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Non-equilibrium Statistical Approach to Friction Models

A geometric approach to the friction phenomena is presented. It is based on the holographic view which has recently been popular in the theoretical physics community. We see the system in one-dimension-higher space. The heat-producing phenomena are most widely treated by using the non-equilibrium statistical physics. We take 2 models of the earthquake. The dissipative systems are here formulated from the geometric standpoint. The statistical fluctuation is taken into account by using the (generalized) Feynman's path-integral.


INTRODUCTION
The system we consider consists of the huge number of particles (blocks) and the size of the constituent particles is the mesoscopic-scale. It is lager than 50 nm =5 × 10 −8 m and is far bigger than the atomic scale (∼ 10 −10 m). It is smaller than or nearly equal to the optical microscope scale (∼ 10 −6 m) in the branches such as the soft-matter physics, the nano-science physics and the biophysics. The larger end of the mesoscopic (length) scale depends on each phenomenon. For the earthquake it is about 10 −4 m.
The physical quantities, such as velocity, energy and entropy, are the statistically-averaged ones. It is not obtained by the deterministic way like the classical (Newton) mechanics. Renormalization phenomenon occurs not from the quantum effect but from the statistical fluctuation due to the uncertainty caused by the following facts. Firstly each particle obeys the Newton's law with different initial conditions. The total number of particles, N, is so large that we do not or can not observe the initial data. Usually we do not have interest in the trajectory of every particle and do not observe it. We have interest only in the macroscopic quantities: total energy and total entropy are the most important ones. Secondly, in the real system, the size and shape differ particle by particle. We regard the randomness as a part of fluctuation. Finally the models, presented in the following, contains discrete parameters (t n in Sec.2 and y n in Sec.3). As far as the discreteness is kept (in the case that we no not take the continuous limit), the quantities determined by the minimal principle include the inevitable ambiguity which is regarded as a part of fluctuation.
We exploit the computational step number n instead of (usual) time. The step flow is given by the discrete Morse flows theory [1,2]. In the first model (Sec.2), we adopt this step-wise approach for the time-development. The time variable is introduced as t n = nh (h : time-interval unit). In the second model (Sec.3), we take the approach for the space-propagation. The position variable is introduced as y n = na (a : space-interval unit). The non-equilibrium dissipative system is recently formulated using the discrete Morse flows theory combined with the (generalized) path-integral [3,4].

SPRING-BLOCK MODEL
We treat the movement of a block which is pulled by the spring which moves at the constant speedV. The block moves on the surface with friction. This is called the spring-block (SB) model. We adopt the discrete Morse flows method to treat this non-equilibrium system [1,2]. We take the following n-th energy function to define the step(n) flow.
where η is the friction coefficient and m is the block mass. h is the 'time' interval parameter.
x is the position of the block. The potential V(x) has two terms: one is the harmonic oscillator with the spring constant k, and the other is the linear term of x with a new parameter¯ (the natural length of the spring).V is the velocity (constant) with which the front-end of the spring moves. K 0 n is a constant which does not depend on x. The n-th step x n is determined by the energy minimum principle: δK n (x)| x=x n = 0 with the pre-known position at the (n-1)-th, x n−1 , and that at the (n-2)-th, x n−2 .
This is the ordinary one for the spring-block model. See Fig.1.
The graph of movement (x n , eq. (2)) is shown in Fig.2. From the graph, we see this system starts with the stickslip motion and reaches the steady state as n → ∞. Fig.3 shows the energy change as the step flows. It shows the energy changes in the similar way to the movement. The physical dimensions of the parameters in (3) are listed as Now we consider N copies of the one body system (2). N is sufficiently large, for example, 10 23 (1 mol). We are modeling the present statistical system as follows. The N particles are "moderately" interacting each other in such way that each particle almost independently moves except that energy is exchanged. The interaction is not so strong as to break the dynamics (2). We use Feynman's path-integral method in order to take the statistical average of this N-copies system. The statistical ensemble measure will be given explicitly.
From the energy expression (1), we can read the metric (geometry) of this mechanical system.
[Statistical Ensemble 1a] The first choice of the metric in the 3D (t,X,P) manifold is the Dirac-type one: where Fig.4. (The momentum variable P is the "extra" coordinate in the holographic view. ) The length between 0 ≤ t n = nh ≤ β and the ensemble measure are given by where the free energy F is defined. (e −βF is the partition function.) α is a parameter ('string tension') with the dimension of [Statistical Ensemble 1b] The second choice of the metric is the standard type: Then the statistical ensemble is given by using the following length: Note that w(t) decouples from y(t). The last expression is, whenV = 0,¯ = 0, the same as the partition function of the system of N(= β/h) harmonic oscillators with the frequency k/ηh [6]. The minimal path of (9), by changing y n → y, nh → t and using the variation y → y + δy, we obtain Comparing with (3), we notice 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 takeV = 0,¯ = 0.
[Statistical Ensemble 2] The first two statistical ensembles are based on the line in the 3D manifold (t,X,P). We present here another type which is based on the surface in the 3D space. First the 3D metric (Dirac type) is re-expressed in the following general form.
(ds 2 ) D ≡ 2V 1 (X)dt 2 + dX 2 + dP 2 ≡ e 1 G I J (X)dX I dX J , where we have introduced the dimensionless coordinates X I . Here we introduce the following surface to define a "path" (surface) in the 3D space.
where the radius parameter r is chosen to have the dimension of √ ML. See Fig.5. Then we can define the induced metric g i j on the 2D surface.
(ds 2 ) D on-path = 2V 1 (X)dt 2 + dX 2 + dP 2 on-path Using the (dimensionless) surface area A, the third partition function e −βF is given by where α is the (dimensionless) "string" constant and here is a model parameter. It is determined by fitting the present result with the experimental data. (Note 2V 1 r 2 is dimensionless. )

BURRIDGE-KNOPOFF MODEL
Let us take the following n-th energy function to define Burridge-Knopoff (BK) model in the step(n) flow method.
whereẋ n = dx n (t)/dt. t is the time variable. I 0 n is a constant term which does not depend on x(t). We consider the system of N particles (blocks) which distribute over the (1-dim) space {y}. The coordinate y is taken to be periodic: y → y + 2L. The particles (blocks) are moving around the equilibrium points {P n | n = 1, 2, · · · , n − 1, N} where P N ≡ P 0 (periodic boundary condition). The point P n is located at y = y n ≡ na (Na = 2L) where a is the 'lattice-spacing'. N(= 2L/a) is a huge number and the present system constitutes the statistical ensemble. The nth particle's position at t, x n (t) (deviation from the equilibrium point P n ) is determined by the energy minimal principle δI n (x)| x=x n = 0 with the pre-known movement of the (n-1)-th particle, x n−1 (t), and that of the (n-2)-th, where 0 ≤ t ≤ β, and F(ẋ n−1 ) and G(ẋ n−1 ) are some functions ofẋ n−1 . This recursion relation determines the space-propagation. We assume the system is periodic in time: t → t + β. This is Burridge-Knopoff model (Fig.6).
(See the recent review article ref. [5] for the use of BK model in the earthquake phenomena. ) G is newly introduced in the present paper.
In the continuous space limit: the step flow equation (16) reduces to Note that, in the third term of the above equation, there appears the velocity-gradient ∂ 2 x ∂y∂t . (When the system slowly oscillates and G is expanded as G(ẋ) = η + c 1ẋ + · · · , the first constant is the viscosity. ) The physical dimensions of the parameters in (18) are listed as Using the periodicity in time, we can read the metric (geometry) of this mechanical system.
where we assume ∆ẋ n−1 = ∆ẋ n .ṽ n is the longitudinal strain. The full(complete) metric, G I J , is constructed from (20) as where we use dṽ = d(∂x/∂y) = (∂v/∂y)dt. X I are the dimensionless coordinates. Note that we have here done the natural replacement: dy 2 → −dt 2 + dy 2 and the addition of ma 2 dv 2 . (For the damped harmonic oscillator limit, F = G = 0, V = K = 0, dy = 0, the line element reduces to −mdx 2 + kx 2 dt 2 + ma 2 dv 2 [6]. ) G I J (X) is the metric in 4D manifold (X I ). Note that both d 1 and d 2 have the same dimension of Length. d 1 (large scale) comes from V, whereas d 2 (small scale) from K. The 2 coordinates, y and x, both describe the position in the space, but their roles are different. In the terminology of the general relativity, the former is the general coordinate and the latter the local coordinate. (The momentum coordinate X 3 = v/d 3 is the "extra" coordinate in the holographic view.) Now we take the following map from the 2D space {(t, y)| 0 ≤ t ≤ β, 0 ≤ y ≤ 2L} to the 4D space (t, y, x, v). (The 4D space is called "target space" in the string-theory community. ) x =x(t, y), v =v(t, y), dx = ∂x ∂t dt + ∂x ∂y dy, dv = ∂v ∂t dt + ∂v ∂y dy.
This map expresses a 2D surface in the 4D space (Fig.7). On the surface, the line element (21) reduces to where ∂G ∂t = dG(v) dv ∂v ∂t and i = 0, 1. The number of blocks N is a huge number, the present system statistically fluctuates. We regard the ensemble as the collection of possible surfaces (x(t, y),v(t, y)) in the 4D space (t, y, x, v). The probability for each surface (system configuration) is specified by its area A(x,v) as follows. Using the (dimensionless) surface area A, the partition function e −βF is given by where α is a dimensionless model parameter. F is the free energy. α is determined by comparing the present result and the experimental data. The minimum area surface, which gives the main contribution to the above quantity, is given by the following equation.  x(t) = e −η t/2V {(η 2 /2ω 2 − 1)(sin Ωt)/Ω + (η /ω 2 ) cos Ωt} −¯ +V(t − η /ω 2 ) ,Ω = (1/2) 4ω 2 − η 2 = 9.99 , 0 ≤ t ≤ 2 , x(0) = −¯ ,ẋ(0) = 0.