## 2 Analytical Model
### Statistical and Thermodynamic Properties of the Skyrmion Diameters Population
#### 2.2.4 Configurational Entropy of Skyrmion Diameters Distribution
This condition is obtained via the normalization to unity (in place of the normalization to the particles density\(n=N/V\) for ideal gas) resulting in \(C_{\text{sky}}^{\text{av2D}}\ =\ \frac{1}{\pi^{\frac{3}{2}}}\Big{(}\frac{a}{k_{\text{B}}T} \Big{)}^{\frac{1}{2}}\frac{1}{<D_{\text{sky}}^{\text{2D}}>}\). Substituting \(C_{\text{sky}}^{\text{av2D}}\) and \(A_{\text{s}}\) into \(H_{0}^{\text{2D}}\) of Equation (13) yields:  
\[H_{0}^{\text{2D}}\ =\ -\frac{1}{2}\Bigg{(}\text{ln}\left(\frac{16\,\pi k_{\text{B}} T}{a<D_{\text{sky}}^{\text{2D}}>^{2}}\right)+1\Bigg{)}. \tag{14}\]  
The configurational skyrmion entropy of the 2D skyrmions population at thermodynamic equilibrium is calculated as \(S=-k_{\text{B}}\,H_{0}\):  
\[S^{\text{2D}}(T)\ =\ \frac{1}{2}k_{\text{B}}\,\,\text{ln}\left(\frac{16\,\pi k_{ \text{B}}T}{a<D_{\text{sky}}^{\text{2D}}>^{2}}\right)+S_{0} \tag{15}\]  
with \(S_{0}=1/2k_{\text{B}}\) and \(<D_{\text{sky}}^{\text{2D}}>\) calculated by means of Equation (6). \(S^{\text{2D}}\) exhibits a logarithmic dependence and the logarithm argument depends on the ratio between the thermal energy \(k_{\text{B}}T\) and the average skyrmion energy \(a<D_{\text{sky}}^{\text{2D}}>^{2}\). For a comparison, the corresponding configurational entropy computed in [13] for the 3D skyrmion diameters population reads:  
\[S^{\text{3D}}(T)\ =\ k_{\text{B}}\Bigg{[}\text{ln}\left(\frac{(k_{\text{B}}T)^{ \frac{3}{2}}+2(k_{\text{B}}T)^{\frac{1}{2}}a<D_{\text{sky}}^{\text{3D}}>^{2}} {a^{\frac{3}{2}}<D_{\text{sky}}^{\text{3D}}>^{2}t}\right)+\frac{1}{2}\Bigg{(} \frac{3k_{\text{B}}T+2a\ <D_{\text{sky}}^{\text{3D}}>^{2}}{k_{\text{B}}T+2a\ <D_{\text{sky}}^{\text{3D}}>^{2}}\Bigg{)}\Bigg{]}+S_{0} \tag{16}\]  
with \(S_{0}=k_{\text{B}}\) (2 ln 2 + 1/2 ln \(\pi\)). Looking at Equation (15) one notes that, as within the 3D model (Equation (16)), the configurational entropy has a geometric, thermal, and magnetic parameters dependence. However, while \(S^{\text{3D}}\) includes a logarithmic contribution and a fractional contribution having different dependences on \(T\), \(S^{\text{2D}}\) is proportional only to a logarithmic contribution and has no explicit dependence on the thickness \(t\) of the ferromagnetic material hosting the skyrmion. This results in a configurational entropy that is much smaller than the one calculated according to the 3D model. This difference highlights also the role played by the thickness \(t\) of the magnetic skyrmion (see Equation (16)) in the 3D model. In both cases, the configurational entropy depends on the size of the skyrmion via the average skyrmion diameter, has a dependence on the thermal energy even though with different functional forms decreasing with decreasing temperature, and is affected by the magnetic parameters by means of the coefficient \(a\). The dependence on the skyrmion size in both the 2D and 3D frameworks confirms the link with the skyrmion breathing mode independently of the considered dimensionality. As for \(S^{\text{3D}}\) the thermal dependence becomes critical below \(T=1\) K because \(S^{\text{2D}}\) diverges negatively violating [PERSON]'s third principle of thermodynamics (see the next subsection for more details). Finally, also \(S^{\text{2D}}\) expressed in Equation (15) can be considered a general result because other skyrmion textures would lead to energy profiles that can be approximated via the harmonic approximation in the vicinity of their minimum. 2.5 Behavior of \(<\)Dsky\({}^{\text{2D}}\)\(>\), \(\sigma_{\text{Dsky}}^{\text{2D}}\), and S\({}^{\text{2D}}\) at Low Temperatures  
It is interesting to derive the low-temperature behavior (0 \(\div\) 50 K) of the average skyrmion diameter, the standard deviation and the configurational entropy [13]. The low-temperature relations obtained are still classical because they are based on relations derived within the classical 2D distribution of diameters at equilibrium.