We employ the generalized bracket formalism of nonequilibrium thermodynamics by Beris and Edwards to derive Lorentz-covariant time-evolution equations for an imperfect fluid with viscosity, dilatational viscosity, and thermal conductivity. Following closely the analysis presented by Öttinger (Physica A, 259, 1998, 24–42; Physica A, 254, 1998, 433–450) to the same problem but for the GENERIC formalism, we include in the set of hydrodynamic variables a covariant vector playing the role of a generalized thermal force and a covariant tensor closely related to the velocity gradient tensor. In our work here, we derive first the nonrelativistic equations and then we proceed to obtain the relativistic ones by elevating the thermal variable to a four-vector, the mechanical force variable to a four-by-four tensor, and by representing the Hamiltonian of the system with the time component of the energy-momentum tensor. For the Poisson and dissipation brackets we assume the same general structure as in the nonrelativistic case, but with the phenomenological coefficients in the dissipation bracket describing friction to heat and viscous effects being properly constrained for the resulting dynamic equations to be manifest Lorentz-covariant. The final relativistic equations are identical to those derived by Öttinger but the present approach seems to be more general in the sense that one could think of alternative forms of the phenomenological coefficients describing friction that could ensure Lorentz-covariance.

中文翻译：

---

来自非平衡热力学单发生器支架形式的相对论流体力学

我们采用 Beris 和 Edwards 的非平衡热力学的广义括号形式来推导具有粘度、膨胀粘度和导热系数的不完美流体的洛伦兹协变时间演化方程。紧随 Öttinger (Physica A, 259, 1998, 24–42; Physica A, 254, 1998, 433–450) 对同一问题提出的分析，但对于 GENERIC 形式主义，我们在流体动力学变量集中包含一个协变矢量扮演广义热力和与速度梯度张量密切相关的协变张量的角色。在我们这里的工作中，我们首先推导非相对论方程，然后通过将热变量提升为四向量，将机械力变量提升为四乘四张量，并通过表示的哈密顿量来获得相对论方程具有能量动量张量的时间分量的系统。对于泊松和耗散括号，我们假设与非相对论情况相同的一般结构，但耗散括号中描述摩擦热和粘性效应的唯象系数被适当约束，以使所得动态方程明显洛伦兹协变。最终的相对论方程与奥廷格导出的方程相同，但目前的方法似乎更通用，因为人们可以想到描述摩擦力的唯象系数的替代形式，从而确保洛伦兹协方差。