Equation In Phase Space Research Articles

Wigner Function and Non-classicality for Oscillator Systems

The Schrodinger equation in phase space is used to calculate the Wigner function for oscillator systems. The first is two oscillators including dissipative effects. In this case, the non-classicality of the states is studied by the non-classicality indicator of the Wigner function, which is calculated as a function of the dissipation parameter. The second oscillator system is non-linear pendulum. Analytical results are derived and the Wigner function is analyzed.

Quantum Physics in Phase Space: an Analysis of Simple Pendulum

In this work we present a brief review about quantum mechanics in phase space. The approach discussed is based in the notion of symplectic structure and star-operators. In this sense, unitary representations for the Galilei group are construct, and the Schrodinger equation in phase space is derived. The connection between phase space amplitudes and Wigner function is presented. As a new result we solved the Schrodinger equation in phase space for simple pendulum. PACS Numbers: 11.10.Nx, 11.30.Cp, 05.20.Dd

Quantum mechanics on phase space: The hydrogen atom and its Wigner functions

Symplectic quantum mechanics (SQM) considers a non-commutative algebra of functions on a phase space Γ and an associated Hilbert space HΓ, to construct a unitary representation for the Galilei group. From this unitary representation the Schrödinger equation is rewritten in phase space variables and the Wigner function can be derived without the use of the Liouville–von Neumann equation. In this article the Coulomb potential in three dimensions (3D) is resolved completely by using the phase space Schrödinger equation. The Kustaanheimo–Stiefel(KS) transformation is applied and the Coulomb and harmonic oscillator potentials are connected. In this context we determine the energy levels, the amplitude of probability in phase space and correspondent Wigner quasi-distribution functions of the 3D-hydrogen atom described by Schrödinger equation in phase space.

Evolution of axis ratios from phase space dynamics of triaxial collapse

We investigate the evolution of axis ratios of triaxial haloes using the phase space description of triaxial collapse. In this formulation, the evolution of the triaxial ellipsoid is described in terms of the dynamics of eigenvalues of three important tensors: the Hessian of the gravitational potential, the tensor of velocity derivatives and the deformation tensor. The eigenvalues of the deformation tensor are directly related to the parameters that describe triaxiality, namely, the minor to major and intermediate to major axes ratios ($s$ and $q$) and the triaxiality parameter $T$. Using the phase space equations, we evolve the eigenvalues and examine the evolution of the PDF (probability distribution function) of the axes ratios as a function of mass scale and redshift for Gaussian initial conditions. We find that the ellipticity and prolateness increase with decreasing mass scale and decreasing redshift. These trends agree with previous analytic studies but differ from numerical simulations. However, the PDF of the scaled parameter ${\tilde q} = (q-s)/(1-s)$ follows a universal distribution over two decades in mass range and redshifts which is in qualitative agreement with the universality for conditional PDF reported in simulations. We further show using the phase space dynamics that, in fact, ${\tilde q}$ is a phase space invariant and is conserved individually for each halo. These results, demonstrate that the phase space analysis is a useful tool that provides a different perspective on the evolution of perturbations and can be applied to more sophisticated models in the future.

A commuting-vector-field approach to some dispersive estimates

We prove the pointwise decay of solutions to three linear equations: (i) the transport equation in phase space generalizing the classical Vlasov equation, (ii) the linear Schrodinger equation, (iii) the Airy (linear KdV) equation. The usual proofs use explicit representation formulae, and either obtain $L^1$---$L^\infty$ decay through directly estimating the fundamental solution in physical space, or by studying oscillatory integrals coming from the representation in Fourier space. Our proof instead combines "vector field" commutators that capture the inherent symmetries of the relevant equations with conservation laws for mass and energy to get space-time weighted energy estimates. Combined with a simple version of Sobolev's inequality this gives pointwise decay as desired. In the case of the Vlasov and Schrodinger equations we can recover sharp pointwise decay; in the Schrodinger case we also show how to obtain local energy decay as well as Strichartz-type estimates. For the Airy equation we obtain a local energy decay that is almost sharp from the scaling point of view, but nonetheless misses the classical estimates by a gap. This work is inspired by the work of Klainerman on $L^2$---$L^\infty$ decay of wave equations, as well as the recent work of Fajman, Joudioux, and Smulevici on decay of mass distributions for the relativistic Vlasov equation.

Topographic–baroclinic instability and formation of Kuroshio current loop

Using time-series figures of sea-level anomaly and geostrophic currents from merged absolute dynamic topography, we analyzed the formation and evolution of the Kuroshio current loop (KCL). The main results are as follows. Perturbation origins of the KCLs are in three areas (eastern, western, and southern) surrounding the Hengchun Submarine Ridge. There are two basic types of KCL formation, i.e., “Kuroshio bend pushing” and “Kuroshio Branch rewinding”, plus their combination. The KCLs propagate westward at 1.6–4.5 cm/s. There are two forms of KCL evolution into a shed eddy. The first is such that the northern KCL section initially divides to become an eddy joining the Kuroshio Branch current, which then separates from that current to become a shed eddy. The second form is such that the northern and southern sections of the KCL are separated almost simultaneously in westward elongated process. To understand the KCL formation mechanism, we derive linear equations in phase space from the governing equations in σ-coordinates, ultimately obtaining two groups of analytical solutions for interactions between waves, topography, and the basic current field. The solutions lead to the following results. The KCL propagates westward with the group velocity of the Kuroshio center region. The Kuroshio generally sweeps over the Hengchun Submarine Ridge, especially in winter, such that there is topographic–baroclinic instability. The analytical solutions effectively reveal the dynamic mechanism of the two basic types of KCL formation.

Quantum mechanics on phase space and the Coulomb potential

Symplectic quantum mechanics (SMQ) makes possible to derive the Wigner function without the use of the Liouville–von Neumann equation. In this formulation of the quantum theory the Galilei Lie algebra is constructed using the Weyl (or star) product with Qˆ=q⋆=q+iħ2∂p,Pˆ=p⋆=p−iħ2∂q, and the Schrödinger equation is rewritten in phase space; in consequence physical applications involving the Coulomb potential present some specific difficulties. Within this context, in order to treat the Schrödinger equation in phase space, a procedure based on the Levi-Civita (or Bohlin) transformation is presented and applied to two-dimensional (2D) hydrogen atom. Amplitudes of probability in phase space and the correspondent Wigner quasi-distribution functions are derived and discussed.

Expansion of a wedge of non-ideal gas into vacuum

We study the problem of expansion of a wedge of non-ideal gas into vacuum in a two-dimensional bounded domain. The non-ideal gas is characterized by a van der Waals type equation of state. The problem is modeled by standard Euler equations of compressible flow, which are simplified by a transformation to similarity variables and then to hodograph transformation to arrive at a second order quasilinear partial differential equation in phase space; this, using Riemann variants, can be expressed as a non-homogeneous linearly degenerate system provided that the flow is supersonic. For the solution of the governing system, we study the interaction of two-dimensional planar rarefaction waves, which is a two-dimensional Riemann problem with piecewise constant data in the self-similar plane. The real gas effects, which significantly influence the flow regions and boundaries and which do not show-up in the ideal gas model, are elucidated; this aspect of the problem has not been considered until now.

Schrödinger like equation for wavelets

An explicit phase space representation of the wave function is build based on a wavelet transformation. The wavelet transformation allows us to understand the relationship between s − ordered Wigner function, (or Wigner function when s = 0), and the Torres-Vega-Frederick’s wave functions. This relationship is necessary to find a general solution of the Schrödinger equation in phase-space.

Active matter beyond mean-field: ring-kinetic theory for self-propelled particles.

Recently, Hanke et al. [Phys. Rev. E 88, 052309 (2013)] showed that mean-field kinetic theory fails to describe collective motion in soft active colloids and that correlations must not be neglected. Correlation effects are also expected to be essential in systems of biofilaments driven by molecular motors and in swarms of midges. To obtain correlations in an active matter system from first principles, we derive a ring-kinetic theory for Vicsek-style models of self-propelled agents from the exact N-particle evolution equation in phase space. The theory goes beyond mean-field and does not rely on Boltzmann's approximation of molecular chaos. It can handle precollisional correlations and cluster formation, which are both important to understand the phase transition to collective motion. We propose a diagrammatic technique to perform a small-density expansion of the collision operator and derive the first two equations of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. An algorithm is presented that numerically solves the evolution equation for the two-particle correlations on a lattice. Agent-based simulations are performed and informative quantities such as orientational and density correlation functions are compared with those obtained by ring-kinetic theory. Excellent quantitative agreement between simulations and theory is found at not-too-small noises and mean free paths. This shows that there are parameter ranges in Vicsek-like models where the correlated closure of the BBGKY hierarchy gives correct and nontrivial results. We calculate the dependence of the orientational correlations on distance in the disordered phase and find that it seems to be consistent with a power law with an exponent around -1.8, followed by an exponential decay. General limitations of the kinetic theory and its numerical solution are discussed.

Critical collapse in the spherically symmetric Einstein-Vlasov model

We solve the coupled Einstein-Vlasov system in spherical symmetry using direct numerical integration of the Vlasov equation in phase space. Focusing on the case of massless particles we study critical phenomena in the model, finding strong evidence for generic type I behaviour at the black hole threshold that parallels what has previously been observed in the massive sector. For differing families of initial data we find distinct critical solutions, so there is no universality of the critical configuration itself. However we find indications of at least a weak universality in the lifetime scaling exponent, which is yet to be understood. Additionally, we clarify the role that angular momentum plays in the critical behaviour in the massless case.

Towards a quantitative kinetic theory of polar active matter

A recent kinetic approach for Vicsek-like models of active particles is reviewed. The theory is based on an exact Chapman-Kolmogorov equation in phase space. It can handle discrete time dynamics and "exotic" multi-particle interactions. A nonlocal mean-field theory for the one-particle distribution function is obtained by assuming molecular chaos. The Boltzmann approach of Bertin et al., Phys. Rev. E 74, 022101 (2006) and J. Phys. A: Math. Theor. 42, 445001 (2009), is critically assessed and compared to the current approach. In Boltzmann theory, a collision starts when two particles enter each others action spheres and is finished when their distance exceeds the interaction radius. The average duration of such a collision, $\tau_0$, is measured for the Vicsek model with continuous time-evolution. If the noise is chosen to be close to the flocking threshold, the average time between collisions is found to be roughly equal to $\tau_0$ at low densities. Thus, the continuous-time Vicsek-model near the flocking threshold cannot be accurately described by a Boltzmann equation, even at very small density because collisions take so long that typically other particles join in, rendering Boltzmann's binary collision assumption invalid. Hydrodynamic equations for the phase space approach are derived by means of a Chapman-Enskog expansion. The equations are compared to the Toner-Tu theory of polar active matter. New terms, absent in the Toner-Tu theory, are highlighted. Convergence problems of Chapman-Enskog and similar gradient expansions are discussed.

Testing the origin of cosmological magnetic fields through the large-scale structure consistency relations

We study the symmetries of the post-recombination cosmological magnetohydrodynamical equations which describe the evolution of dark matter, baryons and magnetic fieldsin a self-consistent way. This is done both at the level of fluid equations and of Vlasov-Poisson-Maxwell equations in phase space. We discuss some consistency relations for the soft limit of the (n+1)-correlator functions involving magnetic fields and matter overdensities. In particular, we stress that any violation of suchconsistency relations at equal-time would point towards an inflationary origin of the magnetic field.

Função de Wigner-80 anos e as origens da geometria não-comutativa

O conceito de espaços não-comutativos tem origem com a formulação de Wigner da mecânica quântica no espaço de fase, em 1932. Em paralelo, Heisenberg foi o primeiro a propor relações de não-comutação entre as componentes do operador de posição. Essa possibilidade ganha formulação matemática com Snyder, ao estudar representações do grupo de De Sitter (4+1). Uma síntese desses trabalhos é o conceito de geometria não-comutativa, estabelecida a partir do produto de Moyal, que aparece no formulismo de Wigner. Além disso, este tipo de não-comutatividade é reencontrada em certos limites das teorias de cordas, gerando expectativas da mensuração da não-comutatividade espacial na física de altas energias. Neste trabalho, apresentamos uma revisão pedagógica sobre teorias físicas em espaços não-comutativos, a partir de uma perpectiva histórica. Destacamos as teorias de representação de grupos de simetria no espaço de fase, apontando dois aspectos de interesse, mas pouco conhecidos: (a) a noção de amplitude de probabilidade e a representação da equação de Schrödinger no espaço de fase (usualmente a representação no espaço de fase da mecânica quântica é construída através da matrix de densidade e da equação de Liouville-von Neumann); e (b) um trabalho de Dirac de 1930, onde foi introduzida pela primeira vez uma formulação da física quântica no espaço de fase.

Tunneling of an energy eigenstate through a parabolic barrier viewed from Wigner phase space

We analyze the tunneling of a particle through a repulsive potential resulting from an inverted harmonic oscillator in the quantum mechanical phase space described by the Wigner function. In particular, we solve the partial differential equations in phase space determining the Wigner function of an energy eigenstate of the inverted oscillator. The reflection or transmission coefficients $R$ or $T$ are then given by the total weight of all classical phase space trajectories corresponding to energies below, or above the top of the barrier given by the Wigner function.

Phase-space equation for wave equations

Transforming a space-time function into the phase space of position and wave number offers considerable insight into the nature of the function, and also has many practical applications. If the function is governed by a wave equation then the common procedure is to solve the wave equation and then calculate the phase space distribution function for the solution. We show that significant advantages ensue if one transforms the original differential equation into a phase space differential equation. We give a number of examples and show that phase space equations are often more revealing than the original equation and lead to new approximation methods. [Work supported by ONR.]

The Noncommutative Harmonic Oscillator Based on Symplectic Representation of Galilei Group

We study symplectic unitary representations for the Galilei group and derive the Schrödinger equation in phase space. Our formalism is based on the noncommutative structure of the star product. Guided by group theoretical concepts, we construct a physically consistent phase-space theory in which each state is described by a quasi-probability amplitude associated with the Wigner function. As applications, we derive the Wigner functions for the 3D harmonic oscillator and the noncommutative oscillator in phase space.

Methods of analytical mechanics for exact Cosserat elastic rod dynamics

Thin elastic rod mechanics with background of a kind of single molecule such as DNA and other engineering object has entered into a new developing stage. In this paper the vector method of exact Cosserat elastic rod dynamics is transformed into the form of analytical mechanics with the arc length and time as its independent variables, whose aims are to find new tools for studying rod mechanics and to develop the area of applications of classical analytical mechanics. Based on the plane cross-section assumption, a cross-section of the rod is taken as an object. Basic formulas on deformation and motion of the section are given. After defining virtual displacement of a cross-section and its equivalent variation rule, a differential variational principle such as d’Alembert-Lagrange one is established, from which dynamical equations of thin elastic rod are expressed as Lagrange equations or Nielsen equations under the condition of linear elasticity of the rod. For the rod statics when there exist conserved quantities, Lagrange equation which makes use of these quantities is derived and its first integral is discussed. Finally integral variational principle is derived from differential one, and expressed as Hamilton principle under the condition of linear elasticity. Hamilton canonical equations in phase space with 3×6 dimensions are also derived. All of the results have formed the method of analytical mechanics of dynamics of an exact Cosserat elastic rod, so that the further problems such as symmetry and conserved quantities, and numerical simulation of the rod dynamics may be further studied.

Quantum behavior derived as an essentially stochastic phenomenon

We study the behavior of the atomic matter immersed in a zero-point radiation field, which is taken as the source of the endless stochastic motion of electrons. The starting point is a statistical phase-space equation for a particle in interaction with the field. Under certain natural approximations, the system reaches a stationary condition; the corresponding reduced description obtained in configuration space is just the Schrödinger quantum mechanics. Quantum nonlocality is revisited from this point of view and is shown to appear as an artifact of the reduced description. The theory presented also serves to explain why usual quantum mechanics is not amenable to a phase-space formulation.

Solving the stationary Liouville equation via a boundary element method

Energy distributions of linear wave fields are, in the high frequency limit, often approximated in terms of flow or transport equations in phase space. Common techniques for solving the flow equations in both time dependent and stationary problems are ray tracing and level set methods. In the context of predicting the vibro-acoustic response of complex engineering structures, related methods such as Statistical Energy Analysis or variants thereof have found widespread applications. We present a new method for solving the transport equations for complex multi-component structures based on a boundary element formulation of the stationary Liouville equation. The method is an improved version of the Dynamical Energy Analysis technique introduced recently by the authors. It interpolates between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. We demonstrate that the method can be used to efficiently deal with complex large scale problems giving good approximations of the energy distribution when compared to exact solutions of the underlying wave equation.