Phase Space Evolution Research Articles

Numerical study on the stability of weakly collisional plasma in E×B fields

Plasma stability in weakly collisional plasmas in the presence of E×B fields is studied with numerical simulations. Different types of ion-neutral collisions are considered in a fully magnetized regime. We study the influence of ion-neutral collisions and the role of collision types on the stability of plasma. It is found that the stability of plasma depends on the type of ion-neutral collisions, with the plasma being unstable for charge exchange collisions, and stable for the elastic scattering. The analysis focuses on the temporal evolution of the velocity phase space, RMS values of the potential fluctuations, and coherent structures in potential densities. For the unstable case, we observe growth and propagation of electrostatic waves. Simulations are performed with a three-dimensional electrostatic particle in cell code.

Phase-Flow Interpretation of Magnetization Relaxation in Nanomagnets

This paper is devoted to the study of the probability of relaxation of a bistable nanomagnet from high energy to one of its two energy minima. The evolution in phase space of a Gibbs ensemble of replicas of the nanomagnet is analyzed in the limit of small Gilbert damping and in the absence of thermal noise. It is shown that when an external magnetic field is applied along the intermediate anisotropy axis, the relaxation probability exhibits a saw-tooth dependence on the field amplitude, periodically reaching the extreme values 0 or 1, in correspondence to which relaxation is fully insensitive to randomness in initial conditions.

Dynamics of One-dimensional Self-gravitating Systems Using Hermite-Legendre Polynomials

The current paradigm for understanding galaxy formation in the universe de- pends on the existence of self-gravitating collisionless dark matter. Modeling such dark matter systems has been a major focus of astrophysicists, with much of that effort directed at computational techniques. Not surprisingly, a comprehensive understanding of the evolution of these self-gravitating systems still eludes us, since it involves the collective nonlinear dynamics of many-particle systems inter- acting via long-range forces described by the Vlasov equation. As a step towards developing a clearer picture of collisionless self-gravitating relaxation, we analyze the linearized dynamics of isolated one-dimensional systems near thermal equi- librium by expanding their phase space distribution functions f(x,v) in terms of Hermite functions in the velocity variable, and Legendre functions involving the position variable. This approach produces a picture of phase-space evolution in terms of expansion coefficients, rather than spatial and velocity variables. We obtain equations of motion for the expansion coefficients for both test-particle dis- tributions and self-gravitating linear perturbations of thermal equilibrium. This development presents the opportunity to avoid time-consuming N-body simula- tions that are limited by statistical uncertainty and provides a powerful analysis tool for understanding the relaxation to equilibrium.Â

Shot-noise triggered electron beam micro-bunching in SASE-FELs

In this paper, the evolution of shot-noise micro-bunching in a SASE-FEL is investigated. First, a simple one-dimensional model for phase-space evolution of electrons is presented. Then, for an e-beam with random distribution of electrons (i.e. with shot-noise), the start-up and growth of micro-bunching is calculated. It is shown that shot-noise micro-bunching starts in random positions within the e-beam and grows in amplitude and extent. The final micro-bunching pattern contains distinct regions each with a coherent micro-bunching but with no phase relation with that of the others. Such distinct micro-bunched regions show themselves in the so-called spiky pattern of output radiation.

Hydrodynamic shock wave studies within a kinetic Monte Carlo approach

We introduce a massively parallelized test-particle based kinetic Monte Carlo code that is capable of modeling the phase space evolution of an arbitrarily sized system that is free to move in and out of the continuum limit. Our code combines advantages of the DSMC and the Point of Closest Approach techniques for solving the collision integral. With that, it achieves high spatial accuracy in simulations of large particle systems while maintaining computational feasibility. Using particle mean free paths which are small with respect to the characteristic length scale of the simulated system, we reproduce hydrodynamic behavior. To demonstrate that our code can retrieve continuum solutions, we perform a test-suite of classic hydrodynamic shock problems consisting of the Sod, the Noh, and the Sedov tests. We find that the results of our simulations which apply millions of test-particles match the analytic solutions well. In addition, we take advantage of the ability of kinetic codes to describe matter out of the continuum regime when applying large particle mean free paths. With that, we study and compare the evolution of shock waves in the hydrodynamic limit and in a regime which is not reachable by hydrodynamic codes.

Phase-Space Dynamics of Ionization Injection in Plasma-Based Accelerators

The evolution of beam phase space in ionization injection into plasma wakefields is studied using theory and particle-in-cell simulations. The injection process involves both longitudinal and transverse phase mixing, leading initially to a rapid emittance growth followed by oscillation, decay, and a slow growth to saturation. An analytic theory for this evolution is presented and verified through particle-in-cell simulations. This theory includes the effects of injection distance (time), acceleration distance, wakefield structure, and nonlinear space charge forces, and it also shows how ultralow emittance beams can be produced using ionization injection methods.

Dynamics of one-dimensional self-gravitating systems using Hermite–Legendre polynomials

The current paradigm for understanding galaxy formation in the universe depends on the existence of self-gravitating collisionless dark matter. Modeling such dark matter systems has been a major focus of astrophysicists, with much of that effort directed at computational techniques. Not surprisingly, a comprehensive understanding of the evolution of these self-gravitating systems still eludes us, since it involves the collective nonlinear dynamics of many-particle systems interacting via long-range forces described by the Vlasov equation. As a step towards developing a clearer picture of collisionless self-gravitating relaxation, we analyze the linearized dynamics of isolated one-dimensional systems near thermal equilibrium by expanding their phase space distribution functions f(x,v) in terms of Hermite functions in the velocity variable, and Legendre functions involving the position variable. This approach produces a picture of phase-space evolution in terms of expansion coefficients, rather than spatial and velocity variables. We obtain equations of motion for the expansion coefficients for both test-particle distributions and self-gravitating linear perturbations of thermal equilibrium. N-body simulations of perturbed equilibria are performed and found to be in excellent agreement with the expansion coefficient approach over a time duration that depends on the size of the expansion series used.

Relational evolution of observables for Hamiltonian-constrained systems

Evolution of systems in which Hamiltonians are generators of gauge transformations is a notion that requires more structure than the canonical theory provides. We identify and study this additional structure in the framework of relational observables (``partial observables''). We formulate necessary and sufficient conditions for the resulting evolution in the physical phase space to be a symplectomorphism. We give examples which satisfy those conditions and examples which do not. We point out that several classic positions in the literature on relational observables contain an incomplete approach to the issue of evolution and false statements. Our work provides useful clarification and opens the door to studying correctly formulated definitions.

Bouncing cosmologies in massive gravity on de Sitter

In the framework of massive gravity with a de Sitter reference metric, we study homogeneous and isotropic solutions with positive spatial curvature. Remarkably, we find that bounces can occur when cosmological matter satisfies the strong energy condition, in contrast to what happens in classical general relativity. This is due to the presence in the Friedmann equations of additional terms, which depend on the scale factor and its derivatives and can be interpreted as an effective fluid. We present a detailed study of the system using a phase space analysis. After having identified the fixed points of the system and investigated their stability properties, we discuss the cosmological evolution in the global physical phase space. We find that bouncing solutions are generic. Moreover, depending on the solutions, the cosmological evolution can lead to an asymptotic de Sitter regime, a curvature singularity or a determinant singularity.

Scalar Wigner theory for polarized light in nonlinear Kerr media

A scalar Wigner distribution function for describing polarized light is proposed in analogy with the treatment of spin variables in quantum kinetic theory. The formalism is applied to the propagation of circularly polarized light in nonlinear Kerr media and an extended phase space evolution equation is derived along with invariant quantities. We further consider modulation instability as well as the extension to partially coherent fields.

Relativistic harmonic oscillator, the associated equations of motion, and algebraic integration methods

We consider the relativistic generalization of the harmonic oscillator problem by addressing different questions regarding its classical aspects. We treat the problem using the formalism of Hamiltonian mechanics. A Lie algebraic technique is used to solve the associated Liouville equations, yielding the phase-space evolution of an ensemble of relativistic particles, subject to a ``harmonic'' potential. The nonharmonic distortion of the spatial and momentum distributions due to the intrinsic nonlinear nature of the relativistic contributions is discussed. We analyze the relativistic dynamics induced by two types of Hamiltonian, which can be ascribed to those of harmonic oscillator type. Finally, we briefly discuss the quantum aspects of the problem by considering possible strategies for the solution of the associated Salpeter equation.

Quantum mechanical version of the classical Liouville theorem

In terms of the coherent state evolution in phase space, we present a quantum mechanical version of the classical Liouville theorem. The evolution of the coherent state from |z〉 to |sz − rz*〉 corresponds to the motion from a point z (q,p) to another point sz − rz* with |s|2 − |r|2 = 1. The evolution is governed by the so-called Fresnel operator U(s,r) that was recently proposed in quantum optics theory, which classically corresponds to the matrix optics law and the optical Fresnel transformation, and obeys group product rules. In other words, we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space, which seems to be a combination of quantum statistics and quantum optics.

Accelerating piston action and plasma heating in high-energy density laser plasma interactions

In the field of high-energy density physics (HEDP), lasers in both the nanosecond and picosecond regimes can drive conditions in the laboratory relevant to a broad range of astrophysical phenomena, including gamma-ray burst afterglows and supernova remnants. In the short-pulse regime, the strong light pressure (>Gbar) associated ultraintense lasers of intensity I > 1018 W/cm2 plays a central role in many HEDP applications. Yet, the behavior of this nonlinear pressure mechanism is not well-understood at late time in the laser–plasma interaction. In this paper, a more realistic treatment of the laser pressure ‘hole boring’ process is developed through analytical modeling and particle-in-cell simulations. A simple Liouville code capturing the phase space evolution of ponderomotively-driven ions is employed to distill effects related to plasma heating and ion bulk acceleration. Taking into account these effects, our results show that the evolution of the laser-target system encompasses ponderomotive expansion, equipartition, and quasi-isothermal expansion epochs. These results have implications for light piston-driven ion acceleration scenarios, and astrophysical applications where the efficiencies of converting incident Poynting flux into bulk plasma flow and plasma heat are key unknown parameters.

Phase space evolution of pairs created in strong electric fields

We study the process of energy conversion from overcritical electric field into electron–positron–photon plasma. We solve numerically Vlasov–Boltzmann equations for pairs and photons assuming the system to be homogeneous and anisotropic. All the 2-particle QED interactions between pairs and photons are described by collision terms. We evidence several epochs of this energy conversion, each of them associated to a specific physical process. Firstly pair creation occurs, secondly back reaction results in plasma oscillations. Thirdly photons are produced by electron–positron annihilation. Finally particle interactions lead to completely equilibrated thermal electron–positron–photon plasma.

Fuzzy topology, quantization and gauge invariance

Quantum space-time with Dodson-Zeeman topological structure is studied. In its framework the states of massive particle m correspond to elements of fuzzy set called fuzzy points. Due to their weak (partial) ordering, m space coordinate x acquires principal uncertainty σ x . Quantization formalism is derived from consideration of m evolution in fuzzy phase space with minimal number of additional assumptions. Particle’s interactions on fuzzy manifold are studied and shown to be gauge invariant.

Electron-ion energy partition when a charged particle slows in a plasma: Theory

The preceding paper [Brown, Preston, and Singleton Jr., Phys. Rev. E 86, 016406 (2012)] presented precise results for the partition of the initial energy E(0) of a fast particle into the ions and electrons--E(I)/E(0) and E(e)/E(0)--when the fast particle slows in a plasma whose ion and electron temperatures may differ. As emphasized in that paper, this is an important problem because nuclear fusion reactions, such as those that occur in an inertial confinement fusion capsule, involve ion temperatures that run away from the electron temperatures. As also noted in the preceding paper, a precise evaluation entails the use of a well-defined Fokker-Planck equation for the phase-space evolution of initially fast projectile particles. When the plasma has differing ion and electron temperatures, the projectiles must slow into a "schizophrenic" final ensemble of particles that has neither the electron nor the ion temperature. This is not a simple Maxwell-Boltzmann distribution since the electrons are not in thermal equilibrium with the ions. Thus, detailed calculations are required for the solution of the problem. These we provide here for a weakly to moderately coupled plasma. The Fokker-Planck equation holds to first subleading order in the dimensionless plasma coupling constant, which translates to computing to order n ln n (leading) and n (subleading) in the plasma density n. The energy partitions for a background plasma in thermal equilibrium have been previously computed, but the order n terms have not been calculated, only estimated. The "schizophrenic" final ensemble of slowed particles gives a new mechanism to bring the electron and ion temperatures together. The rate at which this new mechanism brings the electrons and ions in the plasma into thermal equilibrium will be computed.

Manifestation of a nonclassical Berry phase of an electromagnetic field in atomic Ramsey interference

The Berry phase acquired by an electromagnetic field undergoing an adiabatic and cyclic evolution in phase space is a purely quantum-mechanical effect of the field. However, this phase is usually accompanied by a dynamical contribution and cannot be manifested in any light-beam interference experiment because it is independent of the field state. We here show that such a phase can be produced using an atom coupled to a quantized field and driven by a slowly changing classical field, and it is manifested in atomic Ramsey interference oscillations. We also show how this effect may be applied to one-step implementation of multiqubit geometric phase gates, which is impossible by previous geometric methods. The effects of dissipation and fluctuations in the parameters of the pump field on the Berry phase and visibility of the Ramsey interference fringes are analyzed.

Experimental demonstration of a slippage-dominant free-electron laser amplifier

We report the first experimental demonstration of a slippage-dominant free-electron laser (FEL) amplifier using a 140-fs full width at half maximum broadband seed laser pulse. The evolution of the longitudinal phase space of a laser seeded FEL amplifier in the slippage-dominant regime was experimentally characterized. We observed, for the first time, that the pulse duration of the FEL is primarily determined by the slippage between the seed laser and the electron beam. With a ± 1% variation in the electron-beam energy, we demonstrated reasonably good longitudinal coherence and a ± 2% spectral tuning range. The experimentally observed temporal and spectral evolution of the slippage-dominant FEL was verified by the numerical simulations.

Phase Space Evolution and Discontinuous Schrödinger Waves

The problem of Schrödinger propagation of a discontinuous wavefunction – diffraction in time – is studied under a new light. It is shown that the evolution map in phase space induces a set of affine transformations on discontinuous wavepackets, generating expansions similar to those of wavelet analysis. Such transformations are identified as the cause for the infinitesimal details in diffraction patterns. A simple case of an evolution map, such as SL(2) in a two-dimensional phase space, is shown to produce an infinite set of space-time trajectories of constant probability. The trajectories emerge from a breaking point of the initial wave.

Macroscopic Time Evolution and MaxEnt Inference for Closed Systems with Hamiltonian Dynamics

MaxEnt inference algorithm and information theory are relevant for the time evolution of macroscopic systems considered as problem of incomplete information. Two different MaxEnt approaches are introduced in this work, both applied to prediction of time evolution for closed Hamiltonian systems. The first one is based on Liouville equation for the conditional probability distribution, introduced as a strict microscopic constraint on time evolution in phase space. The conditional probability distribution is defined for the set of microstates associated with the set of phase space paths determined by solutions of Hamilton's equations. The MaxEnt inference algorithm with Shannon's concept of the conditional information entropy is then applied to prediction, consistently with this strict microscopic constraint on time evolution in phase space. The second approach is based on the same concepts, with a difference that Liouville equation for the conditional probability distribution is introduced as a macroscopic constraint given by a phase space average. We consider the incomplete nature of our information about microscopic dynamics in a rational way that is consistent with Jaynes' formulation of predictive statistical mechanics. Maximization of the conditional information entropy subject to this macroscopic constraint leads to a loss of correlation between the initial phase space paths and final microstates. Information entropy is the theoretic upper bound on the conditional information entropy, with the upper bound attained only in case of the complete loss of correlation. In this alternative approach to prediction of macroscopic time evolution, maximization of the conditional information entropy is equivalent to the loss of statistical correlation. In accordance with Jaynes, irreversibility appears as a consequence of gradual loss of information about possible microstates of the system.