# American Institute of Mathematical Sciences

March  2017, 10(1): 263-298. doi: 10.3934/krm.2017011

## On the classical limit of a time-dependent self-consistent field system: Analysis and computation

 1 Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, USA 2 Department of Mathematics, Institute of Natural Sciences and MOE Key Lab in Scientific, and Engineering Computing, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai, 200240, China 3 Department of Mathematics, Statistics, and Computer Science, M/C 249, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607, USA 4 Department of Mathematics, Duke University, Box 90320, Durham NC 27708, USA

Dedicated to Peter Markowich on the occasion of his 60th birthday

Received  October 2015 Revised  January 2016 Published  November 2016

Fund Project: This work was partially supported by NSF grants DMS-1522184 and DMS-1107291: NSF Research Network in Mathematical Sciences KI-Net: Kinetic description of emerging challenges in multiscale problems of natural sciences. C.S. acknowledges support by the NSF through grant numbers DMS-1161580 and DMS-1348092.

We consider a coupled system of Schrödinger equations, arising in quantum mechanics via the so-called time-dependent self-consistent field method. Using Wigner transformation techniques we study the corresponding classical limit dynamics in two cases. In the first case, the classical limit is only taken in one of the two equations, leading to a mixed quantum-classical model which is closely connected to the well-known Ehrenfest method in molecular dynamics. In the second case, the classical limit of the full system is rigorously established, resulting in a system of coupled Vlasov-type equations. In the second part of our work, we provide a numerical study of the coupled semi-classically scaled Schrödinger equations and of the mixed quantum-classical model obtained via Ehrenfest's method. A second order (in time) method is introduced for each case. We show that the proposed methods allow time steps independent of the semi-classical parameter(s) while still capturing the correct behavior of physical observables. It also becomes clear that the order of accuracy of our methods can be improved in a straightforward way.

Citation: Shi Jin, Christof Sparber, Zhennan Zhou. On the classical limit of a time-dependent self-consistent field system: Analysis and computation. Kinetic & Related Models, 2017, 10 (1) : 263-298. doi: 10.3934/krm.2017011
##### References:

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Dedicated to Peter Markowich on the occasion of his 60th birthday

##### References:
The diagram of semi-classical limits: the iterated limit and the classical limit
Reference solution: $\Delta x=\Delta y= \frac{2\pi}{32768}$ and $\Delta t=\frac{0.4}{4096}$. Upper picture: fix $\Delta y= \frac{2\pi}{32768}$ and $\Delta t=\frac{0.4}{4096}$, take $\Delta x=\frac{2\pi}{16384}$, $\frac{2\pi}{8192}$, $\frac{2\pi}{4096}$, $\frac{2\pi}{2048}$, $\frac{2\pi}{1024}$, $\frac{2\pi}{512}$, $\frac{2\pi}{256}$, $\frac{2\pi}{128}$, $\frac{2\pi}{64}$, $\frac{2\pi}{32}$, $\frac{2\pi}{16}$, $\frac{2\pi}{8}$. Lower Picture: fix $\Delta x= \frac{2\pi}{32768}$ and $\Delta t=\frac{0.4}{4096}$, take $\Delta y=\frac{2\pi}{16384}$, $\frac{2\pi}{8192}$, $\frac{2\pi}{4096}$, $\frac{2\pi}{2048}$, $\frac{2\pi}{1024}$, $\frac{2\pi}{512}$, $\frac{2\pi}{256}$, $\frac{2\pi}{128}$, $\frac{2\pi}{64}$, $\frac{2\pi}{32}$, $\frac{2\pi}{16}$, $\frac{2\pi}{8}$. These results show that, when $\delta=O(1)$ and $\varepsilon \ll 1$, the meshing strategy $\Delta x= O(\delta)$ and $\Delta y=O(\varepsilon )$ is sufficient for obtaining spectral accuracy.
Reference solution: $\Delta x=\frac{2\pi}{512}$, $\Delta y= \frac{2\pi}{16348}$ and $\Delta t=\frac{0.4}{4096}$. SSP2: fix $\Delta x=\frac{2\pi}{512}$, $\Delta y= \frac{2\pi}{16348}$, take $\Delta t=\frac{0.4}{1024}$, $\frac{2\pi}{512}$, $\frac{2\pi}{256}$, $\frac{2\pi}{128}$, $\frac{2\pi}{64}$, $\frac{2\pi}{32}$, $\frac{2\pi}{16}$, $\frac{2\pi}{8}$. These results show that, when $\delta=O(1)$ and $\varepsilon \ll 1$, the SSP2 method is unconditionally stable and is second order accurate in time
Fix $\Delta t=0.05$. For $\varepsilon =1/64$, $1/128$, $1/256$, $1/512$, $1/1024$, $1/2048$ and $1/{4096}$, $\Delta x=2\pi\varepsilon/16$, respectively. The reference solution is computed with the same $\Delta x$, but $\Delta t={\varepsilon }/{10}$. These results show that, $\varepsilon$-independent time steps can be taken to obtain accurate physical observables, but not accurate wave functions
$\varepsilon=\frac{1}{512}$. First row: position density and current density of $\varphi^e$;
$\varepsilon=\frac{1}{2048}$. First row: position density and flux density of $\varphi^e$; second row: position density and current density of $\psi^e$
Fix $\Delta$ t=0.005. For $\varepsilon=\frac{1}{256}$, $\frac1{512}$, $\frac1{1024}$, $\frac1{2048}$, $\frac1{4096}$, $\Delta x=\frac{\varepsilon}{8}$, respectively. The reference solution is computed with the same $\Delta x$, but $\Delta t=\frac{0.54\varepsilon}{4}$. These results show that, $\varepsilon$-independent time steps can be taken to obtain accurate physical observables, but not accurate wave functions.
Fix $\varepsilon=\frac{1}{256}$ and $\Delta t=\frac{0.4 \varepsilon}{16}$. Take $\Delta x= \frac{2\pi\varepsilon}{32}$, $\frac{2\pi\varepsilon}{16}$, $\frac{2\pi\varepsilon}{8}$, $\frac{2\pi\varepsilon}{4}$, $\frac{2\pi\varepsilon}{2}$ and $\frac{2\pi\varepsilon}{1}$ respectively. The reference solution is computed with the same $\Delta t$, but $\Delta x=\frac{2\pi\varepsilon}{64}$. These results show that, when $\delta=\varepsilon \ll 1$, the meshing strategy $\Delta x= O(\varepsilon )$ and $\Delta y=O(\varepsilon )$ is sufficient for obtaining spectral accuracy
Fix $\varepsilon=\frac{1}{1024}$ and $\Delta x=\frac{2 \pi}{16}$. Take $\Delta t= \frac{0.4}{32}$, $\frac{0.4}{64}$, $\frac{0.4}{128}$, $\frac{0.4}{256}$, $\frac{0.4}{512}$ and $\frac{0.4}{1024}$, respectively. The reference solution is computed with the same $\Delta x$, but $\Delta t=\frac{0.4}{8192}$. These results show that, when $\delta=\varepsilon \ll 1$, the SSP2 method is unconditionally stable and is second order accurate in time
Fix $\Delta t=\frac{0.4}{64}$. For $\delta=\frac1{256}$, $\frac1{512}$, $\frac1{1024}$, $\frac1{2048}$, $\frac1{4096}$, $\Delta x=2\pi\varepsilon/16$, respectively. The reference solution is computed with the same $\Delta x$, but $\Delta t=\frac{\delta}{10}$. These results show that, $\delta$-independent time steps can be taken to obtain accurate physical observables and classical coordinates, but not accurate wave functions
Fix $\delta=\frac{1}{1024}$ and $\Delta x=\frac{2 \pi}{16}$. Take $\Delta t= \frac{0.4}{32}$, $\frac{0.4}{64}$, $\frac{0.4}{128}$, $\frac{0.4}{256}$, $\frac{0.4}{512}$ and $\frac{0.4}{1024}$, respectively. The reference solution is computed with the same $\Delta x$, but $\Delta t=\frac{0.4}{8192}$. These results show that, the SVSP2 method is unconditionally stable and is second order accurate in time
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