Fermions from classical statistics

Quantum particles and classical particles are described in a common setting of classical statistical physics. The property of a particle being “classical” or “quantum” ceases to be a basic conceptual difference. The dynamics differs, however, between quantum and classical particles. We describe position, motion and correlations of a quantum particle in terms of observables in a classical statistical ensemble. On the other side, we also construct explicitly the quantum formalism with wave function and Hamiltonian for classical particles. For a suitable time evolution of the classical probabilities and a suitable choice of observables all features of a quantum particle in a potential can be derived from classical statistics, including interference and tunneling. Besides conceptual advances, the treatment of classical and quantum particles in a common formalism could lead to interesting cross‐fertilization between classical statistics and quantum physics.

We compare the classical and quantum mechanical position-space probability densities for a particle in an asymmetric infinite well. In an idealized system with a discontinuous step in the middle of the well, the classical and quantum probability distributions agree fairly well, even for relatively small quantum numbers, except for anomalous cases which arise due to the unphysical nature of the potential. We are able to derive upper and lower bounds on the differences between the quantum and classical results. We also qualitatively discuss the momentum-space probability densities for this system using intuitive ideas about the amount of time a classical particle spends in various parts of the well. This system provides an excellent example of a non-trivial, but tractable, quantum mechanical bound state problem where the correlations between the amplitude and curvature of quantum mechanical wavefunctions can be easily compared to classical intuition about particle motion, with quantitative success, but also warning of possible surprises in non-physical limiting cases.

The deficiencies existing in Geotechnical parameters distribution fitting using the classical probability distribution are analyzed and the stochastic weighting method is employed to improve the the small samples of Geotechnical parameters into big samples, thus sloving the problems caused by the small sample. The maximum entropy principle and the big samples improving from small sample are used to generate probability density functions of Geotechnical parameters. It is more rational and scientific with the method proposed in the paper through the Kolmogorov-Smirnov Test. It is demonstrated through the example that the method presented in this paper not only successfully escapes the reliance of the classical probability distribution functions, but also make the computation result more closed to the fact due to the data coming from the big sample improving from the small sample, which has important engineering significance.

When a quantum system is divided into two local subsystems, measurements on the two subsystems can exhibit correlations beyond those possible in a classical joint probability distribution; these are partially explained by entanglement, and more generally by a wider class of measures such as the quantum discord. In this paper, I introduce a simple thought experiment defining a measure of quantum correlations, which I call the accord, and write the result as a minimax optimization over unitary matrices. I find the exact result for pure states as a simple function of the Schmidt coefficients and provide a complete proof, and I likewise provide and prove the result for several classes of mixed states, notably including all states of two qubits and the experimentally relevant case of a pure state mixed with colorless noise. I demonstrate that for two qubit states the accord provides a tight lower bound on the discord; for Bell diagonal states it is also an upper bound on entanglement.

Analytical results for the classical and semiclassical coordinate probability distribution function of anharmonic oscillators

It is shown by use of Milne's nonlinear transform of a one-dimensional Schr\"odinger eigenproblem that the wave function bears the classical statistical information of the density of time spent by the system at any point of the configuration space, as already suggested by White [Phys. Rev. 37, 1416 (1931)].

Predicting and modeling sports events has always been one of the main applications of probability and statistics. From analyzing the likelihood of a team's victory based on historical performance data can be used to estimate the probability of a particular player scoring a goal. This paper focuses on the practical use of probability in football matches, summarizing several innovative prediction models based on classical probability distributions. It demonstrates the accuracy of the bivariate Poisson distribution model in predicting football scores, especially the number of draws through examples. And visually illustrate the precise capture of goals and winning moments in football matches by the In-game win probability model which based on Bayesian statistical framework. The paper also introduces the broad integration prospects of probability in combination with other disciplines in sports events. These interdisciplinary approaches have significant implications for not only helps optimize team organization and management like tactical improvement and risk control, but also plays a key role in promoting the development of related industries such as sports betting and media broadcasting.

In the probability representation of quantum mechanics, quantum states are represented by a classical probability distribution, the marginal distribution function (MDF), whose time dependence is governed by a classical evolution equation. We find and explicitly solve, for a wide class of Hamiltonians, equations for the Green function of such an equation, the so-called classical propagator. We elucidate the connection of the classical propagator to the quantum propagator for the density matrix and to the Green function of the Schr\"odinger equation. Within the new description of quantum mechanics we give a definition of coherence solely in terms of properties of the MDF and we test the definition recovering well known results. As an application, the forced parametric oscillator is considered. Its classical and quantum propagator are found, together with the MDF for coherent and Fock states.

We demonstrate that the Wigner function of a pure quantum state is a wave\nfunction in a specially tuned Dirac bra-ket formalism and argue that the Wigner\nfunction is in fact a probability amplitude for the quantum particle to be at a\ncertain point of the classical phase space. Additionally, we establish that in\nthe classical limit, the Wigner function transforms into a classical\nKoopman-von Neumann wave function rather than into a classical probability\ndistribution. Since probability amplitude need not be positive, our findings\nprovide an alternative outlook on the Wigner function's negativity.\n

Despite various parallels between quantum states and ordinary information, quantum no-go-theorems have convinced many that there is no realistic framework that might underly quantum theory, no reality that quantum states can represent knowledge about. This paper develops the case that there is a plausible underlying reality: one actual spacetime-based history, although with behavior that appears strange when analyzed dynamically (one time-slice at a time). By using a simple model with no dynamical laws, it becomes evident that this behavior is actually quite natural when analyzed “all-at-once” (as in classical action principles). From this perspective, traditional quantum states would represent incomplete information about possible spacetime histories, conditional on the future measurement geometry. Without dynamical laws imposing additional restrictions, those histories can have a classical probability distribution, where exactly one history can be said to represent an underlying reality.

The dynamics of quantum and classical probability distributions are computed for a model of two coupled rotors (or pendulums), with emphasis on the $\ensuremath{\hbar}$ scaling of the quantum-classical (QC) differences. This scaling is not the same for coarse-grained and fine-grained quantities. The QC differences in the averages of observables scale as ${\ensuremath{\hbar}}^{2∕3}$ for chaotic states, but as ${\ensuremath{\hbar}}^{2}$ for regular states. The QC differences in probability distributions scale as ${\ensuremath{\hbar}}^{1∕3}$ for chaotic states. No simple scaling is found for those differences in regular states, although their overall magnitudes are similar to those in chaotic states. QC differences arise first in the short-wavelength regime, and subsequently spread to all wavelengths. A satisfactory classical limit is obtained without invoking environmental decoherence, which in any case would be effective in suppressing only short-wavelength QC differences.

We study the extent to which ψ-epistemic models for quantum measurement statistics-models where the quantum state does not have a real, ontic status-can explain the indistinguishability of nonorthogonal quantum states. This is done by comparing the overlap of any two quantum states with the overlap of the corresponding classical probability distributions over ontic states in a ψ-epistemic model. It is shown that in Hilbert spaces of dimension d≥4, the ratio between the classical and quantum overlaps in any ψ-epistemic model must be arbitrarily small for certain nonorthogonal states, suggesting that such models are arbitrarily bad at explaining the indistinguishability of quantum states. For dimensions d=3 and 4, we construct explicit states and measurements that can be used experimentally to put stringent bounds on the ratio of classical-to-quantum overlaps in ψ-epistemic models, allowing one in particular to rule out maximally ψ-epistemic models more efficiently than previously proposed.

We review the methods and use of random quantum states with particular emphasis on recent theoretical developments and applications in various fields. The guiding principle of the review is the idea that random quantum states can be understood as classical probability distributions in the Hilbert space of the associated quantum system. We show how this central concept connects questions of physical interest that cover different fields such as quantum statistical physics, quantum chaos, mesoscopic systems of both non-interacting and interacting particles, including superconducting and spin–orbit phenomena, and stochastic Schrodinger equations describing open quantum systems.

Due to various uncertainties, most of geotechnical parameters are small samples, which causes much trouble when the probability distribution of geotechnical parameter is fitted using traditional distributions. This paper uses stochastic weighted method to improve the small samples of Geotechnical parameters into big samples, thus solving the problems caused by the small samples. Meanwhile, the probability density function of geotechnical parameter is derived based on maximum entropy principle, the advantage of presented method is verified through Kolmogorov-Smirnov Test. Case study shows that the proposed method not only overcomes the dependence of conventional fitting methods on classical probability distributions, but also the fitting more close to the fact because the data come from the big sample improved by geotechnical parameters, which has important engineering significance.

The analysis of a set of experimental data obtained by an independent team of researchers confirms that neuronal differentiation or synapse formation do occur in time and space with fractal dimension. The interacting cells create first a dynamic system with its own attractor, (i.e., a fragment of time and space where the dynamic processes occur and where no further evolution of the system is possible at all owing to the action of the intrasystemic forces unless some extrasystemic forces act upon it). This attractor is then modified in the active manner by the differentiating cells until the system attains a degenerated stationary state and differentiation ends. The fractal structure of the system is also lost in the course of tumor progression. Our data indicate that the cellular system can attain the degenerated stationary state, leaving the attractor with a fractal dimension directly or undergoing diversification into many attractors and going through the areas of deterministic chaos. Since evolution of the cellular system is driven by the cooperative dynamic processes, as reflected by the changes of the mean fractal dimension between the intervals of the Gompertzian curve, it is likely that cells differentiate into neurons and create synapses with a conjugated probability and non-Gaussian distribution rather than with the classical probability and the Gaussian distribution. These findings can help to optimize features of artificial neural networks. They also define a simple in vitro biological model for biophysical and biochemical studies on natural neural networks.