Work statistics in classical and quantum driven billiards

纳米系统的热力学描述在两个主要方面不同于传统的热力学。首先，能量的典型周转是每个自由度的平均热能的量级。因此，像功、热和熵这样的热力学量在相同的重复实验之间波动，必须用它们的概率分布来表征。其次，量子效应可能会影响动力学，热力学必须以一致的方式纳入这些效应。对于功和热，两个关键的热力学量不是状态变量，而是依赖于系统的完整历史。由于没有经典轨道的量子对应物，小量子系统的热和功的定义已经被证明是非常困难的，并且仍然是一个很大程度上悬而未决的问题。在第一个资助期的相应项目中，我们仔细研究了具有四次势的周期驱动非谐振子中的经典和量子工作的统计。其目的是特别是从文献中的经典可积动力学系统的确定性混沌扩展以前的结果。但是，虽然周期驱动的四次振子可以维持规则和混沌运动，我们发现，它是前者，而不是后者，占主导地位的功能分布。因此，在新的项目中，我们的目标是比较经典和量子工作的系统具有明显的混沌经典行为。作为合适的测试系统，我们确定了两个特殊版本的移动壁台球。静态台球是经典和量子混沌的传统示例系统。他们的动态变化与移动壁的研究要少得多。我们的第一个系统是一个圆形台球，变形为Bunimovich体育场。第二个是一个正方形，它产生了两个墙壁的正弦调制，将其转化为涟漪台球。这两种情况都允许非常精确的数值表征。与此同时，它们与具有移动壁的一维系统有关，对于该一维系统，随时间变化的薛定谔方程可以精确求解。结合分析和数值计算技术，我们将因此调查的经典和量子工作分布之间的详细关系，在两个范式系统与强混沌经典动力学。

The thermodynamic description of nanoscopic systems differs in two major respects from traditional thermodynamics. Firstly, the typical turnover in energy is of the order of the average thermal energy per degree of freedom. Consequently, thermodynamic quantities like work, heat, and entropy fluctuate between identical repetitions of an experiment and must be characterized by their proper probability distributions. Secondly, quantum effects are likely to influence the dynamics and the thermodynamics has to incorporate these effects in a consistent way. With work and heat two key thermodynamic quantities are no state variables but depend on the complete history of the system. Because there is no quantum counterpart for classical trajectories the definition of heat and work for small quantum systems has proven to be notoriously difficult and is still a largely open question.In the corresponding project from the first funding period we have scrutinized the statistics of classical and quantum work in a periodically driven anharmonic oscillator with a quartic potential. The intention was in particular to extend previous results from the literature for systems with classically integrable dynamics to those showing deterministic chaos. But although the periodically driven quartic oscillator may sustain both regular and chaotic motion we found that it is the former rather than the latter that dominates the features of the work distribution. Accordingly, in the new project we aim on the comparison of classical and quantum work for systems with pronounced chaotic classical behaviour. As suitable test systems we identified two special version of billiards with moving walls. Static billiards are traditional example systems for both classical and quantum chaos. Their dynamic variants with moving walls are much less investigated. Our first system is a circle billiard that deforms to a Bunimovich stadium. The second one is a square that develops a sinusoidal modulation of two walls transforming it to a ripple billiard. Both cases allow a very accurate numerical characterization. At the same time they are related to one-dimensional systems with moving walls for which the time-dependent Schrödinger equation may be solved exactly. Combining analytical and numerical techniques we will therefore investigate the detailed relation between classical and quantum work distributions in two paradigmatic systems with strongly chaotic classical dynamics.