Classical Statistical Theory Research Articles

Pairwise correlations in a three-partite quantum system from a prequantum random field

Prequantum classical statistical field theory (PCSFT) is a model that provides the possibility to represent the averages of quantum observables (including correlations of observables on subsystems of a composite system) as averages with respect to fluctuations of classical random fields. In view of the PCSFT terminology, quantum states are classical random fields. The aim of our approach is to represent all quantum probabilistic quantities by means of classical random fields. We obtain the classical-random-field representation for pairwise correlations in three-partite quantum systems. The three-partite case (surprisingly) differs substantially from the bipartite case. As an important first step, we generalized the theory developed for pure quantum states of bipartite systems to the states given by density operators.

The Statistical Origins of Quantum Mechanics

It is shown that Schrödinger's equation may be derived from three postulates. The first is a kind of statistical metamorphosis of classical mechanics, a set of two relations which are obtained from the canonical equations of particle mechanics by replacing all observables by statistical averages. The second is a local conservation law of probability with a probability current which takes the form of a gradient. The third is a principle of maximal disorder as realized by the requirement of minimal Fisher information. The rule for calculating expectation values is obtained from a fourth postulate, the requirement of energy conservation in the mean. The fact that all these basic relations of quantum theory may be derived from premises which are statistical in character is interpreted as a strong argument in favor of the statistical interpretation of quantum mechanics. The structures of quantum theory and classical statistical theories are compared, and some fundamental differences are identified.

An analogue of the Heisenberg uncertainty relation in prequantum classical field theory

Prequantum classical statistical field theory (PCSFT) is a model that provides the possibility of representing averages of quantum observables, including correlations of observables on subsystems of a composite system, as averages with respect to fluctuations of classical random fields. PCSFT is a classical model of wave type. For example, ‘electron’ is described by electronic field. In contrast to quantum mechanics (QM), this field is a real physical field and not a field of probabilities. An important point is that the prequantum field of , for example, an electron contains the irreducible contribution of the background field vacuum fluctuations. In principle, the traditional QM-formalism can be considered as a special regularization procedure: subtraction of averages with respect to vacuum fluctuations. In this paper, we derive a classical analogue of the Heisenberg–Robertson inequality for dispersions of functionals of classical (prequantum) fields. The PCSFT Robertson-like inequality provides a restriction on the product of classical dispersions. However, this restriction is not so rigid as in QM.

A Transductive Learning Approach to Process Fault Identification

The problem of fault identification is considered using recent developments in machine learning that allow the use of unlabeled data to optimally define decision boundaries that separate data belonging to different categories. Traditionally, fault identification uses either hardware redundancy or software redundancy for trouble-shooting the source of faults in a system. Unfortunately, this imposes a data redundancy cost on such systems. Instead of performing model inversion that requires an accurate model, in transduction estimates of the values of a function at specified points are required, instead of learning a general rule on the entire input domain. Transductive learning is motivated from similar arguments underlying state-of-the-art classification and regression methods such as support vector machines. However, transduction is more fundamental as it is a step used in proving learning error bounds in classical statistical learning theory. Use of transduction allows a flexible ordering of the classes of functions from which a model is selected and, therefore, the error bounds are provably tight. The potential of the proposed framework is assessed using data from metallurgical process systems. It is shown that for higher dimensional and large multiclass systems, the proposed framework gives better performances with respect to classification error minimization.

Prequantum Classical Statistical Field Theory: Schrödinger Dynamics of Entangled Systems as a Classical Stochastic Process

The idea that quantum randomness can be reduced to randomness of classical fields (fluctuating at time and space scales which are essentially finer than scales approachable in modern quantum experiments) is rather old. Various models have been proposed, e.g., stochastic electrodynamics or the semiclassical model. Recently a new model, so called prequantum classical statistical field theory (PCSFT), was developed. By this model a “quantum system” is just a label for (so to say “prequantum”) classical random field. Quantum averages can be represented as classical field averages. Correlations between observables on subsystems of a composite system can be as well represented as classical correlations. In particular, it can be done for entangled systems. Creation of such classical field representation demystifies quantum entanglement. In this paper we show that quantum dynamics (given by Schrodinger’s equation) of entangled systems can be represented as the stochastic dynamics of classical random fields. The “effect of entanglement” is produced by classical correlations which were present at the initial moment of time, cf. views of Albert Einstein.

Memory effects in recurrent and extreme events

A dynamical approach to recurrent and extreme events is developed focusing on the role of correlations and memory in the structure of the probability distributions and their low-order moments. The procedure is illustrated on homogeneous first and second order Markov chains, non-Markovian and nonhomogeneous processes and deterministic dynamical systems. Substantial differences with classical statistical theory as applied to independent identically distributed random variables are identified.

Description of Composite Quantum Systems by Means of Classical Random Fields

Recently a new attempt to go beyond QM was performed in the form of so-called prequantum classical statistical field theory (PCSFT). In this approach quantum systems are described by classical random fields, e.g., the electron field or the neutron field. Averages of quantum observables arise as approximations of averages of classical variables (functionals of “prequantum fields”) with respect to fluctuations of fields. For classical variables given by quadratic functionals of fields, quantum and prequantum averages simply coincide. In this paper we generalize PCSFT to the composite quantum system. The main discovery is that, opposite to a rather common opinion, a composite system can be described by the Cartesian product of state spaces (like in classical physics) and not by the tensor product of them (like in conventional QM). A natural interpretation of the quantum pure state for a composite system is proposed: it is the nondiagonal block in the covariance matrix of the random field describing a composite system. The interpretation of a pure state due to Dirac and von Neumann seems to be an artifact of the conventional mathematical description of micro-systems. PCSFT provides a new possibility for interpretation of entanglement.

A microcontact model developed for sphere- and cylinder-based fractal bodies in contact with a rigid flat surface

The present study examines sphere- and cylinder-based fractal bodies in contact with a smooth and rigid flat surface. The generalized Weierstrass–Mandelbrot function (the W–M function) is employed in combination with the radius–vector function method (the (r, θ) method) to derive the formulae of fractal surface asperities formed on a smooth spherical surface and a smooth cylindrical surface. The size distribution functions used to calculate the contact load and the real contact area of an apparent area were derived for the three deformation regimes of microcontacts. The non-Gaussian probability density functions for two contact forms (cylindrical and spherical) were developed by combining the probability density functions developed for all small elements in the contact area using the superposition technique. In the present model, the deformation behavior demonstrated in the elastic, elastoplastic, and fully plastic deformation regimes is consistent with that described by classical (statistical) theories. Both the contact load and the real contact area results predicted by the present fractal model at various minimum separations show very good agreement with the results obtained using the statistical method. However, the results also exhibited properties very different from the results predicted by the unmodified fractal model, which showed the microcontact behavior demonstrated in the three deformation regimes to be exactly opposite to that predicted by the classical theories.

Classical spin systems and the quantum stabilizer formalism: General mappings and applications

We present general mappings between classical spin systems and quantum physics. More precisely, we show how to express partition functions and correlation functions of arbitrary classical spin models as inner products between quantum stabilizer states and product states, thereby generalizing mappings for some specific models established in our previous work [M. Van den Nest et al., Phys. Rev. Lett. 98, 117207 (2007)]. For Ising- and Potts-type models with and without external magnetic field, we show how the entanglement features of the corresponding stabilizer states are related to the interaction pattern of the classical model, while the choice of product states encodes the details of the interaction. These mappings establish a link between the fields of classical statistical mechanics and quantum information theory, which we utilize to transfer techniques and methods developed in one field to gain insight into the other. For example, we use quantum information techniques to recover well known duality relations and local symmetries of classical models in a simple way and provide new classical simulation methods to simulate certain types of classical spin models. We show that in this way all inhomogeneous models of q-dimensional spins with pairwise interaction pattern specified by a graph of bounded tree width can be simulated efficiently. Finally, we show relations between classical spin models and measurement-based quantum computation.

DPRESS: Localizing estimates of predictive uncertainty.

BackgroundThe need to have a quantitative estimate of the uncertainty of prediction for QSAR models is steadily increasing, in part because such predictions are being widely distributed as tabulated values disconnected from the models used to generate them. Classical statistical theory assumes that the error in the population being modeled is independent and identically distributed (IID), but this is often not actually the case. Such inhomogeneous error (heteroskedasticity) can be addressed by providing an individualized estimate of predictive uncertainty for each particular new object u: the standard error of prediction su can be estimated as the non-cross-validated error st* for the closest object t* in the training set adjusted for its separation d from u in the descriptor space relative to the size of the training set.The predictive uncertainty factor γt* is obtained by distributing the internal predictive error sum of squares across objects in the training set based on the distances between them, hence the acronym: Distributed PRedictive Error Sum of Squares (DPRESS). Note that st* and γt*are characteristic of each training set compound contributing to the model of interest.ResultsThe method was applied to partial least-squares models built using 2D (molecular hologram) or 3D (molecular field) descriptors applied to mid-sized training sets (N = 75) drawn from a large (N = 304), well-characterized pool of cyclooxygenase inhibitors. The observed variation in predictive error for the external 229 compound test sets was compared with the uncertainty estimates from DPRESS. Good qualitative and quantitative agreement was seen between the distributions of predictive error observed and those predicted using DPRESS. Inclusion of the distance-dependent term was essential to getting good agreement between the estimated uncertainties and the observed distributions of predictive error. The uncertainty estimates derived by DPRESS were conservative even when the training set was biased, but not excessively so.ConclusionDPRESS is a straightforward and powerful way to reliably estimate individual predictive uncertainties for compounds outside the training set based on their distance to the training set and the internal predictive uncertainty associated with its nearest neighbor in that set. It represents a sample-based, a posteriori approach to defining applicability domains in terms of localized uncertainty.

Cumulative and averaging fusion of beliefs

The problem of fusing beliefs in the Dempster–Shafer belief theory has attracted considerable attention over the last two decades. The classical Dempster’s rule has often been criticised, and many alternative rules for belief fusion have been proposed in the literature. We show that it is crucial to consider the nature of the situation to be modelled and to select the appropriate fusion operator as a function thereof. In this paper, we present the cumulative rule and the averaging rule of belief fusion, which represent generalisations of the subjective logic consensus operator for independent and dependent opinions respectively. The generalised operators are applicable to the combination of general basic belief assignments (bbas). These rules, which can be directly derived from classical statistical theory, produce results that correspond well with human intuition.

Born's rule for momentum measurement from prequantum field theory

Recently a new attempt to go beyond quantum mechanics (QM) was done in the form of so called prequantum classical statistical field theory (PCSFT). In this approach QM is considered as incomplete theory – in total accordance with Einstein [1]. PCSFT is (one of possible) completions of QM. Hidden variables are given by classical fields. (It is again in accordance with Einstein.) These prequantum fields fluctuate on time and space scales which are essentially finer than scales of QM, cf. 't Hooft's attempt to go beyond QM [2, 3]. However opposite to Einstein, PCSFT does not assign results of quantum observations directly to hidden variables. Bohr's idea that the whole experimental arrangement should be taken into account is basic for PCSFT-based measurement theory. This theory for measurements of momentum of quantum particles is presented in this paper.

Model Uncertainty in Sociological Research: An Application to Religion and Economic Growth

Model uncertainty is pervasive in quantitative research. Classical statistical theory assumes that only one (true) model is applied to a sample of data. In practice, however, researchers do not know which exact model specification is best. Modern computing power allows researchers to estimate a huge number of plausible models, yet only a few of these estimates are published. The result is a severe asymmetry of information between analyst and reader. The applied modeling process produces a much wider range of estimates than is suggested by the usual standard errors or confidence intervals. I demonstrate this using the work of Barro and McCleary on religion and economic growth. Small, sensible changes in their model specification produce large changes in the results: the results are inconsistent across time, and the instrumental variables strategy suffers from a weak instrument set. Also, the observed relationship between religiosity and economic growth does not hold in the West; it is largely a feature of Asian and African countries and of countries whose data is poor quality. In short, empirical findings should be evaluated not just by their significance but also by their robustness to model specification. I conclude with suggestions for incorporating model uncertainty into practice and improving the transparency of social science research.

Detection Model Based on Representation of Quantum Particles by Classical Random Fields: Born’s Rule and Beyond

Recently a new attempt to go beyond quantum mechanics (QM) was presented in the form of so called prequantum classical statistical field theory (PCSFT). Its main experimental prediction is violation of Born's rule which provides only an approximative description of real probabilities. We expect that it will be possible to design numerous experiments demonstrating violation of Born's rule. Moreover, recently the first experimental evidence of violation was found in the triple slits interference experiment, see \cite{WWW}. Although this experimental test was motivated by another prequantum model, it can be definitely considered as at least preliminary confirmation of the main prediction of PCSFT. In our approach quantum particles are just symbolic representations of "prequantum random fields," e.g., "electron-field" or "neutron-field"; photon is associated with classical random electromagnetic field. Such prequantum fields fluctuate on time and space scales which are essentially finer than scales of QM, cf. `t Hooft's attempt to go beyond QM \cite{H1}--\cite{TH2}. In this paper we elaborate a detection model in the PCSFT-framework. In this model classical random fields (corresponding to "quantum particles") interact with detectors inducing probabilities which match with Born's rule only approximately. Thus QM arises from PCSFT as an approximative theory. New tests of violation of Born's rule are proposed.

Tests with correct size when instruments can be arbitrarily weak

This paper applies classical exponential-family statistical theory to develop a unifying framework for testing structural parameters in the simultaneous equations model under the assumption of normal errors with known reduced-form variance matrix. The results can be divided into the limited-information and full-information categories. In the limited-information model, it is possible to characterize the entire class of similar tests in a model with only one endogenous explanatory variable. In the full-information framework, this paper proposes a family of similar tests for subsets of endogenous variables’ coefficients. For both limited- and full-information models, there exist power upper bounds for unbiased tests. When the model is just-identified, the Anderson–Rubin, score, and (pseudo) conditional likelihood ratio tests are optimal. When the model is over-identified, the (pseudo) conditional likelihood ratio test has power close to the power envelope when identification is strong.

Statistical analysis for rounded data

When random variables do not take discrete values, observed data are often the rounded values of continuous random variables. Errors caused by rounding of data are often neglected by classical statistical theories. While some pioneers have identified and made suggestions to rectify the problem, few suitable approaches were proposed. In this paper, we propose an approximate MLE (AMLE) procedure to estimate the parameters and discuss the consistency and asymptotic normality of the estimates. For our illustration, we shall consider the estimates of the parameters in AR ( p ) and MA ( q ) models for rounded data.

An analytical model and parametric study of electrical contact resistance in proton exchange membrane fuel cells

This paper presents an analytical model of the electrical contact resistance between the carbon paper gas diffusion layers (GDLs) and the graphite bipolar plates (BPPs) in a proton exchange membrane (PEM) fuel cell. The model is developed based on the classical statistical contact theory for a PEM fuel cell, using the same probability distributions of the GDL structure and BPP surface profile as previously described in Wu et al. [Z. Wu, Y. Zhou, G. Lin, S. Wang, S.J. Hu, J. Power Sources 182 (2008) 265–269] and Zhou et al. [Y. Zhou, G. Lin, A.J. Shih, S.J. Hu, J. Power Sources 163 (2007) 777–783]. Results show that estimates of the contact resistance compare favorably with experimental data by Zhou et al. [Y. Zhou, G. Lin, A.J. Shih, S.J. Hu, J. Power Sources 163 (2007) 777–783]. Factors affecting the contact behavior are systematically studied using the analytical model, including the material properties of the two contact bodies and factors arising from the manufacturing processes. The transverse Young's modulus of chopped carbon fibers in the GDL and the surface profile of the BPP are found to be significant to the contact resistance. The factor study also sheds light on the manufacturing requirements of carbon fiber GDLs for a better contact performance in PEM fuel cells.

Grey Relation between Nonlinear Characteristic and Dynamic Uncertainty of Rolling Bearing Friction Torque

The rolling bearing friction torque which is characterized by its uncertainty and nonlinearity affects heavily the dynamic performance of a system such as missiles, spacecrafts and radars, etc. It is difficult to use the classical statistical theory to evaluate the dynamic evaluation of the rolling bearing friction torque for the lack of prior information about both probability distribution and trends. For this reason, based on the information poor system theory and combined with the correlation dimension in chaos theory, the concepts about the mean of the dynamic fluctuant range (MDFR) and the grey relation are proposed to resolve the problem about evaluating the nonlinear characteristic and the dynamic uncertainty of the rolling bearing friction torque. Friction torque experiments are done for three types of the rolling bearings marked with HKTA, HKTB and HKTC separately; meantime, the correlation dimension and MDFR are calculated to describe the nonlinear characteristic and the dynamic uncertainty of the friction torque, respectively. And the experiments reveal that there is a certain grey relation between the nonlinear characteristic and the dynamic uncertainty of the rolling bearing friction torque, viz. MDFR will become the nonlinear increasing trend with the correlation dimension increasing. Under the condition of fewer characteristic data and the lack of prior information about both probability distribution and trends, the unitive evaluation for the nonlinear characteristic and the dynamic uncertainty of the rolling bearing friction torque is realized with the grey confidence level of 87.7%-96.3%.

Effective convergence of the two-particle irreducible1/Nexpansion for nonequilibrium quantum fields

The $1/N$ expansion of the two-particle irreducible effective action offers a powerful approach to study quantum field dynamics far from equilibrium. We investigate the effective convergence of the $1/N$ expansion in the $O(N)$ model by comparing results obtained numerically in $1+1$ dimensions at leading, next-to-leading and next-to-next-to-leading order in $1/N$ as well as in the weak coupling limit. A comparison in classical statistical field theory, where exact numerical results are available, is made as well. We focus on early-time dynamics and quasiparticle properties far from equilibrium and observe rapid effective convergence already for moderate values of $1/N$ or the coupling.

A new interval mapping approach based on a general sib-pair regression model with a modified Wald test

Various regression models based on sib-pair data have been developed for mapping quantitative trait loci (QTL) in humans since the seminal paper published in 1972 by Haseman and Elston. Fulker and Cardon [D.W. Fulker, L.R. Cardon, A sib-pair approach to interval mapping of quantitative trait loci, Am. J. Hum. Genet. 54 (1994) 1092–1103] adapted the idea of interval mapping [E.S. Lander, D. Botstein, Mapping Mendelian factors underlying quantitative traits using RFLP linkage maps, Genetics 121 (1989) 185–199] to the Haseman–Elston regression model in order to increase the power of QTL mapping. However, in the interval mapping approach of Fulker and Cardon, the statistic for testing QTL effects does not obey the classical statistical theory and hence critical values of the test can not be appropriately determined. In this article, we consider a new interval mapping approach based on a general sib-pair regression model. A modified Wald test is proposed for the testing of QTL effects. The asymptotic distribution of the modified Wald test statistic is provided and hence the critical values or the p -values of the test can be well determined. Simulation studies are carried out to verify the validity of the modified Wald test and to demonstrate its desirable power.