Fractional Degrees Of Freedom Research Articles

Topological Quantum Synchronization of Fractionalized Spins.

The gapped symmetric phase of the Affleck-Kennedy-Lieb-Tasaki model exhibits fractionalized spins at the ends of an open chain. We show that breaking SU(2) symmetry and applying a global spin-lowering dissipator achieves synchronization of these fractionalized spins. Additional local dissipators ensure convergence to the ground state manifold. In order to understand which aspects of this synchronization are robust within the entire Haldane-gap phase, we reduce the biquadratic term, which eliminates the need for an external field but destabilizes synchronization. Within the ground state subspace, stability is regained using only the global lowering dissipator. These results demonstrate that fractionalized degrees of freedom can be synchronized in extended systems with a significant degree of robustness arising from topological protection. A direct consequence is that permutation symmetries are not required for the dynamics to be synchronized, representing a clear advantage of topological synchronization compared to synchronization induced by permutation symmetries.

Fractional Degrees of Freedom at Infinite Coupling in Large N_{f} QED in 2+1 Dimensions.

I consider quantum electrodynamics with many electrons in 2+1 space-time dimensions at finite temperature. The relevant dimensionless interaction parameter for this theory is the fine structure constant divided by the temperature. The theory is solvable at any value of the coupling, in particular for very weak (high temperature) and infinitely strong coupling (corresponding to the zero temperature limit). Concentrating on the photon, each of its physical degrees of freedom at infinite coupling only contributes half of the free-theory value to the entropy. These fractional degrees of freedom are reminiscent of what has been observed in other strongly coupled systems (such as N=4 supersymmetric Yang-Mills theory), and bear similarity to the fractional quantum Hall effect, potentially suggesting connections between these phenomena. The results found for (2+1)-dimensional QED are fully consistent with the expectations from particle-vortex duality.

Stochastic thermodynamics of nonharmonic oscillators in high vacuum.

We perform an analytic study on the stochastic thermodynamics of a small classical particle trapped in a time-dependent single-well potential in the highly underdamped limit. It is shown that the nonequilibrium probability density function for the system's energy is a Maxwell-Boltzmann distribution (as in equilibrium) with a closed form time-dependent effective temperature and fractional degrees of freedom. We also find that the solvable model satisfies the Crooks fluctuation theorem, as expected. Moreover, we compute the average work in this isothermal process and characterize analytically the optimal protocol for minimum work. The optimal protocol presents an initial and a final jump which correspond to adiabatic processes linked by a smooth exponential time-dependent part for all kinds of single-well potentials. Furthermore, we argue that this result connects two distinct relevant experimental setups for trapped nanoparticles, the levitated particle in a harmonic trap, and the free particle in a box, as they are limiting cases of the general single-well potential and display the time-dependent optimal protocols. Finally, we highlight the connection between our system and an equivalent model of a gas of Brownian particles.

Testing for Homogeneity in Meta-Analysis I. The One-Parameter Case: Standardized Mean Difference

Meta-analysis seeks to combine the results of several experiments in order to improve the accuracy of decisions. It is common to use a test for homogeneity to determine if the results of the several experiments are sufficiently similar to warrant their combination into an overall result. Cochran's Q statistic is frequently used for this homogeneity test. It is often assumed that Q follows a chi-square distribution under the null hypothesis of homogeneity, but it has long been known that this asymptotic distribution for Q is not accurate for moderate sample sizes. Here, we present an expansion for the mean of Q under the null hypothesis that is valid when the effect and the weight for each study depend on a single parameter, but for which neither normality nor independence of the effect and weight estimators is needed. This expansion represents an order O(1/n) correction to the usual chi-square moment in the one-parameter case. We apply the result to the homogeneity test for meta-analyses in which the effects are measured by the standardized mean difference (Cohen's d-statistic). In this situation, we recommend approximating the null distribution of Q by a chi-square distribution with fractional degrees of freedom that are estimated from the data using our expansion for the mean of Q. The resulting homogeneity test is substantially more accurate than the currently used test. We provide a program available at the Paper Information link at the Biometrics website http://www.biometrics.tibs.org for making the necessary calculations.

FRACTAL CHARACTERIZATIONS OF MAX STATISTICAL DISTRIBUTION IN GENETIC ASSOCIATION STUDIES

Two noninteger parameters are defined for MAX statistics, which are maxima of d simpler test statistics. The first parameter, dMAX, is the fractional number of tests, representing the equivalent numbers of independent tests in MAX. If the d tests are dependent, dMAX< d. The second parameter is the fractional degrees of freedom k of the chi-square distribution [Formula: see text] that fits the MAX null distribution. These two parameters, dMAXand k, can be independently defined, and k can be noninteger even if dMAXis an integer. We illustrate these two parameters using the examples of MAX2 and MAX3 statistics in genetic case-control studies. We speculate that k is related to the amount of ambiguity of the model inferred by the test. In the case-control genetic association, tests with low k (e.g. k = 1) are able to provide definitive information about the disease model, as versus tests with high k (e.g. k = 2) that are completely uncertain about the disease model. Similar to Heisenberg's uncertain principle, the ability to infer disease model and the ability to detect significant association may not be simultaneously optimized, and k seems to measure the level of their balance.

The Effect of Simulated Cataracts on Drivers’ Hazard Perception Ability

We investigated the extent to which simulated cataracts slow a driver's ability to anticipate potential traffic hazards, a skill that has been found to correlate with crash risk. In previous studies, we found a significant correlation between contrast sensitivity and hazard perception in a sample of older drivers. The present study allowed us to determine the causal direction of this relationship. This is important as it provides a better understanding of the mechanisms underlying the higher crash risk of drivers with cataracts. One hundred eighty-six drivers with normal vision completed a validated video-based hazard perception driving test, designed to measure hazard anticipation response times in dynamic scenes. They also completed a change detection task based on traffic hazards, which was designed to measure object detection times in static scenes. Participants undertook the tasks wearing either mild or moderate cataract simulation goggles, or wearing goggle frames without lenses. Participants wearing moderate simulated cataract goggles were slower than the control group in both the hazard perception test, t(98.50) = -3.71, p < 0.001, and the hazard change detection task, t(124) = -13.86, p < 0.001. Participants with the mild simulated cataract goggles were slower than the control group in the hazard change detection task, t(114) = -4.04, p < 0.001, but not the hazard perception test, t(114) = -1.33, p = 0.19. Moderate levels of simulated cataract slowed drivers' ability to detect and anticipate traffic hazards enough to warrant road safety concerns, despite the fact that the vision of participants wearing the cataract goggles still complied with the minimum legal standard required for driving.

GARCH-type Models with Generalized Secant Hyperbolic Innovations

GARCH-type models have been analyzed assuming various nongaussian distributions of errors. In general, the asymmetric generalized Student-t random variable seems to be the distribution which better captures the nonnormality features of financial data. However, a drawback of this distribution is represented by the technical dificulties due to the evaluation of moments, especially in the case of fractional degrees of freedom. In this paper we propose to model high frequency time series returns using GARCH-type models with a generalized secant hyperbolic (GSH) distribution. The main advantage of the GSH distribution over the Student-t distribution is that all the moments are finite for each value of the shape parameter. The distribution is symmetric with respect to the mean, but we show that it is still possible to obtain the density in a closed form introducing a skewness parameter according to the method proposed by Fernandez and Steel. We use a Monte Carlo experiment to validate this distribution in the context of GARCH models with maximum likelihood estimates of parameters. Finally, we show an application to log returns of a stock index.

On the importance of zero-point effects in molecular level classical simulations of water

We discuss the fundamental difficulties involved in comparing energetic results obtained via classical simulations of bulk water with the observed values. Emphasis is placed on the difference between quantum and classical dynamics, and correction techniques, which can be used to emulate quantum effects in a classical system, are investigated. We present molecular dynamics simulation results for liquid water using the ‘Thole-type’ all atom polarizable water model, which has previously been shown to give reasonable results for both ice Ih and small water clusters. We employ expressions for the density of states power spectrum in the liquid in either atomic or rigid-body coordinates that are appropriate for rigid molecule simulations. It is demonstrated that the atomic power spectra can be written as a linear combination of the center of mass and rotational power spectra via the use of the ‘coupling matrix’ of linear coefficients. This approach allows us to introduce the concept of ‘fractional degrees of freedom’ (DOF) for nuclei in rigid molecule simulation. Within this framework, it is illustrated that in a rigid water molecule the oxygen and hydrogen atoms have 2.82 and 1.59 DOF, respectively (for the TIP4P geometry). Within our suggested approach, we finally demonstrate that Debye–Waller factors can be obtained from the coupling matrix and show that quantum corrections to the structure can be accounted for by raising the temperature of the system in a classical simulation by approximately 50°, a result consistent with previous suggestions.

A Program for F Distributions with Fractional Degrees of Freedom

A FORTRAN 77 program is described that evaluates the F distribution, providing probability values given a computed F statistic and critical values of F given a probability value. The program is accurate for all F distributions and allows the user to input positive fractional degrees of freedom.

On the distribution of the likelihood ratio test statistic for a mixture of two normal distributions

A likelihood ratio statistic can be used to test whether a random sample is taken from a single normal distribution or from a mixture of two normal distributions with a common variance. The asymptotic distribution of the likelihood ratio test statistic G2 (minus twice the difference in the logc likelihoods) in this situation is often assumed to be χ2 with two degrees of freedom. In this study simulation is used to investigate the distribution of G2 for sample sizes up to 256,000. We determine several approximations to the empirical distribution for sample sizes between 50 and 500, including one that does not require the computation of a χ2 distribution with fractional degrees of freedom. We conclude that although the asymptotic distribution of G2 is not a χ2 with two degrees of freedom, this does appear to be a good approximation to the upper 15% of the distribution

Design of the Likelihood Ratio Chart Based on the x2 -approximation with Fractional Degrees of Freedom

Design of the Likelihood Ratio Chart Based on the x² -approximation with Fractional Degrees of Freedom

An adjusted likelihood-ratio approach to the behrens-fisher problem

In many situations it is necessary to test the equality of the means of two normal populations when the variances are unknown and unequal. This paper studies the celebrated and controversial Behrens-Fisher problem via an adjusted likelihood-ratio test using the maximum likelihood estimates of the parameters under both the null and the alternative models. This procedure allows the significance level to be adjusted in accordance with the degrees of freedom to balance the risk due to the bias in using the maximum likelihood estimates and the risk due to the increase of variance. A large scale Monte Carlo investigation is carried out to show that -2 InA has an empirical chi-square distribution with fractional degrees of freedom instead of a chi-square distribution with one degree of freedom. Also Monte Carlo power curves are investigated under several different conditions to evaluate the performances of several conventional procedures with that of this procedure with respect to control over Type I errors and ...

Probability Content of Regions Under Spherical Normal Distributions, II: The Distribution of the Range in Normal Samples

Numerous investigations, both theoretical and numerical, have been made of the distribution of the range in normal samples. One of the first investigators was Student [1] who examined the distribution on an empirical basis. Somewhat earlier, Tippett [2] presented tables and charts for the mean, standard deviation, and of the measures of skewness and kurtosis $(\beta_1 \text{and} \beta_2)$ for the range; further studies of the moments were made by E. S. Pearson [3], Hartley and Pearson [6] and by Ruben [7]. Tables of the moment constants are available in [6] and in Pearson and Hartley's book of statistical tables [10] (Tables 20 and 27), while, more recently, tables of the moment constants were provided by Harter and Clemm [11]. Approximations to the probability integral and percentage points of the distribution were suggested by E. S. Pearson [4], Cox [12], Patnaik [14] and Tukey [15]. Pearson's approximations were based on Pearsonian distributions of Type I and VI, while Cox used the Gamma Function (i.e., a multiple of $\chi^2$ with fractional degrees of freedom) and Patnaik a multiple of $\chi$ with fractional degrees of freedom as the basis for their approximations. These last two approximations have been compared by Pearson [5]. An approximation of a different type was derived by Johnson [16] who gave a series expansion for the probability integral suitable for low sample sizes and low values of the range. The behavior of the distribution for large sample size has been studied by Gumbel [17], [18], [19], Elfving [20], Cox [13] and Harley and Pearson [21]. For theoretical studies of the exact distribution reference is made to McKay and Pearson [22], Pillai [23], [24] and Cadwell [25] (see also Hartley [26]). Finally, tables of the probability integral and percentage points of the distribution have been provided by Hartley and Pearson [27], [10] (Tables 23 and 22), as well as by Harter and Clemm [11]. In the present paper, the following new results relating to the range distribution will be obtained: (i) The latter function may be expressed as the product of the sample size and the probability content of a certain parallelotope relative to a hyperspherical normal distribution, and (ii) the function can be evaluated as an infinite series involving the even moments of a sum of independent truncated normal variables. The distribution function may also be directly related to the moment generating function of the square of the sum of these truncated variables.