A Numerical Study of Quantum Entropy and Information in the Wigner–Fokker–Planck Equation for Open Quantum Systems

. 2024 Mar 14;26(3):263. doi: 10.3390/e26030263

Algorithm 1 Euler–Maruyama for the Wigner–Fokker–Planck equation (harmonic potential)

1:
Define:
2:
$L \leftarrow 1.0$
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$σ_{q} \leftarrow \frac{L}{\sqrt{2}}$
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$σ_{p} \leftarrow \frac{1}{\sqrt{2} L}$
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$δ_{t} \leftarrow 0.01$
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$T o t a l_T i m e \leftarrow 50 .$
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$N u m O f T i m e S t e p \leftarrow round {\frac{T o t a l_T i m e}{δ_{t}}}$
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$N u m O f P a r t i c l e s \leftarrow 10^{4}$
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$μ_{1} \leftarrow 0 .$
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$μ_{2} \leftarrow 0 .$
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$μ \leftarrow [\begin{matrix} μ_{1} & μ_{2} \end{matrix}]$
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$D_{q q} = 1 ., D_{p p} = 1 .$
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$γ = 1 .$
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$D \leftarrow [\begin{matrix} D_{q q} & 0 \\ 0 & D_{p p} \end{matrix}]$
15:
Arrays Initialization:
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$q \leftarrow zeros [N u m O f T i m e S t e p, N u m O f P a r t i c l e s]$
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$p \leftarrow zeros [N u m O f T i m e S t e p, N u m O f P a r t i c l e s]$
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Initial Conditions:
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$q [1, :] \leftarrow normrnd (μ [1], σ_{q}, [1, N u m O f P a r t i c l e s])$
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$p [1, :] \leftarrow normrnd (μ [2], σ_{p}, [1, N u m O f P a r t i c l e s])$
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Update:
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for each $i \in N u m O f T i m e S t e p$ do
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for each $j \in N u m O f P a r t i c l e s$ do
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$ϵ \leftarrow mvnrnd (μ, 2 D δ_{t})$
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$q [i + 1, j] \leftarrow q [i, j] + p [i, j] δ_{t} + ϵ [1]$
26:
$p [i + 1, j] \leftarrow p [i, j] + (- q [i, j] - γ p [i, j]) δ_{t} + ϵ [2]$
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end for
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end for