# Superadiabatic Forces via the Acceleration Gradient in Quantum Many-Body Dynamics

^{*}

## Abstract

**:**

**2015**, 143, 174108) to a one-dimensional Hooke’s helium model atom. The physical dynamics are described on the one-body level beyond the density-based adiabatic approximation. We show that gradients of both the microscopic velocity and acceleration field are required to correctly describe the effects due to interparticle interactions. We validate the proposed analytical forms of the superadiabatic force and transport contributions by comparison to one-body data from exact numerical solution of the Schrödinger equation. Superadiabatic contributions beyond the adiabatic approximation are important in the dynamics and they include effective dissipation.

## 1. Introduction

## 2. Theoretical Background

## 3. Hooke’s Atom

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Sample Availability: Not available. |

**Figure 1.**Time evolution of (

**a**) the reduced density ${n}^{*}=n\sqrt{\hslash /m\omega}$ and (

**b**) the reduced current density ${J}^{*}=J/\omega $ as a function of the reduced box coordinate ${x}^{*}=x\sqrt{m\omega /\hslash}$ for $\u03f5/\hslash \omega =0.5$ and ${\alpha}^{*}=1$. For comparison, at reduced time ${t}^{*}\equiv t\omega =3$ the curves for ${\alpha}^{*}=4$ are also shown (dashed line). Illustration of the total force density and the different contributions according to Equation (3) for ${t}^{*}=0.05$ (

**c**) and ${t}^{*}=2$ (

**d**) given in reduced units as ${F}^{*}=F/m{\omega}^{2}$. The encircled arrows indicate the force direction at selected space points. The total size of the box is $30\sqrt{\hslash /m\omega}$.

**Figure 2.**Reduced kinetic stress tensor ${\tau}^{*}=\tau /\sqrt{m{\omega}^{3}\hslash}$ (blue solid line) and its splitting into ideal (red dashed line) and excess (solid yellow line) parts, Equation (6), and the theoretical prediction (yellow circles), Equation (8), for interaction lengths ${\alpha}^{*}=0.2$ (

**a**), ${\alpha}^{*}=1$ (

**b**) and ${\alpha}^{*}=4$ (

**c**). Internal (green solid line), adiabatic (blue dashed line) and superadiabatic (yellow solid line) force densities, cf. Equation (7), and theoretical prediction (yellow circles), Equation (9), in reduced units, i.e., ${F}^{*}=F/m{\omega}^{2}$ for interaction lengths ${\alpha}^{*}=0.2$ (

**d**), ${\alpha}^{*}=1$ (

**e**) and ${\alpha}^{*}=4$ (

**f**). Data taken at ${t}^{*}=3$. The model parameters and reduced units are identical to those in Figure 1. The insets in panels (

**a**,

**c**,

**d**) are enlarged views of the superadiabatic terms. The inset in panel (

**e**) shows the splitting of the superadiabatic force density (circles) into ${F}_{v}$ (pink) and ${F}_{a}$ (blue).

**Figure 3.**Scaled amplitudes ${C}_{0}^{*}={C}_{0}\sqrt{m\omega /{\hslash}^{3}}$ (yellow), ${C}_{1}^{*}={C}_{1}\sqrt{m\omega /{\hslash}^{3}}$ (violet) and ${C}_{2}^{*}={C}_{2}\sqrt{m{\omega}^{3}/{\hslash}^{3}}$ (blue) of the functional approximations for ${\tau}_{\mathrm{exc}}$, ${F}_{v}$ and ${F}_{a}$, cf. Equations (8), (10) and (11), as a function of the scaled time ${t}^{*}=t\omega $ for three different values of the interaction length, as indicated. Note the different scales (left and right vertical axes). The data set ${C}_{1}^{*}$ (${\alpha}^{*}=4$) has been scaled with a factor $0.1$, as indicated, for a better visualization.

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**MDPI and ACS Style**

Brütting, M.; Trepl, T.; de las Heras, D.; Schmidt, M.
Superadiabatic Forces via the Acceleration Gradient in Quantum Many-Body Dynamics. *Molecules* **2019**, *24*, 3660.
https://doi.org/10.3390/molecules24203660

**AMA Style**

Brütting M, Trepl T, de las Heras D, Schmidt M.
Superadiabatic Forces via the Acceleration Gradient in Quantum Many-Body Dynamics. *Molecules*. 2019; 24(20):3660.
https://doi.org/10.3390/molecules24203660

**Chicago/Turabian Style**

Brütting, Moritz, Thomas Trepl, Daniel de las Heras, and Matthias Schmidt.
2019. "Superadiabatic Forces via the Acceleration Gradient in Quantum Many-Body Dynamics" *Molecules* 24, no. 20: 3660.
https://doi.org/10.3390/molecules24203660