Fluctuation and Noise Letters

The Stefan–Boltzmann (SB) law relates the radiant emittance of an ideal black-body cavity at thermal equilibrium to the fourth power of the absolute temperature $T$ as $q=\sigma T^4$, with $\sigma =5.67\times 10^{-8}$Wm${}^{-2}$K${}^{-4}$ the SB constant, first estimated by Stefan to within $11\%$ of the present theoretical value. The law is an important achievement of modern physics since, following Planck [Ueber das Gesetz der Energieverteilung im Normalspectrum [On the law of distribution of energy in the normal spectrum], Ann. Phys. 4 (1901) 553–563], its microscopic derivation implies the quantization of the energy related to the electromagnetic field spectrum. Somewhat astonishing, Boltzmann presented his derivation in 1878 making use only of electrodynamic and thermodynamic classical concepts, apparently without introducing any quantum hypothesis (here called first Boltzmann paradox). By contrast, the Boltzmann derivation implies two assumptions not justified within a classical approach, namely: (i) the zero value of the chemical potential and (ii) the internal energy of the black body with a finite value and dependent from both temperature and volume. By using Planck [Ueber das Gesetz der Energieverteilung im Normalspectrum [On the law of distribution of energy in the normal spectrum], Ann. Phys. 4 (1901) 553–563] quantization of the radiation field in terms of a gas of photons, the SB law received a microscopic interpretation free from the above assumptions that also provides the value of the SB constant on the basis of a set of universal constants including the quantum action constant $h$. However, the successive consideration by Planck [Uber die Begründung des Gesetzes der schwarzen Strahlung [On the grounds of the law of black body radiation], Ann. Phys. 6 (1912) 642–656] concerning the zero-point energy contribution was found to be responsible of another divergence of the internal energy for the single photon mode at high frequencies. This divergence is of pure quantum origin and is responsible for a vacuum-catastrophe, to keep the analogy with the well-known ultraviolet catastrophe of the classical black-body radiation spectrum, given by the Rayleigh–Jeans law in 1900. As a consequence, from a rigorous quantum-mechanical derivation we would expect the divergence of the SB law (here called second Boltzmann paradox). Here, both the Boltzmann paradoxes are revised by accounting for both the quantum-relativistic photon gas properties, and the Casimir force.

An optimal $H_{\infty }$ criterion for the stability investigation of non linear digital filters with saturation non linearity and external interference is proposed in this paper. The proposed criterion can be used for overflow oscillation-free realization of the non linear digital system implemented with fixed wordlength machines like digital computers or special purpose embedded systems. The proposed approach ascertains the local asymptotic stability of non linear digital filtering schemes without external interference. Further, by employing $H_{\infty }$ performance criterion the conditions for local stability and performance measurement are proposed with external interference. In the proposed work, the stability conditions are proposed in linear matrix inequality (LMI) by exploiting energy-based Lyapunov theory, Lipschitz inequality, and saturation arithmetic. The proposed criterion is shown to be less conservative which will improve the operating region and obtain a better estimate of $H_{\infty }$ performance index compared to the existing local criterion. Moreover, the proposed criterion can be used in selecting the appropriate wordlength of a digital system for tolerable values of interference energy, $H_{\infty }$ performance level, and fixed-point resolution. This will be an additional advantage of this work compared with existing global criteria. Sufficient examples to show the efficacy of the proposed technique are also provided.

An improved Piecewise Tri-stable Stochastic Resonance (PTSR) system is investigated to enhance the low performance of Classical Tri-stable Stochastic Resonance (CTSR) system for weak signal extraction. Firstly, the proposed PTSR system is theoretically compared with CTSR system. Secondly, based on adiabatic approximation theory, the analytical expressions for the Mean First Pass Time (MFPT) and the output Signal-to-Noise Ratio (SNR) of the system are derived. Afterwards, a PTSR-EMD system that combines PTSR system with empirical mode decomposition (EMD) is proposed. Finally, the above three systems are applied to the detection of harmonic signals and actual fault signals, and the superior performance of PTSR-EMD system over PTSR system and CTSR system are confirmed.

The family of visibility algorithms provides a new point of view to describe time series by transforming them into networks. In this paper, we propound a new visibility algorithm named Temporal Vector Visibility graph $({\mathrm{\text{TVV}}}_g)$, which maps the multivariate time series to a directed network. Computed by the $ingoing$ and $outgoing$ degree distributions obtained from the ${\mathrm{\text{TVV}}}_g$, these statistics such as the Kullback–Leibler divergence (KLD), the normalized Shannon entropy and the statistical complexity measure, are introduced to assess the complexity of time series. Furthermore, we also apply the Multivariate Multiscale Entropy Plane (MMEP) to evaluate the complexity of multivariate time series. The experimental results of eight different types of time series verify the effectiveness of our method. Subsequently, this method is employed to explore the complexity characteristics of financial time series and classify different stock markets. Our research reveals that this method is capable of investigating the physical structures of financial time series.

In this paper, we propose to analyze multiple cross-correlation between six stock market indexes (Dow Jones Industrial Average, Nasdaq Composite, DAX Performance, FTSE 100, S&P/TSX Composite and Nikkei 225) based on the multiple detrended cross-correlation coefficient, ${\mathrm{\text{DMC}}}_x^2$. Due to its importance and long existence, we will have as the base index, acting as dependent variable, the Dow Jones Industrial Average, which will be compared with the other five indexes in all possible combinations. In addition to calculating the ${\mathrm{\text{DMC}}}_x^2$ for all combinations with Dow Jones, we will also infer the relative strength of each index in relation to the whole, and this will be done here through an unprecedented map of relative strength. With this map it will be possible to verify that the Nikkei 225 index has less weight for all time-scales and also all combinations, and that there is better distribution in the Nasdaq and DAX combination. Finally, we present more possibilities for this application throughout the paper.

The information theoretically secure Kirchhoff-law-Johnson-noise (KLJN) key exchange scheme, similarly to quantum key distribution (QKD), is also potentially vulnerable against clock attacks, where Eve takes over the control of clock synchronization in the channel. This short note aims to introduce a time synchronization protocol scheme for Alice and Bob, which is resistant against arbitrary time delay attacks, both symmetric and asymmetric ones. We propose and explore various ways of clock synchronization for the KLJN system and propose an ultimate protocol that preserves time and hardware integrity under arbitrary attacks.

In this paper, the symmetrical Kullback–Leibler divergence is proposed to capture the dynamical features of various complex time series more accurately. The symmetrical Kullback–Leibler divergence is modified from the Kullback–Leibler divergence to make it possess the properties of a normal distance measure. A new multidimensional scaling (MDS) method with the proposed symmetrical Kullback–Leibler divergence is established, which is an efficient clustering and visualization technique. The proposed method also includes the probability distribution of the weighted-permutation patterns, which not only provides a modified time series quantification process with more possible patterns, but also facilitates the reduction of error when extracting the hidden information from time series. The upper and lower bounds of the proposed divergence are also presented. Through simulation and reality-based data experiments, it is affirmed that the MDS based on this novel dissimilarity measure is capable of presenting more distinctive and rational clustering results by comparing it with the MDS methods based on conventional dissimilarities.

The geometric process (GP) plays an important role in the reliability theory and life span models. It has been used extensively as a stochastic model in many areas of application. Therefore, the parameter estimation problem is very crucial in a GP. In this study, the parameter estimation problem for GP is discussed under the assumption that $X_1$ has a Lindley distribution with parameter $\theta$. The maximum likelihood (ML) estimators of $a,\mu$ and ${\sigma }^2$ of the GP and their asymptotic distributions are derived. A test statistic is developed based on ML estimators for testing whether $a=1$ or not. The same problem is also studied by using Bayesian methods. Bayes estimators of the unknown model parameters are obtained under squared error loss function (SELF) using uniform and gamma priors on the ratio $a$ and $\theta$ parameters. It is not possible to obtain Bayes estimators in explicit forms. Therefore, Markov Chain Monte Carlo (MCMC), Lindley (LD), and Tierney–Kadane (T–K) methods are used to estimate the parameters $a,\mu$ and ${\sigma }^2$ in GP. The efficiencies of the ML estimators are compared with Bayes estimators via an extensive Monte Carlo simulation study. It is seen that the Bayes estimators perform better than the ML estimators. Two real-life examples are also presented for application purposes. The first data set concerns the coal mining disaster. The second is the number of COVID-19 patients in Turkey.

In this paper, a novel underdamped tri-stable stochastic resonance (NUTSR) is presented in order to overcome the low out-performance of weak signal reinforcement in the classical tri-stable stochastic resonance (CTSR). The two systems are compared using the input and output amplitude as a measure when noiseless. NUTSR improves the output saturation characteristics and has more outstanding signal enhancement capability. In the background of Gaussian noise, the expressions of mean first-pass time (MFPT), steady-state probability density (SPD) and signal-to-noise ratio (SNR) are derived. The fourth-order Runge–Kutta algorithm and genetic algorithm (GA) is used for numerical simulations. Then the numerical simulation results of the two systems are compared comprehensively, and the theoretical deduction and numerical simulation results of NUTSR are also compared. Moreover, the two systems are applied in two bearing fault types of 6205-2RS JEM SKF and HRB 6205-2Z. Finally, the feasibility of NUTSR is verified under different noise, which is simulated numerically under symmetric trichotomous that NUTSR has better noise immunity and the increase in signal amplitude is more pronounced.

Aiming at the deficiency of weak signal amplification ability in traditional bistable system and combining with the advantages of tristable system and asymmetric system, three novel of combined asymmetric tristable stochastic resonance (SR) systems are proposed by combining the classical bistable potential model with the Gaussian potential model in this paper, namely, asymmetric well depth stochastic resonance (AWDSR), asymmetric well width stochastic resonance (AWWSR) and asymmetric well depth width stochastic resonance (AWDWSR). The stochastic resonance phenomena of the above three systems under $\alpha$ noise with the influence of parameters are analyzed, respectively. Through empirical mode decomposition and adaptive genetic algorithm to optimize the parameters for weak signal extraction, it can be found that this method can achieve stronger SR performance. After the bearing fault diagnosis of the proposed systems, the detection results of the three asymmetric systems are obviously better than the classical tristable stochastic resonance (CTSR) and symmetric stochastic resonance (SSR), and the performance advantage of AWDWSR is the greatest. This can provide some reference value for practical engineering application.