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.)
2
Outlines of this talk
Two systems with long-range interaction
(1) Self-gravitating N-body system
A short review of our previous work.
Extremal of generalized entropy by Tsallis
polytrope
f (x, v)  A(0   ) q /(1q)
Taruya and Sakagami,
PRL90(2003)181101,
Physica A322(2003)285
generalization of B-G dist.
  12 v2  ( x)
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
A System with N Particles (stars) (N>>1)
Particles interact with Newtonian gravity each other
2
H 
i
Gmi m j
pi
 
2mi i j i | x i  x j |
Typical example of a system
under long-range force
Key word
Negative Specific Heat
Long-term (thermodynamic) instability ( t  t relax )
Gravothermal instability
Antonov 1962
Lynden-Bell & Wood 1968
3
Maxwell-Boltzmann distribution as
6
Equilibrium
BoltzmannGibbs entropy
S BG    d 3x d 3v f ( x, v ) ln f ( x, v )
Extremization
0   S BG    M    E
Maxwell-Boltzmann distribution
l – reE/GM 2
f ( x, v)  exp(  ) ;
Large
lcrit = 0.335
  v2 / 2  (x)
D  rc / r
re
e
rc
r
e
Dcrit = 709
Adiabatic wall
D= rc/ re
(perfectly reflecting boundary)
A naïve generalization of BG statistics
7
Thermostatistical treatment by generalized entropy
q-entropy

q
Sq   1  d 3x d 3v p(x, v)  p(x, v)
q1

BG limit q→1
One-particle distribution function
identified with escort distribution
f ( x , v)  M
Power-law distribution
Tsallis, J.Stat.Phys.52 (1988) 479
{ p( x, v)}q
3
3
q
d
x
d
v
{
p
(
x
,
v
)}

p  p  p
f (x, v)  A(0   )
q /(1q )
0   Sq    M    E
  v 2 / 2  (x)
9
Energy-density contrast relation for stellar polytrope
f (x, v)  A(0   ) q /(1q)
n=6
Polytropic equation of state
P( r )  Kn r 11 / n ( r )
(e.g., Binney & Tremaine 1987)
Polytrope
index
1
1
n

1 q 2
n→∞
BG limit
Stellar polytrope as quasi-equilibrium state
Energy-density contrast
relation for stellar polytrope
n=6
stable
10
unstable
unstable state appears at n>5
(gravothermal instability)
Survey results of group (A)
11
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.
Run n3A : N-body simulation
Density profile
Initial cond:
12
Stellar polytrope
4
(n=3,D=10 )
One-particle distribution function
Fitting to stellar polytropes is quite good until t ~ 30 trh,i.
1 2
v   ( x)
2
Overview of the results in sel-gravitating system
13
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.
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
Antoni and Ruffo,
PRE 52(1995)2361
A naïve generalization of BG statistics
16
Thermostatistical treatment by generalized entropy
q-entropy

q
1


Sq  
d dp h( , p)  h( , p)
q1 

BG limit q→1
One-particle distribution function
identified with escort distribution
Tsallis, J.Stat.Phys.52 (1988) 479
{h( , p )}q
f ( , p )  N
q
d

d
p
h
(

,
p
)}

Power-law distribution
(polytrope)
h  h  h
f ( , p)  A(0   )
q /(1q )
  p 2 / 2   ( )
Physical quantities by 1-particle distribution
number
N   d dp f ( , p)
energy
E   d dp 12 p 2  ( ) f ( , p)
1
magnetization M  ( M x , M y )   d dp cos  , sin   f ( , p )
N
1
 ( )   d ' dp ' 1  cos(    ' ))  f ( ' , p' )
potential
N
( )  1  M cos(   )
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Polytropic distribution function
f ( , p)  A0   
n1/ 2
18
Chavanis and Campa, Eur.Phys.J. B76(2010)581
Taruya and Sakagami, unpublished
polytrope index BG limit q→1
n →∞
1 2
q
1
  2 p  ( ) ( )  1  M cos n 

1 q 2
0  (n  1)Tphys  1  M 2
M 
n
d

cos

(


1

M
cos

)
0

n
d

(


1

M
cos

)
0

Tphys  2 K / N 
1
N
For given U and n, this eq. self-consistently
determines magnitization M.
2
d

dp
p
f ( , p)

U  E / N  12  Tphys  1  M 2 
As for defenition of Tphys,
Abe, Phys.Lett. A281(2001)126
Thermal equilibrium,ＢＧ Ｌｉｍｉｔ n→∞
Antoni and Ruffo,
PRE 52(1995)2361
19
inhomogeneous
state M  0
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 Tphys (1)
U  E / N  0.69
N=10000, 10 samples of simulations
Initial cond. : Water Bag
Spatially homogeneous
M 0
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Time evolution of M and Tphys (2)
U  E / N  0.69
N=10000, 10 samples of simulations
Initial cond. : Water Bag
M 0
Spatially inhomogeneous
21
Three stages of evolution of M and Tphys
U  E / N  0.69
N=10000, 10 samples of simulations
nearly equilibrium
transient state
quasi-stationary state (qss)
M 0
The evolution over three stages are totally well described by
sequences of polytropes.
22
Tphys – U relation at equilibrium
U  E / N  0.62
Long term behavior
N=1000, single sample
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early stage behavior
Theoretical prediction for
thermal equilibrium. n  
Tphys – U relation at qss
U  E / N  0.62
Long term behavior
N=1000, single sample
early stage behavior
Tphys at early stage of qss
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Tphys – U relation at qss
U  E / N  0.62
Long term behavior
N=1000, single sample
25
polytrope
n=0.5
early stage behavior
Tphys at early stage of qss
are explained by polytrope
with n=0.5.
Evolutionary track on the polytrope sequence U  E / N  0.26
69
N=10000, 10 samples of simulations
U  E / N  0.69
n  0.90
U  0.69
t  60000
n  0.90
f by simulation
prediction by
polytrope
Evolutionary track on the polytrope sequence
U  E / N  0.69
N=10000, 10 samples of simulations
n  1.58
t  300300
n  1.58
Even in qss, polytrope index n and distribution f change.
27
Evolutionary track on the polytrope sequence
U  E / N  0.69
N=10000, 10 samples of simulations
t  600800
n  2.69
n  2.69
28
Evolutionary track on the polytrope sequence
U  E / N  0.69
N=10000, 10 samples of simulations
t  700300
n  3.95
n  3.95
29
Evolutionary track on the polytrope sequence
U  E / N  0.69
N=10000, 10 samples of simulations
t  800800
n  6.46
n  6.46
30
Evolutionary track on the polytrope sequence
U  E / N  0.69
N=10000, 10 samples of simulations
t  1000800
n  13.9
n  13.9
31
Evolutionary track on the polytrope sequence
U  E / N  0.69
N=10000, 10 samples of simulations
t  1800800
n  58 .6
n  58 .6
32
Evolutionary track on the polytrope sequence
U  E / N  0.69
N=10000, 10 samples of simulations
t  2500100
n  66 .4
33
Evolutionary track on the polytrope sequence
U  E / N  0.69
N=10000, 10 samples of simulations
t  3000700
n  597
n  597
34
35
Failure of single polytrope description due to
Thermodynamical instability (negative specific heat).
Stellar polytrope as quasi-equilibrium state
Energy-density contrast
relation for stellar polytrope
n=6
stable
36
unstable
unstable state appears at n>5
(gravothermal instability)
37
self-gravitating system
Sel-similar core-collapse in Fokker-Planck eq.
Halo could not catch up with
core collapse.
CV  0
Heat flow
core halo
CV  0
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
f ( )  A  0   

n1
 c  0   
38
n2 

Black dots: Numerical Self-Similar sol.
by Heggie and Stevenson
f ( )
Magenta lines: fitting by double polytrope

n1  9.538
n2  16.73
c  3.59
r
r
39
Thermodynamical Instability due to
negative specific heat
After a short time, single polytrope fails to
describe the simulated distribution.
t  5000
n 1
We prepare the initial state as
Polytrope with U=0.6, n=1.
40
A description by double polytropes might work.
double polytrope
t  5000
single polytrope
t  5000
41
A description by double polytropes might work.
double polytrope
t  5000
parameters
n1  0.95, n2  1.2, l1  0.62
coexistence conditions
(preliminary)
T  l1 T1  l2 T2
1  l1  l2
U1  U2
U1  12 T1  12 l1 (1  M12 )
U2  12 T2  12 l2
Summary and Discussion
42
(1) Polytrope (Extremal of Generalized Entropy)
describes evolution along quasi-stationary and transient
states to 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.
43
How to determine polytropic index n.

2
 
2


2n  1
2
2
2


(
n

2
)
T

(
2
n

3
)
l
M
T

l
(
1

M
)
phys
phys
2
4( n  2 )
44
45
Polytrope による準定常状態の記述の限界
Fokker-Planck eq. によるCore-Collapse の解析
46
self-similar evolution
CV  0
Heat flow
core halo
CV  0
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.
r
Heggie and Stevenson,
MN 230 p.223 (1988)
47

power law envelope
ln r
r r 2.21
n  9 .7
ln r
isothermal core
fitting of self-similar sol. with double polytrope
f ( )  A  0   

n1
 c  0   
48
n2 

Black dots: Numerical Self-Similar sol.
by Heggie and Stevenson
f ( )
Magenta lines: fitting by double polytrope

n1  9.538
n2  16.73
c  3.59
r
r
49
2D ＨＭＦモデル
Antoni&Torcini PRE 57(1998) R6233
Antoni, Ruffo&Torcini, PRE 66(2003) 025103R
1 N 2
H   ( px ,i  p 2y ,i )
2 i 1
1

2N
  3  cos(x  x )  cos( y  y )  cos(x  x ) cos( y  y )
N
i
j
i
j
i
j
i
j
i, j
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
50
Magnetization
T-U curve Boltzmann case
U  1.95
T
U
(energy)
N  10 , N  9 10
4
4
Thermal equilibrium
Transient st
polytrope ?
t/N
Vlasov phase ?
dist.
func.
dist.
func.
vx
vx
51
Initial : polytrope U=1.9
52
Negative specific heat
Magnetization
dist.
func.
p
t
(logarithmic)
Initial: polytrope U=1.7
53
positive specific heat
Magnetization
dist.
func.
p
t
(logarithmic)
Initial WB U=1.9
54
Negative specific heat
Magnetization
dist.
func.
p
t
(logarithmic)
Initial WB U=1.7
55
positive specific heat
Magnetization
dist.
func.
p
t
(logarithmic)
56
How to derive the evolution equation
for polytropic index, q or n.
self-gravitating systems
Kinetic-theory approach
57
For a better understanding of the quasi-equilibrium states,
Fokker-Planck (F-P) model for stellar dynamics
orbit-averaged F-P eq.
  f ( ) 
 ( ) 






 t 
  t
;   16 2G2m2 ln 
  ln f ( )  ln f ( ) 
 ( )   d  f ( ) f ( ) min[ ( ), ( )] 









16 2
3/ 2
2


 ( ) 
dr
r
2
[



(
r
)]
phase space volume
3 
Complicated, but helpful for semi-analytic understanding
Generalized Variational Principle for F-P eq.
58
Glansdorff & Prigogine (1971)
Local potential
Inagaki & Lynden-Bell (1990)
2
 f 0 

  ln f  ln f  
 ( f , f 0 )   d 
ln
f




d

d

f
f
min(

,

)



0 0
0 0 

4
  
 t 
 
Variation
w.r.t. f

( f , f0 )
0
 f
ff
F-P equation for f 0
0
f0
fixed
Absolute minimum at a solution
f0
   ( f , f 0 )   ( f 0 , f 0 )  0
Application:
Takahashi & Inagaki (1992); Takahashi (1993ab)
The evolution eq. for “n” from generalized
variational principle
59
Assuming stellar polytropes with time-varying polytrope index as
transient state,
trial function
f ( )  A(t )[ 0 (t )   ]n (t )3 / 2

( f , f 0 )
0
n
ff
( A and  0 are the functions of n)
function of n, E , M
0
 l 

n(t )  
  e  n
 l    
l       ln f

 d ( )  e   n    n   e   
n
n
e
e

2
 l    ln f 
l    ln f  

 d f ( ) e   n    n   e  
n
n
e
e

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.
60
Summary and Discussion
(1) Polytrope (Extremum of Generalized Entropy)
Transient states to 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. ?
61
62
63
Summary
（１） 重力多体系
（２） 準定常状態
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
ポリトロープ状態の系列で記述できる
P( r )  K n r
11 / n
(r)
n
1
1

1 q 2
(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
(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？
64