Kyoto university masa aki sakagami
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Non-equilibrium evolution of Long-range Interacting Systems and Polytrope. Kyoto University Masa-aki SAKAGAMI. Collaboration with. T.Kaneyama  ( Kyoto U. ) A.Taruya (RESCEU, Tokyo U.). Outlines of this talk. Two systems with long-range interaction. (1) Self-gravitating N-body system.

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Kyoto University Masa-aki SAKAGAMI

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Kyoto university masa aki sakagami

Non-equilibrium evolution of Long-range Interacting Systems and Polytrope

Kyoto University

Masa-aki SAKAGAMI

Collaboration with

T.Kaneyama (Kyoto U.)

A.Taruya(RESCEU, Tokyo U.)


Kyoto university masa aki sakagami

Outlines of this talk

Two systems with long-range interaction

(1) Self-gravitating N-body system

Taruya and Sakagami,

PRL90(2003)181101,

Physica A322(2003)285

A short review of our previous work.

Extremal of generalized entropy by Tsallis

polytrope

generalization of B-G dist.

It well describes non-equilibrium evolution of the system.

(2) 1-dim. Hamiltonian Mean-Field model (HMF model)

We show another example where non-equilibrium evolution of

the system moves along a sequence of polytrope.

Thermodynamical instability implies a superposition of polytrope.

(negative specific heat)


Self gravitating n body system

Antonov 1962

Lynden-Bell & Wood 1968

Long-term (thermodynamic) instability

Self-gravitating N-body System

A System with N Particles (stars) (N>>1)

Particles interact with Newtonian gravity each other

Typical example of a system under long-range force

Key word

Negative Specific Heat

Gravothermal instability


Maxwell boltzmann distribution as equilibrium

2

l= – reE/GM

Boltzmann-Gibbs entropy

Extremization

Maxwell-Boltzmann distribution

= 0.335

Large

re

= 709

D= rc/re

Maxwell-Boltzmann distribution as Equilibrium

Adiabatic wall

(perfectly reflecting boundary)


A na ve generalization of bg statistics

q-entropy

BG limit q→1

A naïve generalization of BG statistics

Thermostatistical treatment by generalized entropy

Tsallis, J.Stat.Phys.52 (1988) 479

One-particle distribution function identified with escort distribution

Power-law distribution


Kyoto university masa aki sakagami

Energy-density contrastrelation for stellar polytrope

n=6

Polytropic equation of state

(e.g., Binney & Tremaine 1987)

Polytrope index

n→∞

BG limit


Stellar polytrope as quasi equilibrium state

n=6

stable

unstable

unstable state appears at n>5

(gravothermal instability)

Stellar polytrope as quasi-equilibrium state

Energy-density contrast

relation for stellar polytrope


Survey results of group a

Survey results of group (A)

The evolutionary track keeps the directionincreasing the polytrope index “n”.

Once exceeding the critical value “Dcrit“, central density rapidly increases toward the core collapse.


Run n3a n body simulation

Initial cond:

Stellar polytrope

(n=3,D=10 )

4

Density profile

One-particle distribution function

Fitting to stellar polytropes is quite good until t ~ 30 trh,i.

Run n3A : N-body simulation


Overview of the results in sel gravitating system

Stellar polytropes are not stable in timescale of two-body relaxation.

However, focusing on their transients, we found :

Quasi-equilibrium property

Transient states approximately follow a sequence of stellar polytropes with gradually changing polytrope index “n”.

Quasi-attractive behavior

Even starting from non-polytropic states, system soon settles into a sequence of stellar polytropes.

Overview of the results in sel-gravitating system


Kyoto university masa aki sakagami

Application of generalized entropy and polytrope to another long-range interacting system,

1-dimensional Hamiltonian mean-field model

1D-HMF model

Antoni and Ruffo,

PRE 52(1995)2361


Kyoto university masa aki sakagami

Antoni and Ruffo,

PRE 52(1995)2361


A na ve generalization of bg statistics1

q-entropy

BG limit q→1

A naïve generalization of BG statistics

Thermostatistical treatment by generalized entropy

Tsallis, J.Stat.Phys.52 (1988) 479

One-particle distribution function identified with escort distribution

Power-law distribution

(polytrope)


Physical quantities by 1 particle distribution

Physical quantities by 1-particle distribution

number

energy

magnetization

potential


Polytropic distribution function

Polytropic distribution function

Chavanis and Campa, Eur.Phys.J. B76(2010)581

Taruya and Sakagami, unpublished

polytrope index

BG limit q→1

n →∞

For given U and n, this eq. self-consistently

determines magnitization M.

As for defenition of Tphys,

Abe, Phys.Lett. A281(2001)126


Kyoto university masa aki sakagami

Antoni and Ruffo,

PRE 52(1995)2361

Thermal equilibrium,BG Limit n→∞

inhomogeneous

state

homogeneous state

generalized to seqences of polytrope, describing non-eqilibrium (?)

n=1000

n=10

n=4

n=2

n=0.5

n=1000

n=10

n=4

n=2

n=0.5


Time evolution of m and t phys 1

Time evolution of M and Tphys (1)

N=10000, 10 samplesof simulations

Initial cond. : Water Bag

Spatially homogeneous


Time evolution of m and t phys 2

Time evolution of M and Tphys (2)

N=10000, 10 samplesof simulations

Initial cond. : Water Bag

Spatially inhomogeneous


Three stages of evolution of m and t phys

Three stages of evolution of M and Tphys

N=10000, 10 samplesof simulations

nearly equilibrium

transient state

quasi-stationary state (qss)

The evolution over three stages are totally well described by

sequences of polytropes.


Kyoto university masa aki sakagami

Tphys – U relation at equilibrium

N=1000, single sample

Long term behavior

early stage behavior

Theoretical prediction for

thermal equilibrium.


Kyoto university masa aki sakagami

Tphys – U relation at qss

N=1000, single sample

Long term behavior

early stage behavior

Tphys at early stage of qss


Kyoto university masa aki sakagami

Tphys – U relation at qss

N=1000, single sample

Long term behavior

polytrope

n=0.5

early stage behavior

Tphys at early stage of qss

are explained by polytrope

with n=0.5.


Kyoto university masa aki sakagami

Evolutionary track on the polytrope sequence

N=10000, 10 samplesof simulations

f by simulation

prediction by

polytrope


Kyoto university masa aki sakagami

Evolutionary track on the polytrope sequence

N=10000, 10 samplesof simulations

Even in qss, polytrope index n and distribution f change.


Kyoto university masa aki sakagami

Evolutionary track on the polytrope sequence

N=10000, 10 samplesof simulations


Kyoto university masa aki sakagami

Evolutionary track on the polytrope sequence

N=10000, 10 samplesof simulations


Kyoto university masa aki sakagami

Evolutionary track on the polytrope sequence

N=10000, 10 samplesof simulations


Kyoto university masa aki sakagami

Evolutionary track on the polytrope sequence

N=10000, 10 samplesof simulations


Kyoto university masa aki sakagami

Evolutionary track on the polytrope sequence

N=10000, 10 samplesof simulations


Kyoto university masa aki sakagami

Evolutionary track on the polytrope sequence

N=10000, 10 samplesof simulations


Kyoto university masa aki sakagami

Evolutionary track on the polytrope sequence

N=10000, 10 samplesof simulations


Kyoto university masa aki sakagami

Failure of single polytrope description due to

Thermodynamical instability (negative specific heat).


Stellar polytrope as quasi equilibrium state1

n=6

stable

unstable

unstable state appears at n>5

(gravothermal instability)

Stellar polytrope as quasi-equilibrium state

Energy-density contrast

relation for stellar polytrope


Kyoto university masa aki sakagami

self-gravitating system

Sel-similar core-collapse in Fokker-Planck eq.

Halo could not catch up with

core collapse.

Heat flow

core

halo

When self-similar core collapse takes place,

polytrope could not fit distribution function.

H.Cohn Ap.J 242 p.765 (1980)


Fitting of self similar sol with double polytrope

fitting of self-similar sol. with double polytrope

Black dots: Numerical Self-Similar sol.

by Heggie and Stevenson

Magenta lines: fitting by double polytrope


Kyoto university masa aki sakagami

Thermodynamical Instability due to

negative specific heat

After a short time, single polytrope fails to

describe the simulated distribution.

We prepare the initial state as

Polytrope with U=0.6, n=1.


Kyoto university masa aki sakagami

A description by double polytropes might work.

double polytrope

single polytrope


Kyoto university masa aki sakagami

A description by double polytropes might work.

double polytrope

parameters

coexistence conditions (preliminary)


Summary and discussion

Summary and Discussion

  • Polytrope(Extremal of Generalized Entropy)

describes evolution along quasi-stationary and transient

statesto thermal equilibrium

Self-gravitating system, 1D-HMF

(Long-range interaction)

(2) Break down of single polytrope description

(due to negative specific heat)

implies superposition of polytropes.


Kyoto university masa aki sakagami

How to determine polytropic index n.


Polytrope

Polytropeによる準定常状態の記述の限界

self-similar evolution

Fokker-Planck eq.によるCore-Collapseの解析

Heat flow

core

halo

self-similar core collapse が

始まると polytrope で fit できない

H.Cohn Ap.J 242 p.765 (1980)


Self similar sol of f p eq

Self-similar sol. of F-P eq.

Heggie and Stevenson,

MN 230 p.223 (1988)

power law envelope

isothermal core


Fitting of self similar sol with double polytrope1

fitting of self-similar sol. with double polytrope

Black dots: Numerical Self-Similar sol.

by Heggie and Stevenson

Magenta lines: fitting by double polytrope


Kyoto university masa aki sakagami

2D HMFモデル

Antoni&Torcini PRE 57(1998) R6233

Antoni, Ruffo&Torcini, PRE 66(2003) 025103R

Interaction by Mean-field: Long-range interacting system

2D HMF have the effect of energy transfer due to 2-body scattering process.

Negative specific heat in some range of energy


Kyoto university masa aki sakagami

T-U curveBoltzmann case

Thermal equilibrium

T

Magnetization

Transient st

polytrope ?

U (energy)

t / N

Vlasov phase ?

dist.

func.

dist.

func.


Initial polytrope u 1 9

Initial : polytrope U=1.9

Negative specific heat

dist.

func.

Magnetization

t (logarithmic)


Initial polytrope u 1 7

Initial: polytrope U=1.7

positive specific heat

dist.

func.

Magnetization

t (logarithmic)


Initial wb u 1 9

Initial WB U=1.9

Negative specific heat

dist.

func.

Magnetization

t (logarithmic)


Initial wb u 1 7

Initial WB U=1.7

positive specific heat

dist.

func.

Magnetization

t (logarithmic)


Kyoto university masa aki sakagami

How to derive the evolution equation

for polytropic index, q or n.

self-gravitating systems


Kinetic theory approach

orbit-averaged F-P eq.

Kinetic-theory approach

For a better understanding of the quasi-equilibrium states,

Fokker-Planck (F-P) model for stellar dynamics

phase space volume

Complicated, but helpful for semi-analytic understanding


Generalized variational principle for f p eq

Glansdorff & Prigogine (1971)

Inagaki & Lynden-Bell (1990)

Application:

Takahashi & Inagaki (1992); Takahashi (1993ab)

Generalized Variational Principle for F-P eq.

Local potential

Variation w.r.t. f

F-P equation for

fixed

Absolute minimum at a solution


The evolution eq for n from generalized variational principle

The evolution eq. for “n” from generalized variational principle

Assuming stellar polytropes with time-varying polytrope index as transient state,

trial function

function of


Semi analytic prediction evolution of n

Semi-analytic prediction: evolution of “n”

Time evolution of polytrope index “n” fitted to N-body simulations

Time-scale of quasi-equilibrium states is successfully reproduced from semi-analytic approach based on variational method.


Summary and discussion1

Summary and Discussion

  • Polytrope(Extremum of Generalized Entropy)

  • Transient statesto thermal equilibrium

Self-gravitating system, 2D-HMF

Negative specific heat

Long-range interaction

(2) Generalized variational princple for F-P eq.

Evolution eq. for polytropic index

Works in Progress:

Short-range attracting interaction

negative specific heat

Polytrope ?

Superposition of Boltzmann dist. ?


Summary

Summary

長距離力(引力)

比熱が負

small system

(1) 重力多体系

非平衡進化

準定常状態が存在

(2) 準定常状態

ポリトロープ状態の系列で記述できる

(3) ポリトロープ指数 n の時間発展

一般化された変分原理

Fokker-Planck eq.

ポリトロープ状態: Trial func

指数 n の時間発展方程式

が導出できる


Work in progress

Work in Progress

(1) ポリトロープ    準定常状態: 他の例はあるか

2次元HMFモデルの解析

(2) ポリトロープ    準定常状態: 長距離相互作用が本質?

Yukawa 型相互作用での解析

(3) ポリトロープによる準定常状態の記述の限界

ポリトロープは core collapse 前しか適用できない?


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