RIEMANNIAN GEOMETRY CRITERION FOR CHAOS

IN COLLECTIVE DYNAMICS OF NUCLEI

Pavel Stránský, Pavel Cejnar

XXI Nuclear Physics Workshop, Kazimierz Dolny, Poland 26th September 2014

Institute of Particle and Nuclear PhycicsFaculty of Mathematics and PhysicsCharles University in Prague, Czech Republic

www.pavelstransky.cz

1. Curvature-based Method- flat X curved space, embedding of a Hamiltonian system into a curved space- estimating the onset of chaos based on overall geometric properties

2. Model- classical dynamics of the Geometric Collective Model (GCM) of atomic nuclei

3. Results and discussion- full map of classical chaos in the GCM- instability predicted by the curvature-based criterion- relation between the curvature-based method and the onset of chaos determined numerically

P. Stránský, P. Cejnar, Study of curvature-based criterion for chaos in Hamiltonian systems with two degrees of freedom submitted to Journal of Physics A: Mathematical and Theoretical

RIEMANNIAN GEOMETRY CRITERION FOR CHAOS

IN COLLECTIVE DYNAMICS OF NUCLEI

1. Curvature-based criterion for chaos in Hamiltonian systems

Geometrical embedding

Hamiltonian of the motion in the flat Euclidean space with a

potential:

Why embedding: •Riemannian geometry brings in the notion of curvature that could help clarify the sources of instability, and in the same time quantify the amount of chaos in non-ergodic systems

Bridge: •The equations of motion (Hamilton, Newton) correspond with the geodesic equation

L. Casetti, M. Pettini, E.D.G. Cohen, Phys. Rep. 337, 237 (2000)M. Pettini, Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics (Springer New York, 2007)

Potential

Trajectory

x

y

Hamiltonian of the free motion in a curved

space:

Geodesic

curvature effects

topological effects

Geodesics & Maps- Generalization of a straight line- Describe a ”free motion” in a curved

space- “Shortest path” between two points

Visualisation of a curved space - mapping onto the flat space

Paris -> Paris -> MexicoMexico

PraguePrague

Flat space(dynamics)

Curved space(geometry)

Potential energyTimeForcesCurvature of the potential

MetricArc-lengthChristoffel’s symbolsRiemannian tensorRicci tensorScalar curvature

TrajectoriesHamiltonian equations of motion

GeodesicsGeodesic equation

Tangent dynamics equation

Equation of the geodesic deviation (Jacobi equation)

Lyapunov exponent

Various ways of the geometric embedding1. Jacobi metric

- conformal metric

- arc-length

- nonzero scalar curvature

L. Casetti, M. Pettini, E.D.G. Cohen, Phys. Rep. 337, 237 (2000)

(negative only when V < 0)

2. Eisenhart metric- space with 2 extra dimensions

- arc-length equivalent with time- only one nonzero Christoffel’s symbol and

3. Israeli method (Horwitz et al.)

vanishing scalar curvature

L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007)

- conformal “metric”

- metric incompatible connection form

metric compatible connection

- arc-length proportional to time

Curvature and instabilityBesides solving the equation for the geodesic deviation, can one deduce something about the instability only from the curvature?

1. Riemannian tensorDifficult, the number of components grows with the 4th power of

dimension2. Scalar curvature

• R = const Equation of the geodetic deviation

Equation of motion for •harmonic oscillator with frequency•exponential divergence with Lyapunov exponent

(isotropic manifold)

• R < 0 Unstable motion with estimated Lyapunov exponent

• dim = 2

stable R > 0

unstable R > 0

Equation of motion of a harmonic oscillator with its length (stiffness) modulated in timeUnstable if the frequency is in resonance with any of the frequency of the Fourier expansion, even if R(s) > 0 on the whole manifold: Parametric

instabilityThe metric-compatible connection is required!

Curvature and instabilityBesides solving the equation for the geodesic deviation, can one deduce something about the instability only from the curvature?

3. Israeli method

Using the Israeli connection form, the equation of the geodesic deviation reads as

- projector into a direction orthogonal to the velocity

Stability matrix

Conjecture: A negative eigenvalue of the Stability matrix inside the kinematically accessible area induces instability of the motion.

L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007)

Example of unstable configuration

Kinematically accessible area

Negative lower eigenvalue of

Negative higher eigenvalue of

y

x

Properties of the stability matrix

1. When is big enough, becomes the Hessian matrix for the tangent dynamics2. Eigenvalues can only decrease within the kinematically accessible domain

The size of the negative eigenvalue region can only grow with energy, or remain the same

3. The lower eigenvalue is continuous on the boundary of the accessible domain

f = 2

condition for inflexion points of the curve

4. The lower eigenvalue is zero on the boundary when

concave

convex

The curvature-based criterion for the onset of chaos can be partly

translated into the language of the shape of the equipotential

contours.

concave potential surface - dispersing

convex potential surface - focusing

Instability thresholdScenario A - Penetration

- region of negative , which exists outside the accessible region, starts overlapping with it at some energy E- equipotential contours undergoes the convex-concave transition

Instability thresholdScenario A - Penetration

- region of negative , which exists outside the accessible region, starts overlapping with it at some energy E- equipotential contours undergoes the convex-concave transition

Instability thresholdScenario B - Creation

Scenario A - Penetration

- region of negative , which exists outside the accessible region, starts overlapping with it at some energy E

- region of negative eventually appears somewhere inside the accessible region at some energy E

- all the equipotential contours convex- equipotential contours undergoes the convex-concave transition - necessary condition

2. Model (Geometric collective model of nuclei)

Surface of homogeneous nuclear matter:

Monopole deformations = 0

(even-even nuclei – collective character of the lowest excitations)

- Does not contribute due to the incompressibility of the nuclear matter

Dipole deformations = 1- Related to the motion of the center of mass- Zero due to momentum conservation

- “breathing” mode

Geometric collective model

T…Kinetic term V…Potential

Neglect higher order terms

Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles)

Corresponding tensor of momenta

Quadrupole deformations = 2

G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969)

4 external parameters

neglect

Geometric collective model

Surface of homogeneous nuclear matter:

T…Kinetic term V…Potential

Neglect higher order terms

Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles)

Corresponding tensor of momenta

Scaling properties

4 external parametersAdjusting 3 independent scalesenergy

(Hamiltonian)

1 “shape” parameter

size (deformation)

time

1 “classicality” parametersets absolute density of quantum spectrum (irrelevant in classical case)

P. Stránský, M. Kurian, P. Cejnar, Phys. Rev. C 74, 014306 (2006)

neglect

(order parameter)

Geometric collective model

Surface of homogeneous nuclear matter:

Quadrupole deformations = 2

Principal Axes System (PAS)

Shape variables:

Shape-phase structure

Spherical ground-state shape

V

V

B

A

C=1

Phase separatrix

We focus only on the nonrotating case J = 0!

grey lines – equivalent dynamical configurations

covers all inequivalent configurations of the GCM

Deformed ground-state shape

control parameter

Hamiltonian

It describes:

Motion of a star around a galactic centre, assuming the motion is cylindrically

symmetric(Hénon-Heiles model)

Collective motion of an atomic nucleus

(Bohr model)

… but also (for example):

3. Results and discussion(Israeli geometry method applied to GCM)

Complete map of classical chaos in the GCMIntegrable Integrable limitlimit

Veins ofVeins of regularityregularity

chaotichaoticc

regularegularr

control parameter

Sh

ap

e-p

hase

Sh

ap

e-p

hase

tr

ansi

tion

transi

tion

P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 046202

““ Arc

of

re

gula

rity

”A

rc o

f

regula

rity

”

Integrable Integrable limitlimitdeformed shape

spherical shape

Saddle point / local maximum and minimum

convex-concave border transition (stability / instability)

regularegularr

(mexican hat potential)

Complete map of classical chaos in the GCMIntegrable Integrable limitlimit

Veins ofVeins of regularityregularity

chaotichaoticc

regularegularr

control parameter

Sh

ap

e-p

hase

Sh

ap

e-p

hase

tr

ansi

tion

transi

tion

P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 046202

Integrable Integrable limitlimitdeformed shape

spherical shape

Saddle point / local maximum and minimum

convex-concave border transition (stability / instability)

regularegularr

““ Arc

of

re

gula

rity

”A

rc o

f

regula

rity

”

(mexican hat potential)

Integrable Integrable limitlimit

Veins ofVeins of regularityregularity

chaotichaoticc

regularegularr

control parameter

Sh

ap

e-p

hase

Sh

ap

e-p

hase

tr

ansi

tion

transi

tion

P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 046202

““ Arc

of

re

gula

rity

”A

rc o

f

regula

rity

”

Integrable Integrable limitlimitdeformed shape

spherical shape

regularegularr

Saddle point / local maximum and minimum

convex-concave border transition (stability / instability)

Stability (Application of the curvature-based method)

Stability matrix S

Kinematically accessible area

Negative lower eigenvalue of

Negative higher eigenvalue of

Stability-instability transition, as predicted by the Israeli methodLow-energy region where the regular

harmonic approximation is valid

Stable-unstable transition according to the geometric criterion

saddle point of the potential

Local maximum of the potential – sharp minimum of regularity

concave-convex transition of the equipotential contour

“Regular vein” – strongly pronounced local maximum of regularity

(a)

(b)

(c)

(e)

(g)

(h)

The Riemannian geometry indicator gives a good estimate on the stability. However, it does not capture the full richness of the dynamics of a Hamiltonian system.

Conclusions:1. The curvature-based criterion for the onset of chaos gives a fast indicator of

stability of a Hamiltonian system without the need of solving equations of motion. In 2D systems it exactly corresponds to the convex-concave change in equipotential contours (Scenario A).

2. This criterion, although only approximate, works well in many physical systems:L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007) [2D Toda lattice]Y.B. Zion and L. Horwitz, Phys. Rev. E 76, 046220 (2007) [3D Yang-Mills system]Y.B. Zion and L. Horwitz, Phys. Rev. E 78, 036209 (2008) [2D coupled HO, 2D quartic oscillator]J. Li and S. Zhang, J. Phys. A: Math. Theor. 43, 125105 (2010) [Dicke model]A list of counterexamples (unbound systems) is given inX. Wu, J. Geom. Phys. 59, 1357 (2009).

A detailed discussion of the GCM and the Creagh-Whelan model is given in our paper submitted recently to J. Phys. A: Math. Theor.

3. The complete study of the dynamics in the GCM shows rough coincidence of the criterion and the numerically determined inset of chaos, although some deviations are observed (chaotic dynamics penetration into stable region, completely regular dynamics appearing in unstable region, instability predicted for the integrable configuration).

The Riemannian geometry indicator gives a good estimate on the stability. However, it does not capture the full richness of the dynamics of a Hamiltonian system.

THANK YOU FOR YOUR ATTENTIONAnd special thanks to all organizers of this

inspiring Nuclear Physics Workshop.

Conclusions:1. The curvature-based criterion for the onset of chaos gives a fast indicator of

stability of a Hamiltonian system without the need of solving equations of motion. In 2D systems it exactly corresponds to the convex-concave change in equipotential contours (Scenario A).

2. This criterion, although only approximate, works well in many physical systems:L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007) [2D Toda lattice]Y.B. Zion and L. Horwitz, Phys. Rev. E 76, 046220 (2007) [3D Yang-Mills system]Y.B. Zion and L. Horwitz, Phys. Rev. E 78, 036209 (2008) [2D coupled HO, 2D quartic oscillator]J. Li and S. Zhang, J. Phys. A: Math. Theor. 43, 125105 (2010) [Dicke model]A list of counterexamples (unbound systems) is given inX. Wu, J. Geom. Phys. 59, 1357 (2009).

A detailed discussion of the GCM and the Creagh-Whelan model is given in our paper submitted recently to J. Phys. A: Math. Theor.

3. The complete study of the dynamics in the GCM shows rough coincidence of the criterion and the numerically determined inset of chaos, although some deviations are observed (chaotic dynamics penetration into stable region, completely regular dynamics appearing in unstable region, instability predicted for the integrable configuration).

The Riemannian geometry indicator gives a good estimate on the stability. However, it does not capture the full richness of the dynamics of a Hamiltonian system.

THANK YOU FOR YOUR ATTENTIONAnd special thanks to all organizers of this

inspiring Nuclear Physics Workshop.

Conclusions:1. The curvature-based criterion for the onset of chaos gives a fast indicator of

stability of a Hamiltonian system without the need of solving equations of motion. In 2D systems it exactly corresponds to the convex-concave change in equipotential contours (Scenario A).

2. This criterion, although only approximate, works well in many physical systems:L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007) [2D Toda lattice]Y.B. Zion and L. Horwitz, Phys. Rev. E 76, 046220 (2007) [3D Yang-Mills system]Y.B. Zion and L. Horwitz, Phys. Rev. E 78, 036209 (2008) [2D coupled HO, 2D quartic oscillator]J. Li and S. Zhang, J. Phys. A: Math. Theor. 43, 125105 (2010) [Dicke model]A list of counterexamples (unbound systems) is given inX. Wu, J. Geom. Phys. 59, 1357 (2009).

A detailed discussion of the GCM and the Creagh-Whelan model is given in our paper submitted recently to J. Phys. A: Math. Theor.

3. The complete study of the dynamics in the GCM shows rough coincidence of the criterion and the numerically determined inset of chaos, although some deviations are observed (chaotic dynamics penetration into stable region, completely regular dynamics appearing in unstable region, instability predicted for the integrable configuration).