## 2 Friction Motions and Friction Law  
Through extensive experiments, [PERSON] [26] and [PERSON] [35] demonstrated that rock friction is influenced by both sliding velocity (\(V\)), a state variable (\(\theta\)), and normal stress (\(\sigma\)). [PERSON] interpreted \(D_{c}\) as representing the slip necessary to renew surface contacts; in that case, the ratio \(\frac{D_{c}}{V}\) defines the average contact lifetime \(\theta\). They proposed that the frictional shear stress \(\tau\) could be described as  
\[\tau=\overline{\sigma}\left[\mu_{0}+aln\left(\frac{V}{V_{0}}\right)+bln\left( \frac{V_{0}\theta}{D_{c}}\right)\right] \tag{1}\]  
where \(\overline{\sigma}\) is the effective normal stress equal to the difference between the total normal stress (\(\sigma_{\text{n}}\)) and pore fluid pressure (\(P_{l}\)), and \(\mu_{0}\) is the steady-state friction coefficient at a reference sliding velocity of \(V_{0}\).  
The meanings of the parameters \(a\), \(b\) and \(D_{\text{c}}\) can be illustrated by an idealized velocity stepping test (Figure 1f) of a spring block system (Figure 1e). If the block is sliding at a steady velocity of \(V_{0}\) and the velocity is suddenly increased to \(V_{1}\) (a sudden change requires an infinitely stiff spring), the friction suddenly increases by \(\text{aln}\left(\frac{V_{1}}{V_{0}}\right)\). The friction stress then decays by an amount \(\text{bln}\left(\frac{V_{1}}{V_{0}}\right)\) roughly exponentially over a characteristic distance \(D_{\text{c}}\). If (\(a-b\)) > 0, the new steady-state shear stress is greater than the old one, and the behavior is said to be \"velocity strengthening\". If (\(a-b\)) < 0, the new steady-state shear stress is below the old one, and the behavior is \"velocity weakening\".  
The decay of the shear stress is governed by the evolution of the state variable \(\theta\). This is described by either the aging law [26]  
\[\frac{d\theta}{dt}=1-\frac{V\theta}{D_{c}} \tag{2}\]  
or the slip law ([PERSON], 1983) [35]  
\[\frac{d\theta}{dt}=-\frac{V\theta}{D_{c}}\ln\left(\frac{V\theta}{D_{c}}\right) \tag{3}\]  
The main difference between the two descriptions is that for zero velocity, \(\theta\) increases linearly with time for the aging law, but for the slip law, it changes only with non-zero velocity. Nevertheless, both laws give the same value for the steady state \(\theta\) = 0, \(\theta_{\text{ss}}\)= \(\frac{D_{c}}{V}\) and linearize to the same expression. Notwithstanding this difference, the two evolution laws (2a) and (2b) also have several common features. First, under a steady state, the state variable is a constant given by \(\theta_{\text{ss}}\)= \(\frac{D_{c}}{V}\) and in particular, \(\theta_{\text{s}}\)= \(\frac{D_{c}}{V}\). Second, the steady-state frictional strength, Equation (1) becomes  
\[\mu_{\text{ss}}=\mu_{0}+(a-b)ln\left(\frac{V_{\text{ss}}}{V_{0}}\right), \tag{4}\]  
and the behavior is \"velocity weakening\" or \"velocity strengthening\" depending on the sign of (\(a-b\)).  
The spring block system is an adequate representation of experiments in which slip occurs roughly simultaneously over the frictional surface within the precision of measurements. However, for a slip in a continuum, as in the earth, the stiffness depends on the length of the slipping zone [52; 53] and, in this case, the results of the spring block system are only an analog to earthquake behavior. While the classical RSF framework effectively describes fault slip behavior at low-to-intermediate velocities and over a broad range of geological conditions, it becomes increasingly limited at seismic slip rates (\(V>1\) m/s), especially under hydrothermal conditions. At such high velocities, additional dynamicweakening mechanisms must be considered to account for the dramatic reductions in friction observed in laboratory experiments and inferred from field observations [54; 55]. However, the present review primarily focuses on experimental studies conducted under the slower slip rate (\(V<1\) mm/s) relevant to earthquake nucleation conditions.