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摩擦モデルの非平衡統計力学アプローチ
Shoichi Ichinose
ichinose@u-shizuoka-ken.ac.jp
University of Shizuoka

理化学研究所研究会: 熱場の量子論とその応用 2016 ,8/22-8/24

Shoichi Ichinose (Univ. of Shizuoka)

摩擦モデルの非平衡統計力学アプローチ

理化学研究所研究会: 熱場の量子論とその応用 2/ 10

1. Introduction

Sec 1. Introduction: a. Dissipative Model.

Figure: 1 The spring-block model, (2 ).

Shoichi Ichinose (Univ. of Shizuoka)

摩擦モデルの非平衡統計力学アプローチ

理化学研究所研究会: 熱場の量子論とその応用 2/ 10

1. Introduction

Sec 1. Introduction: b. Dissipative Model.
Frictional Forcesκ sgn(ẋ)(stick slip, present work),
1 + 2α|ẋ|m : block mass M, k : spring const. MT −2 ,frictional parameters α = 2.5 TL−1 , κ = 1.0 LT −2 ,ℓ̄ : block length L, V̄ : Velocity of spring top LT −1 .

Fri = −η ẋ (rain drop, Fig.1), −m

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 modelof earthquakes’
Shoichi Ichinose (Univ. of Shizuoka)

摩擦モデルの非平衡統計力学アプローチ

理化学研究所研究会: 熱場の量子論とその応用 2/ 10

1. Introduction

Sec 1. Introduction: d. Energy with Dissipation
The classical equation of the dissipative block (Stick-Slip).
ẍ +κ sgn(ẋ)+ ω 2 x = ω 2 (V̄ t − ℓ̄).
1 + 2α|ẋ|This has been solved numerically by Runge-Kutta method(Continuous Time Method).
Energy conservation equation :∫ t
1 2 ω2 2κ |ẋ|
2d t̃
H[ẋ, x] ≡ ẋ + x + ω ℓ̄x +220 1 + 2α|ẋ|4th()∫ t2ω
1− ω 2 V̄
=ẋ 2 + x 2 + ω 2 ℓ̄x |t=0 = E0 .
t̃ ẋd t̃
220
5th(2)(3)

Three types of energy: 1 [4th] Dissipative energy ( hysteresis); 2 [5th]External work ( hysteresis); 3 Others. 0 ≤ t̃ ≤ t.
Shoichi Ichinose (Univ. of Shizuoka)

摩擦モデルの非平衡統計力学アプローチ

理化学研究所研究会: 熱場の量子論とその応用 2/ 10

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model a.Discrete Morse FlowTheoryn-th Energy Function
κ sgn(xn−1 − xn−2 )x1 + 2α|xn−1 − xn−2 |/hmkx 2+ 2 (x − 2xn−1 + xn−2 )2 , V (x) =+ k ℓ̄x,2h2Kn (x) = V (x) − hnk V̄ x + m(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)

摩擦モデルの非平衡統計力学アプローチ

理化学研究所研究会: 熱場の量子論とその応用 2/ 10

Sec 2. Spring-Block Model

Sec 2. Spring-Block Model b. Variat. Principle
Minimal Energy Priciple δKn (x)/δx|x=xn = 0.
k1(xn + ℓ̄ − nhV̄ ) + 2 (xn − 2xn−1 + xn−2 ) +mh√κ sgn(xn−1 − xn−2 )k,=0, ω≡1 + 2α|xn−1 − xn−2 |/hm(5)where n = 2, 3, 4, · · · , N − 1, N. Discrete Morse Flow Theory
Recursion relation among n-th, (n-1)-th and (n-2)-th
Kn (xn )/m ≡ En : DMF energy .
Parameters: V̄ = 0.1, ℓ̄ = 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 = −ℓ̄, (x1 − x0 )/h = 0.
Shoichi Ichinose (Univ. of Shizuoka)

摩擦モデルの非平衡統計力学アプローチ

理化学研究所研究会: 熱場の量子論とその応用 2/ 10Sec 2. Spring-Block Model

Sec 2. SB Model e. Movement xn , DMF result
ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100,
3’outSBd.dat’
2.5

21.5

10.5

0-0.5

-1-1.501020304050

Figure: 4 Movement, xn . The DMF solution (5) correctly reproduces thecontinuous-time solution: Stick region and slip region appear.

Shoichi Ichinose (Univ. of Shizuoka)摩擦モデルの非平衡統計力学アプローチ

理化学研究所研究会: 熱場の量子論とその応用 2/ 10

Sec 2. Spring-Block Model

Sec 2. SB Model g. Dissipative Energy, DMF result
ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100,
4’EneSB.dat’
32

10-1-2-3-405101520253035404550Figure: 6 Dissipative Energy. Stick intervals: 2 energy states ±ϵ for eachstick region. ϵ is ’quantized’. Slip intervals: connect −ϵ of a stick region
to +ϵ′ of the next stick one.
Shoichi Ichinose (Univ. of Shizuoka)

摩擦モデルの非平衡統計力学アプローチ

理化学研究所研究会: 熱場の量子論とその応用 2/ 10

Sec 2. Spring-Block Model

Sec 2. SB Model o. Frictional Force, DMF result
ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100,100000’FriForce.dat’ using 1:2

50000

0-50000-100000
05101520253035404550

Figure: 14 Frictional Force Total force Fn ≡ (En − En−1 )/(xn − xn−1 );Spring force Fnsp = ω 2 ∗ (Vnh − xn − ℓ̄); Friction force Frin ≡ Fn − Fnsp .
Fluctuating step-interval and steady one are repeatedly occurring. Theinterval distribution is similar to velocity-ratio (p.) and frictional energy(q.).
Shoichi Ichinose (Univ. of Shizuoka)

摩擦モデルの非平衡統計力学アプローチ

理化学研究所研究会: 熱場の量子論とその応用 2/ 10

Sec 2. Spring-Block Model

Sec 2. SB Model q. Frictional Energy, DMF result
ht= 0.00250,om= 1.00,eta= 0.000,size= 1.000Vel= 0.100,N1= 20000,NoutWidth= 100,’EQmom.dat’
20

100-10-20

05101520253035404550

Figure: 17 Frictional Energy FriE (n) ≡ Frin ∗ (xn − xn−1 ). Energy is’quantized’ in the fluctuating regions. The interval distribution is not thestick-slip one.
Shoichi Ichinose (Univ. of Shizuoka)

摩擦モデルの非平衡統計力学アプローチ

理化学研究所研究会: 熱場の量子論とその応用 2/ 10