关于无界系统存在经典轨道抗磁性可能性的注记--Bohr-van Leeuwen定理
A note on the possibility of classical orbital diamagnetism for an unbounded system --the Bohr-van Leeuwen Theorem
摘要：
最近[EPL，86，（2009）17001]，我们模拟了在外部施加磁场的情况下，带电粒子在球体表面的经典朗之万动力学，并发现了长时间极限下的轨道抗磁矩的有限值。这个结果是令人惊讶的，因为它似乎违反了经典的玻尔-范列文定理的经典抗磁性的情况下。这确实受到了一些工作者的质疑[EPL，89，（2010）37001]，他们验证了从我们的朗之万方程导出的福克-普朗克（FP）方程，通过在FP方程中设置d/dt=0获得的稳态经典正则密度满足。由于正则密度不包含磁场，他们得出结论，反磁矩必须为零。这篇笔记的目的是表明这个论点和结论是无效的-而不是设置d/dt=0，必须首先获得FP方程的基本含时解，然后计算抗磁矩的期望值，最后考虑其长时间限制（即，$t \to \infty$）。这确实符合我们对动力学的数值模拟。这确实是如此，通过考虑更简单的解析可解问题，即对于一个无界平面，上述程序可以精确地进行。然后，我们发现，抗磁矩的期望值的极限值确实是非零的，但通过设置d/dt=0得到的稳态FP方程是由正则密度满足。不可否认，球的精确解析解是不可用的。但是，对于无界二维平面的情形所得到的精确解充分说明了我们的观点。我们还提出了一些新的结果，其他有限的，但无界的表面，如旋转椭球。
英文摘要：
Recently [EPL, 86, (2009) 17001], we had simulated the classical Langevin dynamics of a charged particle on the surface of a sphere in the presence of an externally applied magnetic field, and found a finite value for the orbital diamagnetic moment in the long-time limit. This result is surprising in that it seems to violate the classic Bohr-van Leeuwen Theorem on the absence of classical diamagnetism. It was indeed questioned by some workers [EPL, 89, (2010) 37001] who verified that the Fokker-Planck (FP) equation derived from our Langevin equation, was satisfied by the classical canonical density in the steady state, obtained by setting d/dt=0 in the FP equation. Inasmuch as the canonical density does not contain the magnetic field, they concluded that the diamagnetic moment must be zero. The purpose of this note is to show that this argument and the conclusion are invalid -- instead of setting d/dt=0 one must first obtain the fundamental time-dependent solution for the FP equation, and then calculate the expectation value of the diamagnetic moment, and finally consider its long-time limit (i.e., $t \to \infty$). This would indeed correspond to our numerical simulation of the dynamics. That this is indeed so is shown by considering the simpler analytically solvable problem, namely that for an unbounded plane for which the above procedure can be carried out exactly. We then find that the limiting value for the expectation of the diamagnetic moment is indeed non-zero, and yet the steady-state FP equation obtained by setting d/dt=0 is satisfied by the canonical density. Admittedly, the exact analytical solution for the sphere is not available. But, the exact solution obtained for the case of the unbounded 2D-plane illustrates our point all right. We also present some further new results for other finite but unbounded surfaces such the ellipsoids of revolution.
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