Geometric Approach to Dissipative Systems and Discrete Morse Flow - PowerPoint PPT Presentation

. . 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 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 / 46

1. Introduction Sec 1. Introduction: a.Dissipative Model. Figure: 1 The spring-block model, (7). 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 / 46

1. Introduction Sec 1. Introduction: b.Dissipative Model. Frictional Force − m κ sgn(˙ x ) Fri = − η ˙ x (rain drop) , x | (stick slip motion) , 1 + 2 α | ˙ k : spring const. MT − 2 , m : block mass M , frictional parameters α = 2 . 5 TL − 1 , κ = 1 . 0 LT − 2 , ¯ ¯ V : Velocity of spring top LT − 1 ℓ : block length L , . (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 of earthquakes’ 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 / 46

1. Introduction Sec 1. Introduction: c.Friction Forces. Fric Fric x x stick slip rain drop Figure: 2 Friction Forces [left] rain drop [right] stick slip model. 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 / 46

1. Introduction Sec 1. Introduction: d.Energy with Dissipation The classical equation of the dissipative block (Stick-Slip). x + κ sgn(˙ x ) x | + ω 2 x = ω 2 ( ¯ V t − ¯ ¨ ℓ ) . (2) 1 + 2 α | ˙ This has been solved numerically by Runge-Kutta method (Continuous Time Method). Energy conservation equation : ∫ t ∫ t x 2 + ω 2 x , x ] ≡ 1 κ | ˙ x | 2 x 2 + ω 2 ¯ t − ω 2 ¯ x | d ˜ ˜ xd ˜ H [˙ 2 ˙ ℓ x + V t ˙ t 1 + 2 α | ˙ 0 0 x 2 + ω 2 ( 1 ) 2 x 2 + ω 2 ¯ = 2 ˙ ℓ x | t =0 = E 0 . (3) Three types of energy: 1 [4th] Dissipative energy ( hysteresis); 2 [5th] x = dx (˜ t ) / d ˜ t , 0 ≤ ˜ External work ( hysteresis); 3 Others. ˙ t ≤ t . 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 / 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) 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 / 46

1. Introduction Sec 1. Introduction: f.Continuous vs DMF H H t n-2 n-1 n ∆ t+ t ∆ t 0 Continuous Theory Discrete Morse Flow Figure: 3 [left] Continuous Theory [right] Discrete Morse Flow. 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 / 46

Sec 2. Spring-Block Model Sec 2. Spring-Block Model a.Discrete Morse Flow Theory n-th Energy Function κ sgn( x n − 1 − x n − 2 ) K n ( x ) = V ( x ) − hnk ¯ V x + m 1 + 2 α | x n − 1 − x n − 2 | / h x 2 h 2 ( x − 2 x n − 1 + x n − 2 ) 2 , V ( x ) = kx 2 + m + k ¯ ℓ x , (4) 2 x : general position L, x n − 1 : ( n − 1)th , x n − 2 : ( n − 2)th , h : 1 step interval T 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 / 46

Sec 2. Spring-Block Model Sec 2. Spring-Block Model b.Variat. Principle Minimal Energy Priciple δ K n ( x ) /δ x | x = x n = 0. V ) + 1 k m ( x n + ¯ ℓ − nh ¯ h 2 ( x n − 2 x n − 1 + x n − 2 ) + √ κ sgn( x n − 1 − x n − 2 ) k 1 + 2 α | x n − 1 − x n − 2 | / h = 0 , ω ≡ 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 × 10 4 ( h · N = 50 Total Step Length(’Time’)) Initial condition: x 0 = − ¯ ℓ, ( x 1 − x 0 ) / h = 0 . See 2.e Movement, 2.f Velocity 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 / 46

Sec 2. Spring-Block Model Sec 2. Spring-Block Model c.Continuous Limit Continuous Limit h → 0 , n → large , nh = t n → t , x n − 1 − x n − 2 x , x n − 2 x n − 1 + x n − 2 → ˙ → ¨ x , (6) h 2 h ℓ ) − m κ sgn(˙ x ) x = k ( ¯ V t − x − ¯ m ¨ . (7) 1 + 2 α | ˙ x | This is the spring-block model. See Fig.1. DMF Numerical Result The graph of movement ( x n , 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. 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 / 46

Sec 2. Spring-Block Model Sec 2. Spring-Block Model d. Three Types of Energy Three types of energy DMF-energy E n ≡ K n ( x n ) / m = MAR n + EXT n + DIS n , β sgn( x n − 1 − x n − 2 ) 1 . DissipativeEnergy : DIS n = 1 + 2 α | x n − 1 − x n − 2 | / hx n , EXT n = − hn ω 2 ¯ 2 . ExternalEnergy : V x n 3 . MarkovianEnergy : MAR n = 1 mV ( x n ) + 1 2 h 2 ( x n − 2 x n − 1 + x n − 2 ) 2 . (8) 1 and 2 do not have hysteresis effect. Fig.8, 6, 13, 11 show the energy change as the step flows. 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 / 46

Sec 2. Spring-Block Model Sec 2. Spring-Block Model e.Movement, DMF ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, 3 ’outSBd.dat’ 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 0 10 20 30 40 50 Figure: 4 Spring-Block Model, Movement, x n The step-wise solution (5) correctly reproduces the continuous-time solution: stick-slip motion 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 / 46

Sec 2. Spring-Block Model Sec 2. Spring-Block Model f.Velocity, DMF ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, 1 ’VelSB.dat’ 0.5 0 -0.5 -1 0 5 10 15 20 25 30 35 40 45 50 Figure: 5 Spring-Block Model, Velocity, x n − x n − 1 / h The step-wise solution (5) correctly reproduces the continuous-time solution: 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 / 46

Sec 2. Spring-Block Model Sec 2. Spring-Block Model g.Dissipative Energy, DMF ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, 4 ’EneSB.dat’ 3 2 1 0 -1 -2 -3 -4 0 5 10 15 20 25 30 35 40 45 50 Figure: 6 Dissipative Energy, DMF DIS n of (8). Stick interval: 2 energy states ± ϵ for each stick region. ϵ is ’quantized’. Slip interval: connect − ϵ of a stick region to + ϵ ′ of next stick one. 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 / 46

Sec 2. Spring-Block Model Sec 2. Spring-Block Model h.Dissipative Energy, Cont. Time om= 1.00,NoutWidth= 40,N1= 20000,Vel= 0.10,el= 1.00,dt= 0.0025al= 2.50,be= 1.00 1 ’DisEne.dat’ 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 ∫ t x | ) d ˜ Figure: 7 Dissipative Energy, Cont.Time, 0 κ | ˙ x | / (1 + 2 α | ˙ t of (3). 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 / 46

Sec 2. Spring-Block Model Sec 2. Spring-Block Model i.DMF-Energy, DMF ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100, 5 ’DMFene.dat’ 0 -5 -10 -15 0 5 10 15 20 25 30 35 40 45 50 Figure: 8 Spring-Block Model, DMF Energy, DMF , E n of (8). 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 / 46