Supersymmetry in a classical world: new insights …aws/_Garching_2015_07_17_2.pdfSupersymmetry in a...

Supersymmetry in a classical world: new insights on stochastic dynamics from

topological field theory

1

Igor Ovchinnikov

Excellence Cluster Universe, Garching2015/7/17

University of California at Los Angeles

• Ubiquitous Chaotic Long-Range Order and State-of-Art Theory of Stochastic Differential Equations (SDEs) and Dynamical Systems (DS) Theory

• The Cohomological Theory of SDEs (ChT-SDE): Extended Hilbert Space; Topological Supersymmetry and Its Spontaneous Breaking (Chaos); Ergodicity and Time Reversal Symmetry Breaking; Phase Diagram and Noise-Induced Chaos.

• Example: Healthy Brain is at the Phase of Noise-Induced Chaos• Conclusion

Plan of the Talk

2Excellence Cluster Universe, Garching2015/7/17

3

Power-law (or scale free) statistics (e.g, Richter Scale) of highly nonlinear processes or events such as earthquakes

1/f, pink, or flicker noise (long-term memory effect)- algebraic power-spectra

Butterfly Effect, i.e., infinite memory of perturbations/init.cond.

Mysterious Chaotic Long-Range Order


4

Darwin Theory of Evolution (1859)

Punctuated Equilibria(Eldredge, Gauld, 1972)

Biological evolution



5

Econodynamics




6

Neurophysics Econodynamics




Geophysics


Magneto-hydrodynamics

Aerodynamics

Traffic


Soft condensed matter

Hydrodynamics

Flocking

Internet

7



Geophysics

Astrophysics


Magneto-hydrodynamics

Aerodynamics

Traffic


Soft condensed matter

Hydrodynamics

Flocking

Internet

8



9

Astrophysics:… a wide range of phenomena in astrophysics, such as planetary magnetospheres, solar flares, cataclysmic variable stars, accretion disks, black holes and gamma-ray bursts, and also to phenomena in galactic physics and cosmology…

Mysterious Chaotic Long-Range Order: Astrophysics


Hamilton dynamics

Least Action Principle,

probabilisticdeterministic

Quantum Theory

General Form O(partial)DEs


Euler Equations

- Hard Condensed Matter- Atomic and Molecular Physics- Elementary Particle Physics

Low-Energy Effective Equations

of Motion

General Form S(partial)DEs

Quantization

- Sort Condensed Matter- Hydro-,Aero- … dynamics- Everything “non-quantum”,

i.e., above the scale of quantum degeneracy/coherence

- All Sciences other than Physics

Generality of S(partial)DEs in Physics

))(()( txFtx

NoisetxFtx ))(()(

0)(/ txS

minS

/ˆ iSeU

Hamilton dynamics



Quantum Theory



Euler Equations


Low-Energy Effective Equations

of Motion


Quantization



- All Sciences other than Physics

Generality of S(partial)DEs in Physics

Domain of observation of CLRO

))(()( txFtx

NoisetxFtx ))(()(

0)(/ txS

minS

/ˆ iSeU

Theory of SDEs is older than the quantum theory and general relativity.Theories of Brownian motion: Smoluchowskii(1906), Einstein (1905), even earlier works


State-of-Art theory of S(partial)DEs

Complexity

13

Modern Picture: Open Questions


Turbulence

Non-equilibrium

dynamics

Chaotic

dynamics

ErgodicThermodynamic

equilibrium

Ergodic theory of

chaos

?

Theory of SDEs is older than the quantum theory and general relativity.Theories of Brownian motion: Smoluchowskii(1906), Einstein (1905), even earlier works

14

Accidental pictureCLRO is a “critical” phenomenon - some excitation has zero gap because the DS is at a “phase transition”Contradiction with ubiquity of CLROSelf-Organized Criticality: postulation of existence of mysterious tendency of self-fine-tuning into the phase transition into ordinary chaos

Symmetry breaking picture CLRO is a symmetry breaking phenomenon, and CLRO is a result of the Goldstone theorem. Requirement from ubiquity of CLROAll stochastic systems must possess such a symmetry and it must be a supersymmetry


Chaotic Long-Range Order

Chaotic Long-Range Order: Potential Origin

)())(()2())(()( 2/1 ttxetxFtx a

a

DS’s variables from

some topological

manifold, X, called

phase space


ChT-SDE: Stochastic Evolution

)())(()2())(()( 2/1 ttxetxFtx a

a

Temperature

A set of vector fields,

vielbeins, If (in)dependent

on x(t), it is said that noise

is (additive)multiplicative

Flow vector field


some topological

manifold, X, called

phase space



)())(()2())(()( 2/1 ttxetxFtx a

a

TemperatureGuassian white

Noise variables

A set of vector fields,

vielbeins, If (in)dependent

on x(t), it is said that noise

is (additive)multiplicative

Flow vector field


some topological

manifold, X, called

phase space

Possible generalizations: any noise; partial differential equations; flow

vector field and vielbeins with explicit time/space dependence and

“integral” or temporary (and spatially) nonlocal dependence on x…



( ')x t ' 2( ) ( ( '), )ttx t M x t

' 3( ) ( ( '), )ttx t M x t

' 1( ) ( ( '), )ttx t M x t



)())(()2())(()( 2/1 ttxetxFtx a

a



)())(()2())(()( 2/1 ttxetxFtx a

a

))'(,()( ),()'(:)(

maps offamily - estrajectoridependent Noise

'' txMtxtxtxM tttt

( ')x t ' 2( ) ( ( '), )ttx t M x t

' 3( ) ( ( '), )ttx t M x t

' 1( ) ( ( '), )ttx t M x t



)())(()2())(()( 2/1 ttxetxFtx a

a

))'(,()( ),()'(:)(


'' txMtxtxtxM tttt

( ')t

t'moment Timet't moment Time

( , )t

Temporal evolution

( ')t 2( , )t

3( , )t

1( , )t

- Hilbert space is the exterior algebra

- Consideration of generalized (not only total) probability

distributions is a mathematical necessity

1

1

( ) 1 ( ) ( )...( !) ( ) ... ( )k

k

k k i i ki ik x dx dx X

- GPDs in the coordinate-free setting = differential or k-forms:

(1) 1 2 1 2 1 2( ) ( | ) ( | )p x p x x dx p x x dx

( )

0

( ) ( )D

k

k

X X

A k-form is naturally coupled to k-

dimensional submanifolds (k-chains)

Examples: conditional and total probability

distributions

(2) 1 2( ) ( )p x p x dx dx

Meaning: in local coordinates

where k-chain belongs the

hyperplane 1,..,k Dx x consts

is the probability to find

within the chain, given

the other variables are known

( )

( ) ( ) 1

k

k k

cp

1,.., kx x

(2)c

(1)c


ChT-SDE: Hilbert Space

1

1

( ) 1 ( ) ( )...( !) ( ) ... ( )k

k

k k i i ki ik x dx dx X

- GPDs in the coordinate-free setting = differential or k-forms:


Standard Conditional Probability Distribution

11 1arg

arg arg

1

1

1 1

1

( ... ) ( .

Example: standard definition of Cond.Prob.Density on

In coordinate-free

.. | ) ( )

setti

... .

ng

..

... .( ) .

( . . |

.

..

k D kD ktot cond m

D km m

kc

D

k D k D

ond cond

D

k

P x x P x x P

P P

P P

x x x x

x x

x

x x

xx

d d

1

1 1arg

) ...

( ... ) .

.

.

.

.

k k

D Dtot cond m tot

D

D

dx dx

P

x

dxP P P x x dx

Quantum theory

Pr

( ) ( ) ( )Total

obabilityFunction

x x TPF x

ChT-SDE

Unthermalized, Thermalized, All vars.unstable vars. stable vars.

1 1 1( ) ... ( ) ... ( ) ...k k D Dx dx dx x dx dx TPF x dx dx

Bra-ket “factorization” of total probability density


ChT-SDE: Hilbert Space



)())(()2())(()( 2/1 ttxetxFtx a

a

))'(,()( ),()'(:)(


'' txMtxtxtxM tttt

( ')t


( , )t

Temporal evolution

( ')t 2( , )t

3( , )t

1( , )t



)())(()2())(()( 2/1 ttxetxFtx a

a

))'(,()( ),()'(:)(


'' txMtxtxtxM tttt

1

1

( )

...( ') ( ( ')) ( ') ... ( ')k

k

iik

i it x t dx t dx t


)(...)()))'(((),( 1

1

)(

...

*

' tdxtdxtxMt k

k

iik

iitt

))(,()'( ' txMtx tt )by Pullback (M t't

Temporal evolution

Variable substitution

( ')t 2( , )t

3( , )t

1( , )t



)())(()2())(()( 2/1 ttxetxFtx a

a

))'(,()( ),()'(:)(


'' txMtxtxtxM tttt

)by inducedpullback )(

)(ˆ ),'(ˆ)(

is),( GPDs,for evolution Stochastic

*

'

*

'''

(MM

MMtMt

XΩψ

t'ttt

Noisetttttt

( ')t 2( , )t

3( , )t

1( , )t



)())(()2())(()( 2/1 ttxetxFtx a

a

))'(,()( ),()'(:)(


'' txMtxtxtxM tttt


)(ˆ ),'(ˆ)(


*

'

*

'''

(MM

MMtMt

XΩψ

t'ttt

Noisetttttt

( ')t ( )t



)())(()2())(()( 2/1 ttxetxFtx a

a

))'(,()( ),()'(:)(


'' txMtxtxtxM tttt


)(ˆ ),'(ˆ)(


*

'

*

'''

(MM

MMtMt

XΩψ

t'ttt

Noisetttttt

H

LLLHeMaa eeF

Htt

tt

ˆ - mally,infiniteseor

ˆˆˆˆ ,ˆ

t

ˆ)'(

'

( ')t ( )t

)())(()2())(()( 2/1 ttxetxFtx a

a

H

LLLHeMaa eeF

Htt

tt


ˆˆˆˆ ,ˆ

t

ˆ)'(

'

2ijˆ ˆ g

noise induced metr

Noise indiced diffusion

.

c

..

i

a ae e i j

ij i ja a

L Lx x

g e e

"Average" flow, Lie or

physical derivative along F



))'(,()( ),()'(:)(


'' txMtxtxtxM tttt


)(ˆ ),'(ˆ)(


*

'

*

'''

(MM

MMtMt

XΩψ

t'ttt

Noisetttttt

( ')t ( )t

No Approximations!



)())(()2())(()( 2/1 ttxetxFtx a

a

))'(,()( ),()'(:)(


'' txMtxtxtxM tttt


)(ˆ ),'(ˆ)(


*

'

*

'''

(MM

MMtMt

XΩψ

t'ttt

Noisetttttt

H

LLLHeMaa eeF

Htt

tt


ˆˆˆˆ ,ˆ

t

ˆ)'(

'

( ) ( ) ( )

Stratonovich Interp. Weyl symmetrization

D i i j D

t i i a j aP x d x F e e P x d x

( ')t ( )t


ChT-SDE: Supersymmetry

t

Stochastic evolution on exterior algebra

The Fokker-Planck operator can be given explicitely supersymmetric form

ˆ ˆ ˆ ˆ ˆ, .

ˆˆˆ

ˆ ˆ

where ˆ ˆ ˆ,

a a

a a

F e e

F e e

H H L L L

H [d,d],

d i Θi L

interior multiplication, and the use of

Cartan formula, has been made with being exterior derivative.

Also,

ˆ ˆˆ ˆ

ˆ ˆ

is a sup ersymmetry of the modelˆ

0,

F

F F

i

L [d,i ] d

[d,H]

d

,

1 1

1 1

1 1

( ) ( )... ...

( ) ( )

Exterior differentive:

is very fundamental to albegraic topology

-) It is the matter of Stokes' theorem,

ˆ ( ) ..

,

. ( )

...

ˆ

k k

k k

k k

k i i k j i ii i i ij

k k

c c

d x dx dx x dx dx dxx

d

2

where is the boundary operator.

-) Its cohomology is De Rahm cohomology

is the algebraic representative of "boundary" operator

It is nilpotent, , "boundary of a boundary" is empty

ˆ

ˆ 0

d

d

*'

commutes with any pullback , and thus with the evolution o

Interpretation: continuous-time dynamics conserv

ˆ ˆ ˆ ˆ[ ,

es the "concept of boundary

( )] 0 [ , ]pe to 0r

"

ra ttd d M d M

*'( )ttM

d̂

d̂A Poincare dual of a chain

*'( )ttM


ChT-SDE: Meaning of Supersymmetry Operator

1 2 2 1

1 2 2 1

1

1

( )...

Properties of wedge product

are those for anticommuting or fermionic fields

Differential forms can be given as functions

of bosonic and fermionic field

.

s

k

i i i i

i i i i

k ii i

dx dx dx dx

dx

1

1

1

( )...

In these terms, exterior derivative has the form

.. ...

ˆ

k k

k

i k i ii i

i

i

dx

dx


ChT-SDE: Meaning of Supersymmetry Operator



t



ˆ ˆ ˆ ˆ ˆ, .

ˆˆˆ

ˆ ˆ

where ˆ ˆ ˆ,

a a

a a

F e e

F e e

H H L L L

H [d,d],

d i Θi L



Also,

ˆ ˆˆ ˆ

ˆ ˆ


0,

F

F F

i

L [d,i ] d

[d,H]

d

,

t



ˆ ˆ ˆ ˆ ˆ, .

ˆˆˆ

ˆ ˆ

where ˆ ˆ ˆ,

a a

a a

F e e

F e e

H H L L L

H [d,d],

d i Θi L



Also,

ˆ ˆˆ ˆ

ˆ ˆ


0,

F

F F

i

L [d,i ] d

[d,H]

d

,

n n

n n

Eigenstates of are either

-) supersymmetric singlets: but

All have zero eigenvalues !

-) or non-supersymmetric double

ˆ

ˆ ˆ0 somethin

ts: and

g

ˆ 0

H

d d

d



- Eigenvalues are either real or complex conjugate pairs (Ruelle-Pollicott resonances of DS theory)

- Eigensystem

FP operator is real and thus p

is complete, bi-orthogonal:

suedo-Hermitian

ˆ ˆ,n nH n E n n H E

- Real parts of eigenvalues are bounded from be

, 1,

For physical models with positive definite noise-metric

From

low

- All eigenstates are either supersymmetr

sup

ic

ersymm

singlets or non-

etry

s

knnn n n k n

upersymmetric doublets

- All non-zero eigenvalues correspond to non-supersymmetric pairs

- There always exist a supersymmetric state of the steady-state total probability distribution,

. ., the state ofi e "thermodynamic equili brium".

nIm E

nRe E

nIm E

nRe E

nIm E

nRe E

d- and PT reversal

symmetries broken

d-symmetry broken


ChT-SDE: Possible FP Spectra

Quantum theory

ˆ( )t iH

nIm E

nRe E

nIm E

nRe E

nIm E

nRe E


ChT-SDE: Possible FP Spectra

Quantum theory

ˆ( )t iH

Pathintegralrepresentation

Contribution only from

Physical meaning Value

Partition Function Ground states (in large time limit)

Stochastic number of periodic solutions (for some models)

For spectrum bChaotic behavior

Witten Index Supersymmetric states only

Partition function ofnoise (up to a topological factor)

Euler characteristic of

X

,Q

APBCD e

ˆ tHZ Tre

ˆ ˆ

W ( 1)F tHTr e

,Q

PBCD e

| |

2 gt EtZ e

Both have physical meaning only if the entire exterior algebra is the Hilbert space !

d- and PT reversal

symmetries broken

d-symmetry broken

Exp. decayExp. growth


ChT-SDE: Ground States and Ergodicity

nIm E

nRe E

nIm E

nRe E

nIm E

nRe E

d- and PT reversal

symmetries broken

d-symmetry broken Quantum theory

ˆ( )t iH

1 1 2 1ˆ ˆ ˆ ˆ1 ( ) ( ) ( ) ( )

1 1 1

11

ˆ ˆ ˆ ˆ( ).

Spectra a and b: correlators in the long t

.. ( ) ...

ime limit reduce to those of the ground states nly

ˆ

o

k k kt t H t t H t t H t t Hk k t t k

tg g

O t O t Z Tre O e e O e

N g O

1 1

1

ˆ ˆ( ) ( )1 1 1

Interpretation: ergodi

where in Heisenberg picture

For spectra c: one can use standard trick of

city property

quantum t

ˆ( )... ( ) ,

ˆ ˆ(

heory

(a little Wick rotation o

)

) t

k k

t t H t t H

t O t g

O t e O e

make theory "ergodic"

In order to study response of the system, introduce probing fields

T

( ( )) ( ( )) ( ) ( ( ))

ˆˆ ˆ ˆ ˆ ˆ

he evolution operator tran

( ) ( ) ( )

sforms as

b

i i b ib

b bf

F x t F x t t f x t

H H t H t L H t d

T

T

0

1 1

Chron.Ordering

ground

ˆ- ( )

1 1

statesI

1

1 0n

The response can be charactized by response correlators ( )

ˆ, ,

( ) ˆ ˆˆ ˆ, ( ) ... , ( )( ).... ( )

b

t

b bk k

f

H d

t

tt g f f kb bt

gk

Z Tr e

ZLim Z N g d t d t

t t

T

1

Heisenberg picture

ground state

11

s

ˆ ˆˆ ˆ ˆ ˆ[ , ] (, where )... , ( )b bk

g f f kg

g

N g d A g A t d t

All vanish by the definition of supersymmetric

(ground) states

Interpretation: forgets perturbations/initial

conditions in the long-time limit

Some do not vanish because the ground states are not

supersymmetric

Interpretation: remembers perturbations/initial conditions

even in the infinitely long time limit – the butterfly effect !

External perturbation


ChT-SDE: Butterfly Effect

1

0For perturbation of the (for (m ) )b bt t t

1 0 the infinitesemal shift of the tr( ( )) ajectoryf x t

Perturbed trajectory

Unperturbed trajectory


ChT-SDE: Expectation Values and Statistics

ˆ ˆ1 ( ) 1

0

Expectation value of an operator, , at moment

For wide class of operators including

where the "ergodic" probability density

ˆ ,

ˆ ˆ ˆ( ) lim

ˆ=

ˆ ( ) ( )

(

T t H tHT gn gT

gX

g

O t

O t Z n e Oe n N g O g

O O

O O x P x

P 1 1

Statistics works no matter if the supersymme

) ( ) ( )

try is broken or

.

n

.

ot

. Dg g gg

x N x x dx dx

U(x),P(x) U(x),P(x)-?

- Forgets initial conditions

- Thermalizes to a steady-state total probability distribution (the ground state)

- Never forgets initial conditions because the variable is not stable

- The steady-state total probability distribution is meaningless . The ground state is

Langevin 1D dynamics


ChT-SDE: Deterministic Ground States

( )g P x dx1g 1g ( )g P x dx

( ) 1xg g P dx ( ) 1g P x dxg

The ground state is not a distribution in unstable variables



homoclinic tangle

Ket – Poincare dual of global unstable manifoldsBra- Poincare dual of global stable manifolds

g

g

Saddle

chain ofboundary of dual Poincarechain of dual Poincareˆ d

Integrable (non-chaotic) flows: the supersymmetric ground states are Poincare duals of global unstable manifolds

Non-integrable (chaotic) flows: the non-supersymmetric ground states are Poincare duals of global unstable manifolds modified by a functional dependence on position

Langevin ODE on 2D torus

Langevin function is the height in the “3rd“ direction

1 , circle, 1-form

1 , circle, 1-form

0 , point, 2-form

2 ,circle, 1-form

3 entire space,

constant function

0 , entire space,

constant function

2 , cirlce, 1 form

3 , point, 2 form

0 1

1 2

( 1) 3 3 ( 1) 2 2

( 1) 1 1 ( 1) 0 (Euler characteristic of t0 )0 orus

W



Lorenz model

Unstable manifold

(picture by Hinke Osinga)

Type I

Params.

TE

C

Physical Laplacians do not break supersymmetry.

With the increase of noise temperature, the susy will eventually get restored – strong enough noise destroys chaotic long-range order


ChT-SDE: Stochastic Phase Diagram

, Temp.

High temp. ˆ ˆ ˆ ˆ ˆ: ˆ a a a aF e e e eH L L L L L

TE - Thermodynamic equilibrium

C - "Ordinary" chaos

Type II

Params.

TE

CN N

• Weak noise introduces exponentially weak overlap or noise-induced tunneling processes (instantons) between “local” ground states on different attractors.

• When tunneling processes break susy, Goldstinos representing parameters (modulii) are gapless. Result: power-law or scale free statistics.

• Mostly “regular” behavior along attractors interrupted by sudden unpredictable processes with scale-free statistics. Exists on the border of “ordinary” chaos. Typical description of Self-Organized Criticality!

• At higher temperatures, N-C is a crossover because external observer can not tell between different tunneling events


ChT-SDE: Stochastic Phase Diagram

, Temp.

Noise-induced Chaos

TE - Thermodynamic equilibrium

C - "Ordinary" chaos

N - Noise-Induced Chaos

Experimental evidence: neuroavalanches exhibit power-law statistics. This suggests brain has its susy broken by these processes – the N-phase


Example: Neurodynamics

A. Levina, J.M. Herrmann, and T. Geisel (2006).

Dynamical Synapses Give Rise to a Power-Law

Distribution of Neuronal Avalanches

Numerical Evidence:Neuroavalanches exhibit power-law for a range of parameters.

Recovery of consciousness goes

through a set of phase transitions

coma

consciousanesthetic


unconscious


Atkinson and Shiffrin model: short-term memory and long-term memory



coma

consciousanesthetic


unconscious

Scale:~<1minPhysics: fast electrochemical dynamics of light ionsSDEs: well know, e.g., Hudgkin-Huxley model

Scale:~>1min Physics: bio-chemical and mechanical rewiring of synapses


Atkinson and Shiffrin model: short-term memory and long-term memory



coma

consciousanesthetic


unconscious

Scale:~<1minPhysics: fast electrochemical dynamics of light ionsSDEs: well know, e.g., Hudgkin-Huxley model

Scale:~>1min Physics: bio-chemical and mechanical rewiring of synapses

Determined by the

same neurodynamical

phenomena


coma

consciousanesthetic


unconscious


The ChT-SDE picture

-) Conscious brain is in the N-phase. One can tell between neuroavalanches-this is weak-noise regime.

-) C-phase must correspond to neurodynamical phenomenon of “seizure” (like in epilepsy) – neurons fire non-

stop, non-integrable flow. Again, because the N-C transition is sharp, the brain is in the weak-noise regime

-) Coma is in the TE-phase where there is no chaotic dynamical memory. Therefore, chaotic dynamical memory

is the short-term neurodynamical memory.

-) Existence of short-term memory is a necessary condition for being conscious. Consciousness is within the N-

phase

Params.

Temp.

TE-phase N-phase C-phase

Seizure


Anti-anestheticconsciousunconscious

coma


anesthetic

The ChT-SDE picture


ChT-SDE: Low-Energy Effective Theory

Model Symmetry broken

Order Parameter Goldstone-Nambu Particle

Low-Energy Effective Theory

Ferromagnet O(3) Local Magnetization Spinwave LLG Equation

Crystal Spatial translation

Local Stress Transverse sound

Theory of Elasticity

Supersonductor (Global) U(1) “Wavefunction” of Bose-Condensate of

Cooper pairs

Zero sound GL Theory

… … …

Chaotic DS Top. Susy Unstable variables Goldstinos ?

Example: 2D vortex “dominated” turbulence- Order parameter: (half-of) spatial positions of vortices- Goldtinos are supersymmetric partners of

position of vortices- LEET: could it be Schwartz-type TFT? If yes,

does the concepts of braiding, topological quantum computing etc. apply somehow?

Time

The newly found approximation-free theory of stochastic dynamics

• reveals the mathematical origin of the ubiquitous dynamical/chaotic long-range – the spontaneous breakdown of topological supersymmetry that all natural/stochastic dynamical systems possess. “Chaos” (absence of order) is a misnomer in a certain sense because dynamical chaos is a low-symmetry or “ordered” phase.

• clarifies the concepts of thermodynamic equilibrium and ergodicity

• demystifies the controversial concept of self-organized criticality

• shows that dynamical properties of chaotic DSs can not be described by statistics. Low-energy effective theories of chaotic models, such as turbulent water, are those of gapless goldstinos/fermions – supersymmetric partners of unthermalized variables

• because of its multidisciplinary character and widest applicability, has a potential to bring together specialists in DSs theory, statistics, and topological field theories and/or algebraic topology that will result in cross fertilization of these mathematical disciplines

• has a potential to bring the studies of non-quantum systems to the new level of mathematical rigor and beauty, i.e., the level of supertsymmetric quantum (field) theories


Conclusion


Supersymmetry in Classical World…

Ergodic Dynamics

Infinitely long dynamics with the global ground state. Periodic B.C.

Transients (weak noise)

Finite-time dynamics starting and ending at different points; composite instanton

Examples: crumbling paper, Barkhausen jumps in ferromagnets, glasses, cascades and chain reactions of various types …


ChT-SDE: Transient Processes

Examples: turbulent water, brain …

outin

out

Time of

observation

d-symmetry is intrinsically broken on instantons –crackles must (and they do) exhibit power-law statistics

Transients can be though of being in the N-phase only in a sense of equivalent “ergodic” theory.

It is in this sense, that earthquakes are in N-phase

Tectonic

plates

Tectonic

“rings”

Additional Slides





Where to Search?

))(()( txFtx

-) Hamilton/Classical, Conservative dynamics

-) Electric circuitry

-) Magnetodynamics (LLG)

-) Schrodinger equation

-) (Magneto-)Hydro-, Bio-, Econo-, Neuro- Dynamics

Everything in Nature !

Flow vector field: Phase Portrait

System’s variables, points from a topological manifold called the phase space.


2ˆ( / 2 )i p m V

Where to Search?

))(()( txFtx

/

/

x H p

p H x

-) Hamilton/Classical, Conservative dynamics

-) Electric circuitry

-) Magnetodynamics (LLG)

-) Schrodinger equation

-) (Magneto-)Hydro-, Bio-, Econo-, Neuro- Dynamics

Everything in Nature !

Flow vector field: Phase Portrait

System’s variables, points from a topological manifold called the phase space.


Where to Search?

))(()( txFtx

))(2/ˆ( 2 xVmpi xdtdxVmpiS d

t ))(2/ˆ(( 2

dtxpHxpS ))(( /

/

x H p

p H x

Hamilton dynamics



Quantum Theory



Euler Equations


Low-Energy Eff. Theories


Quantization

Where to Search?



))(()( txFtx

NoisetxFtx ))(()(

0)(/ txS

minS

/ˆ iSeU

Deterministic Chaos is the major discovery within DS theory-) “the three-body problem” by Poincare (1887) -) Numerical rediscovery by Lorenz (1963) and others.

In hydrodynamics, chaos is known as turbulence

Deterministic chaos in ODEs– non-integrability in the sense of DS theory

This definition does not lead to the explanation of the Butterfly Effect

ODE is the domain of the DSs Theory

Vs.


Where to Search?

( ) ( , ) ( ) 2 ( )a

t ax t F x t e x t

temperature

SDE = ODE + Noise

trajectoryFlow

Diffusion

Deterministic Probabilistic

SDEODE

Initial point

Initial point

Flow

Average trajectory

noise

A set of vector fields


Where to Search?

Chaos

Thermodynamic

equilibrium

All natural DS’s

-) Ground state has unstable or “unthermalized” variables (positive Lyapunov exponents), in which it is not a probability distribution

-) The “butterfly effect”

-) This is an ordered (or low-symmetry) phase –opposed to the semantics of word “chaos”

-) Statistics not applicable (example, replica trick)

-) The stationary total probability distribution is (among) the ground state(s). Note: not the issue of its existence!

-) Thermodynamics, statistics are applicable (Markov chains)

-) Forgets initial conditions/perturbations


ChT-SDE: Butterfly Effect

Comparison with Supersymmetry in Quantum Theory:

-) Hadron Collider – the primary goal is to find supersymmetry in quantum

theory of elementary particles

ChT-SDE: supersymmetry exists (at least) everywhere else from quantum

theory

-) Supersymmetry (if exists) in quantum world must be spontaneously broken.

Prioblems, however, with theory of susy breaking – susy’s hard to break.

ChT-SDE: Chaos is the spontaneously broken susy.

In particular, all life forms are DS’s with spontaneously broken susy


65

Stephen Hawking: “it is in complexity that I think the most important developments of the next millennium will be."

Werner Heisenberg: "When I meet God, I am going to ask him two questions: Why relativity and why turbulence? I really believe he will have an answer for the first."

Richard Feynman: “turbulence -the most important unsolved problem of classical physics"

Importance Emphasized by Classics


ChT-SDE: Previous Approach

66

' '

( ) ( ) ( ) ( )

( ) (

If , what would be according to the SDE ?

This is how it derived:

No

) ,

( ) ( ) ( )

( )

w

Furth

( ) (1 / 2) ( ) ... ( )

,

:

er

D

D

Ns

i i j Di ij

Ns

i

f t f x P xt d x f t t

x t t x t x

f t t f x x P xt d x

f x x f x x x f x P xt d x

x

i i j

''jF( )= F ( F )F2) ( ( ) ...

( 0, 1 / 2)

ˆ ˆ: ( ) 1 ( ) ... ( )

ˆ

Ito, Stratonovich,

Thus Fokker-Planck Eq

( )

uation is:

i i

D

t

t x x t x t x x

f x tH P xt d x

P H P

F1/ 2( ) ( ( )) (2 ) ( ( )) ( ) ( )at ax t F x t e x t t t

,

,

*,

*,

( ) ( ( )) ( )

( ( ) )

( ) ( )

( ) ( )

Dt t t

Ns

t t t

Dt t t

Ns

Dt t t Ns

f t t f M x P xt d x

M x x

f x M P xt d x

f x M P xt d x

( )t

1 1 n n nt t t

n

1n

1n

time

' '

1a b abn n nnNs t tt


67

Accidental pictureCLRO is a “critical” phenomenon - some excitation has zero gap because the DS is at a phase transition

Chaos

All stochastic DSs

Symmetry breaking picture: CLRO is a symmetry breaking phenomenon - a gapless excitation is the Goldstone particle.

Self-Organized Criticality (Bak&Tang&Wiesenfeld, 1987): proposition to believe that there is a mysterious force that self-tunes parameters of some SDEs to the phase transition into “chaos”:1) A mystery explain by another mystery. Technically, this is just a

green light for using well-developed RG approach.2) If this is true, where does the CLRO come from in chaotic DSs?3) There was no definition of chaos for stochastic dynamics


Long-Range Order = Gapless Excitation

Ubiquity of CLRO

ContradictionAll DSs must possess this (super)symmetry

Requirement


68

Accidental pictureCLRO is a “critical” phenomenon - some excitation has zero gap because the DS is at a phase transition

Chaos

All stochastic DSs

Symmetry breaking picture: CLRO is a symmetry breaking phenomenon - a gapless excitation is the Goldstone particle.

Self-Organized Criticality (Bak&Tang&Wiesenfeld, 1987): proposition to believe that there is a mysterious force that self-tunes parameters of some SDEs to the phase transition into “chaos”:1) A mystery explain by another mystery. Technically, this is just a

green light for using well-developed RG approach.2) If this is true, where does the CLRO come from in chaotic DSs?3) There was no definition of chaos for stochastic dynamics


Long-Range Order = Gapless Excitation

Ubiquity of CLRO

ContradictionAll DSs must possess a (super)symmetry

Requirement

Such a supersymmetry does exist !-) Langevin SDEs: Parisi&Sourlas (1979) -) Its topological meaning: Witten (1982) -> ChTs (1988).-) Pseudo-Hermitian Ev. Op. Mostafazadeh (2002)-) Stochastic Evolution Op.: Ruelle (1987)-) ChT-SDE for all SDEs (2013)

Chaos

All stochastic DSs


Cohomological or Witten-

type topological field theory,

Parisi-Sourlas Quantization

Pseudo-Hermitian

quantum theory

Dynamical

systems theory

ChT-SDE – approximation-free theory of SDEs

Probability,

Statistics,

Thermodynamics


Geneology of ChT-SDE

ChT-SDE – dynamical generalization of statistics, or

stochastic generalization of DS theory

Mathematical Foundation: Cohomological or Supersymmetric Field Theories

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