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Resonance states of a Liouvillian
and
Hofstadter’s butterfly type of singular spectrum of
a collision operator for a protein chain
Tomio Yamakoshi Petrosky
Quic kTime™ and a
TIFF (Unc ompres sed) dec ompres sor
are needed to see this pic ture.
Center for Complex Quantum Systems, University of Texas at Austin
Naomichi Hatano (University of Tokyo)
Kazuk Kanki (Osaka Prefecture University)
Satoshi Tanaka (Osaka Prefecture University)
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Alien Baltan
August 10, 2010
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Asteroid belt
Eigenvalue Problem of Liouville-von Neumann Operators
(Classical and Quantum)
Still in poorly understood situation especially for unstable systems:
Resonance states for the Louvillian
Eigenvalue problem of the collision operators in Kinetic equations
(complex eigenvalues and irreversibility)

Other interesting properties (reversible dynamics):
continuous spectrum, discrete spectrum, band structure, level repulsion, …
Instability due to the resonance
1) Resonance States of Quantum Liouvillian for Protein Chain
Myoglobyn
-helix
Primary Structure: One-dimensional Molecular Chain
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2) Level Repulsion and threefold degeneracy of eigenstates
of the Liouvillian in the Kirkwood gaps in the asteroid belt
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I. Resonance States of Quantum Liouvillian for Protein Chain
・Band spectrum in the relaxation modes for momentum disturibution
Characteristic behavior of 1D system:
・No classical limit just like the 1D classical gas
・Rational-irrationality dependence of the spectrum in a physical
parameter (Fractal structure)
⇒ Similarity to Hofstadter’s butterfly of a Hamiltonian spectrum
for a 2D tight-binding model in a magnetic field

R


reversible process

irreversible process


i
 LH 
t
Liouville equation
1
 [H, ]
LH   

i{H, }P.B.

(Q.M.)
(C.M.)
Intrinsic degeneracy
H0    

L0   
1
H
0

1
     H0  (   )  
L0    L0   
0
Dispersion equation and Resonance states of Liouvillian
H  H0  gV  LH  L0  gLV
L0 |  )  l |  )
H0    
LH | F )  Z | F )
H  
E 
P    , Q 1 P

P |  )( |, Q  1 P
 equation
Dispersion
E    g 2  VQ
1
QV 
E  Q HQ

Operator equation


1
2
Z P | F )  l  g P LV Q
Q LV P P | F )
Z

Q
L
Q



 H 
The collision operator
Kinetic equation

Complex eigenvalues: Transport coefficient in irreversible process
H  H vib  H ph  gH 
Exciton

Dimensionless Hamiltonian
p 
p
2J

q
q 
0
 cos( p), q 
 2B sin  
2 
2J
2J
B
c d
2J
ratio of band width
Reduced density matrix
f0 (P,t )  P f (t ) P
L0 P0  0  l0  0
Momentum distribution function
Kinetic equation for the momentum distribution (weak coupling case)

The collision operator in P0 subspace: iKP(0)


Planck distribution
for the phonon
resonance
Eigenvalue problem of the collision operator
KP    ,
  0
H-theorem

R
1

sin 1B
 rational

 irrational
: resonance condition
1
l
R  sin B 

m
1
R

R


3
8
q
2
1
6

7
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9
4
6
2m =10
1
5 0
m=6
R
t>0
t<0

Rich structure of the entropy production
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Alien Baltan
type-I
type-II
type-I
type-II
type-II
Band structure of the spectrum
type-I
!
t-1/2 power law decay
Spectra of the collision operator
Spectra of the collision operator
type-II
II. Level Repulsion and threefold degeneracy of eigenstates
of the Liouvillian in the Kirkwood gaps in Asteroid belt
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Band spectrum of the Liouvillian
in the restrict problem of three bodys
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Keplar’s third law
  a3 / 2
a : semi - major axis
Hamiltonian of the restricted three-body problem
1  2 p2 
1  
H  pr  2  p 

2 
r 
r1
r2
r1  r 2  2rcos  2
r2  r 2  2(1 )rcos  (1  )2

The synodic coordinate
Y'
Y
1


 r


 

r
2
X'
r
MJ

 0.95 103
MS  MJ
1 

 X
The sidereal coordinate
Delaunay’s variables:
(L,l),
(G,g)
1
H   2  G  V (L,G,l, g;  )
2L

1 1  
V (L,G,l, g; )  

r
r1
r2




L a
l : mean anomaly
G : angler momentum of the asteroid
g : argument of the perihelion
in the rotating system
G 2
  1   : eccentricity
L 
l  u   sin u : Keplar' s Equation
u : eccentric anomaly
a(1 2 )
  cos f
r  a(1  cos u ) =
cos u =
1  cos f
1  cos f
f    g : true anomaly

cos f   
2(1  )
2




 J ' (n ) cos(nl )
n
n1
sin f  2 1 2  J n' 
(n ) sin (nl )
n1
 
1 1
  Pn (cos  )  for  1, r ~ 1
r 
r1 r n 0
n

 r 
1
1

Pn (cos  )

 for  1, 0  r  1
r2 1  n 0
1  

n
H  H0  V  LH  L0  LV
H0 1
l 
 3
L L
H0
g 
 1
G


L0  i l  i g
l
g
L0 e
i(nlmg)
 n
 i(nlmg )
  3  me
L
 

m, n  0,1,2,
Dispersion relation for m-n mode:
  m,n (L)
a
L
5.2AU
n
m, n (L)  3  m

L
m : n  3 :1
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
7:3

2 :1
a


5:2
Threefold degeneracy
mn0
The degenerate perturbation theory for resonances
3 : 1
0:0
3 :1
0 L 31 0    
 * 
 1   1 
0:0
L -3-1 0 L 312  2 
0 L * 0    
3 :1

31
 3   3 
3 : 1




Mode coupling
(Selection rule)
Saturn’s ring?
2 :1
L 31 ~ a 3 / 21

3 :1

a ~ 102 AU 103 AU
  0,  L 31  L 31
5:2
2
Level repulsion

a 5 / 20 a 9 / 21
!
2



7 
:3

2
a13 / 2
(for   103 ,   0.1)
m:n
a
m
1
2 mn2

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Evidence of the threefold degeneracy
Summary (Instability due to the resonance)
・Rich fractal structure of the resonance spectrum of the Liouvillian in a protein chain,
Rational-irrationality dependence of the spectrum, Band structure
・Rich structure of the entropy production,
"Complex Spectrum Representation of the Liouvillan and Kinetic Theory in Nonequilibrium Physics,"
T. Petrosky, Prog. Theor. Phys. 123, 395 (2010)
"Hofstadter’s butterfly type of singular spectrum of a collision operator for a model of molecular chains,"
T. Petrosky, N. Hatano, K. Kanki, and S. Tanaka, Prog. Theor. Phys. Supplement, No. 184, 457 (2010).
・Quantum analogy of the spectrum of the Liouvillian in classical system,
・Analysis of classical systems in terms of “states” in stead of “trajectories,”
・The degenerate perturbation theory for resonance effect, level repulsion, band
structure, … (classical quantization?)
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Eigenvalue Problem for Hamiltonian
Hofstadter’s butterfly
2D tight-binding model + Magnetic field （reversible process）
Dispersion relation:
E(k)  2E 0 (cos kx a  cos ky a)
k  p  eA/c


(dimensionless) rational or irrational
Hofstadter’s butterfly


Band spectrum for fixed value of R
Davydov’s interaction
R
 Hint    Bn Bn (un 1  un1)
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n
Type I

 of the exciton
Band width
(kBT)/(2J) = 1
Hint    B Bn (un 1  un )

n
n
(kBT)/(2J) = 1
Type II

p2
( p) 
,  (q)  c | q | case
2m
(kBT)/(2J) = 0.1
Resonance states of a Liouvillian
and
Hofstadter’s butterfly type of singular spectrum of
a collision operator for a protein chain
Tomio Yamakoshi Petrosky
Quic kTime™ and a
TIFF (Unc ompres sed) dec ompres sor
are needed to see this pic ture.
Center for Complex Quantum Systems, University of Texas at Austin
Naomichi Hatano (University of Tokyo)
Kazuk Kanki (Osaka Prefecture University)
Satoshi Tanaka (Osaka Prefecture University)
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Ç ™Ç ±Ç ÃÉsÉN É`ÉÉÇ ¾å©Ç ÈÇ žÇ ½Ç …Ç ÕïKóvÇ ­Ç •
ÅB
Alien Baltan
SPQS2010, August 2, 2010
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Asteroid belt