Phase transitions driven by L\'evy stable noise: exact solutions and stability analysis of nonlinear fractional Fokker-Planck equations

Phase transitions and effects of external noise on many body systems are one of the main topics in physics. In mean field coupled nonlinear dynamical stochastic systems driven by Brownian noise, various types of phase transitions including nonequilibrium ones may appear. A Brownian motion is a special case of L\'evy motion and the stochastic process based on the latter is an alternative choice for studying cooperative phenomena in various fields. Recently, fractional Fokker-Planck equations associated with L\'evy noise have attracted much attention and behaviors of systems with double-well potential subjected to L\'evy noise have been studied intensively. However, most of such studies have resorted to numerical computation. We construct an {\it analytically solvable model} to study the occurrence of phase transitions driven by L\'evy stable noise.Comment: submitted to EP

Fractional Kinetics for Relaxation and Superdiffusion in Magnetic Field

We propose fractional Fokker-Planck equation for the kinetic description of relaxation and superdiffusion processes in constant magnetic and random electric fields. We assume that the random electric field acting on a test charged particle is isotropic and possesses non-Gaussian Levy stable statistics. These assumptions provide us with a straightforward possibility to consider formation of anomalous stationary states and superdiffusion processes, both properties are inherent to strongly non-equilibrium plasmas of solar systems and thermonuclear devices. We solve fractional kinetic equations, study the properties of the solution, and compare analytical results with those of numerical simulation based on the solution of the Langevin equations with the noise source having Levy stable probability density. We found, in particular, that the stationary states are essentially non-Maxwellian ones and, at the diffusion stage of relaxation, the characteristic displacement of a particle grows superdiffusively with time and is inversely proportional to the magnetic field.Comment: 15 pages, LaTeX, 5 figures PostScrip

Enhancing the Approach to Forecasting the Dynamics of Socio-Economic Development during the COVID-19 Pandemic

This study reveals the approach to scaling socio-economic indicators to ensure economic security through regional budget expenditures to the GRP ratio example. Indicator choice is conditioned by the necessity to determine the degree of the federal center's rational influence on the regional strategic goals of sustainable development. The study aims to develop and test the system for assessing the dynamics of Russian socio-economic development based on the authors' interpretation of the scaling factor values. The main research method is scaling, which provides additional perspectives reflected by preserving proportions when changing the target parameters. The new method's effectiveness is confirmed by calculating the scaling factor. Its value interpretation gives a tool for assessing the effectiveness of the strategy development system and its economic security. The study's relevance is due to adaptation to global transformations based on the management system's capability to act under various crisis scenarios and make anti-crisis decisions important for the Russian economy. The findings improve the basis for implementing a sustainable strategic planning system and strengthening national security in the COVID-19 pandemic. The findings make it possible to predict the further evolution of the relationships between indicator groups in order to increase the role of per capita budgetary expenditures in GRP. Doi: 10.28991/esj-2022-SPER-08 Full Text: PD

Using PPO Models to Predict the Value of the BNB Cryptocurrency

This paper identifies hidden patterns between trading volumes and the market value of an asset. Based on open market data, we try to improve the existing corpus of research using new, innovative neural network training methods. Dividing into two independent models, we conducted a comparative analysis between two methods of training Proximal Policy Optimization (PPO) models. The primary difference between the two PPO models is the data. To showcase the drastic differences the PPO model makes in market conditions, one model uses historical data from Binance trading history as a data sample and the trading pair BNB/USDT as a predicted asset. Another model, apart from purely price fluctuations, also draws data on trading volume. That way, we can clearly illustrate what the difference can be if we add additional markers for model training. Using PPO models, the authors conduct a comparative analysis of prediction accuracy, taking the sequence of BNB token values and trading volumes on 15-minute candles as variables. The main research question of this paper is to identify an increase in the accuracy of the PPO model when adding additional variables. The primary research gap that we explore is whether PPO models specifically trained on highly volatile assets can be improved by adding additional markers that are closely linked. In our study, we identified the closest marker, which is a trading volume. The study results show that including additional parameters in the form of trading volume significantly reduces the model's accuracy. The scientific contribution of this research is that it shows in practice that the PPO model does not require additional parameters to form accurately predicting models within the framework of market forecasting. Doi: 10.28991/ESJ-2023-07-04-012 Full Text: PD

Excitation of surface plasmon-polaritons in metal films with double periodic modulation: anomalous optical effects

We perform a thorough theoretical analysis of resonance effects when an arbitrarily polarized plane monochromatic wave is incident onto a double periodically modulated metal film sandwiched by two different transparent media. The proposed theory offers a generalization of the theory that had been build in our recent papers for the simplest case of one-dimensional structures to two-dimensional ones. A special emphasis is placed on the films with the modulation caused by cylindrical inclusions, hence, the results obtained are applicable to the films used in the experiments. We discuss a spectral composition of modulated films and highlight the principal role of ``resonance'' and ``coupling'' modulation harmonics. All the originating multiple resonances are examined in detail. The transformation coefficients corresponding to different diffraction orders are investigated in the vicinity of each resonance. We make a comparison between our theory and recent experiments concerning enhanced light transmittance and show the ways of increasing the efficiency of these phenomena. In the appendix we demonstrate a close analogy between ELT effect and peculiarities of a forced motion of two coupled classical oscillators.Comment: 24 pages, 17 figure

An improvement of the Berry--Esseen inequality with applications to Poisson and mixed Poisson random sums

By a modification of the method that was applied in (Korolev and Shevtsova, 2009), here the inequalities $$\rho(F_n,\Phi)\le\frac{0.335789(\beta^3+0.425)}{\sqrt{n}}$$ and $$\rho(F_n,\Phi)\le \frac{0.3051(\beta^3+1)}{\sqrt{n}}$$ are proved for the uniform distance $\rho(F_n,\Phi)$ between the standard normal distribution function $\Phi$ and the distribution function $F_n$ of the normalized sum of an arbitrary number $n\ge1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $\beta^3$. The first of these inequalities sharpens the best known version of the classical Berry--Esseen inequality since $0.335789(\beta^3+0.425)\le0.335789(1+0.425)\beta^3<0.4785\beta^3$ by virtue of the condition $\beta^3\ge1$, and 0.4785 is the best known upper estimate of the absolute constant in the classical Berry--Esseen inequality. The second inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry--Esseen inequality for Poisson random sums to 0.3051 which is strictly less than the least possible value of the absolute constant in the classical Berry--Esseen inequality. As a corollary, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.Comment: 33 page

Beam propagation in a Randomly Inhomogeneous Medium

An integro-differential equation describing the angular distribution of beams is analyzed for a medium with random inhomogeneities. Beams are trapped because inhomogeneities give rise to wave localization at random locations and random times. The expressions obtained for the mean square deviation from the initial direction of beam propagation generalize the "3/2 law".Comment: 4 page

Theory of Systematic Computational Error in Free Energy Differences

Systematic inaccuracy is inherent in any computational estimate of a non-linear average, due to the availability of only a finite number of data values, N. Free energy differences (DF) between two states or systems are critically important examples of such averages in physical, chemical and biological settings. Previous work has demonstrated, empirically, that the ``finite-sampling error'' can be very large -- many times kT -- in DF estimates for simple molecular systems. Here, we present a theoretical description of the inaccuracy, including the exact solution of a sample problem, the precise asymptotic behavior in terms of 1/N for large N, the identification of universal law, and numerical illustrations. The theory relies on corrections to the central and other limit theorems, and thus a role is played by stable (Levy) probability distributions.Comment: 5 pages, 4 figure

Cation Ordering and Superstructures in Natural Layered Double Hydroxides

Layered double hydroxides (LDHs) constitute an important group of materials with many applications ranging from catalysis and absorption to carriers for drug delivery, DNA intercalation and carbon dioxide sequestration. The structures of LDHs are based upon double brucite-like hydroxide layers [M2+nM3+m(OH)2(m+n)]m+, where M2+ = Mg2+, Fe2+, Mn2+, Zn2+, etc.; M3+ = Al3+, Fe3+, Cr3+, Mn3+, etc. Structural features of LDHs such as cation ordering, charge distribution and polytypism have an immediate influence upon their properties. However, all the structural studies on synthetic LDHs deal with powder samples that prevent elucidation of such fine details of structure architecture as formation of superstructures due to cation ordering. In contrast to synthetic materials, natural LDHs are known to form single crystals accessible to single-crystal X-ray diffraction analysis, which provides a unique possibility to investigate 3D cation ordering in LDHs that results in formation of complex superstructures, where 2D cation order is combined with a specific order of layer stacking (polytypism). Therefore LDH minerals provide an indispensable source of structural information for modeling of structures and processes happening in LDHs at the molecular and nanoscale levels