Electronic transport through chaotic quantum dots exhibits universal behaviour which can be understood through the semiclassical approximation. Within the approximation, transport moments reduce to codifying classical correlations between scattering
trajectories. These can be represented as ribbon graphs and we develop an algorithmic combinatorial method to generate all such graphs with a given genus. This provides an expansion of the linear transport moments for systems both with and without time reversal symmetry. The computational implementation is then able to progress several orders higher than previous semiclassical formulae as well as those derived from an asymptotic expansion of random matrix results. The patterns observed also suggest a general form for the higher orders.
To study electronic transport through chaotic quantum dots, there are two main theoretical approachs. One involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging. The other treats the tra
nsport in the semiclassical approximation and studies correlations among sets of classical trajectories. There are established evaluation procedures within the semiclassical evaluation that, for several linear and non-linear transport moments to which they were applied, have always resulted in the agreement with random matrix predictions. We prove that this agreement is universal: any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory. The equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs (maps) with certain properties and exhibiting systematic cancellations among their contributions. Remaining trajectory sets can be identified with primitive (palindromic) factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices. The equivalence is proved for systems with and without time reversal symmetry.
Most theories of homogeneous nucleation are based on a Fokker-Planck-like description of the behavior of the mass of clusters. Here we will show that these approaches are incomplete for a large class of nucleating systems, as they assume the effectiv
e dynamics of the clusters to be Markovian, i.e., memoryless. We characterize these non-Markovian dynamics and show how this influences the dynamics of clusters during nucleation. Our results are validated by simulations of a three-dimensional Ising model with locally conserved magnetization.
In many systems, the time scales of the microscopic dynamics and macroscopic dynamics of interest are separated by many orders of magnitude. Examples abound, for instance nucleation, protein folding, and chemical reactions. For these systems, direct
simulation of phase space trajectories does not efficiently determine most physical quantities of interest. The last decade has seen the advent of methods circumventing brute force simulation. For most dynamical quantities, these methods all share the drawback of systematical errors. We present a novel method for generating ensembles of phase space trajectories. By sampling small pieces of these trajectories in different phase space domains and piecing them together in a smart way using equilibrium properties, we obtain physical quantities such as transition times. This method does not have any systematic error and is very efficient; the computational effort to calculate the first passage time across a free energy barrier does not increase with the height of the barrier. The strength of the method is shown in the Ising model. Accurate measurements of nucleation times span almost ten orders of magnitude and reveal corrections to classical nucleation theory.
The time evolution of the strength of the Earths virtual axial dipole moment (VADM) is analyzed by relating it to the Fokker-Planck equation, which describes a random walk with VADM-dependent drift and diffusion coefficients. We demonstrate first tha
t our method is able to retrieve the correct shape of the drift and diffusion coefficients from a time series generated by a test model. Analysis of the Sint-2000 data shows that the geomagnetic dipole mode has a linear growth time of 13 to 33 kyr, and that the nonlinear quenching of the growth rate follows a quadratic function of the type [1-(x/x0)^2]. On theoretical grounds, the diffusive motion of the VADM is expected to be driven by multiplicative noise, and the corresponding diffusion coefficient to scale quadratically with dipole strength. However, analysis of the Sint-2000 VADM data reveals a diffusion which depends only very weakly on the dipole strength. This may indicate that the magnetic field quenches the amplitude of the turbulent velocity in the Earths outer core.
