A new dynamical approach to black hole thermodynamics

在过去十年左右的时间里，动力系统理论与统计力学的联系在数学上取得了重大进展。最值得注意的是，汉斯·亨里克·拉夫（Hans Henrik Rugh）在1997年发表了一篇开创性的论文，提出了一种新的动力学方法来研究微正则系综中的热力学。在这种方法中，哈密顿动力系统的温度被计算为一个特定函数在能量面上的时间平均值。函数本身是由哈密顿函数的导数得到的。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，其中时间平均值被视为等同于微正则系综上的空间平均值。然而，直到最近，还没有一个明确的温度数学公式来反映这一点。新公式不仅提供了一种计算温度的算法，而且在动力系统理论和哈密顿系统的统计力学之间提供了长期寻求的联系。此外，数学形式现在已经扩展到除了能量以外的其他守恒量（例如角动量）。****我的目标之一是将这些来自统计力学的新思想和计算算法应用于黑洞热力学。黑洞非常适合这种形式，因为它们是由三个守恒量描述的哈密顿动力系统：质量（能量）M、电荷Q和角动量j。这些守恒量中的每一个都可以表示为表面积分，并有一个相关的热力学变量，可以计算为时间平均值。特别地，黑洞的温度将被计算为仅由哈密顿量计算的函数的时间平均值。与最近对自由能的计算一样，温度可以在引力坍缩的情况下进行数值计算。这将是对黑洞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. **************