A new dynamical approach to black hole thermodynamics

在过去的十年左右，在动力系统理论如何 与统计力学有关。最值得注意的是1997年Hans Henrik Rugh的开创性论文，提出了微正则系综中热力学的新动力学方法。在这种方法中，哈密顿动力系统的温度被计算为在能量表面上评估的特定函数的时间平均值。函数本身是从哈密顿量的导数中得到的。长期以来，人们一直认为哈密顿动力系统表现出某种遍历性，其中时间平均被视为等同于微正则系综上的空间平均。然而，直到最近，一个明确的数学公式的温度，反映这是失踪。新公式不仅提供了一种计算温度的算法，而且还揭示了动力系统理论与哈密顿系统统计力学之间长期寻求的联系。此外，数学形式主义现在已经扩展到除了能量之外的其他守恒量（例如角动量）。 我的目标之一是应用这些新的想法和计算算法源于统计力学黑洞（BH）热力学。黑洞非常适合这种形式主义，因为它们是由三个守恒量描述的哈密顿动力学系统：质量（能量）M，电荷Q和角动量J。特别地，BH的温度将被计算为仅从哈密尔顿算子评估的函数的时间平均。与最近的自由能计算一样，在引力坍缩的情况下，温度可以用数值计算。这将是对BH热力学的一个新贡献。

In the last decade or so, there has been major mathematical advances in how dynamical systems theory connects with statistical mechanics. Most notably, there is the 1997 seminal paper by Hans Henrik Rugh that presents a new dynamical approach to thermodynamics in the microcanonical ensemble. In this approach, the temperature of a Hamiltonian dynamical system is computed as a time average of a particular function evaluated on the energy surface. The function itself is obtained from derivatives of the Hamiltonian. It has long been assumed that Hamiltonian dynamical systems exhibit some sort of ergodicity, where time-averages are viewed as being equivalent to space-averages over the microcanonical ensemble. However, until recently, an explicit mathematical formula for the temperature that reflected this was missing. The new formula not only provides an algorithm by which to compute the temperature but furnishes a long-sought connection between dynamical systems theory and the statistical mechanics of Hamiltonian systems. Moreover, the mathematical formalism has now been extended to other conserved quantities besides the energy (e.g. angular momentum). One of my objectives is to apply these new ideas and computational algorithms stemming from statistical mechanics to black hole (BH) thermodynamics. Black holes are ideally suited for this formalism as they are Hamiltonian dynamical systems described by three conserved quantities: mass (energy) M, charge Q and angular momentum J. Each of these conserved quantities can be expressed as a surface integral and has an associated thermodynamic variable that can be calculated as a time-average. In particular, the temperature of a BH would be computed as a time-average of a function evaluated from the Hamiltonian only. As with recent computations of the free energy, the temperature could be evaluated numerically in a gravitational collapse scenario. This would be a novel contribution to BH thermodynamics.