the strength distribution. In Sec. \[sec:results\], we apply the local scaling dimension to the IS GQR strength function in $^{208}$Pb. Detail of damping process is studied in Sec. \[sec:damping\], where the physical origin for the difference of the fluctuation property of the strength function between $^{40}$Ca and $^{208}$Pb is also discussed. Finally, Sec. \[sec:conclusion\] is devoted to conclusion.

Local Scaling Dimension {#sec:lsd}
=======================

We briefly explain the local scaling dimension. See Refs. [@aiba; @aiba2] for details.

The strength function is expressed as [@Bohr-Mottelson2] $$S(E)=\sum_i S_i\delta(E-E_i+E_0).
\label{defstr}$$ Here $E_i$ and $S_i$ denote the energy and the strength of exciting the $i$th energy level, respectively. Strengths are normalized as $\sum_i S_i=1$.

To study the fluctuation at each energy scale, we consider binned distribution of the strength by dividing whole energy interval under consideration into $N$ bins with length $\epsilon$. Strength contained in $n$th bin is denoted by $p_n$, $$p_n\equiv\sum_{i\in n{\rm th~ bin}}S_i.
\label{defp}$$ To characterize the distribution of the binned strengths, we introduce the moments of $p_n$, which are called in literature the partition function $\chi_m(\epsilon)$ defined by $$\chi_m(\epsilon)\equiv \sum_{n=1}^N p_n^m \\
                 =N\langle p_n^m\rangle.
\label{partition}$$ Finally, by extending the idea of the generalized fractal dimensions [@hentschel; @halsey] to non-scaling cases in a 