We present numerical calculations of the charge and statistics, as extracted from Berry phases, of the Laughlin quasi-particles, near filling fraction 1/3, and for system sizes of up to 200 electrons. For the quasi-holes our results confirm that the charge and statistics parameter are e/3 and 1/3, respectively. For the quasi-electron charge we find a slow convergence towards the expected value of −e/3, with a finite size correction for N electrons of approximately −0.13e/N. The statistics parameter for the quasi-electrons has no well defined value even for 200 electrons, but might possibly converge to 1/3. Most noteworthy, it takes on the same sign as for the quasi-holes, due to terms that have previously been ignored. The anyon model works well for the quasi-holes, but requires singular two-anyon wave functions for modelling two Laughlin quasi-electrons. ‡ Supported by The Norwegian Research Council.It is now widely accepted that the fractional quantum Hall effect arises due to quasi-particle excitations created when the filling fraction of the lowest Landau level moves away from its preferred values. The quasi-particles bind to impurities and thereby ensure that the plateaus in the conductivity are formed. Laughlin examined the filling fractions 1/m, with m an odd integer, and argued that the quasi-particles have charge ±e/m (−e is the electron charge) [1]. He showed that these values for the charge imply a conductivity plateau at the value e 2 /(mh), as observed. He also offered explicit trial wave functions describing the ground state as well as the quasi-particle excitations for the 1/m case. Later, Haldane [2] and Halperin [3] examined the hierarchy structure of the Hall states, and suggested that the quasi-particles obey fractional statistics, i.e. that they are anyons [4].Arovas, Schrieffer and Wilczek derived the charge and statistics of the Laughlin quasiholes in Ref. [5]. They examined the Berry phase [6] corresponding to one quasi-hole encircling the origin and interpreted this as an Aharonov-Bohm phase [6,7]. The charge was then found to be e/m, in confirmation of Laughlin's result. They also considered a pair of quasi-holes encircling one another, and related the two-particle contribution of the Berry phase to the quasi-hole charge. Interpreting this two-particle contribution as an anyon interchange phase they found the anyon statistics parameter to have the value 1/m, equal to the fraction of the elementary charge. The Laughlin quasi-electrons were examined along the same lines in Ref. [8]. Within the approximations used, the results imply that the charge and statistics parameter have the values −e/m and −1/m, respectively.The statistics satisfied by the quasi-particles in the quantum Hall system was also examined in Ref. [9], where the exclusion statistics parameter [10] was considered. The results did not rely on any specific trial wave function but rather on state counting based on numerical simulations for interacting electrons on a sphere. The value of the onedimensional exclusion statistics p...

We present Monte Carlo studies of charge expectation values and charge fluctuations for quasi-particles in the quantum Hall system. We have studied the Laughlin wave functions for quasi-hole and quasi-electron, and also Jain's definition of the quasi-electron wave function. The considered systems consist of from 50 to 200 electrons, and the filling fraction is 1/3. For all quasi-particles our calculations reproduce well the expected values of charge; −1/3 times the electron charge for the quasi-hole, and 1/3 for the quasi-electron. Regarding fluctuations in the charge, our results for the quasi-hole and Jain quasi-electron are consistent with the expected value zero in the bulk of the system, but for the Laughlin quasi-electron we find small, but significant, deviations from zero throughout the whole electron droplet. We also present Berry phase calculations of charge and statistics parameter for the Jain quasi-electron, calculations which supplement earlier studies for the Laughlin quasi-particles. We find that the statistics parameter is more well behaved for the Jain quasi-electron than it is for the Laughlin quasi-electron. † Supported by The Norwegian Research Council.

We examine the anyon representation of the Laughlin quasi-holes, in particular the onedimensional, algebraic aspects of the representation. For the cases of one and two quasiholes an explicit mapping to anyon systems is given, and the connection between the holestates and coherent states of the fundamental algebras of observables is examined. The quasi-electron case is discussed more briefly, and some remaining questions are pointed out. †

Predrill assessment of the probability that a potential drilling spot holds hydrocarbons (HC) is of vital importance to any oil company. Of equally great value is the assessment of hydrocarbon volumes and distributions. We have developed a methodology that uses seismic data to find the probability that a vertical earth profile contains hydrocarbons and the probability distribution of hydrocarbon volumes. The method combines linearized amplitude variation with offset (AVO) inversion and stochastic rock models and predicts the joint probability distribution of the combined lithology and fluid for the entire profile. We use a Bayesian approach and find the solution of the inverse problem by Markov chain Monte Carlo simulation. The stochastic simulation benefits from a new and tailored simulation algorithm. The computational cost of finding the full joint probability distribution is relatively high and implies that the method is best suited to the investigation of a few potential drilling spots. We applied the method to a case with well control and to two locations in a prospect: one in the center and one at the outskirts. At the well location, we identify the two reservoir zones and obtain volumes that fit the log data. At the prospect, we obtain significant increases in HC probability and volume in the center, whereas there are decreases at the outskirts. Despite the large noise components in the data, the risked volumes in the center changed by a factor of three. We have designed an algorithm for computing the joint distribution of lithology, fluid, and elastic parameters for a full vertical profile. As opposed to what can be done with pointwise approaches, this allows us to calculate success probability and HC volumes.

A multigrid Markov mesh model for geological facies is formulated by defining a hierarchy of nested grids and defining a Markov mesh model for each of these grids. The facies probabilities in the Markov mesh models are formulated as generalized linear models that combine functions of the grid values in a sequential neighborhood. The parameters in the generalized linear model for each grid are estimated from the training image. During simulation, the coarse patterns are first laid out, and by simulating increasingly finer grids we are able to recreate patterns at different scales. The method is applied to several tests cases and results are compared to the training image and the results of a commercially available snesim algorithm. In each test case, simulation results are compared qualitatively by visual inspection, and quantitatively by using volume fractions, and an upscaled permeability tensor. When compared to the training image, the method produces results that only have a few percent deviation from the values of the training image. When compared with the snesim algorithm the results in general have the same quality. The largest computational cost in the multigrid Markov mesh is the estimation of model parameters from the training image. This is of comparable CPU time to that of creating one snesim realization. The simulation of one realization is typically ten times faster than the estimation.

This paper describes a methodology that predicts rock properties of a vertical profile of the earth and its associated uncertainty from seismic amplitudes. The methodology can be used to predict the probability of hydrocarbon presence, reservoir column and quality. It is well suited for pre-drill spot tests. The predictions are based on a Bayesian approach, which defines a prior stochastic model for sand beds in a background of shale, and potential fluid contacts. The rock properties are defined by a stochastic rock physics model, which also accounts for the vertical continuity of the rock properties within a lithology and fluid class. A tailored Markov chain Monte Carlo algorithm efficiently explores possible earth models to find a representative set that is consistent with seismic data as well as the prior stochastic model. We have applied the methodology to a case from the North Sea.

No abstract

No abstract