Hoffmann K H, Andresen B, Salamon P
Institut für Physik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Jun;87(6):062106. doi: 10.1103/PhysRevE.87.062106. Epub 2013 Jun 5.
The problem of effectively adiabatic control of a collection of classical harmonic oscillators sharing the same time-dependent frequency is analyzed. The phase differences between the oscillators remain fixed during the process. This fact leads us to adopt the coordinates: energy, Lagrangian, and correlation, which have proved useful in a quantum description and which have the advantage of treating both the classical and quantum problem in one unified framework. A representation theorem showing that two classical oscillators can represent an arbitrary collection of classical or quantum oscillators is proved. An invariant, the Casimir companion, consisting of a combination of our coordinates, is the key to determining the minimum reachable energy. We present a condition for two states to be connectable using one-jump controls and enumerate all possible switchings for one-jump effectively adiabatic controls connecting any initial state to any reachable final state. Examples are discussed. One important consequence is that an initially microcanonical ensemble of oscillators will be transformed into another microcanonical ensemble by effectively adiabatic control. Likewise, a canonical ensemble becomes another canonical ensemble.
分析了对具有相同随时间变化频率的一组经典谐振子进行有效绝热控制的问题。在该过程中，振子之间的相位差保持固定。这一事实促使我们采用能量、拉格朗日量和关联这些坐标，它们在量子描述中已被证明是有用的，并且具有在一个统一框架中处理经典和量子问题的优点。证明了一个表示定理，即两个经典振子可以表示任意一组经典或量子振子。一个由我们的坐标组合而成的不变量——卡西米尔伴随量，是确定可达到的最小能量的关键。我们给出了两个状态可通过单跳控制连接的条件，并列举了将任何初始状态连接到任何可达到的最终状态的单跳有效绝热控制的所有可能切换。讨论了一些例子。一个重要的结果是，一组初始处于微正则系综的振子将通过有效绝热控制转变为另一个微正则系综。同样，一个正则系综会变成另一个正则系综。
