What is macro? self-organized structures in physical systems

What is macro? self-organized structures in physical systems Z. Yoshida U. Tokyo 2016.5.24 Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 1 / 15

Outline Self-organization without blueprints Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 2 / 15

Outline Self-organization without blueprints Creation by space (not by matter) Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 2 / 15

Outline Self-organization without blueprints Creation by space (not by matter) Topological constraints and foliation Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 2 / 15

Outline Self-organization without blueprints Creation by space (not by matter) Topological constraints and foliation Macro-hierarchy as leaves in phase space Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 2 / 15

Outline Self-organization without blueprints Creation by space (not by matter) Topological constraints and foliation Macro-hierarchy as leaves in phase space Self-organization of vortexes Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 2 / 15

Outline Self-organization without blueprints Creation by space (not by matter) Topological constraints and foliation Macro-hierarchy as leaves in phase space Self-organization of vortexes Cluster vs Vortex Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 2 / 15

Outline Self-organization without blueprints Creation by space (not by matter) Topological constraints and foliation Macro-hierarchy as leaves in phase space Self-organization of vortexes Cluster vs Vortex Self-organized confinement in magnetosphere Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 2 / 15

Outline Self-organization without blueprints Creation by space (not by matter) Topological constraints and foliation Macro-hierarchy as leaves in phase space Self-organization of vortexes Cluster vs Vortex Self-organized confinement in magnetosphere Possibility of advanced fusion Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 2 / 15

Self-organization with blueprints Figure: Hierarchical structures are programed to emerge by gene. (The picture of cell is by http://www.rkm.com.au/cell/plant/) Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 3 / 15

Self-organization without blueprints Figure: Self-organization in physical systems: Vortexes are chiral structures spontaneously created without programs. The picture of M51 spiral galaxy is by K. Okano; http://www.asahi-net.or.jp/ rt6k-okn/galaxy/m51aom.jpg Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 4 / 15

Self-organization without blueprints Figure: Self-organization in physical systems: Vortexes are chiral structures spontaneously created without programs. The picture of M51 spiral galaxy is by K. Okano; http://www.asahi-net.or.jp/ rt6k-okn/galaxy/m51aom.jpg Elements are just simple. The magic is played by space-time. Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 4 / 15

Physicist s view of pendulum Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 5 / 15

Duality of matter and space-time Why chiral? Symmetry breaking in the matter Symmetry breaking in the space Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 6 / 15

Duality of matter and space-time Why chiral? Symmetry breaking in the matter Symmetry breaking in the space Here, we put the rattle back into the perspective of skewed space, and explain the chirality as the consequence of the distorted geometry. Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 6 / 15

Duality of matter and space-time Why chiral? Symmetry breaking in the matter Symmetry breaking in the space Here, we put the rattle back into the perspective of skewed space, and explain the chirality as the consequence of the distorted geometry. We start with the Moffatt-Tokieda equation [Proc. Royal Soc. Edinburgh 138A (2008), 361]. We cast it into a Hamiltonian formalism; the underlying Lie algebra is of Bianchi Type VI. The chirality of the rattleback motion is caused by the skewed space in which the rattle back lives. Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 6 / 15

Moffatt-Tokieda equation of rattle back The governing equation of P= pitching, R= rolling, and S= spin is P R P d R = λp R. (1) dt S 0 S We assume that λ is a positive constant number. Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 7 / 15

Moffatt-Tokieda equation of rattle back The governing equation of P= pitching, R= rolling, and S= spin is P R P d R = λp R. (1) dt S 0 S We assume that λ is a positive constant number. 0.4 0.2 0.2 10 20 30 40 50 t 0.4 Figure: Typical solution of spin reversal: Reproduced from Fig. 1 of Moffatt-Tokieda (2008) Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 7 / 15

Casimir invariant and the orbits The rattle back motion is the cross-section of the energy H := 1 2 Z 2 = 1 ( P 2 + R 2 + S 2) 2 and the Casimir invariant C := PR λ. (2) Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 8 / 15

Figure: The slice of the energy contour (H = constant) by a distorted knife (the Casimir leaf) yields a skewed orbit. (ZY, T. Tokieda and J.P. Morrison, 2016 IUTAM Symposium, Venice, Italy) Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 8 / 15 Casimir invariant and the orbits The rattle back motion is the cross-section of the energy H := 1 2 Z 2 = 1 ( P 2 + R 2 + S 2) 2 and the Casimir invariant C := PR λ. (2)

What is macro? Micro = canonical Macro = noncanonical foliation = topological constraints Figure: When the dynamics is topologically constrained to a skewed leaf, the effective energy may have complex distribution, branching out variety of structures. (Fig. 1 of http://dx.doi.org/10.1080/23746149.2015.1127773) Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 9 / 15

Self-organization of vortexes Two different types of naturally-made structures: Clusters: star, nebula, Debye shield, etc. Vortexes: spiral galaxy, magnetosphere, typhoon, tornado, etc. Two-different types of interactions: Forces due to energy: gravity (Newtonian), electrostatic force, etc. Forces due to space-time: magnetic force, Coriolis force, etc. Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 10 / 15

Creation by topological constraints How can magnetic field confine a plasma, despite the fact that f e E/k BT? Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 11 / 15

Creation by topological constraints How can magnetic field confine a plasma, despite the fact that f e E/k BT? Confinement occurs on a macro hierarchy. Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 11 / 15

Creation by topological constraints How can magnetic field confine a plasma, despite the fact that f e E/k BT? Confinement occurs on a macro hierarchy. Magnetized particles = quasi-particle involving micro degree of freedom. Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 11 / 15

Creation by topological constraints How can magnetic field confine a plasma, despite the fact that f e E/k BT? Confinement occurs on a macro hierarchy. Magnetized particles = quasi-particle involving micro degree of freedom. Quasi-particles reside on a leaf of adiabatic invariants. Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 11 / 15

Boltzmann distribution on a leaf Quasi-particles reside on a leaf of adiabatic invariants. Boltzmann distribution of quasi-particles maximizes entropy with respect to the invariant measure of the symplectic leaf. Figure: (left) Boltzmann distribution on the leaf of µ (magnetic moment). (right) Boltzmann distribution on the leaf of µ and J (bounce action). (Fig. 1 of ZY & S.M. Mahajan, PETP 2014, 073J01) Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 12 / 15

Experimental proof of the self-organized magnetosphere Figure: (left) Theoretical model of planetary (Jovian) magnetosphere. (right) RT-1 laboratory magnetosphere at The University of Tokyo. (http://dx.doi.org/10.1080/23746149.2015.1127773) Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 13 / 15

Problems in the frontier Non-integrable topological constraints: singularities in Poisson algebra non-holonomic constraints Fragility of topological constraints: topological evolution turbulence Application of the self-organizing dynamics: advanced fusion (plasma confinement for D- 3He fusion energy) anti-matter plasma (e-p plasma,, etc.) Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 14 / 15

Summary Macro can be different from the simple sum (direct product) of micros, if topological constraints foliates the phase space. Macro hierarchy is identified as leaves of adiabatic invariants. The Casimir invariants of Hamiltonian systems may be regarded as adiabatic invariants, Skewness of Casimir leaves yields non-trivial structures. Z. Yoshida (U. Tokyo) Self-organization in macro systems 2016/05/24 15 / 15