An exact derivation of relativistic hydrodynamics from an underlying microscopic theory has been shown to be an all-order theory. From the relativistic transport equation of kinetic theory, the full expressions of hydrodynamic viscous fluxes have been derived which turn out to include all orders of out-of-equilibrium derivative corrections. It has been shown, that for maintaining causality, it is imperative that the temporal derivatives must include all orders, which can be resummed in non-local, relaxation operator-like forms and finally ‘integrated in’ introducing newer degrees of freedom. The theory can of course be truncated at any higher spatial orders, but the power over the infinite temporal sum increases correspondingly such that the causality is respected. As a result, the theory truncated at any higher order of spatial gradient, requires newer degrees of freedom for each increasing order.

中文翻译：

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为什么在高阶截断流体动力学理论中会出现较新的自由度？


从基础微观理论中精确推导出相对论流体动力学已被证明是全阶理论。从动力学理论的相对论输运方程中，推导出了流体动力学粘性通量的完整表达式，结果证明包括所有不平衡导数校正的阶数。已经表明，为了保持因果关系，时间导数必须包括所有顺序，这些顺序可以以非局部的、类似松弛算子的形式进行归纳，并最终“集成”引入新的自由度。当然，该理论可以在任何更高的空间阶数上被截断，但对无限时间和的幂量会相应地增加，因此会尊重因果关系。因此，在空间梯度的任何更高阶数处截断的理论都需要为每个递增的阶数提供更新的自由度。