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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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Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
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State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
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Computing time-periodic steady-state currents via the time evolution of tensor network states.

Nils E Strand1, Hadrien Vroylandt1, Todd R Gingrich1

  • 1Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA.

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We introduce a binary tree tensor network (BTTN) approach to calculate steady-state currents in 1D ratchets with volume exclusion. This method efficiently approximates many-body configurations, offering an alternative to trajectory sampling for complex systems.

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Area of Science:

  • Statistical mechanics
  • Condensed matter physics
  • Computational physics

Background:

  • Many-particle systems on lattices present computational challenges.
  • Ratchets with volume exclusion and time-periodic drives exhibit complex dynamics.
  • Traditional trajectory sampling can be computationally intensive for steady-state statistics.

Purpose of the Study:

  • To develop an efficient computational method for steady-state current statistics in 1D ratchets.
  • To approximate the distribution of many-body configurations using binary tree tensor network states.
  • To investigate the effects of both typical and rare trajectories on system behavior.

Main Methods:

  • Utilizing binary tree tensor network (BTTN) states to represent many-body configurations.
  • Employing the Gillespie method for stochastic trajectory evolution.
  • Applying the density matrix renormalization group (DMRG) algorithm for BTTN initialization.
  • Propagating BTTN states in time using the time-dependent variational principle (TDVP) algorithm.

Main Results:

  • The BTTN-TDVP approach successfully computes steady-state current statistics for a 1D ratchet model.
  • The method provides a variational approximation to the distribution over many-body configurations.
  • The approach accounts for the influence of both typical and rare trajectories.

Conclusions:

  • Binary tree tensor network states offer a powerful alternative to trajectory sampling for studying interacting lattice models.
  • The BTTN-TDVP strategy is beneficial for systems with time-dependent driving and volume exclusion.
  • This approach is extendable to other complex interacting lattice models.