We study the intrinsic effects of dimensional reduction on the transport equation of a perfectly two-dimensional Landau–Fermi liquid. By employing the orthogonality condition on the 2D analog of the Fourier–Legendre expansion, we find that the equilibrium and non-equilibrium properties of the fermionic system differ from its three-dimensional counterpart, with the latter changing drastically. Specifically, the modified Landau–Silin kinetic equation is heavily dependent on the solution of a non-trivial contour integral specific to the 2D liquid. We find the solution to this integral and its generalizations, effectively reducing the problem of solving for the collective excitations of a collisionless two-dimensional Landau–Fermi liquid to solving for the roots of some high-degree polynomial. This analysis ultimately lays the mathematical foundation for the exploration of atypical behavior in the non-equilibrium properties of two-dimensional fermionic liquids in the context of the ...

中文翻译：

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二维Landau–Fermi液体参数的Chebyshev多项式展开

我们研究了二维还原对完美二维Landau-Fermi液体的输运方程的内在影响。通过在傅立叶-莱格朗德展开的二维模拟上采用正交性条件，我们发现铁离子系统的平衡和非平衡性质不同于其三维对应物，而后者则发生了巨大变化。具体来说，修改后的Landau-Silin动力学方程式很大程度上取决于二维液体特有的非平凡轮廓积分的解。我们找到了该积分的解决方案及其推广，从而有效地解决了无碰撞二维Landau-Fermi液体的集体激发的求解问题，从而解决了一些高阶多项式的根的求解问题。