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Geometric Approach to Dissipative Systems and Discrete Morse Flow Method

Shoichi Ichinose

ichinose@u-shizuoka-ken.ac.jp University of Shizuoka

YITP Workshop: Strings and Fields 2016 ,8/8-8/12

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

• 1. Introduction

Sec 1. Introduction: a.Dissipative Model.

Figure: 1 The spring-block model, (7).

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

• 1. Introduction

Sec 1. Introduction: b.Dissipative Model.

Frictional Force Fri = −η ˙ x (rain drop), −m κ sgn(˙ x) 1 + 2α|˙ x| (stick slip motion), m : block mass M, k : spring const. MT −2, frictional parameters α = 2.5 TL−1, κ = 1.0 LT −2, ¯ ℓ : block length L, ¯ V : Velocity of spring top LT −1 . (1)

• 1. Burridge and Knopoff, Bull. Seismol. Soc.Am.1967
• 2. Carlson and Langer PRL, PRA 1989

’Mechanical model of an earthquake fault’

• 3. Mori and Kawamura, J. Geoph. Res. 2006

’Simulation study of the one-dimensional Burridge-Knopoff model

• f earthquakes’

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

• 1. Introduction

Sec 1. Introduction: c.Friction Forces.

Fric Fric x x rain drop stick slip

Figure: 2 Friction Forces [left] rain drop [right] stick slip model.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

• 1. Introduction

Sec 1. Introduction: d.Energy with Dissipation

The classical equation of the dissipative block (Stick-Slip). ¨ x + κ sgn(˙ x) 1 + 2α|˙ x| + ω2x = ω2( ¯ V t − ¯ ℓ). (2) This has been solved numerically by Runge-Kutta method (Continuous Time Method). Energy conservation equation : H[˙ x, x] ≡ 1 2 ˙ x2 + ω2 2 x2 + ω2¯ ℓx + ∫ t κ |˙ x| 1 + 2α|˙ x|d˜ t − ω2 ¯ V ∫ t ˜ t ˙ xd˜ t = (1 2 ˙ x2 + ω2 2 x2 + ω2¯ ℓx ) |t=0 = E0 . (3) Three types of energy: 1 [4th] Dissipative energy ( hysteresis); 2 [5th] External work ( hysteresis); 3 Others. ˙ x = dx(˜ t)/d˜ t , 0 ≤ ˜ t ≤ t.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

• 1. Introduction

Sec 1. Intro.: e.Discrete Morse Flow(DMF) Theory

Time should be re-considered, when dissipation occurs. Non-Markovian effect, Hysteresis effect → Step-Wise approach to time-development. Connection between step n, n − 1 and n − 2 is determined by the minimal energy principle. Time is ”emergent” from the principle. Direction of flow (arrow of time) is built in from the beginning. n = 0< 1< 2< · · · New approach to Statistical Fluctuation Discrete Morse Flow Method(Kikuchi, ’91) Holography (AdS/CFT, ’98)

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

• 1. Introduction

Sec 1. Introduction: f.Continuous vs DMF

H H t t+ t ∆ t ∆ n-2 Discrete Morse Flow Continuous Theory n-1 n

Figure: 3 [left] Continuous Theory [right] Discrete Morse Flow.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model a.Discrete Morse Flow Theory

n-th Energy Function Kn(x) = V (x) − hnk ¯ V x + m κ sgn(xn−1 − xn−2) 1 + 2α|xn−1 − xn−2|/h x + m 2h2(x − 2xn−1 + xn−2)2 , V (x) = kx2 2 + k ¯ ℓx, (4) x: general position L, xn−1: (n − 1)th , xn−2: (n − 2)th , h: 1 step interval T

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model b.Variat. Principle

Minimal Energy Priciple δKn(x)/δx|x=xn = 0. k m(xn + ¯ ℓ − nh ¯ V ) + 1 h2(xn − 2xn−1 + xn−2) + κ sgn(xn−1 − xn−2) 1 + 2α|xn−1 − xn−2|/h = 0 , ω ≡ √ k m , (5) where n = 2, 3, 4, · · · , N − 1, N. Recursion relation among n-th, (n-1)-th and (n-2)-th Parameters: ¯ V = 0.1, ¯ ℓ = 1, ω = 1.0, κ = 1.0, α = 2.5 1 Step Interval: h = 2.5 × 10−3, Total Step Number: N = 2 × 104 (h · N = 50 Total Step Length(’Time’)) Initial condition: x0 = −¯ ℓ, (x1 − x0)/h = 0. See 2.e Movement, 2.f Velocity

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model c.Continuous Limit

Continuous Limit h → 0 , n → large , nh = tn → t , xn−1 − xn−2 h → ˙ x , xn − 2xn−1 + xn−2 h2 → ¨ x , (6) m¨ x = k( ¯ V t − x − ¯ ℓ) − m κ sgn(˙ x) 1 + 2α|˙ x| . (7) This is the spring-block model. See Fig.1. DMF Numerical Result The graph of movement (xn, eq.(5)) is shown in Fig.4. Ordinary approach: Solving (7) numerically by Runge-Kutta 4. Same result as ours. The velocity change is shown in Fig.5.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model d. Three Types of Energy

Three types of energy DMF-energy En ≡ Kn(xn)/m = MARn + EXTn + DISn ,

• 1. DissipativeEnergy : DISn =

β sgn(xn−1 − xn−2) 1 + 2α|xn−1 − xn−2|/hxn ,

• 2. ExternalEnergy :

EXTn = −hnω2 ¯ V xn

• 3. MarkovianEnergy :

MARn = 1 mV (xn) + 1 2h2(xn − 2xn−1 + xn−2)2 . (8) 1 and 2 do not have hysteresis effect. Fig.8, 6, 13, 11 show the energy change as the step flows.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model e.Movement, DMF

• 1.5
• 1
• 0.5

0.5 1 1.5 2 2.5 3 10 20 30 40 50 ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, ’outSBd.dat’

Figure: 4 Spring-Block Model, Movement, xn The step-wise solution (5) correctly reproduces the continuous-time solution: stick-slip motion

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model f.Velocity, DMF

• 1
• 0.5

0.5 1 5 10 15 20 25 30 35 40 45 50 ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, ’VelSB.dat’

Figure: 5 Spring-Block Model, Velocity, xn − xn−1/h The step-wise solution (5) correctly reproduces the continuous-time solution:

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model g.Dissipative Energy, DMF

• 4
• 3
• 2
• 1

1 2 3 4 5 10 15 20 25 30 35 40 45 50 ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, ’EneSB.dat’

Figure: 6 Dissipative Energy, DMF DISn of (8). Stick interval: 2 energy states ±ϵ for each stick region. ϵ is ’quantized’. Slip interval: connect −ϵ

• f a stick region to +ϵ′ of next stick one.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model h.Dissipative Energy,

• Cont. Time

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 20 30 40 50

• m= 1.00,NoutWidth= 40,N1= 20000,Vel= 0.10,el= 1.00,dt= 0.0025al= 2.50,be= 1.00

’DisEne.dat’

Figure: 7 Dissipative Energy, Cont.Time, ∫ t

0 κ |˙

x|/(1 + 2α|˙ x|)d˜ t of (3).

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model i.DMF-Energy, DMF

• 15
• 10
• 5

5 5 10 15 20 25 30 35 40 45 50 ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, ’DMFene.dat’

Figure: 8 Spring-Block Model, DMF Energy, DMF , En of (8).

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model j.Hamiltonian, Cont. Time

• 0.5
• 0.495
• 0.49
• 0.485
• 0.48
• 0.475

10 20 30 40 50

• m= 1.00,NoutWidth= 40,N1= 20000,Vel= 0.10,el= 1.00,dt= 0.0025al= 2.50,be= 1.00

’Hamil.dat’

Figure: 9 Spring-Block Model, Hamiltonian, ContTime , H of (3). E0 = −0.5 appears at the beginning.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model j’.Energy Conservation, DMF+Cont. Time

• 0.501
• 0.5008
• 0.5006
• 0.5004
• 0.5002
• 0.5

10 20 30 40 50 60 70 80 90 100 ht= 0.00100,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 50000,NoutWidth= 100, ’EneCon.dat’

Figure: 9’ Spring-Block Model, Energy Conservation, DMF+ContTime , h = 0.001, N = 5 × 104(high resolution), The constant value −0.5001 is different from E0 = −0.5.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model k.MARn, DMF

2 4 6 8 5 10 15 20 25 30 35 40 45 50 ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, ’MarEneSB.dat’

Figure: 10 Spring-Block Model, MARn, DMF , MARn of (8). This is very similar to the next continuous time result.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model l.Mar, Cont. Time

• 1

1 2 3 4 5 6 7 8 10 20 30 40 50

• m= 1.00,NoutWidth= 40,N1= 20000,Vel= 0.10,el= 1.00,dt= 0.0025al= 2.50,be= 1.00

’MarEne.dat’

Figure: 11 Spring-Block Model, Mar, Cont. Time , ˙ x2/2 + ω2x2/2 + ω2¯ ℓx

• f (3).

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model m.External Energy EXTn, DMF

• 15
• 10
• 5

5 5 10 15 20 25 30 35 40 45 50 ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, ’ExtWsb.dat’

Figure: 12 Spring-Block Model, External Energy, DMF, EXTn of (8).

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model n.External Energy Ext,

• Cont. Time
• 10
• 9
• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

10 20 30 40 50

• m= 1.00,NoutWidth= 40,N1= 20000,Vel= 0.10,el= 1.00,dt= 0.0025al= 2.50,be= 1.00

’ExtEne.dat’

Figure: 13 Spring-Block Model,External Energy, ContTime , −ω2 ¯ V ∫ t

0 ˜

t ˙ xd˜ t of (3).

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model o.Frictional Force, DMF

• 100000
• 50000

50000 100000 5 10 15 20 25 30 35 40 45 50 ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, ’FriForce.dat’ using 1:2

Figure: 14 Spring-Block Model,Frictional Force , DMF Total force Fn ≡ (En − En−1)/(xn − xn−1); Spring force F sp

n = ω2 ∗ (Vnh − xn − ¯

ℓ); Friction force Frin ≡ Fn − F sp

n . Fluctuating step-interval and steady one

are repeatedly occurring. The interval distribution is similar to velocity-ratio (p.) and frictional energy (q.).

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model p.vn/vn−1, DMF

• 120
• 100
• 80
• 60
• 40
• 20

5 10 15 20 25 30 35 40 45 50 ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, ’vRatio.dat’

Figure: 15 Spring-Block Model,Velocity ratio, DMF , vn/vn−1 = (xn − xn−1)/(xn−1 − xn−2)

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model p’.Frictional Force,

• Cont. Time
• 1000

1000 2000 3000 4000 5000 6000 7000 8000 9000 5 10 15 20 25 30 35 40 45 50

• m= 1.00,NoutWidth= 40,N1= 20000,Vel= 0.10,el= 1.00,dt= 0.0025al= 2.50,be= 1.00

’FriForce.dat’ using 1:2

Figure: 16 Spring-Block Model,Frictional Force , ContTime

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model q.Frictional Energy, DMF

• 20
• 10

10 20 5 10 15 20 25 30 35 40 45 50 ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, ’EQmom.dat’

Figure: 17 Spring-Block Model, Frictional Energy , DMF EQmom(n) = Frin ∗ (xn − xn−1). Energy is ’quantized’ in the fluctuating

• regions. The interval distribution is not the stick-slip one.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 3. Geometry of SpringBlockModel(Fri = −η ˙ x) : a.Definition of Bulk(t,X,P) Metric

¨ x + η m ˙ x + ω2x = ω2( ¯ V t − ¯ ℓ). (9) η: viscosity, m: mass. Line Element（geometry） of Spring-Block Model, Fri = −η ˙ x ∆sn

2 ≡ 2h2Kn(xn) = 2 dt2V1(Xn) + (∆Xn)2 + (∆Pn)2,

V1(Xn) ≡ k 2{( Xn √ηh)2 + 2¯ ℓ Xn √ηh − 2nh ¯ V Xn √ηh}, dt ≡ h , (10) where Xn ≡ √ηhxn, Pn/√m ≡ hvn = (xn − xn−1),.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 3. Geometry of SBM : b.Metric on-path

From (20), the metric(first choice) in the 3D (t,X,P) is (ds2)D ≡ 2V1(X)dt2 + dX 2 + dP2, R0,0 = − k ηh (¯ ℓ − Vt)2 x(x + 2(¯ ℓ − Vt)), (ds2)D − on-path (X = y(t), P = w(t)) → (2V1(y) + ˙ y 2 + ˙ w 2)dt2, (11) where {(y(t), w(t))|0 ≤ t ≤ β} is a path (line) in the 3D space. We call the metric (ds2)D the Dirac-type one. See Fig.19.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 3. Geometry of SBM c.Path in 3D (t,X,P)

Figure: 18 The path {(y(t), w(t), t)|0 ≤ t ≤ β} of line in 3D bulk space (X,P,t).

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 3. Geometry of SBM : d.1st Statistical Ensemble

Let us consider the statistical system of N copies of the spring-block

• model. They are independently moving except that the energy is

exchanged. Length is, for a path (y(t), w(t)), defined as LD = ∫ β ds|on−path = ∫ β √ 2V1(y) + ˙ y 2 + ˙ w 2dt = h

β/h

∑

n=0

√ 2V1(yn) + ˙ y 2

n + ˙

w 2

n,

e−βα−1F = ∫ ∏

n

dyndwne− 1

α LD ,

dµ = e− 1

αLD ∏

t

DyDw, (12) where the free energy F is defined. (β : total time, α : string tension) All paths are taken into account.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 3. Geometry of SBM : e.(2nd) Metric in SBM

The second choice of the metric is the standard type(S.I.,2010): (ds2)S ≡ 1 dt2[(ds2)D]2 − on-path → (2V1(y) + ˙ y 2 + ˙ w 2)2dt2. (13)

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 3. Geometry of SBM : f.2nd Statistical Ensemble

Length LS = ∫ β ds|on−path = ∫ β (2V1(y) + ˙ y 2 + ˙ w 2)dt = h

β/h

∑

n=0

(2V1(yn) + ˙ y 2

n + ˙

w 2

n),

dµ = e− 1

αLSDyDw,

e−βα−1F = ∫ ∏

n

dyndwne− 1

α LS = (const)

∫

β/h

∏

n=0

dyne− h

α (2V1(yn)+ ˙

y2

n ),(14)

where wn is integrated out.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 3. Geometry of SBM : g. Path Integral Result

• f (14) Free Energy F

The free energy F is obtained as F( ¯ V , ¯ ℓ, ω0, α, β, η, h) = − α 2β ln ω0 πβ sinh ω0 + ηhω0

2 ¯

ℓ β (− ¯ ℓ β + ¯ V ) +ηh ¯ V 2 (cosh ω0 sinh ω0 ω0 − 2 sinh ω0 ω0 + 1 − ω02 3 ) . (15) Taking the values: α = 1 , β = 1 , h = 1 , η = 1 , β √ k/hη = ω0 = 0.881374 , sinh(ω0) = 1, (16) F is shown as

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 3. Geometry of SBM : h.Free Energy ( ¯ V >> ¯ ℓ = 0.01, 0 < ¯ V < 5)

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 1 2 3 4 5 al= 1.00,be= 1.00,h= 1.00,eta= 1.00,omr= 0.88,el= 0.0100 ’FreeE.dat’

Figure: 19 Free Energyfor the rapid pulling velocity. The ¯ V = 0 value is the harmonic oscillator one. (The first term in RHS of (15)).

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 3. Geometry of SBM : i.Free Energy ( ¯ V << ¯ ℓ = 1.0, 0 < ¯ V < 0.01)

• 0.142
• 0.141
• 0.14
• 0.139
• 0.138
• 0.137
• 0.136
• 0.135
• 0.134
• 0.133

0.002 0.004 0.006 0.008 0.01 al= 1.00,be= 1.00,h= 1.00,eta= 1.00,omr= 0.88,el= 1.0000 ’FreeE.dat’

Figure: 20 Free Energyfor the slow pulling velocity.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 3. Geometry of SBM : j.Minimal Path

The minimal path of (14), by changing to the continuous case yn → y, nh → t and using the variation y → y + δy, we obtain −ηh¨ x = k( ¯ V t − x − ¯ ℓ), x = y √ηh . (17)

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 2. Spring-Block Model

Sec 3. Geometry of SBM : k.Euclidean time

1) the viscous term disappeared; 2) the mass parameter m is replaced by ηh; 3) the sign in front of the acceleration-term (inertial-term) is different. By changing to the Euclidean time τ = it, the above equation reduces to the harmonic oscillator when we take ¯ V = 0, ¯ ℓ = 0.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 4. Conclusion

Sec 4. Conclusion a.Geometry of Dissipative Systems

Spring-Block Model (rain drop) ∆sn

2 = 2 dt2V1(Xn) + (∆Xn)2 + (∆Pn)2,

V1(Xn) ≡ k 2{( Xn √ηh)2 + 2¯ ℓ Xn √ηh − 2nh ¯ V Xn √ηh}, (18) where Xn ≡ √ηhxn, Pn/√m ≡ hvn = (xn − xn−1),. Spring-Block Model (stick-slip) ∆sn

2 = dt2ω2{Xn 2 + 2(¯

ℓ − nh ¯ V )Xn} ± 2(dt)2κ 1 + 2α|Pn|Xn + (∆Pn)2 .(19)

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 4. Conclusion

Sec 4. Conclusion b.Statistical Models

Spring-Block Model (rain drop) e−βα−1F = ∫ ∏

n

dyndwne− 1

α LS,

LS = ∫ β ds|on−path = h

β/h

∑

n=0

(2V1(yn) + ˙ y 2

n + ˙

w 2

n),

(20) Spring-Block Model (stick-slip) LS = ∫ β ds|on−path = h

β/h

∑

n=0

{ω2yn

2 + 2ω2(¯

ℓ − ¯ V nh)yn ± 2κ 1 + 2α|wn|yn + ˙ w 2

n},

(21)

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 4. Conclusion

Sec 4. Conclusion c.Burridge-Knopoff Model

Figure: 21 Burridge-Knopoff Model (23)

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 4. Conclusion

Sec 4. Conclusion d.Energy Function

n-th energy function to define Burridge-Knopoff (BK) model in the step(n) flow method. In(x) = −xF(˙ xn−1) + G(˙ xn−1)1 a(x − xn−1)(˙ xn−1 − ˙ xn−2) +m 2 (dx dt )2 − k 2(x − Vt)2 + K 2a2(x − 2xn−1 + xn−2)2 + I 0

n ,

(22) where ˙ xn = dxn(t)/dt. t is the time variable.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 4. Conclusion

Sec 4. Conclusion e.DMF Equation

−md2xn dt2 − F(˙ xn−1) + G(˙ xn−1) ˙ xn−1 − ˙ xn−2 a −k (xn − Vt) + K a2 (xn − 2xn−1 + xn−2) = 0, (23) where 0 ≤ t ≤ β, and F(˙ xn−1) and G(˙ xn−1) are some functions of ˙ xn−1.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 4. Conclusion

Sec 4. Conclusion e’.DMF vs Runge-Kutta

The different results between DMF and Runge-Kutta imply the importance to reexamine the statistical analysis of the Burridge-Knopoff model.

• 1. Slides Sec.2g,i,q: energy looks ’quantized’
• 2. Slide Sec.2j’: energy conservation -0.5001 ̸= E0 = −0.5
• 3. Slide Sec.2o: frictional force fluctuates
• 4. Slide Sec.2q: frictional energy fluctuates

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 4. Conclusion

Sec 4. Conclusion f.Multiple Scales

Multiple scales exist. SB model(rain drop): 1. the natural length of the string ¯ ℓ

• 2. the external velocity ¯

V .

• 3. the spring constant k.
• 4. viscosity η.

The use of dimensionless quantities clarifies the description. The multiple scales indicate the existence of the fruitful phases in the present statistical systems.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

Sec 4. Conclusion

Sec 4. Conclusion g.Finally

The dissipative systems are solved by the minimal principle. The difficulty of the hysteresis effect (non-Markovian effect) is avoided in the present approach. These are the advantage of the discrete Morse flow method. We do not use the ordinary time t, instead, exploit the step number n (tn = nh). Several theoretical proposals for the statistical ensembles, appearing in the friction phenomena, are made. It is necessary to numerically evaluate the models with the proposed ensembles and compare the result with the real data appearing both in the natural phenomena and in the laboratory experiment.

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46

• 5. References

Sec 5. References

• 1. N. Kikuchi, NATO Adv. Sci. Inst. Ser. C: Math. Phys.
• Sci. 332, Kluwer Acad. Pub., 1991, p195-198
• 2. N. Kikuchi, Nonlin. World 131(1994)
• 3. S. Ichinose, ”Velocity-Field Theory, Boltzmann’s

Transport Equation, Geometry and Emergent Time”, arXiv: 1303.6616(hep-th), 39 pages

• 4. S. Ichinose, Trib.Int. 93PA(2016)446,

arXiv:1404.6627(cond-mat)

• 5. H. Kawamura, T. Hatano, N. Kato, S. Biswas and B.K.

Chakrabarti, Rev.Mod.Phys.84(2012)839, arXiv:1112.0148

• 6. S. Ichinose, Proc. 5-th World Tribology Congress

(Torino, Italy, 2013.09.8-13), arXiv:1305.5386

Shoichi Ichinose (Univ. of Shizuoka) Geometric Approach to Dissipative Systems and Discrete Morse Flow Method YITP Workshop: Strings and Fields 2016 ,8/8-8/12 / 46