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The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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Eigenstate fluctuation theorem in the short- and long-time regimes.

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The fluctuation theorem holds even when the initial state is a single energy eigenstate, reconciling canonical ensemble assumptions with quantum pure states in statistical mechanics. This finding applies to both short and long time regimes.

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Area of Science:

  • Statistical Mechanics
  • Quantum Many-Body Systems

Background:

  • The canonical ensemble is fundamental for equilibrium statistical mechanics and fluctuation theorem derivations.
  • Recent advances suggest thermal equilibrium can be described by quantum pure states, like single energy eigenstates (ETH).
  • Reconciling these views is crucial for understanding fluctuation theorems in quantum systems.

Purpose of the Study:

  • To investigate the compatibility of the canonical ensemble and single energy eigenstate descriptions within the fluctuation theorem.
  • To demonstrate the validity of the fluctuation theorem when the initial bath state is a single energy eigenstate.

Main Methods:

  • Theoretical analysis utilizing the eigenstate thermalization hypothesis (ETH) for the long-time regime.
  • Theoretical analysis employing the Lieb-Robinson bound and ETH for the short-time regime.
  • Numerical simulations of hard-core bosons using exact diagonalization and finite-size scaling.

Main Results:

  • The fluctuation theorem is shown to hold in both long- and short-time regimes, even with a single energy eigenstate as the initial bath state.
  • Theoretical proofs for the long- and short-time regimes are independent and complementary, covering the entire time domain.
  • Numerical simulations confirm the fluctuation theorem's validity across different time scales and system sizes.

Conclusions:

  • The fluctuation theorem emerges from unitary dynamics in quantum many-body systems, irrespective of whether the initial state is canonical or a single energy eigenstate.
  • These findings bridge foundational concepts in statistical mechanics and quantum physics.
  • Experimental verification using systems like ultracold atoms is feasible.