## 2 Analytical Model
### Statistical and Thermodynamic Properties of the Skyrmion Diameters Population
#### 2.2.4 Configurational Entropy of Skyrmion Diameters Distribution
For the whole range of temperatures studied (0 \(\div\) 300 K) and for each external bias field, \(a(T)=a_{0}\) +\(c\)\(T\) with \(a_{0}=a\) (\(T\)=0 K) and \(c<0\) a coefficient expressed in J/(m\({}^{2}\)K) [13]. Moreover, at low temperatures, \(D_{\text{0 sky}}\) (\(T\)) = \(D_{\text{0 sky}}\) (\(T\) = 0 K) + \(d\)\(T\). It has been found from numerical calculations that \(<\)\(D_{\text{sky}}\) (\(T\)\(\rightarrow\)0 K)\(>\) = \(D_{\text{0 sky}}\) (\(T\)\(\rightarrow\)0 K) as occurred for the 3D distribution (strictly speaking, it is indeed not possible to define an average diameter for \(T=0\) K). Hence, at low temperatures, the expansion of Equation (6) to the first-order yields:  
\[<D_{\text{sky}}^{\text{2D}}(T)>_{T\to 0\text{ K}}\simeq D_{0\text{ sky}}^{0\text{ }}\left(1+\frac{k_{\text{B}}T}{2\,a_{0}\left(D_{0\text{ sky}}^{0\text{ }}\right)^{2}}\right) \tag{17}\]  
with \(D_{0\text{ sky}}^{0\text{ }}=D_{0\text{sky}}\) (\(T=0\) K) for every amplitude of the external bias field. This expression is similar to Equation (6) but note that \(a(T)\) is replaced by \(a_{0}\) and \(D_{0\text{sky}}\) by \(D_{0\text{ sky}}^{0\text{ }}\). This leads to a purely linear dependence of the skyrmion average diameter on \(T\) at low temperatures at fixed \(a_{0}\) and \(D_{0\text{ sky}}^{0\text{ }}\). At low temperatures (\(T\)\(\rightarrow\)\(0\) K), via a series expansion to the first-order of Equation (9), writing \(a=a_{0}+c\)\(T\), the standard deviation turns out to be:  
\[\sigma_{<D_{\text{sky}}^{\text{2D}}>}^{\text{2D}}(T)_{(T\to 0\text{ K})}\simeq\sqrt{\frac{k_{\text{B}}T}{2a_{0}}}. \tag{18}\]  
Hence, at low temperatures, the standard deviation has a square root proportionality on temperature. In the same way, the configurational entropy \(S^{\text{2D}}\) of Equation (15) expanded to the first-order at low temperatures (for \(T\)\(\rightarrow\)\(0\) K), via Equation (17), reads:  
\[S^{\text{2D}}(T)_{(T\to 0\text{ K})}\simeq\frac{1}{2}k_{\text{B}}\left(\ln\left(\frac{16\,\pi\,k_{ \text{B}}T}{a_{0}\left(D_{0\text{ sky}}^{0\text{ }}\right)^{2}}\right)-\frac{c\,T}{a_{0}}\right)+S_{0} \tag{19}\]  
neglecting the small term \(-\frac{1}{2}\frac{k_{\text{B}}^{2}\,T}{a_{0}\left(D_{0\text{sky}}^{0\text{ }}\right)^{2}}\) resulting from the low-temperature expansion of the average diameter. The first term on the second member has a logarithmic dependence on \(T\), hence giving the divergence of \(S\) for \(T=0\) K. On the other hand, the second term expresses the linear dependence of \(S\) on \(T\) and vanishes for \(T=0\) K. As occurred for \(S^{\text{3D}}\), at low temperatures there is a deviation from the linear behavior due to the logarithmic dependence on \(T\) of the first term. The passage from the 3D to the 2D description does not avoid the unphysical divergent result as \(T\)\(\rightarrow\)\(0\) K. This confirms that this result does not depend on the dimensionality but on the fact that the configurational entropy was computed according to a classical approach showing the same limits of the Sackur-Tetrode entropy equation for the ideal gas and that, as \(T\)\(\rightarrow\)\(0\) K, a quantum approach would be necessary to remove the divergent behavior. Numerically, analogously to the 3D model, the calculations performed according to Equations (6), (9) and (15) for the average diameter, the standard deviation and the configurational entropy, respectively, within the 2D model are well reproduced by the corresponding low-temperature expansions to the first-order of Equations (17)-(19), respectively for \(1\) K \(\leq T\leq 50\) K. The curve of \(<D_{\text{sky}}^{\text{2D}}(T)>_{T\to 0\text{ K}}\) is almost superimposed to that of \(<D_{\text{sky}}^{\text{3D}}(T)>_{T\to 0\text{ K}}\approx D_{0\text{ }\text{sky}}^{0\text{ }}\left(1+\frac{k_{\text{B}}T}{a_{0}\left(D_{0\text{ }\text{sky}}^{