## 2 A Thermodynamic Model of Post Seismic Relaxation  
A post-seismic transient deformation stage, where strain varies with time at a decreasing rate, occurs following the instantaneous deformation during an earthquake. Transient deformation is followed by the return to steady-state deformation, where the strain rate is a constant after some time following the main event. Following [33], we define a relaxation function \(R(t)\), where \(t\) is the time after the main seismic event, as a perturbation of the displacement given by the generic expression:  
\[u(t)=u_{\infty}+S\;R(t) \tag{1}\]  
where \(u_{\infty}\) is the equilibrium value of displacement after a long time and \(S\) is a scaling factor. Notice that \(R(t\rightarrow\infty)\approx 0\), while \(u_{\infty}=u(t\rightarrow\infty)\) is the remaining distortion at the end of the relaxation process. The initial value \(u_{o}\) of the post-seismic deformation immediately after the co-seismic perturbation \(\delta u_{c}\) is given from Equation (1) as \(u_{o}=u(t=0^{+})=u_{\infty}+S\;R_{O}\), where \(R_{0}=R(t=0)\). The scaling factor \(S\) of the relaxation constrains the rheological properties of the medium [34; 35] and depends on the rheology of the crust and upper mantle, as well as on the local conditions around the site (i.e., crustal heterogeneities). In addition, \(S\) depends on the stress released in the medium and reflects the response of the medium (i.e., of the magnitude of the earthquake event) after the co-seismic displacement due to earthquake. In most of the cases a simple relaxation mechanism with a single relaxation time \(\tau\) that corresponds to a relaxation function \(\exp{(-t/\tau)}\) (i.e., not varying logarithmically with time) is the simplest rheological model introduced to explain post-seismic deformation, known as the linear Maxwell model [6]. In this case:  
\[R_{o}=1,\;u_{o}=u_{\infty}+S\;\text{and}\;u(t)=u_{\infty}+S\;e^{\frac{-t}{\tau}} \tag{2}\]  
while the normalized displacement is \(\text{D}(t)=\frac{u(t)}{u_{o}}=\frac{1+\frac{S}{u_{\infty}}e^{\frac{-t}{\tau} }}{1+\frac{S}{u_{\infty}}}\). In Figure 1, we plot the displacement as a function of normalized time \(\frac{t}{\tau}\). In the very early stage, expanding the exponential in Equation (2) leads us to \(u(t)=u_{\infty}+S\big{(}1-\frac{t}{\tau}\big{)}\text{or}\text{D}(t)=1-\frac{ \frac{S}{u_{\infty}}}{1+\frac{S}{u_{\infty}}}\;\frac{t}{\tau}\), a linear expression of normalized displacement with time, which for \(\frac{S}{u_{\infty}}\gg 1\), leads to \(\text{D}=1-(t/\tau)\) with the slope equal to \(1/\tau\), and independent of \(S\). We proceeded to study a post-seismic relaxation process with a spectrum of relaxation times within the range defined by the minimum and maximum values \(\tau_{min}\), \(\tau_{max}\) of relaxation time, respectively. If \(f(\tau)\) is the distribution function of the relaxation time, then:  
\[R(t)=\int_{\tau_{min}}^{\tau_{max}}f(\tau)e^{\frac{-t}{\tau}}d\tau \tag{3}\]  
The expression (3) is based on the assumption that the total macroscopic relaxation observed is a result of the superposition of relaxation processes operating with different relaxation times \(\tau\). This approach could be viewed as a generalization of the [PERSON] body rheological model [8; 36; 37], expanding the two discrete relaxation times to a continuum spectrum in the range values \(\tau_{min}\), \(\tau_{max}\). Assuming a relaxation process with activation energy \(E\), then following [PERSON] law [31], the relaxation time is given by:  
\[\tau=\tau_{0}e^{\frac{E}{R_{B}}\tau} \tag{4}\]  
with \(\tau_{0}\) a constant proexponential factor, \(T\) the temperature, and \(k_{B}\) the Boltzmann constant. If the activation energy is distributed following a distribution function \(N(E)\), then the  
Figure 1: Example of exponential relaxation (see Equation (1)) with \(S=5\;\text{mm}\), \(u_{\infty}=-25\;\text{mm}\) and \(\tau=300\) days. distribution function \(f(\tau)\) for the relaxation times satisfies the expression \(f(\tau)=\)\(N(E)(dE/d\tau)\).