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Poisson's And Laplace's Equation01:25

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The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
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If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...
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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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Second Derivatives and Laplace Operator01:22

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The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
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在散射系统中的量子利亚普诺夫指数.

Pablo D Bergamasco1, Gabriel G Carlo2, Alejandro M F Rivas2

  • 1Departamento de Física, CNEA, Libertador 8250, (C1429BNP) Buenos Aires, Argentina.

Physical review. E
|September 19, 2023
PubMed
概括

我们在开放量子系统中研究了时间外顺序相关因子 (OTOC),揭示了它的衰变速率对混乱敏感,并与经典的利亚普诺夫指数相关. 这项研究探讨了编码和消散的相互作用.

科学领域:

  • 量子动力学 量子动力学是什么?
  • 统计力学 统计力学
  • 混沌理论 混沌理论

背景情况:

  • 时间外顺序相关系数 (OTOC) 是量子混乱的关键指标,在封闭系统中得到了广泛的研究.
  • 在开放量子系统中对OTOC的研究往往将杂乱与脱凝效应分开.
  • 分散过程在量子设备和古典动态系统中很常见.

研究的目的:

  • 在开放系统中研究量子杂乱和脱凝之间的相互作用.
  • 为了解释OTOC在相空间收缩消散存在时的行为.
  • 在量子系统中识别敏感的指标,以区分混沌与正规的行为.

主要方法:

  • 在带有散射的开放量子系统中分析OTOC.
  • 将OTOC衰变速率与经典的利亚普诺夫指数进行比较.
  • 调查OTOC和量子进化运算符固有值之间的关系.
  • 用高斯噪声对经典系统进行建模,以探索对应原理.

主要成果:

  • 消散系统中的OTOC衰变率与经典的利亚普诺夫指数密切相关.
  • 在区分混沌与常规动态方面,OTOC比其他措施更为敏感.
  • OTOC衰变速率是量子进化演算子最长寿命固有值的函数.

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  • 将高斯噪声添加到经典系统中可以恢复OTOC衰变速率,符合对应原理.

    • 结论:

    • 编码和消散的相互作用对于理解开放量子系统中的OTOC行为至关重要.
    • OTOC提供了一个敏感的探测器的混乱在消散量子动力学.
    • 对应原则为将经典噪声和量子OTOC衰变联系起来提供了一个框架.