## 1 Introduction  
Magnetic skyrmions are axisymmetric topological solitons hosted in ferromagnetic materials (see [1, 2] and references therein) stabilized by the Dzyaloshinskii-Moriya interaction (DMI), a relativistic exchange interaction [3, 4]. Magnetic skyrmions are characterized by a skyrmion number (otherwise called topological charge) \(S=1/(4\pi)\int\mathrm{d}^{2}\rho\ \mathbf{m}\cdot(\partial\mathbf{m}/ \partial x\times\partial\mathbf{m}/\partial y)\) where \(\mathbf{m}(\rho)=\mathbf{M}(\rho)/M_{\mathrm{s}}\) is the unit magnetization vector with \(\mathbf{M}\) the magnetization, \(\rho=(x,y)\), and \(M_{\mathrm{s}}\) the saturation magnetization, and \(\partial/\partial x\) and \(\partial/\partial y\) are first partial derivatives. There are two types of DMI, the interfacial DMI (IDMI) stabilizing chiral Neel (hedgehog-like) and the bulk DMI stabilizing Bloch (vortex-like) skyrmions. In early 2000, [PERSON] and [PERSON] developed a phenomenological theory of chiral symmetry breaking in magnetic thin films and multilayers, and they predicted the formation of skyrmions in magnetic thin films and multilayers stabilized by induced DMI interactions [5]. Afterwards, it has been theoretically demonstrated that skyrmion structures can be formed as spontaneous ground states in magnetic metals with DMI interactions without the need of applying external fields or the presence of defects [6]. Recently, several efforts were done to study static properties, such as skyrmion stability in low-dimensional magnetic systems [7], and dynamic properties (see e.g., [8; 9]). However, until recently, not much attention has been paid to the investigation of statistical and statistical thermodynamic properties of magnetic skyrmions. In particular, only during the last years the problem of entropy for magnetic skyrmions was faced. The skyrmion entropy has been estimated from experimental data and its thermodynamic meaning has been used in bulk B20 compounds to give a classification of the magnetization phase transition at the transition temperature [10; 11]. Its inclusion in the Arrhenius law turned out to be determinant to explain the disagreement between the measurement and the calculation of the lifetime of a skyrmion lattice [12]. In a recent paper, the configurational entropy of a skyrmion diameters population was calculated within a three-dimensional (3D) approach showing that the entropy can be expressed as a function of physical and geometrical parameters and temperature. According to the 3D model, the distribution of the skyrmion diameters population represents the analogous of the Maxwell-Boltzmann (MB) distribution of particles for a dilute ideal gas and, on this basis, the configurational entropy, thermal fluctuations of energy and the statistical thermodynamics of chiral skyrmions were investigated (see [13; 14] and references therein). As observed in micromagnetic simulations [13], configurational entropy can be regarded as the number of different skyrmions having the same average energy resulting from the thermal fluctuations that enable the promotion of a number of skyrmions possessing the same average energy. In the 3D analytical model, it was expressed as a quantity proportional to \(<\)ln\(f_{0}\)\(>\) where \(<\ldots>\) denotes the statistical average over the spatial coordinates and \(f_{0}\) is the Gaussian distribution of diameters at thermodynamic equilibrium depending on the skyrmion diameter \(D_{\text{sky}}\) and on its average value. Instead, in the present work the skyrmions population is regarded as a two-dimensional (2D) distribution considering that skyrmions are planar structures. From this point of view, the analogy with the ideal gas is not so strict because the 2D distribution is not, strictly speaking, the direct correspondent of the ideal gas MB distribution.