Showing papers on "Phase space published in 2013"

Dimers and cluster integrable systems

Abstract: We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type - a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph. The sum of Hamiltonians is essentially the partition function of the dimer model. Any graph on a torus gives rise to a bipartite graph on the torus. We show that the phase space of the latter has a Lagrangian subvariety. We identify it with the space parametrizing resistor networks on the original graph.We construct several discrete quantum integrable systems.

Berry curvature and four-dimensional monopoles in the relativistic chiral kinetic equation.

Abstract: We derive a relativistic chiral kinetic equation with manifest Lorentz covariance from Wigner functions of spin-1/2 massless fermions in a constant background electromagnetic field. It contains vorticity terms and a four-dimensional Euclidean Berry monopole which gives an axial anomaly. By integrating out the zeroth component of the 4-momentum p, we reproduce the previous three-dimensional results derived from the Hamiltonian approach, together with the newly derived vorticity terms. The phase space continuity equation has an anomalous source term proportional to the product of electric and magnetic fields (FσρF[over ˜]σρ∼EσBσ). This provides a unified interpretation of the chiral magnetic and vortical effects, chiral anomaly, Berry curvature, and the Berry monopole in the framework of Wigner functions.

Resonant post-newtonian eccentricity excitation in hierarchical three-body systems

Abstract: We study the secular, hierarchical three-body problem to first-order in a post-Newtonian expansion of general relativity (GR). We expand the first-order post-Newtonian Hamiltonian to leading-order in the ratio of the semi-major axis of the two orbits. In addition to the well-known terms that correspond to the GR precession of the inner and outer orbits, we find a new secular post-Newtonian interaction term that can affect the long-term evolution of the triple. We explore the parameter space for highly inclined and eccentric systems, where the Kozai-Lidov mechanism can produce large-amplitude oscillations in the eccentricities. The standard lore, i.e., that GR effects suppress eccentricity, is only consistent with the parts of phase space where the GR timescales are several orders of magnitude shorter than the secular Newtonian one. In other parts of phase space, however, post-Newtonian corrections combined with the three-body ones can excite eccentricities. In particular, for systems where the GR timescale is comparable to the secular Newtonian timescales, the three-body interactions give rise to a resonant-like eccentricity excitation. Furthermore, for triples with a comparable-mass inner binary, where the eccentric Kozai-Lidov mechanism is suppressed, post-Newtonian corrections can further increase the eccentricity and lead to orbital flips even when the timescale of the former is much longer than the timescale of the secular Kozai-Lidov quadrupole perturbations.

Controllability Transition and Nonlocality in Network Control

Abstract: A common goal in the control of a large network is to minimize the number of driver nodes or control inputs. Yet, the physical determination of control signals and the properties of the resulting control trajectories remain widely underexplored. Here we show that (i) numerical control fails in practice even for linear systems if the controllability Gramian is ill conditioned, which occurs frequently even when existing controllability criteria are satisfied unambiguously, (ii) the control trajectories are generally nonlocal in the phase space, and their lengths are strongly anti-correlated with the numerical success rate and number of control inputs, and (iii) numerical success rate increases abruptly from zero to nearly one as the number of control inputs is increased, a transformation we term numerical controllability transition. This reveals a trade-off between nonlocality of the control trajectory in the phase space and nonlocality of the control inputs in the network itself. The failure of numerical control cannot be overcome in general by merely increasing numerical precision--successful control requires instead increasing the number of control inputs beyond the numerical controllability transition.

Nonequilibrium dynamics of photoexcited electrons in graphene : collinear scattering, Auger processes, and the impact of screening

Abstract: We present a combined analytical and numerical study of the early stages (sub-100-fs) of the nonequilibrium dynamics of photoexcited electrons in graphene. We employ the semiclassical Boltzmann equation with a collision integral that includes contributions from electron-electron (e-e) and electron–optical phonon interactions. Taking advantage of circular symmetry and employing the massless Dirac fermion (MDF) Hamiltonian, we are able to perform an essentially analytical study of the e-e contribution to the collision integral. This allows us to take particular care of subtle collinear scattering processes—processes in which incoming and outgoing momenta of the scattering particles lie on the same line—including carrier multiplication (CM) and Auger recombination (AR). These processes have a vanishing phase space for two-dimensional MDF bare bands. However, we argue that electron-lifetime effects, seen in experiments based on angle-resolved photoemission spectroscopy, provide a natural pathway to regularize this pathology, yielding a finite contribution due to CM and AR to the Coulomb collision integral. Finally, we discuss in detail the role of physics beyond the Fermi golden rule by including screening in the matrix element of the Coulomb interaction at the level of the random phase approximation (RPA), focusing in particular on the consequences of various approximations including static RPA screening, which maximizes the impact of CM and AR processes, and dynamical RPA screening, which completely suppresses them.

Unconventional states of bosons with synthetic spin-orbit coupling

Abstract: Spin-orbit coupling with bosons gives rise to novel properties that are absent in usual bosonic systems. Under very general conditions, the conventional ground state wavefunctions of bosons are constrained by the "no-node" theorem to be positive-definite. In contrast, the linear-dependence of spin-orbit coupling leads to complex-valued condensate wavefunctions beyond this theorem. In this article, we review the study of this class of unconventional Bose-Einstein condensations focusing on their topological properties. Both the 2D Rashba and 3D $\vec{\sigma} \cdot \vec{p}$-type Weyl spin-orbit couplings give rise to Landau-level-like quantization of single-particle levels in the harmonic trap. The interacting condensates develop the half-quantum vortex structure spontaneously breaking time-reversal symmetry and exhibit topological spin textures of the skyrmion type. In particular, the 3D Weyl coupling generates topological defects in the quaternionic phase space as an SU(2) generalization of the usual U(1) vortices. Rotating spin-orbit coupled condensates exhibit rich vortex structures due to the interplay between vorticity and spin texture. In the Mott-insulating states in optical lattices, quantum magnetism is characterized by the Dzyaloshinskii-Moriya type exchange interactions.

Disentangling satellite galaxy populations using orbit tracking in simulations

Abstract: Physical processes regulating star formation in satellite galaxies represent an area of ongoing research, but the projected nature of observed coordinates makes separating different populations of satellites (with different processes at work) difficult. The orbital history of a satellite galaxy leads to its present-day phase space coordinates; we can also work backwards and use these coordinates to statistically infer information about the orbital history. We use merger trees from the MultiDark Run 1 N-body simulation to compile a catalog of the orbits of satellite haloes in cluster environments. We parameterize the orbital history by the time since crossing within 2.5 rvir of the cluster centre and use our catalog to estimate the probability density over a range of this parameter given a set of present-day projected (i.e. observable) phase space coordinates. We show that different populations of satellite haloes, e.g. infalling, backsplash and virialized, occupy distinct regions of phase space, and semi-distinct regions of projected phase space. This will allow us to probabilistically determine the time since infall of a large sample of observed satellite galaxies, and ultimately to study the effect of orbital history on star formation history (the topic of a future paper). We test the accuracy of our method and find that we can reliably recover this time within +/-2.58 Gyr in 68 per cent of cases by using all available phase space coordinate information, compared to +/-2.64 Gyr using only position coordinates and +/-3.10 Gyr guessing 'blindly', i.e. using no coordinate information, but with knowledge of the overall distribution of infall times. In some regions of phase space, the accuracy of the infall time estimate improves to +/-1.85 Gyr. Although we focus on time since infall, our method is easily generalizable to other orbital parameters (e.g. pericentric distance and time).

HW/HZ + 0 and 1 jet at NLO with the POWHEG BOX interfaced to GoSam and their merging within MiNLO

Abstract: We present a generator for the production of a Higgs boson H in association with a vector boson V=W or Z (including subsequent V decay) plus zero and one jet, that can be used in conjunction with general-purpose shower Monte Carlo generators, according to the POWHEG method, as implemented within the POWHEG BOX framework. We have computed the virtual corrections using GoSam, a program for the automatic construction of virtual amplitudes. In order to do so, we have built a general interface of the POWHEG BOX to the GoSam package. With this addition, the construction of a POWHEG generator within the POWHEG BOX is now fully automatized, except for the construction of the Born phase space. Our HV + 1 jet generators can be run with the recently proposed MiNLO method for the choice of scales and the inclusion of Sudakov form factors. Since the HV production is very similar to V production, we were able to apply an improved MiNLO procedure, that was recently used in H and V production, also in the present case. This procedure is such that the resulting generator achieves NLO accuracy not only for inclusive distributions in HV + 1 jet production but also in HV production, i.e. when the associated jet is not resolved, yielding a further example of matched calculation with no matching scale.

Toward a theory of astrophysical plasma turbulence at subproton scales

Abstract: We present an analytical study of subproton electromagnetic fluctuations in a collisionless plasma with a plasma beta of the order of unity. In the linear limit, a rigorous derivation from the kinetic equation is conducted focusing on the role and physical properties of kinetic-Alfvand whistler waves. Then, nonlinear fluid-like equations for kinetic-Alfvwaves and whistler modes are derived, with special emphasis on the similarities and differences in the corresponding plasma dynamics. The kinetic-Alfvmodes exist in the lower-frequency region of phase space, ω � k⊥v Ti , where they are described by the kinetic-Alfvsystem. These modes exist both below and above the ion-cyclotron frequency. The whistler modes, which are qualitatively different from the kinetic-Alfv´ modes, occupy a different region of phase space, k⊥v Ti � ω � kzv Te , and they are described by the electron magnetohydrodynamics (MHD) system or the reduced electron MHD system if the propagation is oblique. Here, kz and k⊥ are the wavenumbers along and transverse to the background magnetic field, respectively, and v Ti and v Te are the ion and electron thermal velocities, respectively. The models of subproton plasma turbulence are discussed and the results of numerical simulations are presented. We also point out possible implications for solar-wind observations.

Unconventional states of bosons with the synthetic spin–orbit coupling

Abstract: The spin–orbit coupling with bosons gives rise to novel properties that are absent in usual bosonic systems. Under very general conditions, the conventional ground state wavefunctions of bosons are constrained by the ‘no-node’ theorem to be positive definite. In contrast, the linear dependence of the spin–orbit coupling leads to complex-valued condensate wavefunctions beyond this theorem. In this paper, we review the study of this class of unconventional Bose–Einstein condensations focusing on their topological properties. Both the 2D Rashba and 3D σ · p-type Weyl spin–orbit couplings give rise to Landau-level-like quantization of single-particle levels in the harmonic trap. Interacting condensates develop the half-quantum vortex structure spontaneously breaking the time-reversal symmetry and exhibit topological spin textures of the skyrmion type. In particular, the 3D Weyl coupling generates topological defects in the quaternionic phase space as an SU(2) generalization of the usual U(1) vortices. Rotating spin–orbit-coupled condensates exhibit rich vortex structures due to the interplay between vorticity and spin texture. In the Mott-insulating states in optical lattices, quantum magnetism is characterized by the Dzyaloshinskii–Moriya-type exchange interactions. (Some figures may appear in colour only in the online journal)

Optimal estimation of joint parameters in phase space

Abstract: We address the joint estimation of the two defining parameters of a displacement operation in phase space. In a measurement scheme based on a Gaussian probe field and two homodyne detectors, it is shown that both conjugated parameters can be measured below the standard quantum limit when the probe field is entangled. We derive the most informative Cram\'er-Rao bound, providing the theoretical benchmark on the estimation, and observe that our scheme is nearly optimal for a wide parameter range characterizing the probe field. We discuss the role of the entanglement as well as the relation between our measurement strategy and the generalized uncertainty relations.

Dynamics of a few corotating vortices in Bose-Einstein condensates.

Abstract: We study the dynamics of small vortex clusters with a few (2--4) corotating vortices in Bose-Einstein condensates by means of experiments, numerical computations, and theoretical analysis. All of these approaches corroborate the counterintuitive presence of a dynamical instability of symmetric vortex configurations. The instability arises as a pitchfork bifurcation at sufficiently large values of the vortex system angular momentum that induces the emergence and stabilization of asymmetric rotating vortex configurations. The latter are quantified in the theoretical model and observed in the experiments. The dynamics is explored both for the integrable two-vortex particle system, where a reduction of the phase space of the system provides valuable insight, as well as for the nonintegrable three- (or more) vortex case, which additionally admits the possibility of chaotic trajectories.

A new approach to simulating collisionless dark matter fluids

Abstract: Recently, we have shown how current cosmological N-body codes already follow the fine grained phase-space information of the dark matter fluid. Using a tetrahedral tesselation of the three-dimensional manifold that describes perfectly cold fluids in six-dimensional phase space, the phase-space distribution function can be followed throughout the simulation. This allows one to project the distribution function into configuration space to obtain highly accurate densities, velocities, and velocity dispersions. Here, we exploit this technique to show first steps on how to devise an improved particle-mesh technique. At its heart, the new method thus relies on a piecewise linear approximation of the phase space distribution function rather than the usual particle discretisation. We use pseudo-particles that approximate the masses of the tetrahedral cells up to quadrupolar order as the locations for cloud-in-cell (CIC) deposit instead of the particle locations themselves as in standard CIC deposit. We demonstrate that this modification already gives much improved stability and more accurate dynamics of the collisionless dark matter fluid at high force and low mass resolution. We demonstrate the validity and advantages of this method with various test problems as well as hot/warm-dark matter simulations which have been known to exhibit artificial fragmentation. This completely unphysical behaviour is much reduced in the new approach. The current limitations of our approach are discussed in detail and future improvements are outlined.

A method for revealing phase space structures of general time dependent dynamical systems

Abstract: In this paper we develop new techniques for revealing geometrical structures in phase space that are valid for aperiodically time dependent dynamical systems, which we refer to as Lagrangian descriptors. These quantities are based on the integration, for a finite time, along trajectories of an intrinsic bounded, positive geometrical and/or physical property of the trajectory itself. We discuss a general methodology for constructing Lagrangian descriptors, and we discuss a “heuristic argument” that explains why this method is successful for revealing geometrical structures in the phase space of a dynamical system. We support this argument by explicit calculations on a benchmark problem having a hyperbolic fixed point with stable and unstable manifolds that are known analytically. Several other benchmark examples are considered that allow us the assess the performance of Lagrangian descriptors in revealing invariant tori and regions of shear. Throughout the paper “side-by-side” comparisons of the performance of Lagrangian descriptors with both finite time Lyapunov exponents (FTLEs) and finite time averages of certain components of the vector field (“time averages”) are carried out and discussed. In all cases Lagrangian descriptors are shown to be both more accurate and computationally efficient than these methods. We also perform computations for an explicitly three dimensional, aperiodically time-dependent vector field and an aperiodically time dependent vector field defined as a data set. Comparisons with FTLEs and time averages for these examples are also carried out, with similar conclusions as for the benchmark examples.

When matter matters

Abstract: We study a recently proposed scenario for the early universe:Subluminal Galilean Genesis. We prove that without any other matter present in the spatially flat Friedmann universe, the perturbations of the Galileon scalar field propagate with a speed at most equal to the speed of light. This proof applies to all cosmological solutions — to the whole phase space. However, in a more realistic situation, when one includes any matter which is not directly coupled to the Galileon, there always exists a region of phase space where these perturbations propagate superluminally, indeed with arbitrarily high speed. We illustrate our analytic proof with numerical computations. We discuss the implications of this result for the possible UV completion of the model.

Gauss-Bonnet coupling constant as a free thermodynamical variable and the associated criticality

Abstract: The thermodynamic phase space of Gauss-Bonnet (GB) AdS black holes is extended, taking the inverse of the GB coupling constant as a new thermodynamic pressure $P_{\mathrm{GB}}$. We studied the critical behavior associated with $P_{\mathrm{GB}}$ in the extended thermodynamic phase space at fixed cosmological constant and electric charge. The result shows that when the black holes are neutral, the associated critical points can only exist in five dimensional GB-AdS black holes with spherical topology, and the corresponding critical exponents are identical to those for Van der Waals system. For charged GB-AdS black holes, it is shown that there can be only one critical point in five dimensions (for black holes with either spherical or hyperbolic topologies), which also requires the electric charge to be bounded within some appropriate range; while in $d>5$ dimensions, there can be up to two different critical points at the same electric charge, and the phase transition can occur only at temperatures which are not in between the two critical values.

Free motion around black holes with discs or rings: between integrability and chaos - I

Abstract: We continue the study of time-like geodesic dynamics in exact static, axially and reflection symmetric space–times describing the fields of a Schwarzschild black hole surrounded by thin discs or rings. In the previous paper, the rise (and decline) of geodesic chaos with ring/disc mass and position and with test particle energy was revealed on Poincare sections, on time series of position or velocity and their power spectra, and on time evolution of the orbital ‘latitudinal action’. In agreement with the KAM theory of nearly integrable dynamical systems and with the results observed in similar gravitational systems in the literature, we found orbits of very different degrees of chaoticity in the phase space of perturbed fields. Here we compare selected orbits in more detail and try to classify them according to the characteristics of the corresponding phase-variable time series, mainly according to the shape of the time-series power spectra, and also applying two recurrence methods: the method of ‘average directional vectors’, which traces the directions in which the trajectory (recurrently) passes through a chosen phase-space cell, and the ‘recurrence-matrix’ method, which consists of statistics over the recurrences themselves. All the methods proved simple and powerful, while it is interesting to observe how they differ in sensitivity to certain types of behaviour.

Phase space crystals: a new way to create a quasienergy band structure.

Abstract: A novel way to create a band structure of the quasienergy spectrum for driven systems is proposed based on the discrete symmetry in phase space. The system, e.g., an ion or ultracold atom trapped in a potential, shows no spatial periodicity, but it is driven by a time-dependent field coupling highly nonlinearly to one of its degrees of freedom (e.g., $\ensuremath{\sim}{q}^{n}$). The band structure in quasienergy arises as a consequence of the $n$-fold discrete periodicity in phase space induced by this driving field. We propose an explicit model to realize such a phase space crystal and analyze its band structure in the frame of a tight-binding approximation. The phase space crystal opens new ways to engineer energy band structures, with the added advantage that its properties can be changed in situ by tuning the driving field's parameters.

3-Cocycles, Non-Associative Star-Products and the Magnetic Paradigm of R-Flux String Vacua

Abstract: We consider the geometric and non-geometric faces of closed string vacua arising by T-duality from principal torus bundles with constant H-flux and pay attention to their double phase space description encompassing all toroidal coordinates, momenta and their dual on equal footing. We construct a star-product algebra on functions in phase space that is manifestly duality invariant and substitutes for canonical quantization. The 3-cocycles of the Abelian group of translations in double phase space are seen to account for non-associativity of the star-product. We also provide alternative cohomological descriptions of non-associativity and draw analogies with the quantization of point-particles in the field of a Dirac monopole or other distributions of magnetic charge. The magnetic field analogue of the R-flux string model is provided by a constant uniform distribution of magnetic charge in space and non-associativity manifests as breaking of angular symmetry. The Poincare vector comes to rescue angular symmetry as well as associativity and also allow for quantization in terms of operators and Hilbert space only in the case of charged particles moving in the field of a single magnetic monopole.

The Loss-cone Problem in Axisymmetric Nuclei

Abstract: We consider the problem of consumption of stars by a supermassive black hole (SBH) at the center of an axisymmetric galaxy. Inside the SBH sphere of influence, motion of stars in the mean field is regular and can be described analytically in terms of three integrals of motion: the energy E, the z-component of angular momentum Lz , and the secular Hamiltonian H. There exist two classes of orbits, tubes and saucers; saucers occupy the low-angular-momentum parts of phase space and their fraction is proportional to the degree of flattening of the nucleus. Perturbations due to gravitational encounters lead to diffusion of stars in integral space, which can be described using the Fokker-Planck equation. We calculate the diffusion coefficients and solve this equation in the two-dimensional phase space (Lz , H), for various values of the capture radius and the degree of flattening. Capture rates are found to be modestly higher than in the spherical case, up to a factor of a few, and most captures take place from saucer orbits. We also carry out a set of collisional N-body simulations to confirm the predictions of the Fokker-Planck models. We discuss the implications of our results for rates of tidal disruption and capture in the Milky Way and external galaxies.

When Matter Matters

Abstract: We study a recently proposed scenario for the early universe: Subluminal Galilean Genesis. We prove that without any other matter present in the spatially flat Friedmann universe, the perturbations of the Galileon scalar field propagate with a speed at most equal to the speed of light. This proof applies to all cosmological solutions -- to the whole phase space. However, in a more realistic situation, when one includes any matter which is not directly coupled to the Galileon, there always exists a region of phase space where these perturbations propagate superluminally, indeed with arbitrarily high speed. We illustrate our analytic proof with numerical computations. We discuss the implications of this result for the possible UV completion of the model.

Attractor solutions in scalar-field cosmology

Abstract: Models of cosmological scalar fields often feature “attractor solutions” to which the system evolves for a wide range of initial conditions. There is some tension between this well-known fact and another well-known fact: Liouville’s theorem forbids true attractor behavior in a Hamiltonian system. In universes with vanishing spatial curvature, the field variables ϕ and ϕ˙ specify the system completely, defining an effective phase space. We investigate whether one can define a unique conserved measure on this effective phase space, showing that it exists for m^2ϕ^2 potentials and deriving conditions for its existence in more general theories. We show that apparent attractors are places where this conserved measure diverges in the ϕ-ϕ˙ variables and suggest a physical understanding of attractor behavior that is compatible with Liouville’s theorem.

The Dynamics of Merging Clusters: A Monte Carlo Solution Applied to the Bullet and Musket Ball Clusters

Abstract: Merging galaxy clusters have become one of the most important probes of dark matter, providing evidence for dark matter over modified gravity and even constraints on the dark matter self-interaction cross-section. To properly constrain the dark matter cross-section it is necessary to understand the dynamics of the merger, as the inferred cross-section is a function of both the velocity of the collision and the observed time since collision. While the best understanding of merging system dynamics comes from N-body simulations, these are computationally intensive and often explore only a limited volume of the merger phase space allowed by observed parameter uncertainty. Simple analytic models exist but the assumptions of these methods invalidate their results near the collision time, plus error propagation of the highly correlated merger parameters is unfeasible. To address these weaknesses I develop a Monte Carlo method to discern the properties of dissociative mergers and propagate the uncertainty of the measured cluster parameters in an accurate and Bayesian manner. I introduce this method, verify it against an existing hydrodynamic N-body simulation, and apply it to two known dissociative mergers: 1ES 0657-558 (Bullet Cluster) and DLSCL J0916.2+2951 (Musket Ball Cluster). I find that this method surpasses existing analytic models—providing accurate (10% level) dynamic parameter and uncertainty estimates throughout the merger history. This, coupled with minimal required a priori information (subcluster mass, redshift, and projected separation) and relatively fast computation (~6 CPU hours), makes this method ideal for large samples of dissociative merging clusters.

Cosmological dynamics in f(R) gravity

Abstract: eld description of the theory, which we believe is a more intuitive way of treating the problem. In order to study how the physical solutions evolve in f(R) cosmology, we explore the cosmological dynamics of a range of f(R) models, including models that yield a large hierarchy of scales and are singularity free. We present generic features of the phase-space dynamics in f(R) cosmology. We study the global structure of the phase space in f(R) gravity by compactifying the innite phase space into a nite space via the Poincar e transformation. On the expansion branch of the phase space, the constraint surface has a repeller and a de Sitter attractor; while on the contraction branch, the constraint surface has an attractor and a de Sitter repeller. Generally, the phase currents originate from the repeller and terminate at the corresponding attractor in each space. The trajectories between the repeller and the attractor in the presence of matter density are dierent from those in the vacuum case. The phase analysis techniques developed in this paper are very general, and can be applied to other similar dynamical systems.

Constraint analysis for variational discrete systems

Abstract: A canonical formalism and constraint analysis for discrete systems subject to a variational action principle are devised. The formalism is equivalent to the covariant formulation, encompasses global and local discrete time evolution moves and naturally incorporates both constant and evolving phase spaces, the latter of which is necessary for a time varying discretization. The different roles of constraints in the discrete and the conditions under which they are first or second class and/or symmetry generators are clarified. The (non-) preservation of constraints and the symplectic structure is discussed; on evolving phase spaces the number of constraints at a fixed time step depends on the initial and final time step of evolution. Moreover, the definition of observables and a reduced phase space is provided; again, on evolving phase spaces the notion of an observable as a propagating degree of freedom requires specification of an initial and final step and crucially depends on this choice, in contrast to the continuum. However, upon restriction to translation invariant systems, one regains the usual time step independence of canonical concepts. This analysis applies, e.g., to discrete mechanics, lattice field theory, quantum gravity models and numerical analysis.

Constraint analysis for variational discrete systems

Abstract: A canonical formalism and constraint analysis for discrete systems subject to a variational action principle are devised. The formalism is equivalent to the covariant formulation, encompasses global and local discrete time evolution moves and naturally incorporates both constant and evolving phase spaces, the latter of which is necessary for a time varying discretization. The different roles of constraints in the discrete and the conditions under which they are first or second class and/or symmetry generators are clarified. The (non-) preservation of constraints and the symplectic structure is discussed; on evolving phase spaces the number of constraints at a fixed time step depends on the initial and final time step of evolution. Moreover, the definition of observables and a reduced phase space is provided; again, on evolving phase spaces the notion of an observable as a propagating degree of freedom requires specification of an initial and final step and crucially depends on this choice, in contrast to the continuum. However, upon restriction to translation invariant systems, one regains the usual time step independence of canonical concepts. This analysis applies, e.g., to discrete mechanics, lattice field theory, quantum gravity models, and numerical analysis.

Phase space formalism for quantum estimation of Gaussian states

Abstract: We formulate, with full generality, the asymptotic estimation theory for Gaussian states in terms of their first and second moments. By expressing the quantum Fisher information (QFI) and the elusive symmetric logarithmic derivative (SLD) in terms of the state's moments (and their derivatives) we are able to obtain the noncommutative extension of the well known expression for the Fisher information of a Gaussian probability distribution. Focusing on models with fixed first moments and identical Williamson 'diagonal' states --which include pure state models--, we obtain their SLD and QFI, and elucidate what features of the Wigner function are fundamentally accessible, and at what rates. In addition, we find the optimal homodyne detection scheme for all such models, and show that for pure state models they attain the fundamental limit.

Mathematical theory of Lyapunov exponents

Abstract: This paper reviews some basic mathematical results on Lyapunov exponents, one of the most fundamental concepts in dynamical systems. The first few sectionscontainsomeverygeneralresultsinnonuniformhyperbolictheory.We consider (f ,μ ),where f is an arbitrary dynamical system and μ is an arbitrary invariant measure, and discuss relations between Lyapunov exponents and several dynamical quantities of interest, including entropy, fractal dimension and rates of escape. The second half of this review focuses on observable chaos, characterized by positive Lyapunov exponents on positive Lebesgue measure sets. Much attention is given to SRB measures, a very special kind of invariantmeasuresthatofferawaytounderstandobservablechaosindissipative systems. Paradoxical as it may seem, given a concrete system, it is generally impossible to determine with mathematical certainty if it has observable chaos unless strong geometric conditions are satisfied; case studies will be discussed. The final section is on noisy or stochastically perturbed systems, for which we present a dynamical picture simpler than that for purely deterministic systems. In this short review, we have elected to limit ourselves to finite-dimensional systems and to discrete time. The phase space, which is assumed to be R d or a Riemannian manifold, is denoted by M throughout. The Lebesgue or the Riemannian measure on M is denoted by m, and the dynamics are generated by iterating a self-map of M, written f : M . For flows, the reviewed results