Phase Space Geometry Research Articles

Optimal switching between collective motion states for two agents

We show that steering control can be chosen to give bistability between parallel and anti-parallel collective motion states for a continuous-time kinetic model of two agents moving in the plane with unit speed. Variational methods are used to determine the optimal input to the steering control of one of the agents which leads to switching between these collective states. For any given time interval of switching, such an optimal input is shown to exist and to be unique. The properties of optimal inputs are interpreted by considering the phase space geometry of the Euler–Lagrange equations associated with the optimization.

Nonlinear Geometric Estimation for Satellite Attitude

A novel algorithm for efficient high-accuracy satellite attitude estimation is presented to address the increasing performance requirements of resource-constrained small satellites. The algorithm results from an investigation of the Bayesian nonlinear estimation problem based on the phase-space geometry of Hamiltonian systems. Probability density functions are shown to be conserved properties of deterministic Hamiltonian systems, and appropriate geometric integrators exactly preserve the functions as they evolve in time. Based on these insights, a new iterative filter is derived that conserves the geometric structure and invariant properties of the dynamics and exactly preserves the nonlinear a posteriori probability density function when solving for the state estimate. Comparisons with a benchmark iterative filter demonstrate significantly reduced computational burden, improved state estimation accuracy, and improved constants of motion estimation, particularly in the presence of high nonlinearity and low noise inputs. Based on numerous simulations, the authors conclude that this new method shows promise for improved attitude estimation onboard high-performance, resource-constrained small satellites.

Fuzzy geometry of phase space and quantization of massive fields

The quantum spacetime and the phase space with fuzzy structure are investigated as the possible quantization formalism. In this theory, the state of the nonrelativistic particle m corresponds to the element of fuzzy ordered set (Foset), i.e. the fuzzy point. Due to Foset partial (weak) ordering, the m space coordinate x acquires principal uncertainty σx. It is shown that Schrödinger formalism of quantum mechanics can be completely derived from consideration of m evolution in fuzzy phase space with minimal number of axioms.

Geometry of phase space and solutions of semilinear elliptic equations in a ball

We consider the problem {-Δu=u p +λu in in B, u>0 in B, u=0 on ∂B, where B denotes the unit ball in R N , N > 3, A > 0 and p > 1. Merle and Peletier showed that for p > N+2/N-2 there is a unique value A = λ* > 0 such that a radial singular solution exists. This value is the only one at which an unbounded sequence of classical solutions of (1) may accumulate. Here we prove that if additionally p 0 0 in B, u = 0 on ∂B, where f(s) = s p + s q , 1 < q < p, and p satisfies the same condition as above.

On the geometry and entropy of non-Hamiltonian phase space

We analyse the equilibrium statistical mechanics of canonical, non-canonical andnon-Hamiltonian equations of motion, throwing light on the peculiar geometric structure ofphase space. Some fundamental issues regarding time translation and phase space measureare clarified. In particular, we emphasize that a phase space measure should be defined bymeans of the Jacobian of the transformation between different kinds of coordinates sincesuch a determinant is different from zero in the non-canonical case even if thephase space compressibility is null. Instead, the Jacobian determinant associatedwith phase space flows is unity whenever non-canonical coordinates lead to avanishing compressibility, so its use for defining a measure may not always be correct.To better illustrate this point, we derive a mathematical condition for definingnon-Hamiltonian phase space flows with zero compressibility. The Jacobian determinantassociated with the time evolution in phase space is very useful for analysing timetranslation invariance. The proper definition of a phase space measure is particularlyimportant when defining the entropy functional in the canonical, non-canonical, andnon-Hamiltonian cases. We show how the use of relative entropies can circumvent somesubtle problems that are encountered when dealing with continuous probabilitydistributions and phase space measures. Finally, a maximum (relative) entropyprinciple is formulated for non-canonical and non-Hamiltonian phase space flows.

Dynamical Hierarchy in Transition States of Reactions

We present a partial normalization procedure of Lie canonical perturbation theory to elucidate the phase space geometry of the transition state in the multidimensional phase space for a wide range of energy above the threshold. State selectivity and dynamical correlation along the evolution of reactions will also be discussed.

Born Reciprocity and the Granularity of Spacetime

The Schr\"odinger-Robertson inequality for relativistic position and momentum operators X^\mu, P_\nu, \mu, \nu = 0,1,2,3, is interpreted in terms of Born reciprocity and `non-commutative' relativistic phase space geometry. For states which saturate the Schr\"odinger-Robertson inequality, a typology of semiclassical limits is pointed out, characterised by the orbit structure within its unitary irreducible representations, of the full invariance group of Born reciprocity, the so-called `quaplectic' group U(3,1)xH(3,1) (the semi-direct product of the unitary relativistic dyamical symmetry U(3,1) with the Weyl-Heisenberg group H(3,1)). The example of the `scalar' case, namely the relativistic oscillator, and associated multimode squeezed states, is treated in detail. In this case,it is suggested that the semiclassical limit corresponds to the separate emergence of space-time and matter, in the form of the stress-energy tensor, and the quadrupole tensor, which are in general reciprocally equivalent.

Travelling-wave analysis of a model of the immune response to cancer

In this paper we present a travelling-wave analysis of a mathematical model describing the growth of a solid tumour in the presence of an immune system response. From a modelling perspective, attention is focused upon the attack of tumour cells by tumour infiltrating cytotoxic lymphocytes (TICLs), in a small multicellular tumour, without necrosis and at some stage prior to (tumour-induced) angiogenesis. As we have shown in previous work, for a particular choice of parameters, the underlying reaction–diffusion–chemotaxis system of partial differential equations is able to simulate the well-documented phenomenon of cancer dormancy by depicting spatially heterogeneous tumour cell distributions that are characterized by a relatively small total number of tumour cells. This behaviour is consistent with several immunomorphological investigations. Moreover, the alteration of certain parameters of the model is enough to induce bifurcations into the system, which in turn result in tumour invasion in the form of a standard travelling wave. The work presented in this paper complements the bifurcation analysis undertaken by Matzavinos et al. [Math. Med. Biol. IMA 21 (2004) 1–34] and establishes the existence of travelling-wave solutions for the system under discussion by promoting the understanding of the geometry of an appropriate phase space. To cite this article: A. Matzavinos, M.A.J. Chaplain, C. R. Biologies 327 (2004).

Starting, stopping, and resetting biological oscillators: in search of optimum perturbations

Biological oscillators are commonly subjected to a single brief stimulus to perturb the ongoing rhythm. After stimulation, the oscillator can recover although its phase may be advanced or delayed relative to the original cycle. A single transient perturbation can also stop the ongoing rhythm. Arthur Winfree systematically classified these responses to brief perturbations, as determined by the strength of the stimulus, and the phase within the cycle at which the stimulus was given. A natural question arises from Winfree's work. Are certain stimulus shapes better than others at achieving a desired effect? The present study explores this question using: (1) analysis of phase space geometry, (2) calculus of variations, and (3) analysis of responses to noisy perturbations. Methods 1 and 2 can yield exact solutions, but have limited applicability in biology because they require a detailed mathematical description of the oscillator. Method 3 is applicable to any oscillator without mathematical prerequisites. We validate this method by finding optimum stimuli that start and stop repetitive firing in a neural pacemaker model, and the optimum light stimulus for resetting the circadian rhythm in a model of the human circadian clock. We propose that the elucidation of optimum stimulus shapes may be useful for studying many periodic phenomena in biology and medicine. Optimum stimuli can be used to induce a desired behavior without producing undesirable side effects.

Thermodynamical relations for systems in contact with finite heat baths

We obtain the relations between macroscopic thermodynamical properties and microscopic ensemble averages of a system in contact with a heat bath which has a finite number of degrees of freedom. Our approach is based on the geometry of the accessible phase space of the system. By using a suitable definition of temperature for such kind of system (energy equipartition theorem), it is shown that the relations obtained here have the same dependence on the system's partition function Z 1 as in Boltzmann–Gibbs statistics.

Quantum-mechanical dualities from classical phase space

The geometry of the classical phase space C of a finite number of degrees of freedom determines the possible duality symmetries of the corresponding quantum mechanics. Under duality we understand the relativity of the notion of a quantum with respect to an observer on C . We illustrate this property explicitly in the case when classical phase space is complex n-dimensional projective space CP n . We also provide some examples of classical dynamics on CP n that exhibit these properties at the quantum level.

Twisted Einstein Tensors and Orbifold Geometry

Following recent advances in the local theory of current-algebraic orbifolds, we study various geometric properties of the general WZW orbifold, the general coset orbifold and a large class of (nonlinear) sigma model orbifolds. Phase-space geometry is emphasized for the WZW orbifolds — while for the sigma model orbifolds we construct the corresponding sigma model orbifold action, which includes the previously-known general WZW orbifold action and general coset orbifold action as special cases. We focus throughout on the twisted Einstein tensors with diagonal monodromy, including the twisted Einstein metric, the twisted B field and the twisted torsion field of each orbifold sector. Finally, we present strong evidence for a conjectured set of twisted Einstein equations which should describe those sigma model orbifolds in this class which are also one-loop conformal.

The role of phase space geometry in Heisenberg’s uncertainty relation

Aiming towards a geometric description of quantum theory, we study the coherent states-induced metric on the phase space, which provides a geometric formulation of the Heisenberg uncertainty relations (both the position–momentum and the time–energy ones). The metric also distinguishes the original uncertainty relations of Heisenberg from the ones that are obtained from non-commutativity of operators. Conversely, the uncertainty relations can be written in terms of this metric only, hence they can be formulated for any physical system, including ones with non-trivial phase space. Moreover, the metric is a key ingredient of the probability structure of continuous-time histories on phase space. This fact allows a simple new proof the impossibility of the physical manifestation of the quantum Zeno and anti-Zeno paradoxes. Finally, we construct the coherent states for a spinless relativistic particle, as a non-trivial example by which we demonstrate our results.

Non-Noether symmetries and their influence on phase space geometry

In the present paper some geometric aspects of the concept of non-Noether symmetry are discussed. It is shown that in regular Hamiltonian systems such a symmetry canonically leads to a Lax pair on the algebra of linear operators on cotangent bundle over the phase space. Correspondence between the non-Noether symmetries and other wide spread geometric methods of generating conservation laws such as bi-Hamiltonian formalism, bidifferential calculi and Frölicher–Nijenhuis geometry is considered. It is proved that the integrals of motion associated with the continuous non-Noether symmetry are in involution whenever the generator of the symmetry satisfies a certain Yang–Baxter type equation.

Metric and curvature in gravitational phase space

At a fixed point in spacetime (say, x0), gravitational phase space consists of the space of symmetric matrices {Fab} (corresponding to the canonical momentum πab(x0)) and of symmetric matrices {Gab} (corresponding to the canonical metric gab(x0)), where 1 ⩽ a, b ⩽ n, and, crucially, the matrix {Gab} is necessarily positive definite, i.e. ∑uaGabub > 0 whenever ∑(ua)2 > 0. In an alternative quantization procedure, known as metrical quantization, the first and most important ingredient is the specification of a suitable metric on classical phase space. Our choice of phase-space metrics, guided by a recent study of affine quantum gravity, leads to gravitational phase-space geometries which possess constant scalar curvature and may be regarded as higher-dimensional analogues of the Poincaré plane, which applies when n = 1. This result is important because phase spaces endowed with such symmetry lead naturally, via the procedures of metrical quantization, to acceptable Hilbert spaces of high dimension.

Hamiltonian mappings and circle packing phase spaces

We introduce three area preserving maps with phase space structures which resemble circle packings. Each mapping is derived from a kicked Hamiltonian system with one of the three different phase space geometries (planar, hyperbolic or spherical) and exhibits an infinite number of coexisting stable periodic orbits which appear to ‘pack’ the phase space with circular resonances.

Encoding a qubit in an oscillator

Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry of phase space to protect against errors that shift the values of the canonical variables q and p. In the setting of quantum optics, fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting; however, nonlinear mode coupling is required for the preparation of the encoded states. Finite-dimensional versions of these codes can be constructed that protect encoded quantum information against shifts in the amplitude or phase of a d-state system. Continuous-variable codes can be invoked to establish lower bounds on the quantum capacity of Gaussian quantum channels.

Mel’nikov analysis of a symmetry-breaking perturbation of the NLS equation

The effects of loss of symmetry due to noneven initial conditions on the chaotic dynamics of a Hamiltonian perturbation of the nonlinear Schrödinger equation (NLS) were first numerically studied in [Phys. A 228 (1996)], where it was observed that temporally irregular evolutions can occur even in the absence of homoclinic crossings. In this article, we introduce a symmetry-breaking damped-driven perturbation of the NLS equation in order to develop a Mel’nikov analysis of the noneven chaotic regime. We obtain the following results: (1) spatial symmetry breaking within the chaotic regime causes the wave form to exhibit a more complex dynamics than in [Phys. A 228 (1996)]: center-wing jumping (which characterizes the even chaotic dynamics) about a shifted lattice site alternates (at random times) with the occurrence of modulated traveling wave solutions whose velocity changes sign in a temporally random fashion; (2) we give a heuristic description of the geometry of the full phase space (with no evenness imposed) and compute Mel’nikov type measurements in terms of the complex gradient of the Floquet discriminant. The Mel’nikov analysis yields explicit conditions for the onset of chaotic dynamics which are consistent with the numerical observations; in particular, the imaginary part of the Mel’nikov integral appears to be correlated with purely noneven features of the chaotic wave form.

Geometric phases for corotating elliptical vortex patches

We describe a geometric phase that arises when two elliptical vortex patches corotate. Using the Hamiltonian moment model of Melander, Zabusky, and Styczek [J. Fluid Mech. 167, 95–115 (1986)] we consider two corotating uniform elliptical patches evolving according to the second order truncated equations of the model. The phase is computed in the adiabatic setting of a slowly varying Hamiltonian as in the work of Hannay [J. Phys. A 18, 221–230 (1985)] and Berry [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We also discuss the geometry of the symplectic phase space of the model in the context of nonadiabatic phases. The adiabatic phase appears in the orientation angle of each patch—it is similiar in form and is calculated using a multiscale perturbation procedure as in the point vortex configuration of Newton [Physica D 79, 416–423 (1994)] and Shashikanth and Newton [J. Nonlinear Sci. 8, 183–214 (1998)], however, an extra factor due to the internal stucture of the patch is present. The final result depends on the initial orientation of the patches unlike the phases in the works of Hannay and Berry [J. Phys. A 18, 221–230 (1985)]; [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We then show that the adiabatic phase can be interpreted as the holonomy of a connection on the trivial principal fiber bundle π:T2×S1→S1, where T2 is identified with the product of the momentum level sets of two Kirchhoff vortex patches and S1 is diffeomorphic to the momentum level set of two point vortex motion. This two point vortex motion is the motion that the patch centroids approach in the adiabatic limit.

Phase space geometry and stochasticity thresholds in hamiltonian dynamics

Results of numerical computations of the largest Lyapunov exponent lambda(1)(varepsilon,N) as a function of the energy density varepsilon and the number of particles N are here reported for a Fermi-Pasta-Ulam alpha+beta model. These results show the coexistence at large N of two thresholds: a stochasticity threshold, found before for the alpha model alone, and a strong stochasticity threshold (SST), found before for the beta model alone. Although this coexistence may seem at first sight plausible, it is not obvious a priori that the alpha+beta model superimposes properties of the alpha and beta models independently. The main point of this paper, however, is a geometric characterization of the SST via the mean curvature of the constant energy hypersurfaces in the phase space of the model and the characteristic decay time of its time autocorrelation function tau(c)(varepsilon,N), which correlates with that of lambda(1)(varepsilon,N) for fixed N. This appears to provide important information on the very complicated geometry of the phase space of this simple solidlike model.