## 3 Temperature Distribution \(T(t,r)\) in Glasses under Local Thermal Perturbations
### Dependence of \(T(t,r)\) on the Distribution of Relaxation Times  
The influence of the time dispersion of the dynamic heat capacity is most pronounced at the beginning of the heating process on a nanosecond time scale; see Figure 6b. This effect is significant already at \(\tau_{0}\) about 0.1 \(\upmu\)s, increases with increasing \(\tau_{0}\), and it reaches saturation at \(\tau_{0}\) above \({r_{0}}^{2}/D_{0}\), where \({r_{0}}^{2}/D_{0}\) is about 4 \(\upmu\)s at \(r_{0}=1\)\(\upmu\)m and \(D_{0}=2.6\cdot 10^{-7}\) m\({}^{2}/s\) for borosilicate glass. In fact, in glasses, the relaxation times \(\tau_{0}\) are distributed over a wide range. The parameters of the VFTH relationship (see Equation (5)) can be obtained from measurements of glass transition processes depending on the cooling rate. For example, we use these parameters for borosilicate glass [63], sodium-time-magnesium-silicate glass [83], and sodium-silicate glass [84]. The relaxation parameters of these silicate glasses used for model calculations are collected in Table 2.  
Figure 8: \(T(t_{1},r)\) depending on the distance \(r\) at \(t_{1}=100\) ns (**a**) and time dependence \(T(t,0)\) in the center of the hot zone (**b**) for borosilicate glass at \(r_{0}=1\)\(\upmu\)m, \(\tau_{p}=10\) ps, and \(E_{p}=15\) nJ \(T_{in}=300\) K (\(\tau_{0}=0\), 0.1 \(\upmu\)s, 1 \(\upmu\)s, and 100 \(\upmu\)s—squares, circles, triangles up, triangles down, and crosses). The solution \(T_{AV}(t,r)\) averaged using the distribution function \(H(\tau_{K}(T_{int}))\) obtained from [63] at \(T_{int}=700\) K and 1000 K, which are represented by diamonds and stars, respectively.  
Figure 7: \(T(t_{1},r)\) depending on the distance \(r\) at \(t_{1}=100\) ns (**a**) and time dependence \(T(t,0)\) in the center of the hot zone (**b**) at the same parameters as in Figure 3 (\(\tau_{0}=0\) and 10 \(\upmu\)s—squares and circles). The temperature distribution \(T_{AV}(t,r)\) is averaged using the distribution function \(H(\tau_{K}(T_{int}))\) obtained from [84] at \(T_{int}=700\) K and 1000 K—diamonds and stars, as well as from [83] at \(T_{int}=700\) K and 1000 K, which are represented by triangles facing up and triangles facing down, respectively.  
The dynamic heat capacity \(c_{dyn}(t)\) can be modeled using Equation (4), where \(\tau_{\rm K}(T)=\frac{0.737}{2\pi f_{max}}\) can be obtained from \(f_{max}\) (see Equation (5)) using the relaxation parameters; see Table 2. Thus, we obtain the temperature distribution \(T_{AV}(t,r)\) averaged using the distribution function \(H(\tau_{\rm K}(T_{int}))\), where \(\tau_{K}(T_{int})\) is obtained at several intermediate temperatures between \(T_{in}\) and \(T_{max}\) (where \(T_{max}\) is the maximum value of the temperature \(T(t,0)\) at the center of the hot zone). For example, set \(T_{int}=700\) K and 1000 K and consider the relaxation parameters for borosilicate glass [63], sodium-time-magnesium-silicate glass [83], and sodium-silicate glass [84], see Table 2. The results are presented in Figures 7 and 8. The difference between \(T(t,r)\), obtained for sufficiently large \(\tau_{0}\geq 10\) us, and \(T_{AV}(t,r)\), obtained for \(\tau_{K}(700\) K) and \(\tau_{K}(1000\) K), is insignificant; see Figures 7 and 8. Since the influence of the time dispersion of the dynamic heat capacity reaches saturation at \(\tau_{0}\) about 10 us, the shape of the distribution function \(H(\tau_{0})\) has little effect on the temperature distribution. Thus, the influence of the time dispersion of the dynamic heat capacity on the temperature distribution \(T(t,r)\) can be calculated for a fixed, sufficiently large \(\tau_{0}\).  
The influence of time dispersion of dynamic heat capacity is significant in both borosilicate and sodium-silicate glasses. Now, let us consider the influence of the size of the heating zone \(r_{0}\) on the temperature distribution \(T(t,r)\).