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Spectral properties of simple classical and quantum reset processes.

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Resetting Markovian processes accelerates relaxation to a stationary state by shifting eigenvalues. This study analyzes spectral properties and dynamics of reset processes in classical and quantum systems.

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Area of Science:

  • Physics
  • Quantum Mechanics
  • Statistical Mechanics

Background:

  • Markovian processes are fundamental in describing systems evolving probabilistically over time.
  • Reset dynamics, where a system is periodically returned to an initial state, can alter system behavior.
  • Understanding spectral properties is key to analyzing the stability and convergence of dynamical processes.

Purpose of the Study:

  • To investigate the spectral properties of classical and quantum Markovian processes subjected to random resets.
  • To determine how reset dynamics influence the relaxation rates and stationary states of these processes.
  • To explore the impact of resets on systems exhibiting metastability.

Main Methods:

  • Analysis of the Markov generator's eigenvalues under reset conditions.
  • Derivation of expressions for stationary states and probability currents in reset processes.
  • Application of the framework to classical stochastic processes (random walks, Brownian motion) and quantum models (hopping particle, Ising model).

Main Results:

  • Reset dynamics induce a uniform shift in eigenvalues, accelerating relaxation to the stationary state.
  • The stationary state and probability current of reset processes can be expressed using modes of the reset-free system.
  • Resets can significantly affect the dynamics of systems with metastability.

Conclusions:

  • Random resets provide a general mechanism to control and enhance the convergence of Markovian processes.
  • The spectral shift offers a powerful tool for analyzing and predicting the behavior of reset dynamical systems.
  • The findings have implications for both classical and quantum systems, particularly those with complex dynamics like metastability.