE Hamiltonians. Ergodicity of these ensembles completes the example.

Transport properties of the diluted Lorentz slab.

We study the behavior of a point particle incident on a slab of a randomly diluted triangular array of circular scatterers. Various scattering properties, such as the reflection and transmission probabilities and the scattering time are studied as a function of thickness and dilution. We show that a diffusion model satisfactorily describes the mentioned scattering properties. We also show how some of these quantities can be evaluated exactly and their agreement with numerical experiments. Our results exhibit the dependence of these scattering data on the mean free path. This dependence again shows excellent agreement with the predictions of a Brownian motion model.

Transition from Gaussian-orthogonal to Gaussian-unitary ensemble in a microwave billiard with threefold symmetry.

Recently it has been shown that time-reversal invariant systems with discrete symmetries may display, in certain irreducible subspaces, spectral statistics corresponding to the Gaussian-unitary ensemble (GUE) rather than to the expected orthogonal one (GOE). A Kramers-type degeneracy is predicted in such situations. We present results for a microwave billiard with a threefold rotational symmetry and with the option to display or break a reflection symmetry. This allows us to observe the change from GOE to GUE statistics for one subset of levels. Since it was not possible to separate the three subspectra reliably, the number variances for the superimposed spectra were studied. The experimental results are compared with a theoretical and numerical study considering the effects of level splitting and level loss.

Violation of the zeroth law of thermodynamics in systems with negative specific heat.

We show that systems with negative specific heat can violate the zeroth law of thermodynamics. By both numerical simulations and by using exact expressions for free energy and microcanonical entropy, it is shown that if two systems with the same intensive parameters but with negative specific heat are thermally coupled, they undergo a process in which the total entropy increases irreversibly. The final equilibrium is such that two phases appear; that is, the subsystems have different magnetizations and internal energies at temperatures which are equal in both systems, but that can be different from the initial temperature.

Surprising relations between parametric level correlations and fidelity decay.

Relations among fidelity, cross-form-factor (i.e., parametric level correlations), and level velocity correlations are found both by deriving a Ward identity in a two-matrix model and by comparing exact results, using supersymmetry techniques, in the framework of random matrix theory. A power law decay near Heisenberg time, as a function of the relevant parameter, is shown to be at the root of revivals recently discovered for fidelity decay. For cross-form-factors the revivals are illustrated by a numerical study of a multiply kicked Ising spin chain.

Large-N scaling behavior of the Lipkin-Meshkov-Glick model.

We introduce a novel semiclassical approach to the Lipkin model. In this way the well-known phase transition arising at the critical value of the coupling is intuitively understood. New results--showing for strong couplings the existence of a threshold energy which separates deformed from undeformed states as well as the divergence of the density of states at the threshold energy--are explained straightforwardly and in quantitative terms by the appearance of a double well structure in a classical system corresponding to the Lipkin model. Previously unnoticed features of the eigenstates near the threshold energy are also predicted and found to hold.

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.

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 Onsager's reciprocity relations.

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