Turbulent Crystal and idealized glass
description
Transcript of Turbulent Crystal and idealized glass
![Page 1: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/1.jpg)
TURBULENT CRYSTAL AND IDEALIZED GLASS
Shin-ichi Sasa ( Kyoto University) 2013/07/19
![Page 2: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/2.jpg)
Tokyo life (every morning) Kyoto life (every morning)
![Page 3: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/3.jpg)
Do turbulent crystals exist? David Ruelle, Physica A 113, (1982)
![Page 4: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/4.jpg)
Who is David Ruelle ?
Statistical Mechanics David Ruelle,Benjamin, New York, 1969. 11+219 pp.
Cited by 2689
AbstractA mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed.
On the nature of turbulenceD Ruelle, F Takens - Communications in mathematical physics, 1971 - Springer
Cited by 2634
![Page 5: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/5.jpg)
Do turbulent crystals exist? David Ruelle, Physica A 113, (1982)
AbstractWe discuss the possibility that, besides periodic and quasiperiodic crystals, there exist turbulent crystals as thermodynamic equilibrium states at non-zero temperature. Turbulent crystals would not be invariant under translation, but would differ from other crystals by the fuzziness of some diffraction peaks. Turbulent crystals could appear by breakdown of long range order in quasiperiodic crystals with two independent modulations.
![Page 6: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/6.jpg)
Part I Turbulent crystal
![Page 7: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/7.jpg)
Regular time series
Periodic Quasi-periodic
Time series
Power-Spectrum
tt
![Page 8: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/8.jpg)
Irregular but deterministic time series
Time series
Power-Spectrum
t
Chaos
It can be distinguished from “noise” in experiments !
![Page 9: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/9.jpg)
From time series to patterns
Quasi-periodic motion
Periodic motion
Quasi-periodic pattern
Periodic pattern
Chaotic motion Chaotic pattern
Replace “time” by “space coordinate”
Example: nnnnn K sin2 11
Stationary solution: 0sin2 11 nnnn K
Standard map
Zn
![Page 10: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/10.jpg)
From patterns to equilibrium phases
![Page 11: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/11.jpg)
From periodic patterns to crystal phase
Crystal1) Ground states are generated by periodic repetition of a unit
2) Long-range positional order (Bragg Peak)
3) Translational symmetrybreaking occurs in statistical measure with finite temperature
![Page 12: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/12.jpg)
From quasi-periodic patterns to quasi-crystals phase
Mathematical study of tiling(1961 ~ 1975):Regular but aperiodic tiling !
Experiments (1984)
1) Ground states are generated by non-periodic repetition of two units
2) Long-range positional order (Bragg Peak)
3) Translational symmetrybreaking in statistical measure with finite temperature
![Page 13: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/13.jpg)
Thermodynamic phase associated with chaotic patterns?
1) No long-range positional order (No Bragg Peak)
2) Translational symmetrybreaking in statistical measure with finite temperature
No Bragg peak, whileTranslational symmetry breaking
1) Ground states are described as some irregular patterns
2) They are generated by a rule, and robust with respect to thermal noise(irregularly frozen patterns at finite temperature)
![Page 14: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/14.jpg)
Do turbulent crystals exist? David Ruelle, Physica A 113, (1982)
AbstractWe discuss the possibility that, besides periodic and quasiperiodic crystals, there exist turbulent crystals as thermodynamic equilibrium states at non-zero temperature. Turbulent crystals would not be invariant under translation, but would differ from other crystals by the fuzziness of some diffraction peaks. Turbulent crystals could appear by breakdown of long range order in quasiperiodic crystals with two independent modulations.
![Page 15: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/15.jpg)
Current status of Ruelle’s question
Some constructed “chaotic patterns” with forgetting the stability against thermal noise
2) Translational symmetrybreaking in statistical measure with finite temperature
The heart of the problem is to find the compatibility between the two:
1) No long-range positional order (No Bragg Peak)
Is it possible ?
![Page 16: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/16.jpg)
A possible landscape picture
How to find this phenomenon ?
Typical configurations are classified into several groups each of which consists of configurations with macroscopic overlaps with some special irregular configuration
irregular
irregular
Irregular
Irregular
irregular irregular
irregularirregular
irregular
irregular
![Page 17: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/17.jpg)
The concept of overlap
2'1 i
iiNq
)',( σσ
)(qP
i) Divide the space into boxes each of which can have at most one particle
ii) Define the occupation variable for each site 1i
iif a particle exists
0i otherwise
ii )( σ Particle configuration
iii) Prepare two independent systems
iv) Define the overlap between the two:
v) Look into the distribution function of the overlap:
)()( qqP for the phase without symmetry breaking (like liquid)
![Page 18: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/18.jpg)
Overlaps in “turbulent crystals” )(qP
when two samples belong to different groups, there is no correlation between them
q0q *qq
when two samples belong to the same group, there is correlation between them
Typical configurations are classified into several groups each of which consists of configurations with macroscopic overlaps with some special irregular configuration
![Page 19: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/19.jpg)
Spin glass terminology
One step replica symmetry breaking(1-RSB)
![Page 20: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/20.jpg)
Example of the 1RSB phaseHard-constraint particles on random graphsReferences: Biroli and Mezard, PRL 88, 025501 (2002) and others
The contact number of each particle is less than 2
)ˆ()( qqqP
q7.6c ( Hukushima and Sasa, 2010)
Consistent with the cavity method (Krzakala, Tarzia, Zdeborova, 2007)
![Page 21: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/21.jpg)
This model was proposed as
a lattice model describing the idealized glass in statistical mechanical sense
In order to distinguish it fromidealized glass in the sense of MCT, and idealized glass in the sense of KCM, I call the idealized glass “Pure glass”.
![Page 22: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/22.jpg)
This means …
“Turbulent crystal” by Ruelle may be given by“pure glass in finite dimensions. “
We know many models that exhibit “pure glass” in the mean-field type description
No finite-dimensional model that exhibits “pure glass” has been proposed
(But, recall Bethier’s talk yesterday.)
![Page 23: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/23.jpg)
Problem we would like to solve
Construct a finite-dimensional model that exhibits “pure glass”
Artificial Glass Project
![Page 24: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/24.jpg)
Our first step result:
S. Sasa, Pure Glass in Finite Dimensions, PRL arXiv:1203.2406
20 minutes
![Page 25: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/25.jpg)
Part II MODEL
![Page 26: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/26.jpg)
Guiding principle of model construction An infinite series of “irregular” local minimum configurations generated by a deterministic rule
Statistical behavior of the model on the basis of an energy landscape of LMCs
irregular
irregular
Irregular
Irregular
irregular irregular
irregularirregular
irregular
irregular
![Page 27: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/27.jpg)
128 -states molecule
127,...,1,0
),,,,,,( )7()6()5()4()3()2()1(
1,0)( k
7
1
1)( 2k
kk
State of molecule
7 -spins
)()8( f
An irregular function
81 ,)( kkmark configuration in a unit cube 7,5,4,1 ,1)( kk
例:
Molecule a unit cube in the cubic lattice
![Page 28: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/28.jpg)
Hamiltonian
ii )(σ
Liiiii k 1|),,( 321Cubic lattice
ij
jikVH ),()( σ
Molecule configuration
Hamiltonian
ij NN-pair keij
1),( jikV A mark configuration in the positive k surface of is different from that in the negative k surface of i
j
1or 0),( jikV
Irregular function (choose it with probability ½ and fix it )
3LN
)3,2,1( k
A mark configuration in the positive k surface of is different from that in the negative k surface of i
j
![Page 29: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/29.jpg)
Example of interaction potential
1)',(1 V
29 106'
1)',(2 V 1)',(3 V
1,0),'(1 V 1),'(2 V 1),'(3 V
Choose it with probability ½ and fix it
![Page 30: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/30.jpg)
Statistical mechanicsij
jikVH ),()( σ
)(
)(1)( σσ HeZ
P
Hamiltonian
--- nearest neighbor interaction
--- translational invariant (PBC)
Canonical distribution
![Page 31: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/31.jpg)
Perfect matching configuration (PMC) ( construction of mark configurations )
(1) iteration (cellular automaton)
01 i02 i
03 i1i2i
3i
(0) put randomly in the surface
;do to1for 3 Li ;do to1for 2 Li
;do to1for 1 Li
put if
),,( 321 iiii
return; PMC
133 2
2 LL
0ki
1)8( i
![Page 32: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/32.jpg)
Properties of PMCs #1 typically irregular ! ( not yet proven )
2/3 Li Molecule configuration in the surface
#2 PMCs are local minimum ! (trivial)
32L
![Page 33: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/33.jpg)
Energy distribution of LMCs NHu /)(
σ A
LMCs are irregular
The energy density obeys a Gaussian distribution with dispersion O(1/N)( central limiting theorem ) N >>1
![Page 34: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/34.jpg)
Low temperature limit :D
AA set of configurations that reach the LMC by zero-temperature dynamics
Random Energy Model
*uu BThe minimum of energy density In the thermodynamic limit
σ B)(Condensation transition to a
σ
)exp(||1 NuD
ZP
![Page 35: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/35.jpg)
Part III Numerical experiments
![Page 36: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/36.jpg)
Energy density
NHu /)(ˆ σ
8,9,10,11L
uu ˆ
Free BC
![Page 37: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/37.jpg)
Energy fluctuation
udTduC 2
9,10,11L
7.4max )4.3/( Lu
23 uuLu
![Page 38: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/38.jpg)
Energy fluctuation
)))((( /1/ LLfL cu
A scaling relation:
0.1
21.07.4/1
![Page 39: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/39.jpg)
Thermodynamic transition
First order transition
Latent heat
![Page 40: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/40.jpg)
Nature of the low temperature phase
No Bragg peak
No internal symmetry breaking (e.g. Ising)
Condensation transition :
![Page 41: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/41.jpg)
Distribution of overlap
i
iiNq )',(1
,σ,σ
)',( σσTwo independent systems
Distribution function
);( qP
The overlap between the two
10L Free boundary condition (FBC)
2.1 5.1
![Page 42: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/42.jpg)
Part V Summary
![Page 43: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/43.jpg)
Summary
Turbulent crystals (by Ruelle)
Pure glass in finite dimensions
1-RSB (for spin glasses)
Review:
Question:
Result:
Proposal of a 128-state model
![Page 44: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/44.jpg)
Future problems
Complete theory
Molecular Dynamics simulation model
Laboratory experiments
Further numerical evidences
Simpler model ?
![Page 45: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/45.jpg)
![Page 46: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/46.jpg)
Selection by a boundary configuration
ii )( * Equilibrium configuration in a low temperature
Fix a boundary configuration
~
* *),(~1
iiiN
q
![Page 47: Turbulent Crystal and idealized glass](https://reader035.fdocuments.net/reader035/viewer/2022062520/56815e07550346895dcc57cf/html5/thumbnails/47.jpg)
Dynamics
17.1,165.1,16.1,155.1,15.1
1281))0(),((1)(
iiiq t
NtC