## 2 A Thermodynamic Model of Post Seismic Relaxation
Introducing expression (9) in (1), the very early stage displacement immediately following the co-seismic one is given by the expression:  
\[R_{o}=1,\;u_{o}=u_{\infty}+S\;\text{and}\;u(t)=u_{\infty}+S\;e^{\frac{-t}{\tau}} \tag{2}\]1  
where the slope of Equation (10) is \(F_{1}=S\big{(}\frac{1}{\tau_{min}}-\frac{1}{\tau_{max}}\big{)}\approx S/\tau_ {min}\) for \(\tau_{min}\ll\tau_{max}\) and \(u_{o}=u_{\infty}+SR_{0}\). In the case when \(\tau_{min}\ll\tau_{max}\) and the time \(t\) is such that \(\tau_{min}\ll t\ll\tau_{max}\), we could approximate exponentials such as \(e^{\frac{-t}{\tau_{min}}}\approx 0\) and \(e^{\frac{-t}{\tau_{max}}}\approx 1\) and Equation (8) results to:  
\[R_{o}=1,\;u_{o}=u_{\infty}+S\;\text{and}\;u(t)=u_{\infty}+S\;e^{\frac{-t}{\tau}} \tag{2}\]2  
which, after integration, gives:  
\[R_{o}=1,\;u_{o}=u_{\infty}+S\;\text{and}\;u(t)=u_{\infty}+S\;e^{\frac{-t}{\tau}} \tag{2}\]3  
with \(R_{1}\) an integration constant. Equation (12) reveals that the relaxation function \(R(t)\) exhibits a logarithmic behavior in an intermediate time interval bounded by the relaxation times \(\tau_{min}\) and \(\tau_{max}\) with a slope \(-1\) in a graph of \(R(t)\) versus \(\ln\left(t\right)\). It becomes evident that when multiplying \(R(t)\) with a scale factor \(S\) in Equation (1), there is a modification of the slope of the postseismic relaxation versus \(\ln\left(t\right)\) related to the particular characteristics of earthquake event and the seismotectonic settings and the rheology of the deformed zone as well. Substituting Equation (11) into (1), displacement obtains logarithmic behavior:  
\[R_{o}=1,\;u_{o}=u_{\infty}+S\;\text{and}\;u(t)=u_{\infty}+S\;e^{\frac{-t}{\tau}} \tag{2}\]4  
for \(\tau_{min}\ll\tau\ll\tau_{max}\) with a slope \(F_{2}=S\). Defining \(\tau^{*}\) as the point where the transition from the linear to logarithmic behavior starts, there is a continuation of displacement (as given by Equations (10) and (12), respectively), that permits the estimation of \(R_{1}\) as:  
\[R_{o}=1,\;u_{o}=u_{\infty}+S\;\text{and}\;u(t)=u_{\infty}+S\;e^{\frac{-t}{\tau}} \tag{2}\]5  
In order to estimate the order of magnitude of \(R_{1}\), we calculate its value for two reasonable values of \(\tau^{*}\), as that of \(\tau^{*}=\tau_{min}\) and \(\tau^{*}=2\tau_{min}\). For \(\tau^{*}=\tau_{min}\), (i.e., the transition in the logarithmic range starts in the very early stage of deformation), we have \(R_{1}=\ln\tau_{max}-1+\frac{\tau_{min}}{\tau_{max}}\), which for \(\frac{\tau_{min}}{\tau_{max}}~{}\ll~{}1\), leads to:  
\[R_{o}=1,\;u_{o}=u_{\infty}+S\;\text{and}\;u(t)=u_{\infty}+S\;e^{\frac{-t}{\tau}} \tag{2}\]6  
giving \(R=-\ln\Bigl{(}\frac{t}{\tau_{max}}\Bigr{)}-1\). Since \(R(t)\) is positively defined, the above expression is valid for \(t<\tau_{max}/e\approx 0.37\tau_{max}\). For a more reasonable estimation, we set \(\tau^{*}=2\tau_{min}\), leading to \(R_{1}=\ln\tau_{max}+\ln 2-2(1-\frac{\tau_{min}}{\tau_{max}})\), which for \(\frac{\tau_{min}}{\tau_{max}}\ll 1\), leads to:  
\[R_{o}=1,\;u_{o}=u_{\infty}+S\;\text{and}\;u(t)=u_{\infty}+S\;e^{\frac{-t}{\tau}} \tag{2}\]7  
with \(R=-\ln\Bigl{(}\frac{t}{\tau_{max}}\Bigr{)}+\ln 2-2\). From the positive definition of \(R(t)\), we conclude that the above expression is valid for \(t<\frac{\tau_{max}}{2-\ln 2}~{}\approx 0.76\tau_{max}\). A comparison between Equations (10) and (13) indicates that the slope of the logarithmic part \(F_{2}=S\) is scaled with that of early stage deformation behavior \(F_{1}=S\Bigl{(}\frac{1}{\tau_{min}}-\frac{1}{\tau_{max}}\Bigr{)}\). For \(\tau_{min}\ll\tau_{max}\), this scaling can provide an estimation of \(\tau_{min}\), since \(\frac{F_{2}}{F_{1}}=\tau_{min}\).