We examine the duality relating the equilibrium dynamics of the mean-field p-spin ferromagnets at finite size in the Guerra’s interpolation scheme and the Burgers hierarchy. In particular, we prove that—for fixed p—the expectation value of the order parameter on the first side w.r.t. the generalized partition function satisfies the \(p-1\)-th element in the aforementioned class of nonlinear equations. In the light of this duality, we interpret the phase transitions in the thermodynamic limit of the statistical mechanics model with the development of shock waves in the PDE side. We also obtain the solutions for the p-spin ferromagnets at fixed N, allowing us to easily generate specific solutions of the corresponding equation in the Burgers hierarchy. Finally, we obtain an effective description of the finite N equilibrium dynamics of the \(p=2\) model with some standard tools in PDE side.

我们研究了在Guerra插值方案和Burgers层次结构中有限大小的平均磁场p旋铁磁体的平衡动力学的对偶性。特别地，我们证明了-对于固定的p-阶数参数在第一侧的期望值广义划分函数满足上述非线性方程组中的第\（p-1 \）个元素。根据这种二元性，我们解释了统计力学模型的热力学极限中的相变以及PDE侧的冲击波的发展。我们还获得了固定N下p自旋铁磁体的解决方案，使我们能够轻松地在Burgers层次结构中生成相应方程式的特定解。最后，我们使用PDE端的一些标准工具有效地描述了\（p = 2 \）模型的有限N平衡动力学。