How long does it take a quantum particle to return to its origin? As shown previously under repeated projective measurements aimed to detect the return, the closed cycle yields a geometrical phase which shows that the average first detected return time is quantized. For critical sampling times or when parameters of the Hamiltonian are tuned this winding number is modified. These discontinuous transitions exhibit gigantic fluctuations of the return time. While the general formalism of this problem was studied at length, the magnitude of the fluctuations, which is quantitatively essential, remains poorly characterized. Here, we derive explicit expressions for the variance of the return time, for quantum walks in finite Hilbert space. A classification scheme of the diverging variance is presented, for four different physical effects: the Zeno regime, when the overlap of an energy eigenstate and the detected state is small and when two or three phases of the problem merge. These scenarios present distinct physical effects which can be analyzed with the fluctuations of return times investigated here, leading to a topology-dependent time-energy uncertainty principle.
Returning a system to a desired state under a force field involves a thermodynamic cost, i.e., {\it work}. This cost fluctuates for a small-scale system from one experimental realization to another. We introduce a general framework to determine the work distribution for returning a system facilitated by a confining potential with its minimum at the restart location. The general strategy, based on average over {\it resetting pathways}, constitutes a robust method to gain access to the statistical information of observables from resetting systems. We exploit paradigmatic setups, where explicit c…
Randomly repeated measurements during the evolution of a closed quantum system create a sequence of probabilities for the first detection of a certain quantum state. The related discrete monitored evolution for the return of the quantum system to its initial state is investigated. We found that the mean number of measurements until the first detection is an integer, namely the dimensionality of the accessible Hilbert space. Moreover, the mean first detected return time is equal to the average time step between successive measurements times the mean number of measurements. Thus, the mean first …
Based on the tick-by-tick price changes of the companies from the U.S. and from the German stock markets over the period 1998-99 we reanalyse several characteristics established by the Boston Group for the U.S. market in the period 1994-95, which serves to verify their space and time-translational invariance. By increasing the time scales we find a significantly more accelerated crossover from the power-law (alpha approximately 3) asymptotic behaviour of the distribution of returns towards a Gaussian, both for the U.S. as well as for the German stock markets. In the latter case the crossover i…
Statistics of Extreme Values in Time Series with Intermediate-Term Correlations
本文将讨论具有非指数衰减的有限项相关的时间序列中极值的统计特性。准确地说，它将被认为是数值分析的结果有关的波动的电阻和缺陷分数的极端值的返回间隔显示的粒状结构的电阻器在非平衡稳态。电阻和缺陷分数计算作为时间的函数，通过蒙特卡罗模拟使用电阻器网络的方法。当波动的自关联函数呈现非指数和非幂律衰减时，极值的返回区间的分布是一个拉伸的指数，指数与阈值基本无关。最近，Bunde et al.（2003）和Altmann and Kantz（2005）在长期相关时间序列中发现了极值收益区间的拉伸指数分布。因此，目前的研究结果表明，拉伸指数分布的回报间隔是不是一个长期相关的时间序列的唯一特征。
