and Gravitational Resonant Phase Transitions Bence Kocsis GALNUC - - PowerPoint PPT Presentation

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Jan 15, 2023 105 likes •517 views

Liquid Crystals of Stars and Gravitational Resonant Phase Transitions Bence Kocsis GALNUC ERC Starting Grant Eotvos University, Budapest collaborators: Gergely Mathe, Akos Szolgyen, Adam Takacs, Zacharias Roupas, Scott Tremaine,

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SLIDE 1

Bence Kocsis

“GALNUC” ERC Starting Grant

Eotvos University, Budapest

collaborators:

Gergely Mathe, Akos Szolgyen, Adam Takacs, Zacharias Roupas, Scott Tremaine,

Liquid Crystals of Stars and Gravitational Resonant Phase Transitions

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SLIDE 2

Types of relaxation

  • Collisionless relaxation (mean-field relaxation)
  • coarse-grained phase space density mixes
  • Violent relaxation
  • collisionless Boltzmann eq. + time-dependent

potential

  • Collisional relaxation (thermal relaxation)
  • energy exchange, evolves toward equipartition
  • Resonant relaxation
  • action-angle variables
  • adiabatic invariants: inhibit relaxation
  • resonance between angles: accelerates relaxation
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SLIDE 3

Relaxation hierarchy

  • rbit

in-plane precession reorientation semimajor axis change [1—104 yr ] [104—5 yr] [105–7 yr] [109 yr] adiabatic invariants

semimajor axis eccentricity

  • rbital plane
  • rientation of ellipse

semimajor axis eccentricity

  • rbital plane

semimajor axis eccentricity stationary annulus elliptic wire static spherical point-mass stationary spherical stationary axi-symmetric – –

symmetries

2 phase space componens relax extremely quickly!!

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SLIDE 4

Outside 0.27-0.47 pc Middle 0.13-0.27 pc Inside 0.03-0.13 pc

Liquid crystals of stars to explain

Anisotropy of massive stars

Density map of angular momentum vector directions for massive stars at three different locations

Yelda+ 2014

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SLIDE 5

Outside 0.27-0.47 pc Middle 0.13-0.27 pc Inside 0.03-0.13 pc

Liquid crystals of stars to explain

Anisotropy of massive stars

Density map of angular momentum vector directions for massive stars at three different locations

Cos[ polar angle ] azimuthal angle azimuthal angle azimuthal angle Bartko+ 2009

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SLIDE 6

Vector resonant relaxation Hamiltonian

  • average interaction effects over orbit and in-plane precession
  • only the angular momentum vector directions may change
  • Hamiltonian (see Kocsis & Tremaine 2015 for details)

qij

masses semi-major axes eccentricities Angle between orbit normals

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SLIDE 7

Similar models in condensed matter physics

  • Ferromagnetism
  • Liquid crystals
  • Spin glasses
  • Disordered systems:

random variable coupling coefficients

  • Models with additional constraints
  • “frustrated” state
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SLIDE 8

Results

c.f: the observed distribution

Log[ distance from center ] Cos[ inclination ]

Monte Carlo Markov Chain

RMS inclination [deg] Cos[ inclination ] Log[ distance from center / 4 arcsec ] distance from center [arcsec]

Time evolution Mean field theory

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SLIDE 9

Results

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SLIDE 10

Monte Carlo Markov Chain

  • Assumption

– distribution fills up phase space uniformly in the allowed region

  • Total energy fixed
  • Total angular momentum fixed
  • Method

– Grab two angular momentum vectors Li, Lj – Rotate around sum Li+Lj by random angle – Check if total energy error is within a max tolerance

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SLIDE 11

Monte Carlo Markov Chain

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SLIDE 12

Monte Carlo Markov Chain

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SLIDE 13
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Final state in the simulation

Log(semimajor axis) Cos[ inclination] Log(semimajor axis) T=0 T=500 T=1500 T=1000 T=500 T=0

  • Three snapshots in two simulations

Heavy objects in a disk Light objects spherical Heavy objects spherical Light objects in a disk Initially:

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SLIDE 15

Statistical equilibrium

  • Mean field theory
  • Maier & Saupe (1959)
  • keep only the quadrupole term
  • assume interaction dominated

by stars on same radius

  • self-consistency equation for

quadrupole moment

  • Objects fill up phase space uniformly in the allowed region

Find maximum entropy configuration under constraints

const

tot 

E const

tot 

L           kT E C f L L L . ) ( exp ) (

Phase transition in inclination

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SLIDE 16

Mean field theory

Maximum entropy with constraints

  • Particle number
  • Total resonant-relaxation energy
  • Total angular momentum
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SLIDE 17

Free energy

as a function of order parameters

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SLIDE 18

Cos2[ inclination ] – 1/3

Mean field theory

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SLIDE 19

Phase transition

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SLIDE 20

Phase transition

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SLIDE 21

Probability density

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SLIDE 22

Negative temperature equilibria

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SLIDE 23

Results with higher harmonics

  • Order parameters are the multipole moments.
  • First order phase transition in all order parameters
  • Ordered phase is dominated by quadrupole

Takacs, Kocsis (2018)

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SLIDE 24

Results with higher harmonics

  • Entropy – energy curve is qualitatively similar as in the simple case

Takacs, Kocsis (2018)

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Statistical physics from “molecular dynamics”

  • Statistical equilibria have lower Boltzmann

entropy at intermediate energies

  • Stongly convex feature  phase transition?

Mathe, Kocsis in prep.

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SLIDE 26

Ensemble inequivalence

  • Stability of equilibria are different in the

canonical and microcanonical ensembles

– No phase transition in microcanonical ensemble

  • Microcanonical ensemble of a one-component

system cannot show a phase separation

– This is due to the interaction energy between any two subsystems

  • There may be a phase separation in two

component systems

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SLIDE 27

Summary

  • Resonant effects accelerate relaxation in 2 phase space

components

  • Liquid crystals have similar Hamiltonian
  • Disk at low temperature
  • Spherical phase at high temperature
  • Young stars in the Galactic center show a similar structure
  • First order phase transition

Use this to

– model the inclination distribution of different stellar types – predict the distribution of black holes

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SLIDE 28
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SLIDE 29

Kocsis & Tremaine (2011)

Hierarchy of Interaction Timescales vs. radius

Keplerian orbit around SMBH Precession in plane Re-orientiation

  • f orbital plane

Eccentricity change Semimajor axis change Disk age

Time scale

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SLIDE 30

Vector resonant relaxation Hamiltonian

  • mutual torques lead to relaxation of orbit normal vectors

(angular momenta)

  • self-gravitating system in “inclination”
  • interaction energy

qij

Legendre polinomial constant (depends on e, a) constant

changes with time

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SLIDE 31

Vector resonant relaxation Hamiltonian

  • mutual torques lead to relaxation of orbit normal vectors

(angular momenta)

  • self-gravitating system in “inclination”
  • interaction energy

qij

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SLIDE 32

Evidence for disrupted globular clusters?

A fraction of stars was delivered by infalling globular clusters

Fermi measured excess gamma ray emission from the Galactic bulge

  • Millisecond pulsars match the observed spectrum
  • Millisecond pulsars do not form in the bulge
  • Infalling globular clusters delivered the needed

population (No need to invoke dark matter annihilation to explain the gamma ray excess, just ordinary MSPs) Brandt & Kocsis (2015)

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SLIDE 33

Statistical equilibrium

Maximum entropy with constraints

  • Particle number
  • Total resonant-relaxation energy
  • Total angular momentum
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SLIDE 34

total angular momentum

Phase diagrams

total angular momentum

Axisymmetric states Lopsided (triaxial) states

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SLIDE 35

Statistical equilibrium

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SLIDE 36

Order parameter

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SLIDE 37

For a one-component system:

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SLIDE 38

Free energy

as a function of order parameters

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SLIDE 39

Ensemble inequivalence