1 Non-equilibrium evolution of Long-range Interacting Systems and Polytrope Kyoto University...

1

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.)

2Outlines of this talk

(1) Self-gravitating N-body system

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

A short review of our previous work.

Extremal of generalized entropy by Tsallis

)1/(0 )()v,x( qqAf polytrope generalization of B-G dist.

It well describes non-equilibrium evolution of the system.

Two systems with long-range interaction

)(221 xv

Taruya and Sakagami, PRL90(2003)181101, Physica A322(2003)285

We show another example where non-equilibrium evolution ofthe system moves along a sequence of polytrope.

Thermodynamical instability implies a superposition of polytrope.(negative specific heat)

3

Self-gravitating N-body System

Typical example of a system under long-range force

i ij ji

ji

i i

i

xx

mGm

m

pH

||2

2

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

Particles interact with Newtonian gravity each other

Key word Negative Specific Heat

Gravothermal instabilityGravothermal instability Antonov 1962Lynden-Bell & Wood 1968

Long-term (thermodynamic) instability )( relaxtt

6

D= c e

–

reE

/GM

2 Largere c

e

e/D c

Maxwell-Boltzmann　distribution　as　

Equilibrium

Maxwell-Boltzmann distributionMaxwell-Boltzmann distribution

;)exp()v,( xf )x(2/v2

crit

critD

= 0.335

= 709

)vx(ln)vx(vx 33BG ,f,fddS Boltzmann-Boltzmann-

Gibbs entropyGibbs entropy

ExtremizationEMS BG0

Adiabatic wall(perfectly reflecting boundary)

7A　naïve　generalization　of　BG　statistics

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

Power-law distribution

)1/(0 )()v,x( qqAf )x(2/v2

ppp

Thermostatistical treatment by generalized entropy

q-entropy )v,x()v,x(vx1

1 33q ppdd

qS q

BG limit q→1

q

q

pdd

pMf

)}v,x({vx

)}v,x({)v,x(

33

One-particle distribution function identified with escort distribution

EMS q0

9

n=6

Energy-density contrast relation for stellar polytrope

Polytrope index

n→∞BG limit

)1/(0 )()v,x( qqAf

Polytropic equation of state

(e.g., Binney & Tremaine 1987)

2

1

1

1

qn

)()( /11 rKrP nn

10

n=6 Energy-density contrastrelation for stellar polytrope

stable unstable

unstable state appears at n>5(gravothermal instability)

Stellar　polytrope　as　quasi-equilibrium　state

11Survey　results　of　group　(A)

The evolutionary track keeps the direction increasing the polytrope index “n”.

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

12Run　n3A　:　N-body　simulation

Initial　cond:Stellar　polytrope

((n=3,D=10　　)　　)44

Density profileDensity profile One-particle distribution functionOne-particle distribution function

)(v2

1 2 x

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

13Overview　of　the　results　in　sel-gravitating　system

However, focusing on their transients, we found :

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

Quasi-equilibrium propertyQuasi-equilibrium property

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

Quasi-attractive behaviorQuasi-attractive behavior

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

14

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

15


16A　naïve　generalization　of　BG　statistics

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

Power-law distribution(polytrope)

)1/(0 )(),( qqApf )(2/2 p

hhh

Thermostatistical treatment by generalized entropy

q-entropy ),(),(1

1q phphdpd

qS q

BG limit q→1

q

q

phpdd

phNpf

)},(

)},({),(

One-particle distribution function identified with escort distribution

Physical quantities by 1-particle distribution 17

),( pfdpdN

),()(221 pfpdpdE

),(sin,cos1

),( pfdpdN

MM yx M

)','())'cos(1''1

)( pfdpdN

)cos(1)( M

number

energy

magnetization

potential

Polytropic distribution function 18

2/10),( nApf

2

1

1

q

qn cos1)( M

20 1)1( MTn phys

n

n

Md

MdM

)cos1(

)cos1(cos

0

0

),(/2 21 pfpdpdNKT Nphys

221 1/ MTNEU phys

polytrope index

)(221 p

BG limit q→1　　　　　 n →∞

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

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

As for defenition of Tphys,Abe, Phys.Lett. A281(2001)126

19Thermal equilibrium, ＢＧ　Ｌｉｍｉｔ n→∞

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

n=1000n=10n=4n=2n=0.5

n=1000n=10n=4n=2n=0.5


homogeneous state

inhomogeneousstate 0M

Time evolution of M and Tphys (1) 20

N=10000, 10 samples of simulations

Initial cond. : Water Bag

69.0/ NEU

Spatially homogeneous

0M

Time evolution of M and Tphys (2) 21


Initial cond. : Water Bag

69.0/ NEU

Spatially inhomogeneous

0M

Three stages of evolution of M and Tphys 22

N=10000, 10 samples of simulations69.0/ NEU

0Mquasi-stationary state (qss)

transient state

nearly equilibrium

The evolution over three stages are totally well described by sequences of polytropes.

23N=1000, single sample

62.0/ NEULong term behavior

early stage behavior

Theoretical prediction for thermal equilibrium. n

Tphys – U relation at equilibrium

24



Tphys at early stage of qss

N=1000, single sample Tphys – U relation at qss

25



N=1000, single sample Tphys – U relation at qss

Tphys at early stage of qssare explained by polytropewith n=0.5.

polytropen=0.5

26

69.0/ NEU

60000t

90.0n

Evolutionary track on the polytrope sequence

90.0n

N=10000, 10 samples of simulations69.0/ NEU

69.0U

f by simulation

prediction by polytrope

27

69.0/ NEU

300300t

58.1n


58.1n

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


28

69.0/ NEU

600800t

69.2n


69.2n


29

69.0/ NEU95.3n

700300t

Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations

95.3n

30

69.0/ NEU

800800t

46.6n


46.6n

31

1000800t

9.13n69.0/ NEU


9.13n

32

1800800t

6.58n69.0/ NEU


6.58n

33

2500100t

4.66n69.0/ NEU


34

69.0/ NEU

3000700t

597n


597n

35

Failure of single polytrope description due to Thermodynamical instability (negative specific heat).

36

n=6 Energy-density contrastrelation for stellar polytrope

stable unstable

unstable state appears at n>5(gravothermal instability)

Stellar　polytrope　as　quasi-equilibrium　state

37

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

Heat flowHeat flow

corecore halohalo

0CV

0CV

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

self-gravitating system

When self-similar core collapse takes place,polytrope could not fit distribution function.

Halo could not catch up with core collapse.

38fitting of self-similar sol. with double polytrope

)(f

r

2

01

0)(n

cn

Af

Black dots: Numerical Self-Similar sol. by Heggie and Stevenson

Magenta lines: fitting by double polytrope

59.3

73.16

538.9

2

1

c

n

n

39Thermodynamical Instability due to negative specific heat

5000t

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.

1n

40

5000t

A description by double polytropes might work.

5000t

double polytrope single polytrope

41

5000t

A description by double polytropes might work.

double polytrope parameters

62.0,2.1,95.0 121 nn

coexistence conditions (preliminary)

211

21 UU

)1( 2112

112

11 MTU

221

221

2 TU

2211 TTT

Summary and Discussion42

(1) Polytrope (Extremal of Generalized Entropy)

Self-gravitating system, 1D-HMF

(2) Break down of single polytrope description

(due to negative specific heat)

(Long-range interaction)

implies superposition of polytropes.

describes evolution along quasi-stationary and transient states to thermal equilibrium

43How to determine polytropic index n.

)1()32()2()2(4

12 2222

22

MTMnTnn

nphysphys

46Polytrope による準定常状態の記述の限界

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

Heat flowHeat flow

corecore halohalo

0CV

0CV

Fokker-Planck eq. による Core-Collapse の解析 self-similar evolution

self-similar core collapse が始まると polytrope で fit できない

47

Self-similar sol. of F-P eq.Heggie and Stevenson, MN 230 p.223 (1988)

ln

rln

isothermal core

power law envelope

21.2 r

7.9n

48fitting of self-similar sol. with double polytrope

)(f

r

2

01

0)(n

cn

Af

Black dots: Numerical Self-Similar sol. by Heggie and Stevenson

Magenta lines: fitting by double polytrope

59.3

73.16

538.9

2

1

c

n

n

2D ＨＭＦモデル49

N

jijijijiji

iy

N

iix

yyxxyyxxN

ppH

,

2,

1

2,

)cos()cos()cos()cos(32

1

)(2

1

Antoni&Torcini PRE 57(1998) R6233Antoni, Ruffo&Torcini, PRE 66(2003) 025103R

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

Interaction by Mean-field ：　 Long-range interacting system

Negative specific heat in some range of energy

51

T

U (energy)

95.1U

Mag

neti

zati

on

t / N

T-U curve Boltzmann case

Transient stpolytrope ?

Thermal equilibrium

44 109,10 NN

Vlasov phase ?

xv xv

dist. func.

dist. func.

Initial : polytrope U=1.9 52

t (logarithmic)

Mag

neti

zati

on

dist. func.

p

Negative specific heat

Initial: polytrope U=1.7 53

t (logarithmic)

Mag

neti

zati

on

dist. func.

p

positive specific heat

Initial WB U=1.9 54

t (logarithmic)

Mag

neti

zati

on

dist. func.

p

Negative specific heat

Initial WB U=1.7 55

t (logarithmic)

Mag

neti

zati

on

dist. func.

p

positive specific heat

56

How to derive the evolution equation for polytropic index, q or n.

self-gravitating systems

57Kinetic-theory　approach

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

orbit-averaged F-P eq.

tt

f

)()( ln16; 222 mG

2/322

)]([23

16)( rrdr

)(ln)(ln)](),(min[)()()(

ffffd

For a better understanding of the quasi-equilibrium states,

Complicated, but helpful for semi-analytic understanding

phase space volume

58Generalized　Variational　Principle　for　F-P　eq.

Local potential

ft

fdff ln),( 0

0

2

0000

lnln),min(

4

ff

ffdd

Variation w.r.t. f F-P equation for0),(

0

0 ff

fff

Absolute minimum at a solution 0f

0),(),( 000 ffff

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

Glansdorff & Prigogine (1971)

Inagaki & Lynden-Bell (1990)

0f

0f fixed

59The　evolution　eq.　for　“n”　from　generalized　variational　principle

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

2/3)(0 ])()[()( tnttAf

0),(0

0 ff

ffn

)offunctionstheareand( 0 nA

2

lnln)(

ln)(

)(

nene

nene

ne f

nn

ffd

f

nnd

tn

ee

ee

trial function

function of MEn ,,

60

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

Semi-analytic　prediction:　evolution　of　“n”

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

Summary and Discussion61

(1) Polytrope (Extremum of Generalized Entropy) Transient states to thermal equilibrium 　

Self-gravitating system, 2D-HMF Negative specific heat

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

Evolution eq. for polytropic index

Short-range attracting interaction

negative specific heat

Long-range interaction

Works in Progress:

Polytrope ? Superposition of Boltzmann dist. ?

Summary63

（１）　重力多体系長距離力（引力）比熱が負small system

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

)()( /11 rKrP nn

2

1

1

1

qn

(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 Progress64

(1) ポリトロープ　　　　準定常状態：　他の例はあるか

(2) ポリトロープ　　　　準定常状態：　長距離相互作用が本質？

(3) ポリトロープによる準定常状態の記述の限界

２次元ＨＭＦモデルの解析

Yukawa 型相互作用での解析

ポリトロープは core collapse 前しか適用できない？

1 Non-equilibrium evolution of Long-range Interacting Systems and Polytrope Kyoto University...

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