One-dimensional Phase Space Research Articles

A splitting Fourier pseudospectral method for Vlasov–Poisson–Fokker–Planck system

In this paper, we propose a splitting Fourier pseudospectral method for Vlasov–Poisson–Fokker–Planck (VPFP) system, which describes the motion of charged particles in plasma. The numerical integration for the system is performed by employing the splitting method in time, Fourier Galerkin method in space direction, and Fourier collocation method in phase direction, respectively. The algorithm has spectral accuracy in both space and phase directions and can be implemented efficiently with the fast Fourier transform and technique of diagonalization, respectively. Extensive numerical results in one-dimensional phase space (or 1x×1v) from the proposed numerical method are shown and have proven the good agreement with the theory and previous studies. Numerical algorithm for the VPFP system in two-dimensional phase space (or 2x×2v) has been summarized in Appendix.

Nonlinear random dynamical model for the stock market

A new model of a stock market as a nonlinear random dynamical system with additive noise in three-dimensional phase space is offered. Implementations of this model in the adiabatic approximation possess all the key signs of fractal market, making the model a reasonable evolutionary model for a stock market. The use of adiabatic approximation allows us to model a stock market as a nonlinear random dynamical system with multiplicative noise with the power-law in one-dimensional phase space. This model shows fractality, long memory and 1/f noise in a stock market.

Two-dimensional superintegrable systems from operator algebras in one dimension

We develop new constructions of 2D classical and quantum superintegrable Hamiltonians allowing separation of variables in Cartesian coordinates. In classical mechanics we start from two functions on a one-dimensional phase space, a natural Hamiltonian H and a polynomial of order N in the momentum p . We assume that their Poisson commutator vanishes, is a constant, a constant times H, or a constant times K. In the quantum case H and K are operators and their Lie commutator has one of the above properties. We use two copies of such pairs to generate two-dimensional superintegrable systems in the Euclidean space E2, allowing the separation of variables in Cartesian coordinates. Nearly all known separable superintegrable systems in E2 can be obtained in this manner and we obtain new ones for N = 4.

How mutation alters the evolutionary dynamics of cooperation on networks

Cooperation is ubiquitous at every level of living organisms. It is known that spatial (network) structure is a viable mechanism for cooperation to evolve. A recently proposed numerical metric, average gradient of selection (AGoS), a useful tool for interpreting and visualizing evolutionary dynamics on networks, allows simulation results to be visualized on a one-dimensional phase space. However, stochastic mutation of strategies was not considered in the analysis of AGoS. Here we extend AGoS so that it can analyze the evolution of cooperation where mutation may alter strategies of individuals on networks. We show that our extended AGoS correctly visualizes the final states of cooperation with mutation in the individual-based simulations. Our analyses revealed that mutation always has a negative effect on the evolution of cooperation regardless of the payoff functions, fraction of cooperators, and network structures. Moreover, we found that scale-free networks are the most vulnerable to mutation and thus the dynamics of cooperation are altered from bistability to coexistence on those networks, undergoing an imperfect pitchfork bifurcation.

Complexity of a Microblogging Social Network in the Framework of Modern Nonlinear Science

Recent developments in nonlinear science have caused the formation of a new paradigm called the paradigm of complexity. The self‐organized criticality theory constitutes the foundation of this paradigm. To estimate the complexity of a microblogging social network, we used one of the conceptual schemes of the paradigm, namely, the system of key signs of complexity of the external manifestations of the system irrespective of its internal structure. Our research revealed all the key signs of complexity of the time series of a number of microposts. We offer a new model of a microblogging social network as a nonlinear random dynamical system with additive noise in three‐dimensional phase space. Implementations of this model in the adiabatic approximation possess all the key signs of complexity, making the model a reasonable evolutionary model for a microblogging social network. The use of adiabatic approximation allows us to model a microblogging social network as a nonlinear random dynamical system with multiplicative noise with the power‐law in one‐dimensional phase space.

Thermalization of mini-jets in a quark-gluon plasma

We complete the physical picture for the evolution of a high-energy jet propagating through a weakly-coupled quark-gluon plasma by investigating the thermalization of the soft components of the jet. We argue that the following scenario should hold: the leading particle emits a significant number of mini-jets which promptly evolve via quasi-democratic branchings and thus degrade into a myriad of soft gluons, with energies of the order of the medium temperature T. Via elastic collisions with the medium constituents, these soft gluons relax to local thermal equilibrium with the plasma over a time scale which is considerably shorter than the typical lifetime of the mini-jet. The thermalized gluons form a tail which lags behind the hard components of the jet. We support this scenario, first, via parametric arguments and, next, by studying a simplified kinetic equation, which describes the jet dynamics in longitudinal phase-space. We solve the kinetic equation using both (semi-)analytical and numerical methods. In particular, we obtain the first exact, analytic, solutions to the ultrarelativistic Fokker-Planck equation in one-dimensional phase-space. Our results confirm the physical picture aforementioned and demonstrate the quenching of the jet via multiple branching followed by the thermalization of the soft gluons in the cascades.

Intensity–intensity correlations determined by dimension of quantum state in phase space: P-distribution

We use the P-distribution to show that the familiar values 1, 2 and 3 of the normalized second order correlation function at equal times corresponding to a coherent state, a thermal state and a highly squeezed vacuum are a consequence of the number of dimensions these states take up in quantum phase space. Whereas the thermal state exhibits rotational symmetry and thus extends over two dimensions, the squeezed vacuum factorizes into two independent one-dimensional phase space variables, and in the limit of large squeezing is therefore a one-dimensional object. The coherent state is a point in the phase space of the P-distribution and thus has zero dimensions. The fact that for photon number states the P-distribution is even narrower than that of the zero-dimensional coherent state suggests the notion of ‘negative’ dimensions.

Transition frequencies in the neuronal oscillator

I present dynamical mechanisms responsible for transition frequencies, which are hidden in the homoclinic (HC) to supercritical Andronov-Hopf (AH) bifurcation switching in the two-dimensional Hindmarsh-Rose (2DHR) oscillator. For this purpose, I derive frequency gradients with respect to the input current and timescale in the 2DHR oscillator. The frequency gradients are formulated with phase response curves (PRCs) by expanding the firing frequency with small perturbations of the input current and timescale on the one-dimensional phase space of the 2DHR oscillator. I thus find two different types of striking breaks in the frequency plot with respect to the timescale, the sigmoid functional frequency and the discontinuous frequency. I also show effects of the unique PRC change on these transition frequencies.

Quantum correlations in nanostructured two-impurity Kondo systems

We study the ground-state entanglement properties of nanostructured Kondo systems consisting of a pair of impurity spins coupled to a background of confined electrons. The competition between the Ruderman-Kittel-Kasuya-Yosida-like coupling and the Kondo effect determines the development of quantum correlations between the different parts of the system. A key element is the electronic filling due to confinement. An even electronic filling leads to results similar to those found previously for extended systems, where the properties of the reduced impurity spin subsystem are uniquely determined by the spin-correlation function defining a one-dimensional phase space. An odd filling, instead, breaks spin rotation symmetry unfolding a two-dimensional phase space showing rich entanglement characteristics as, e.g., the requirement of a larger amount of entanglement for the development of nonlocal correlations between impurity spins. We illustrate these results by numerical simulations of elliptic quantum corrals with magnetic impurities at the foci as a case study.

Statistical analysis of complex systems with nonclassical invariant measures

I investigate the problem of finding a statistical description of a complex many-body system whose invariant measure cannot be constructed stemming from classical thermodynamics ensembles. By taking solitons as a reference system and by employing a general formalism based on the Ablowitz-Kaup-Newell-Segur scheme, I demonstrate how to build an invariant measure and, within a one-dimensional phase space, how to develop a suitable thermodynamics. A detailed example is provided with a universal model of wave propagation, with reference to a transparent potential sustaining gray solitons. The system shows a rich thermodynamic scenario, with a free-energy landscape supporting phase transitions and controllable emergent properties. I finally discuss the origin of such behavior, trying to identify common denominators in the area of complex dynamics.

Velocity structure of self-similar spherically collapsed halos

Using a generalized self-similar secondary infall model, which accounts for tidal torques acting on the halo, we analyze the velocity profiles of halos in order to gain intuition for N-body simulation results. We analytically calculate the asymptotic behavior of the internal radial and tangential kinetic energy profiles in different radial regimes. We then numerically compute the velocity anisotropy and pseudo-phase-space density profiles and compare them to recent N-body simulations. For cosmological initial conditions, we find both numerically and analytically that the anisotropy profile asymptotes at small radii to a constant set by model parameters. It rises on intermediate scales as the velocity dispersion becomes more radially dominated and then drops off at radii larger than the virial radius where the radial velocity dispersion vanishes in our model. The pseudo-phase-space density is universal on intermediate and large scales. However, its asymptotic slope on small scales depends on the halo mass and on how mass shells are torqued after turnaround. The results largely confirm N-body simulations but show some differences that are likely due to our assumption of a one-dimensional phase space manifold.

Probability Density Functions for Solute Transport in Random Field

Effective description of flow and transport in irregular porous media, adequate understanding and reliable estimation of the uncertainty, all require stochastic approach. The primary problem is finding the relations between the non-random functionals of the unknown and the given random fields, that is moments, distribution functions, probability density distributions, etc. This paper considers the process of transport of non-reactive admixture in random media and attempts to develop a method for finding the probability density function of the concentration of the solute. We introduce the random functional p(x, t; c), the density distribution function (DDF) of the local random concentration c(x, t) in the one-dimensional phase space of its possible values c, where parameter x is multi-dimensional vector, and parameter t is time. By using the stochastic transport equation in the (x, t) space, one can write the so-called stochastic Liouville equation for the functional p(x, t; c) and this equation bears the form of the transport equation in the (x, t; c) space. The averaging of the new transport equation in the (x, t; c) space leads to equations for P(x, t; c) = 〈 p(x, t; c) 〉 – the probability density function (PDF) for c(x, t) and the corresponding power moments. We present the analysis examples of PDFs for the concentration c(x, t) in several different cases of flow velocity field and initial concentration distribution.

An exact effective Hamiltonian for a perturbed Landau level

Considers the effect of a scalar potential V (x, y) on a Landau level in two dimensions. An exact effective Hamiltonian is derived which describes the effect of the potential on a single Landau level, expressed as a power series in V/Ec, where Ec is the cyclotron energy. The effective Hamiltonian can be represented as a function H (x, p) in a one-dimensional phase space. The function H (x, p) resembles the potential V (x, y): when the area of a flux quantum is much smaller than the square of the characteristic length scale of V, then H approximately=V. Also H (x, p) retains the translational and rotational symmetries of V(x, y) exactly, but reflection symmetries are not retained beyond the lowest order of the perturbation expansion.