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R. A. Serota

Publications and source records attributed to R. A. Serota.

At least 37 records · Page 2Linked to original sources
arXiv2013

Spectral and Parametric Averaging for Integrable Systems

We analyze two theoretical approaches to ensemble averaging for integrable systems in quantum chaos - spectral averaging and parametric averaging. For spectral averaging, we introduce a new procedure - rescaled spectral averaging. Unlike traditional spectral averaging, it can describe the correlation function of spectral staircase and produce persistent oscillations of the interval level number variance. Parametric averaging, while not as accurate as rescaled spectral averaging for the correlation function of spectral staircase and interval level number variance, can also produce persistent oscillations of the global level number variance and better describes saturation level rigidity as a function of the running energy. Overall, it is the most reliable method for a wide range of statistics.

quant-ph↗
arXiv2013

Distribution of Human Response Times

We demonstrate that distributions of human response times have power-law tails and, among closed-form distributions, are best fit by the generalized inverse gamma distribution. We speculate that the task difficulty tracks the half-width of the distribution and show that it is related to the exponent of the power-law tail.

q-bio.NC↗
arXiv2013

A Model for Stock Returns and Volatility

We prove that Student's t-distribution provides one of the better fits to returns of S&P component stocks and the generalized inverse gamma distribution best fits VIX and VXO volatility data. We further argue that a more accurate measure of the volatility may be possible based on the fact that stock returns can be understood as the product distribution of the volatility and normal distributions. We find Brown noise in VIX and VXO time series and explain the mean and the variance of the relaxation times on approach to the steady-state distribution.

q-fin.ST↗
arXiv2012

Distribution Of Wealth In A Network Model Of The Economy

We show, analytically and numerically, that wealth distribution in the Bouchaud-Mézard network model of the economy is described by a three-parameter generalized inverse gamma distribution. In the mean-field limit of a network with any two agents linked, it reduces to the inverse gamma distribution.

q-fin.ST↗
arXiv2012

Global Level Number Variance in Integrable Systems

We study previously un-researched second order statistics - correlation function of spectral staircase and global level number variance - in generic integrable systems with no extra degeneracies. We show that the global level number variance oscillates persistently around the saturation spectral rigidity. Unlike other second order statistics - including correlation function of spectral staircase - which are calculated over energy scales much smaller than the running spectral energy, these oscillations cannot be explained within the diagonal approximation framework of the periodic orbit theory. We give detailed numerical illustration of our results using four integrable systems: rectangular billiard, modified Kepler problem, circular billiard and elliptic billiard.

quant-ph↗
arXiv2011

Elliptic billiard - a non-trivial integrable system

We investigate the semiclassical energy spectrum of quantum elliptic billiard. The nearest neighbor spacing distribution, level number variance and spectral rigidity support the notion that the elliptic billiard is a generic integrable system. However, second order statistics exhibit a novel property of long-range oscillations. Classical simulation shows that all the periodic orbits except two are not isolated. In Fourier analysis of the spectrum, all the peaks correspond to periodic orbits. The two isolated periodic orbits have small contribution to the fluctuation of level density, while non-isolated periodic orbits have the main contribution. The heights of the majority of the peaks match our semiclassical theory except for type-O periodic orbits. Elliptic billiard is a nontrivial integrable system that will enrich our understanding of integrable systems.

quant-ph↗
arXiv2010

Level Repulsion in Integrable Systems

Contrary to conventional wisdom, level repulsion in semiclassical spectrum is not just a feature of classically chaotic systems, but classically integrable systems as well. While in chaotic systems level repulsion develops on a scale of the mean level spacing, regardless of location in the spectrum, in integrable systems it develops on a much longer scale - geometric mean of the mean level spacing and the running energy in the spectrum. We show that at this scale level correlations in integrable systems have a universal dependence on level separation, as well as discuss their exact form at any scale. These correlations have dramatic consequences, including deviations from Poissonian statistics in the nearest level spacing distribution and persistent oscillations of level number variance as a function of the interval width. We illustrate our findings on two models - a rectangular infinite well and a modified Kepler problem - that serve as generic types of hard-wall billiards and potential problems. Our theory is based on the concept of parametric averaging, which allows for a statistical ensemble of integrable systems at a given spectral location (running energy).

quant-ph↗
arXiv2008

Effects of Sample Specific Variations and Fluctuations of Thermal Occupancy on Fluctuations of Thermodynamic Quantities

Standard results for the fluctuations of thermodynamic quantities are derived under the assumption of sampling identical systems that are in different, not fully equilibrated states. These results apply to fluctuations with time in a particular macroscopic body and can be traced to the fluctuations of thermal occupancy. When many identically prepared - but not identical - systems are studied, mesoscopic fluctuations due to variations from sample to sample contribute to the fluctuations of thermodynamic quantities. We study the combined effect of mesoscopic fluctuations and fluctuations of thermal occupancy. In particular, we evaluate the total particle number and specific heat fluctuations in a two-dimensional, non-interacting electron gas in classically integrable and chaotic circumstances.

cond-mat.mes-hall↗
arXiv2007

Quantum jumps of saturation level rigidity and anomalous oscillations of level number variance in the semiclassical spectrum of a modified Kepler problem

We discover quantum Hall like jumps in the saturation spectral rigidity in the semiclassical spectrum of a modified Kepler problem as a function of the interval center. These jumps correspond to integer decreases of the radial winding numbers in classical periodic motion. We also discover and explain single harmonic dominated oscillations of the level number variance with the width of the energy interval. The level number variance becomes effectively zero for the interval widths defined by the frequency of the shortest periodic orbit. This signifies that there are virtually no variations from sample to sample in the number of levels on such intervals.

quant-ph↗
arXiv2005

Mesoscopic fluctuations of off-diagonal matrix elements of the angular momentum and orbital magnetism of free electrons in a rectangular box

We study, analytically and numerically, mesoscopic fluctuations of the off-diagonal matrix elements of the orbital angular momentum between the nearest energy levels $i=(n_{x},n_{y}) $ and $f=(k_{x},k_{y}) $ in a rectangular box with incommensurate sides. In the semiclassical regime, where the level number of $i$ is $\cal{N}\gg 1$, our derivation gives $\left< | \hat{L}_{if}| ^{2}\right> \sim \sqrt{\cal{N}}$. Numerical simulations, using simultaneous ensemble averaging (over the aspect ratios of rectangles) and spectral averaging (over the energy interval), are in excellent agreement with this analytical prediction. Physically, the mean is dominated by the level pairs $k_{x}=n_{x}\pm 1$, $k_{y}=n_{y}\mp 1$. Also in a rectangular box, we investigate the mean orbital susceptibility of a free electron gas and argue that it reduces, up to a coefficient, to the two-level van Vleck susceptibility that involves the last occupied (Fermi) level $i$ and the first unoccupied level $f$. This result is confirmed numerically as well, albeit the effect of fluctuations is much more pronounced for the susceptibility since it is due both to large fluctuations in $\left< | \hat{L}_{if}| ^{2}\right>$ and in level separations $ε_{f}-ε_{i}$ (level bunching).

cond-mat.mes-hall↗
arXiv2003

The Absence Of Saturation Of The Level Number Variance In A Rectangular Box

The variance of the number of levels in an energy interval around a level with large quantum numbers (semiclassical quantization) is studied for a particle in a rectangular box. Sampling involves changing the ratio of the rectangle's sides while keeping the area constant. For sufficiently narrow intervals, one finds the usual linear growth with the width of the interval. For wider intervals, the variance undergoes large, non-decaying oscillations around what is expected to be the saturation value. These oscillations can be explained as a superposition of just a few harmonics that correspond to the shortest periodic orbits in the rectangle. The analytical and numerical results are in excellent agreement.

cond-mat.mes-hall↗
arXiv2001

Level correlations in integrable systems

We derive a simple analytical expression for the level correlation function of an integrable system. It accounts for both the lack of correlations at smaller energy scales and for global rigidity (level number conservation) at larger scales. We apply our results to a rectangle with incommensurate sides and show that they are in excellent agreement with the limiting cases established in the semiclassical theory of level rigidity.

cond-mat.mes-hall↗
arXiv2001

Mesoscopic Fluctuations Of Orbital Magnetic Response In Level-Quantized Metals

We evaluate the distribution function of mesoscopic fluctuations of orbital magnetic response in finite-size level-quantized metal particles and Aharonov-Bohm rings for temperatures smaller than the mean level spacing. We find a broad distribution with the reduced moments much larger than the mean. For strong spin-orbit interaction we find very long tails due to thermal activation of large effective moments of the electrons at the Fermi level.

cond-mat.mes-hall↗
arXiv2000

Polarizability and Absorption of Small Conducting Particles in a Time-Varying Electromagnetic Field

We study small conducting particles and thin films in an oscillating longitudinal electric field. We find the charge, current, and field distribution in the particle, the polarizability and the electric dipole absorption. We account for Thomas-Fermi screening by adding a Fick's diffusion term to Ohm's law. Alternatively, we describe a particle as a dielectric body with a non-local dielectric constant which is derived in a microscopic linear-response theory. We show that both approaches are equivalent.

cond-mat.mes-hall↗
arXiv2000

Permanent magnetic moment in mesoscopic metals with spin-orbit interaction

We argue that at zero temperature an isolated metal particle (or an AB ring) with spin-orbit interaction and odd number of electrons will have a permanent magnetic moment, even in zero magnetic field (flux). In a zero-field-cooled state both the direction and the magnitude of the moment varies from particle to particle and averages to zero. In a field-cooled state it averages to $\sim μ_{B}(k_{F}\ell) ^{1/2}$. We argue that the permanent moment is due to an uncompensated electron in the last occupied (Fermi) level. We introduce an effective single-electron Hamiltonian which accounts for spin-orbit coupling.

cond-mat.mes-hall↗
arXiv1998

Quantum absorption in small metal particles

We evaluate the electric dipole absorption in small metal particles in a longitudinal electric field taking into account the Fermi-Thomas screening. When either the level broadening or the frequency of the field are larger than the mean energy-level spacing, the main contribution to absorption is classical, with quantum corrections. When both the broadening and the frequency are smaller than the mean level spacing, the absorption is manifestly quantum and can be understood in terms of the two-level system.

cond-mat.mes-hall↗