Stationary Distribution Function Research Articles

Dense QCD2 with matrix product states

We study one-flavor SU(2) and SU(3) lattice QCD in (1 + 1) dimensions at zero temperature and finite density using matrix product states and the density matrix renormalization group. We compute physical observables such as the equation of state, chiral condensate, and quark distribution function as functions of the baryon number density. As a physical implication, we discuss the inhomogeneous phase at nonzero baryon density, where the chiral condensate is inhomogeneous, and baryons form a crystal. We also discuss how the dynamical degrees of freedom change from hadrons to quarks through the formation of quark Fermi seas.

On the stability of a particular class of one-dimensional states of dynamic equilibrium of the Vlasov–Poisson electron gas

The one-dimensional problem of the linear stability of dynamic states of local thermodynamic equilibria with respect to small perturbations was studied for the case when the Vlasov–Poisson electron gas contains electrons with a stationary distribution function that is constant in physical space and variable in a continuum of velocities. The absolute instability of all considered one-dimensional dynamic states of any local thermodynamic equilibrium was proved using the direct Lyapunov method. The scope of applicability of the Newcomb–Gardner–Rosenbluth sufficient condition for linear stability was outlined. An a priori exponential estimation was obtained for the rise of small one-dimensional perturbations from below. Analytic counterexamples to the spectral Newсomb–Gardner theorem and the Penrose criterion were constructed. Earnshaw’s theorem was extended from classical mechanics tostatistical one.

The stationary distribution and density function of a stochastic SIRB cholera model with Ornstein–Uhlenbeck process

Cholera is a global epidemic infectious disease that seriously endangers human life. It is disturbed by random factors in the process of transmission. Therefore, in this paper, a class of stochastic SIRB cholera model with Ornstein–Uhlenbeck process is established. On the basis of verifying that the model exists a unique global solution to any initial value, a sufficient criterion for the existence of a stationary distribution of the positive solution of the random model is established by constructing an appropriate random Lyapunov function. Furthermore, under the same condition that there is a stationary distribution, the specific expression of the probability density function of the random model around the positive equilibrium point is calculated. Finally, the theoretical results are verified by numerical model.

A stochastic predator–prey model with distributed delay and Ornstein–Uhlenbeck process: Characterization of stationary distribution, extinction, and probability density function

As the evolution of species relies on not only the current state but also the past information, it is more reasonable and realistic to take delay into an ecological model. This paper deals with a stochastic predator–prey model that considers the distribution delay and assume that the intrinsic growth rate and the death rate in the model are governed by Ornstein–Uhlenbeck process to simulate the random factors in the environment. Based on the existence and uniqueness of the global solution to the model and the boundedness of the order moments of the solution, several conditions are established to analyze the survival of the species. Specifically, a criteria for the existence of the stationary distribution to the stochastic system is established by constructing some suitable Lyapunov functions. And the analytical expression of the probability density function of the model around the quasi‐equilibrium is obtained. Furthermore, the extinction of species in the model is also explored. Finally, numerical simulations are carried out to illustrate the theoretical results obtained in this paper.

Noise-induced bistability and noise-enhanced stability of a stochastic model for resource production–consumption under crowding effect and sigmoidal consumption pattern

A resource production–consumption model with crowding effect and sigmoidal consumption pattern, subject to cross-correlated multiplicative and additive noises, is proposed in this paper to investigate noise-induced phenomena such as noise-induced bistability and noise-enhanced stability. Firstly, the stationary probability distribution and effective potential function are obtained through an approximate Fokker–Planck equation. Subsequently, P-bifurcation and noise-induced bistability are thoroughly investigated. Finally, the mean first passage time is utilized to calculate the average time for the system escaping from its attraction domain, while further exploring noise-enhanced stability. Our findings demonstrate that noise can induce P-bifurcation, especially bistability, regardless of the presence of bistability in the deterministic counterpart. Furthermore, appropriate noise parameters can cause a shift from an outbreak state to a shortage or depletion state. Additionally, both types of noise enhance the stability of the outbreak state when there is negative cross-correlation; however, for positive cross-correlation, they accelerate the transition from an outbreak state to a shortage or depletion state. The results obtained may offer preliminary or supplementary insights for the formulation of resource management policies.

Stationary distribution and density function analysis of stochastic SIQS epidemic model with Markov chain

In this paper, a stochastic SIQS epidemic model perturbed by both white and telephone noises is investigated. By constructing several suitable Lyapunov functions, we obtain sufficient conditions for the existence of ergodic stationary distribution of the positive solution. Moreover, by solving the Fokker–Planck equation, we obtain the exact expression of probability density function around the quasi-equilibrium of the stochastic model. In addition, sufficient conditions for the extinction are established. Finally, the results of this paper are further verified by numerical simulation.

Stationary distribution and density function analysis of SVIS epidemic model with saturated incidence and vaccination under stochastic environments

In this article, we study the dynamical properties of susceptible-vaccinated-infected-susceptible (SVIS) epidemic system with saturated incidence rate and vaccination strategies. By constructing the suitable Lyapunov function, we examine the existence and uniqueness of the stochastic system. With the help of Khas'minskii theory, we set up a critical value [Formula: see text] with respect to the basic reproduction number [Formula: see text] of the deterministic system. A unique ergodic stationary distribution is investigated under the condition of [Formula: see text]. In the epidemiological study, the ergodic stationary distribution represents that the disease will persist for long-term behavior. We focus for developing the general three-dimensional Fokker-Planck equation using appropriate solving theories. Around the quasi-endemic equilibrium, the probability density function of the stochastic system is analyzed which is the main theme of our study. Under [Formula: see text], both the existence of ergodic stationary distribution and density function can elicit all the dynamical behavior of the disease persistence. The condition of disease extinction of the system is derived. For supporting theoretical study, we discuss the numerical results and the sensitivities of the biological parameters. Results and conclusions are highlighted.

Consistent wide-range equation of state of silicon by a unified first-principles method

Using a unified method of the extended first-principles molecular dynamics, we report a wide-range (density $\ensuremath{\rho}=2.329--18.632$ $\mathrm{g}/{\mathrm{cm}}^{3}$, temperature $T=1.00\ifmmode\times\else\texttimes\fi{}{10}^{4}--1.29\ifmmode\times\else\texttimes\fi{}{10}^{8}$ K) equation of state (EOS) and principal Hugoniot of silicon, which agree well with experimental results, and then evaluate the precision of the results of first-principles calculations and a universal database. Moreover, we investigate the effects of finite-temperature exchange-correlation functionals and emphasize their importance and necessity in the warm dense matter regime, although it vanishes in the nondegenerate and fully degenerate limits. Finally, we show the ionic radial distribution function and density of electronic states to study the evolution of warm dense Si towards the classic plasma limit along its principal Hugoniot. The established standard theoretical EOS table of Si provides a benchmark for the development of high-precision first-principles methods and may improve radiation-hydrodynamics simulations of inertial confinement fusion and deepen the understanding of high-energy-density physics.

Stationary distribution and probability density function of a stochastic waterborne pathogen model with saturated direct and indirect transmissions

We investigate the dynamical behavior of deterministic and stochastic waterborne pathogen models with saturated incidence rates. In particular, the indirect transmission via person–water–person contact is half‐saturated and the direct transmission by person–person contact takes the saturated form. Possible equilibrium points of the model are investigated, and their stability criterion is discussed. Basic reproduction number of the model is obtained through the next‐generation matrix method. It has been shown that the disease‐free equilibrium is locally stable when and unstable for . Furthermore, a stochastic Lyapunov method is adopted to establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the proposed stochastic model, which reveals that the infection will persist if . We also obtain an exact expression of the probability density function near the quasi‐endemic equilibrium of stochastic system. Finally, numerical simulations are carried out to support our analytical findings. Results suggest that saturation constants in both direct and indirect transmissions play a positive role in controlling disease. Our results may provide some new insights for the elimination of waterborne disease.

A stochastic predator–prey model with Ornstein–Uhlenbeck process: Characterization of stationary distribution, extinction and probability density function

This paper concerns with a stochastic system modeling the population dynamical behavior of one prey and two predators. In this paper, we adopt a special method to simulate the effect of the environmental interference to the system instead of using the linear functions of white noise, i.e., the growth rate of the prey and the death rates of the two predators are all governed by Ornstein–Uhlenbeck processes, which is more practical and interesting. The dynamical behavior among the three species in the model is discussed in detail. The existence and uniqueness, moment boundedness and long time asymptotic behavior of the unique global solution are investigated. Through rigorous proof and theoretical derivation, some sharp sufficient conditions for the persistence and extinction of the three species have been established. Furthermore, we obtain that when one or two predators die out, the remaining species can be stable in time average under same conditions. Moreover, by constructing some suitable Lyapunov functions, the sufficient conditions for the existence of the stationary distribution are explicitly determined. Under the same parametric restrictions, it is worth noting that we obtain the six-dimensional probability density function of the stochastic one-prey two-predator model around the quasi-equilibrium E∗, which can reflect most statistical properties. Finally, several numerical simulations are carried out to illustrate the theoretical results.

Stationary distribution and probability density function of a stochastic waterborne pathogen model with logistic growth

Waterborne disease threatens public health globally. Previous studies mainly consider that the birth of pathogens in water sources arises solely by the shedding of infected individuals. However, for free-living pathogens, intrinsic growth without the presence of hosts in environment could be possible. In this paper, a stochastic waterborne disease model with a logistic growth of pathogens is investigated. We obtain the sufficient conditions for the extinction of disease and also the existence and uniqueness of an ergodic stationary distribution if the threshold [Formula: see text]. By solving the Fokker–Planck equation, an exact expression of probability density function near the quasi-endemic equilibrium is obtained. Results suggest that the intrinsic growth in bacteria population induces a large reproduction number to determine the disease dynamics. Finally, theoretical results are validated by numerical examples.

Stationary distribution and probability density function analysis of a stochastic Microcystins degradation model with distributed delay.

A stochastic Microcystins degradation model with distributed delay is studied in this paper. We first demonstrate the existence and uniqueness of a global positive solution to the stochastic system. Second, we derive a stochastic critical value $ R_0^s $ related to the basic reproduction number $ R_0 $. By constructing suitable Lyapunov function types, we obtain the existence of an ergodic stationary distribution of the stochastic system if $ R_0^s > 1. $ Next, by means of the method developed to solve the general four-dimensional Fokker-Planck equation, the exact expression of the probability density function of the stochastic model around the quasi-endemic equilibrium is derived, which is the key aim of the present paper. In the analysis of statistical significance, the explicit density function can reflect all dynamical properties of a chemostat model. To validate our theoretical conclusions, we present examples and numerical simulations.

Energy distribution of inelastic gas in a box is dominated by a power law—a derivation in the framework of sample space reducing processes

We use the framework of sample space reducing (SSR) processes as an alternative to Boltzmann equation based approaches to derive the energy and velocity distribution functions of an inelastic gas in a box as an example of a dissipative, driven system. SSR processes do not assume molecular chaos and are characterized by a specific type of eigenvalue equation whose solutions represent stationary distribution functions. The equations incorporate the geometry of inelastic collisions and a driving mechanism in a transparent way. Energy is injected by boosting particles that hit the walls of the container to high energies. The numerical solution of the resulting equations yields approximate power laws over the entire energy region. The exponents decrease with the driving rate from about 2 to below 1.5 and depend on the coefficient of restitution. Results are confirmed with a molecular dynamics simulation in 3D with the same driving mechanism. The numerical solution of the resulting equations yields approximate power laws over the entire energy region. Deviations depend on the details of driving, density, and container.

Exact Shearing Flow Magnetized Hybrid Kinetic Equilibria with Inhomogeneous Temperature

Magnetized plasmas with shearing flows are found in many natural contexts, such as around Earth’s magnetopause. In collisionless plasmas where physical quantities vary on a scale of the order of or larger than ion scales, the hybrid Vlasov−Maxwell description (kinetic ions coupled to a neutralizing electron fluid via electromagnetic fields) represents a suitable approach. When crossing the magnetopause, the ion temperature, density, and direction of magnetic field vary. We derive a form for an exact stationary solution of the hybrid Vlasov−Maxwell equations that represent a magnetized plasma with a quasi-planar shearing flow, variable density and ion temperature, and variable magnetic field direction. A stationary ion distribution function is expressed as a suitable combination of particle constants of motion and evaluated numerically in such a way to obtain configurations with variable density and temperature and two quasi-planar oppositely directed velocity shear layers. Properties of particular configurations are derived from Magnetospheric Multiscale measures during crossings of Earth’s magnetopause. In the first case a quasi-uniformly directed, nearly perpendicular magnetic field is present, while in the second case, going from the magnetosheath to the magnetosphere, the magnetic field makes a wide rotation from one side to the other of the shearing flow plane. In both cases, the ion distribution function departs from a Maxwellian in the shear layers, displaying temperature anisotropy and agyrotropy, with a nonsymmetric behavior in the two shear layers. The configurations considered here can be used as models for Earth’s magnetopause in simulations of the Kelvin–Helmholtz instability.

Design study of a HPGe detector array for β-decay investigation in laboratory ECR plasmas

In the frame of the PANDORA project, a new experimental approach aims at measuring in-plasma β-decay rate as a function of thermodynamical conditions of the environment, namely a laboratory magnetized plasma able to mimic some stellar-like conditions. The decay rates (expected to change dramatically as a function of the ionization state) will be measured as a function of the charge state distribution of the in-plasma ions. The new experimental approach aims at correlating the plasma environment and the decay rate. This can be performed by simultaneously identifying and discriminating—through an innovative multi-diagnostic system working synergically with a γ-ray detection system —the photons emitted by the plasma and γ-rays emitted after the isotope β-decay. In this study, the numerical simulations supporting the design of the γ-ray detector array, including a statistical significance study to check the feasibility of measuring the in-plasma decay rates, are presented. Geant4 simulations focused on the design of the array of γ-ray detectors and the evaluation of total efficiency depending on the detector type and the optimal displacement of detectors around the trap (including collimation systems and shielding). The simulation results showed that, due to technical limitations in the number of apertures that can be created in the magnetic trap, the best compromise is to use 14 HPGe (70% of relative efficiency) detectors surrounding the magnetic trap. The HPGe detectors were chosen for their excellent energy resolution (0.2% @ 1 MeV), since the harsh environment (the background is represented by the intense plasma self-emission) strongly affects the signal-to-background ratio. Once determined the total photopeak efficiency (0.1–0.2%), the sensitivity of the PANDORA experiment was checked in a “virtual experimental run,” by exploring the measurability of isotope decay rates for the first three physical cases of PANDORA: 176Lu, 134Cs and 94Nb. The preliminary results demonstrated the feasibility of the measurement in terms of the signal-to-background ratio and significance that it is possible to reach. The results indicated that experimental run durations could take from several days to 3 months, depending on the isotope under investigation, thus shedding new light on the role of weak interactions in stellar nucleosynthesis.

Stationary distribution, extinction and density function of a stochastic prey-predator system with general anti-predator behavior and fear effect

In this paper, we examine a stochastic prey-predator system with fear effect and general anti-predator behavior. To tackle the impact of stochastic perturbations, we first propose a p-stochastic threshold method to construct several necessary p-Lyapunov functions. Then by defining a quasi-carrying capacity x∗, sufficient conditions are established for the existence and uniqueness of stationary distribution ϖ⋅ of the system. By solving the Fokker-Planck equation, the approximate expression of probability density function of the distribution ϖ⋅ around its quasi-positive equilibrium is further derived. Besides, the extinction of prey and predator populations is studied. Finally, some numerical examples are provided to verify our theoretical results and study two aspects: (i) the impact of anti-predator behavior; (ii) the effect of prey fear.

Enhanced nonlinear conductivity due to hot-electron injection in carbon nanotubes

We have theoretically obtained an expression for the current density in a terahertz field due to hot-electron injection in carbon nanotubes. The injection modifies the stationary distribution function and leads to a qualitative change in the behavior of the current-voltage characteristics and causes absolute negative conductivity. We compared the current-voltage characteristic behavior at different injection rates and observed a drastic change in the current density and absolute negative conductivity values. We propose that carbon nanotubes with hot-electron injection may be useful for high-frequency applications.

Generalization of classical Hillert's grain growth and LSW theories to a wide family of kinetic evolution equations and stationary distribution functions

Coarsening of objects as grains or precipitates belongs to the most relevant processes during the heat treatment and high-temperature application of materials. Based on the appropriate models incorporating the relevant phenomena, the coarsening of objects can be described by evolution equations, which are reflected by stationary radius distribution functions. Such a distribution function for a distinct evolution equation has been derived e.g. in the frame of Lifshitz, Slyozov and Wagner (LSW) theory for the precipitate coarsening. The current evolution equations are in prevailing number of cases based on the mean field approach. A two-parametric family of evolution equations is proposed, which includes e.g. the LSW theory as a special case. The specific phenomena controlling the object evolution can be incorporated by using proper values of parameters used in the evolution equation. An effective procedure is presented how the corresponding stationary radius distribution function can be calculated for each evolution equation from the family. The distribution functions are demonstrated for several combinations of admissible parameters. Basic characteristics of their shapes are also evaluated.

A stochastic SEIRS rabies model with population dispersal: Stationary distribution and probability density function

In this paper, we propose a stochastic SEIRS rabies epidemic models with two patches to investigate the spatial spread of rabies under the influence of environmental noise. Adopting a new technique to construct the stochastic Lyapunov functions, we obtain the sufficient conditions for the existence of an ergodic stationary distribution. In addition, if the movement of dogs is forbidden, we obtain the exact expression of probability density function around a quasi-equilibrium point. Finally, numerical simulations are employed to verify our analytical results and reveal the impact of stochastic perturbations on rabies transmission.

Stationary Distribution, Extinction and Probability Density Function of a Stochastic Vegetation–Water Model in Arid Ecosystems

Stationary Distribution, Extinction and Probability Density Function of a Stochastic Vegetation–Water Model in Arid Ecosystems