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The water inflow rate into a storage tank is not constant but increases over time. Initially, the pump delivers water at a rate of 5 L/min. However, the inflow rate increases by 2 L/min for each additional minute due to rising pressure or system adjustments. This scenario can be described mathematically by a linear function:It is necessary to integrate the inflow rate function to measure the total volume of water added to the tank over time. The total water volume V(t) is obtained by performing...
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Hydrostatic force is a fluid's total force at rest on a surface. For a horizontal surface submerged at a fixed depth, the pressure is constant and calculated as the product of fluid density, gravitational acceleration, and depth. In the case of a vertical dam wall submerged in water, this force is not evenly distributed due to the increasing pressure with depth. This variation arises from the cumulative weight of the water above each point. Integration is used to account for the continuous...
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Evaluating Areas Under Curves with DiscontinuitiesA definite integral is considered improper when the integrand is discontinuous at one of the limits of integration. This occurs when the function is undefined or becomes infinite at an endpoint, making the corresponding region under the curve unbounded. Such behavior is commonly associated with vertical asymptotes at the boundary of the interval. To properly define and evaluate these integrals, a limiting process is used to determine whether a...
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Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
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When evaluating a definite integral whose integrand matches the structure of a composite function, the substitution method provides an efficient way to simplify the calculation. This method is based on reversing the chain rule from differentiation, allowing a complicated expression to be rewritten in a simpler form. When the integrand contains an inner function and its derivative, substitution naturally reduces the complexity of the problem.The core idea of substitution for definite integrals...
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The process of breathing involves the periodic intake and expulsion of air, known as the respiratory cycle, which typically lasts about five seconds. Modeling the volume of air inhaled into the lungs as a function of time provides insight into both the dynamics and efficiency of pulmonary ventilation. This volume is determined by integrating the airflow rate over time, which captures the cumulative effect of air entering the lungs.Sinusoidal Model of AirflowAirflow during respiration is not...
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A Modeling and Simulation Method for Preliminary Design of an Electro-Variable Displacement Pump
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Pumping approximately integrable systems.

Florian Lange1, Zala Lenarčič1, Achim Rosch1

  • 1Institute for Theoretical Physics, University of Cologne, Zülpicher Straße 77a, D-50937 Cologne, Germany.

Nature Communications
|June 10, 2017
PubMed
Summary

Weakly driven integrable quantum systems can generate significant spin and heat currents. This occurs even with minor integrability-breaking effects, by activating approximate conserved quantities for pumping energy.

Area of Science:

  • Quantum physics
  • Many-body systems
  • Statistical mechanics

Background:

  • Weak perturbations can drive systems from equilibrium by utilizing protected degrees of freedom.
  • This principle has enabled the creation of exotic condensates like those of photons and magnons.
  • Integrable quantum systems possess infinite conservation laws, offering unique dynamics.

Purpose of the Study:

  • To develop a theoretical framework for understanding weakly driven integrable quantum systems.
  • To investigate the induction of spin and heat currents in these systems, even with integrability-breaking perturbations.
  • To characterize the resulting steady states and their dependence on perturbation structure.

Main Methods:

  • Development of a theory for weakly driven integrable systems.

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  • Analysis of how pumping activates local and quasi-local conserved quantities.
  • Characterization of steady states using a truncated generalized Gibbs ensemble.

    • Main Results:

    • Pumping can induce substantial spin and heat currents in weakly driven integrable systems.
    • Integrability-breaking perturbations do not prevent current generation; they activate approximate conserved quantities.
    • The steady state is described by a generalized Gibbs ensemble whose parameters depend on perturbation structure, not amplitude or initial state.

    Conclusions:

    • Weakly driven integrable systems offer a novel route to generating directed spin and heat transport.
    • The concept of activating approximate conserved quantities is key to understanding these phenomena.
    • Spin-chain materials driven by terahertz radiation are proposed as experimental platforms for integrability-based pumps.