On global solutions to some non-Markovian quantum kinetic models of Fokker–Planck type

Abstract

In this paper, global well-posedness of the non-Markovian Unruh–Zurek and Hu–Paz–Zhang master equations with nonlinear electrostatic coupling is demonstrated. They both consist of a Wigner–Poisson like equation subjected to a dissipative Fokker–Planck mechanism with time-dependent coefficients of integral type, which makes necessary to take into account the full history of the open quantum system under consideration to describe its present state. From a mathematical viewpoint this feature makes particularly elaborated the calculation of the propagators that take part of the corresponding mild formulations, as well as produces rather strong decays near the initial time ($t=0$) of the magnitudes involved, which would be reflected in the subsequent derivation of a priori estimates and a significant lack of Sobolev regularity when compared with their Markovian counterparts. The existence of local-in-time solutions is deduced from a Banach fixed point argument, while global solvability follows from appropriate kinetic energy estimates.

Appendix

This Appendix is devoted to show the complete positivity of the Brownian dynamics described by the approximation of the HPZ model given in Sect. 2.2. To this purpose we make use of the sufficient and necessary condition derived in [9], which in turn is based on the positivity of a certain $2 \times 2$ matrix $\Psi _t$ describing the evolution of the generator of the Weyl algebra (in a units system in which $\hbar = m = 1$), namely,

$$\begin{aligned} \Psi _t= & {} \left( \begin{array} {cc} - 2 \phi _1(t) &{} - \phi _3(t) \\ - \phi _3(t) &{} - 2 \phi _2(t) \end{array} \right) + \frac{i}{2} \left( \begin{array} {cc} 0 &{} 1 \\ - 1 &{} 0 \end{array} \right) \\&- \, \frac{i}{2} \left( \begin{array} {cc} G_1(t) &{} G_2(t) \\ G_1'(t) &{} G_2'(t) \end{array} \right) \left( \begin{array} {cc} 0 &{} 1 \\ - 1 &{} 0 \end{array} \right) \left( \begin{array} {cc} G_1(t) &{} G_1'(t) \\ G_2(t) &{} G_2'(t) \end{array} \right) . \end{aligned}$$

This positivity property is of course reduced to the positivity of its trace and determinant, which are respectively given by

$$\begin{aligned} {\hbox {tr}}(\Psi _t)= & {} -2 \big ( \phi _1(t) + \phi _2(t)\big ) , \\ {\hbox {det}}(\Psi _t)= & {} 4 \phi _1(t) \phi _2(t) - \phi _3(t)^2 - \frac{1}{4} (1 - F(t))^2 , \end{aligned}$$

with

$$\begin{aligned} F(t) = G_1(t) G_2'(t) - G_1'(t) G_2(t) . \end{aligned}$$

Here,

$$\begin{aligned} G_1(t) = G_2'(t) , \quad G_2(t) = {\mathcal {L}}^{-1} \left[ \frac{1}{s^2 + \Omega ^2 + 2 \int \limits _0^\infty \frac{I(\omega )}{\omega } \, \mathrm{d}\omega + {\mathcal {L}}[D(t)](s)} \right] (t) \end{aligned}$$

are the Green functions describing the evolution of the position and momentum operators of the system, where (${\mathcal {L}}^{-1}$) ${\mathcal {L}}$ denotes the (inverse) Laplace transform and where

$$\begin{aligned} D(t) = 2 \int \limits _0^\infty I(\omega ) \sin (\omega t) \, \mathrm{d}\omega , \end{aligned}$$

$I(\omega )$ being the Drude-Lorentz spectral function defined in (14). Also,

$$\begin{aligned} \phi _1(t)= & {} - \frac{1}{4} \int \limits _0^t \int \limits _0^t D_1(t'-s) G_2(t-t') G_2(t-s) \, \mathrm{d}s \, \mathrm{d}t' , \end{aligned}$$

(62)

$$\begin{aligned} \phi _2(t)= & {} - \frac{1}{4} \int \limits _0^t \int \limits _0^t D_1(t'-s) G_1(t-t') G_1(t-s) \, \mathrm{d}s \, \mathrm{d}t' , \end{aligned}$$

(63)

$$\begin{aligned} \phi _3(t)= & {} - \frac{1}{2} G_2(t) \int \limits _0^t D_1(s) G_2(t-s) \, \mathrm{d}s , \end{aligned}$$

(64)

where

$$\begin{aligned} D_1(t) = 2 \int \limits _0^\infty I(\omega ) \coth \left( \frac{\beta \omega }{2} \right) \cos (\omega t) \, \mathrm{d}\omega . \end{aligned}$$

Step 1: Computation of $G_2$. Under our approximation (cf. Sect. 2.2) we have

$$\begin{aligned} D(t) = \delta \pi \Gamma ^2 e^{-\Gamma t} , \quad D_1(t) = \frac{2\delta \pi \Gamma }{\beta } e^{-\Gamma t} . \end{aligned}$$

Then ${\mathcal {L}}[e^{-\Gamma t}](s) = \frac{\delta \pi \Gamma ^2}{\Gamma + s}$, and thus we can compute $G_2$ as

$$\begin{aligned} G_2(t) = {\mathcal {L}}^{-1} \left[ \frac{s + \Gamma }{s^3 + \Gamma s^2 + (\Omega ^2 + \delta \pi \Gamma )s + \Omega ^2 \Gamma } \right] (t) , \end{aligned}$$

where we used that $\int \limits _0^\infty \frac{I(\omega )}{\omega } \, \mathrm{d}\omega = \frac{\delta \pi \Gamma }{2}$. To compute the inverse Laplace transform we factorize the polynomial in the denominator as

$$\begin{aligned} s^3 + \Gamma s^2 + (\Omega ^2 + \delta \pi \Gamma )s + \Omega ^2 \Gamma = (s-r)(s-z)(s-\overline{z}) , \end{aligned}$$

where r is the unique real root and $z = - \frac{1}{2} \big ( \lambda + i \sqrt{4 \mu - \lambda ^2} \big )$, with

$$\begin{aligned} \lambda = r + \Gamma , \quad \mu = r^2 + \Omega ^2 + (r + \pi \delta ) \Gamma . \end{aligned}$$

After expansion in powers of the parameters $\Omega$ and $\Gamma$ up to fourth order, r reads

$$\begin{aligned} r = r_9 \Gamma ^{\frac{7}{2}} -r_8 \Gamma ^3 + r_7 \Gamma ^{\frac{5}{2}} - r_6 \Gamma ^2 + r_5 \Gamma ^{\frac{3}{2}} + r_4 \Omega ^2 \Gamma ^{\frac{3}{2}} - r_3 \Omega ^2 \Gamma + r_2 \Omega ^2 \Gamma ^{\frac{1}{2}} - r_1 \Omega ^2 \Gamma ^{-\frac{1}{2}} - r_0 \Omega ^2 \end{aligned}$$

with

$$\begin{aligned}&r_0 = \frac{1}{2\pi \delta } , \quad r_1 = \frac{1}{\sqrt{3 \pi \delta }} , \quad r_2 = \frac{5}{6 \sqrt{3 \pi ^3 \delta ^3}} , \quad r_3 = \frac{7}{18 \pi ^2 \delta ^2} , \quad r_4 = \frac{55}{144 \sqrt{3 \pi ^5 \delta ^5}} , \\&r_5 = \frac{1}{12 \sqrt{3\pi \delta }} , \quad r_6 = \frac{1}{24 \pi \delta } , \quad r_7 = \frac{1}{16 \sqrt{3 \pi ^3 \delta ^3}} , \quad r_8 = \frac{7}{288 \pi ^2 \delta ^2} , \quad r_9 = \frac{67}{2592 \sqrt{3 \pi ^5 \delta ^5}} , \end{aligned}$$

while

$$\begin{aligned} \sqrt{4 \mu - \lambda ^2} = - \left( p_8 \Gamma ^{\frac{7}{2}} + p_7 \Gamma ^3 + p_6 \Gamma ^{\frac{5}{2}} - p_5 \Gamma ^2 + p_4 \Gamma + p_3 \Omega ^2 - p_2 \Omega ^2 \Gamma ^{\frac{1}{2}} - p_1 \Omega ^2 \Gamma + p_0 \Omega ^2 \Gamma ^{\frac{3}{2}}\right) ^{\frac{1}{2}} \end{aligned}$$

with

$$\begin{aligned}&p_0 = 2 (r_2 - 3 r_0 r_5 - 3 r_1 r_6) , \quad p_1 = 2 (r_0 + 3 r_1 r_5) , \quad p_2 = 2 r_1 , \quad p_3 = 4 , \quad p_4 = 4 \pi \delta , \\&p_5 = 1 , \quad p_6 = 2 r_5 , \quad p_7 = 3 r_5^2 + 2 r_6 , \quad p_8 = 2(r_7 + 3r_5 r_6) . \end{aligned}$$

We can now decompose

$$\begin{aligned} \frac{s + \Gamma }{s^3 + \Gamma s^2 + (\Omega ^2 + \delta \pi \Gamma )s + \Omega ^2 \Gamma } = \frac{A}{s-r} + \frac{\frac{\mu A - \Gamma }{r} - A s}{(s-z)(s-\overline{z})} \end{aligned}$$

with

$$\begin{aligned} A = \frac{r + \Gamma }{\mu + r(r + \lambda )} = \frac{r+\Gamma }{3r^2 + \Omega ^2 + (2r + \pi \delta ) \Gamma }, \end{aligned}$$

so that $G_2$ becomes

$$\begin{aligned} G_2(t) = A e^{rt} + \frac{(2\mu + \lambda r)A - 2 \Gamma }{r \sqrt{4 \mu - \lambda ^2}} e^{- \frac{\lambda }{2} t} \sin \left( \frac{\sqrt{4 \mu - \lambda ^2}}{2} t\right) - A e^{- \frac{\lambda }{2} t}\cos \left( \frac{\sqrt{4 \mu - \lambda ^2}}{2} t\right) . \end{aligned}$$

Step 2: Computation of $\phi _1$, $\phi _2$, $\phi _3$, and${\hbox {tr}}(\Psi _t)$. Following (62) we have

$$\begin{aligned} \phi _1(t)= & {} - \frac{\pi \delta \Gamma }{2 \beta } \int \limits _0^t \int \limits _0^t e^{-\Gamma (t'-s)} G_2(t-t') G_2(t-s) \, \mathrm{d}s \, \mathrm{d}t' \\= & {} - \frac{\pi \delta \Gamma }{2 \beta } \left( \int \limits _0^t e^{-\Gamma s} G_2(s) \, \mathrm{d}s \right) \left( \int \limits _0^t e^{\Gamma s} G_2(s) \, \mathrm{d}s \right) = - \frac{\pi \delta \Gamma }{2 \beta } \Theta _1(t) , \end{aligned}$$

with

$$\begin{aligned} \Theta _1(t) = \left( \int \limits _0^t G_2(s) \, \mathrm{d}s \right) ^2 + \Gamma ^2 \left[ \left( \int \limits _0^t G_2(s) \, \mathrm{d}s \right) \left( \int \limits _0^t s^2 G_2(s) \, \mathrm{d}s \right) - \left( \int \limits _0^t s G_2(s) \, \mathrm{d}s \right) ^2 \right] . \end{aligned}$$

The computation of $\phi _2$ starting from (63) is analogous, by just changing $G_2$ to $G_1$ (thus, $\Theta _1$ to $\Theta _2$). We then find

$$\begin{aligned} {\hbox {tr}}(\Psi _t) = \frac{\pi \delta \Gamma }{2 \beta } \big ( \Theta _1(t) + \Theta _2(t) \big ) , \end{aligned}$$

with

$$\begin{aligned} \Theta _1(t) + \Theta _2(t)= & {} G_2(t)^2 + (1 - \Gamma ^2) \left( \int \limits _0^t G_2(s) \, \mathrm{d}s \right) ^2 \\&+ \, 2 \Gamma ^2 G_2(t) \left( t \int \limits _0^t G_2(s) \, \mathrm{d}s - \int \limits _0^t s G_2(s) \, \mathrm{d}s \right) \\&+ \, \Gamma ^2 \left( \left( \int \limits _0^t G_2(s) \, \mathrm{d}s \right) \left( \int \limits _0^t s^2 G_2(s) \, \mathrm{d}s \right) -\left( \int \limits _0^t s G_2(s) \, \mathrm{d}s\right) ^2 \right) , \end{aligned}$$

which yields the following eighth degree polynomial

$$\begin{aligned} {\hbox {tr}}(\Psi _t)=\widetilde{e}_8 t^8 + \widetilde{e}_7 t^7 + \widetilde{e}_6 t^6 + \widetilde{e}_5 t^5 + \widetilde{e}_4 t^4 + \widetilde{e}_3 t^3 + \widetilde{e}_2 t^2 + \widetilde{e}_1 t + \widetilde{e}_0 , \end{aligned}$$

where

$$\begin{aligned}&\widetilde{e}_8= \frac{\pi ^3 \delta ^3 \Gamma ^3}{640 \beta } , \quad \widetilde{e}_7= \frac{13 \pi ^{\frac{3}{2}} \delta ^{\frac{3}{2}} \Gamma ^{\frac{7}{2}} }{8640 \sqrt{3} \beta } - \frac{11 \pi ^{\frac{3}{2}} \delta ^{\frac{3}{2}} \Omega ^2 \Gamma ^{\frac{3}{2}} }{720 \sqrt{3} \beta } + \frac{13 \pi ^2 \delta ^2 \Gamma ^3 }{720 \beta } , \\&\widetilde{e}_6 = \frac{173 \sqrt{\pi \delta } \Gamma ^{\frac{7}{2}} }{8640 \sqrt{3} \beta } + \frac{\pi ^3 \delta ^3 \Gamma ^3 }{45 \beta } + \frac{17 \pi \delta \Gamma ^3 }{160 \beta } -\frac{163 \sqrt{\pi \delta } \Omega ^2 \Gamma ^{\frac{3}{2}} }{1440 \sqrt{3} \beta } - \frac{\pi ^2 \delta ^2 \Gamma ^2 }{48 \beta }, \\&\widetilde{e}_5 = \frac{11 \pi ^{\frac{3}{2}} \delta ^{\frac{3}{2}} \Gamma ^{\frac{7}{2}}}{720 \sqrt{3} \beta } + \frac{\sqrt{3} \Gamma ^{\frac{7}{2}}}{32 \beta \sqrt{\pi \delta }} +\frac{\Gamma ^{\frac{7}{2}}}{144 \sqrt{3 \pi \delta } \beta } + \frac{11 \pi ^2 \delta ^2 \Gamma ^3}{60 \beta } +\frac{49 \Gamma ^3}{144 \beta } - \frac{\sqrt{\pi \delta } \Gamma ^{\frac{5}{2}} }{72 \sqrt{3} \beta } \\&\quad - \, \frac{\pi \delta \Gamma ^2}{6 \beta } - \frac{\pi ^{\frac{3}{2}} \delta ^{\frac{3}{2}} \Omega ^2 \Gamma ^{\frac{3}{2}}}{6 \sqrt{3} \beta } -\frac{47 \Omega ^2 \Gamma ^{\frac{3}{2}}}{72 \sqrt{3 \pi \delta } \beta } + \frac{ \Omega ^2 \Gamma }{24 \beta } +\frac{\sqrt{\pi \delta } \Omega ^2 \Gamma ^{\frac{1}{2}} }{12 \sqrt{3} \beta } ,\\&\widetilde{e}_4 = \frac{7 \sqrt{\pi \delta } \Gamma ^{\frac{7}{2}} }{48 \sqrt{3} \beta } + \frac{373 \Gamma ^{\frac{7}{2}}}{1728 \sqrt{3 \pi ^3 \delta ^3} \beta } +\frac{17 \pi \delta \Gamma ^3}{12 \beta } + \frac{167 \Gamma ^3}{324 \pi \delta \beta } -\frac{17 \Gamma ^{\frac{5}{2}}}{144 \sqrt{3 \pi \delta } \beta } - \frac{\pi ^2 \delta ^2 \Gamma ^2}{6 \beta } \\&\quad - \, \frac{7 \Gamma ^2}{12 \beta } - \frac{5 \sqrt{\pi \delta } \Omega ^2 \Gamma ^{\frac{3}{2}}}{6 \sqrt{3} \beta } - \frac{271 \Omega ^2 \Gamma ^{\frac{3}{2}}}{144 \sqrt{3 \pi ^3 \delta ^3} \beta } + \frac{17 \Omega ^2 \Gamma }{54 \pi \delta \beta } +\frac{\pi \delta \Gamma }{8 \beta } + \frac{7 \Omega ^2 \Gamma ^{\frac{1}{2}}}{12 \sqrt{3 \pi \delta } \beta } ,\\&\widetilde{e}_3 = \frac{91 \Gamma ^{\frac{7}{2}}}{1728 \sqrt{3 \pi ^5 \delta ^5} \beta } +\frac{223 \Gamma ^{\frac{7}{2}}}{288 \sqrt{3 \pi \delta } \beta } +\frac{175 \Gamma ^3}{864 \pi ^2 \beta \delta ^2}+\frac{713 \Gamma ^3}{144 \beta } -\frac{5 \sqrt{\pi \delta } \Gamma ^{\frac{5}{2}}}{72 \sqrt{3} \beta } - \frac{7 \Gamma ^{\frac{5}{2}}}{16 \sqrt{3 \pi ^3 \delta ^3} \beta } \\&\quad -\frac{5 \pi \delta \Gamma ^2}{6 \beta } -\frac{25 \Gamma ^2}{24 \pi \delta \beta } -\frac{161\Omega ^2 \Gamma ^{\frac{3}{2}} }{216 \sqrt{3 \pi ^5 \delta ^5} \beta } -\frac{11 \Omega ^2 \Gamma ^{\frac{3}{2}}}{3 \sqrt{3 \pi \delta } \beta } + \frac{\Gamma ^{\frac{3}{2}}}{12 \sqrt{3 \pi \delta } \beta } + \frac{7 \Omega ^2 \Gamma }{12 \pi ^2 \delta ^2 \beta } + \frac{\Omega ^2 \Gamma }{4 \beta } + \frac{\Gamma }{\beta } \\&\quad + \, \frac{\Omega ^2 \Gamma ^{\frac{1}{2}} }{ \sqrt{3 \pi ^3 \delta ^3} \beta } + \frac{\sqrt{\pi \delta } \Omega ^2 \Gamma ^{\frac{1}{2}} }{2 \sqrt{3} \beta } ,\\&\widetilde{e}_2 = - \frac{49 \Gamma ^{\frac{7}{2}}}{216 \sqrt{3 \pi ^7 \delta ^7} \beta } + \frac{31 \Gamma ^{\frac{7}{2}}}{24 \sqrt{3 \pi ^3 \delta ^3} \beta } + \frac{157 \Gamma ^3}{864 \pi ^3 \delta ^3 \beta } + \frac{749 \Gamma ^3}{144 \pi \delta \beta } - \frac{31 \Gamma ^{\frac{5}{2}}}{72 \sqrt{3 \pi ^5 \delta ^5} \beta } - \frac{5 \Gamma ^{\frac{5}{2}}}{12 \sqrt{3 \pi \delta } \beta } \\&\quad - \, \frac{35 \Gamma ^2}{216 \pi ^2 \delta ^2 \beta } - \frac{2 \Gamma ^2}{\beta } - \frac{ 73 \Omega ^2 \Gamma ^{\frac{3}{2}} }{12 \sqrt{3 \pi ^3 \delta ^3} \beta } + \frac{\Gamma ^{\frac{3}{2}}}{3 \sqrt{3 \pi ^3 \delta ^3} \beta } + \frac{13 \Omega ^2 \Gamma }{12 \pi \delta \beta } + \frac{\pi \delta \Gamma }{2 \beta } + \frac{2 \Gamma }{\pi \delta \beta } +\frac{2 \Omega ^2 \Gamma ^{\frac{1}{2}}}{\sqrt{3 \pi \delta } \beta } ,\\&\widetilde{e}_1 = \frac{91 \Gamma ^{\frac{7}{2}}}{864 \sqrt{3 \pi ^5 \delta ^5} \beta } +\frac{175 \Gamma ^3}{432 \pi ^2 \delta ^2 \beta } - \frac{7 \Gamma ^{\frac{5}{2}}}{8 \sqrt{3 \pi ^3 \delta ^3} \beta } -\frac{25 \Gamma ^2}{12 \pi \delta \beta } - \frac{161 \Omega ^2 \Gamma ^{\frac{3}{2}} }{108 \sqrt{3 \pi ^5 \delta ^5} \beta } + \frac{\Gamma ^{\frac{3}{2}}}{6 \sqrt{3 \pi \delta } \beta } \\&\quad + \frac{7 \Omega ^2 \Gamma }{6 \pi ^2 \delta ^2 \beta } +\frac{2 \Gamma }{\beta } + \frac{2 \Omega ^2 \Gamma ^{\frac{1}{2}}}{\sqrt{3 \pi ^3 \delta ^3} \beta } ,\\&\widetilde{e}_0 = - \frac{49 \Gamma ^{\frac{7}{2}}}{216 \sqrt{3 \pi ^7 \delta ^7} \beta } + \frac{157 \Gamma ^3}{864 \pi ^3 \delta ^3 \beta } - \frac{31 \Gamma ^{\frac{5}{2}}}{72 \sqrt{3 \pi ^5 \delta ^5} \beta } - \frac{35 \Gamma ^2}{216 \pi ^2 \delta ^2 \beta } + \frac{\Gamma ^{\frac{3}{2}}}{3 \sqrt{3 \pi ^3 \delta ^3} \beta } +\frac{2 \Gamma }{\pi \delta \beta } . \end{aligned}$$

The polynomial returns positive values for a wide range of choices of the physical parameters. For instance, by choosing $\beta = 0.1 \delta$ and $\Gamma = \delta ^{\frac{3}{4}}$, and varying the oscillator frequency below and above the cut-off frequency as (i) $\Omega = 0.3 \Gamma$, (ii) $\Omega = \Gamma$ and (iii) $\Omega = 1.3 \Gamma$, we find that the trace becomes positive for positive times provided that (i) $0.02< \delta < 0.7$, (ii) $0.02< \delta < 0.96$, and (iii) $0.02< \delta < 1$. Notice that here we have considered as positivity criterium the fact that the minimum of the ($\delta$-dependent) trace function be strictly positive.

Finally, from (64) we compute

$$\begin{aligned} \phi _3(t)= & {} - \frac{\pi \delta \Gamma }{\beta } G_2(t) \\\times & {} \left( \left( \frac{\Gamma ^2}{2} t^2 - \Gamma t + 1 \right) \int \limits _0^t G_2(s) \, \mathrm{d}s + \Gamma (1 - \Gamma t) \int \limits _0^t s G_2(s) \, \mathrm{d}s + \frac{\Gamma ^2}{2} \int \limits _0^t s^2 G_2(s) \, \mathrm{d}s \right) . \end{aligned}$$

Step 3: Computation of ${\hbox {det}}(\Psi _t)$. We first calculate

$$\begin{aligned}&\Delta (t) := 4 \phi _1(t) \phi _2(t) - \phi _3(t)^2 = \frac{\pi ^2 \delta ^2 \Gamma ^2}{\beta ^2} \Big ( \int \limits _0^t G_1(s) \, \mathrm{d}s \Big )^2 \Big ( \int \limits _0^t G_2(s) \, \mathrm{d}s \Big )^2 \\&- \frac{\pi ^2 \delta ^2 \Gamma ^2}{\beta ^2} G_2(t)^2 \left( (1-2 \Gamma t) \Big ( \int \limits _0^t G_2(s) \, \mathrm{d}s\Big )^2 + 2 \Gamma \left( \int \limits _0^t G_2(s) \, \mathrm{d}s \right) \ \left( \int \limits _0^t s G_2(s) \, \mathrm{d}s \right) \right) , \end{aligned}$$

and use that $G_1 = G_2'$ to certify that

$$\begin{aligned} \Delta (t) = \frac{2 \pi ^2 \delta ^2 \Gamma ^3}{\beta ^2} G_2(t)^2 \left( t \left( \int \limits _0^t G_2(s) \, \mathrm{d}s \right) ^2 - \left( \int \limits _0^t G_2(s) \, \mathrm{d}s \right) \left( \int \limits _0^t s G_2(s) \, \mathrm{d}s \right) \right) . \end{aligned}$$

Finally, the determinant can be expressed as

$$\begin{aligned} {\hbox {det}}(\Psi _t)= & {} \Delta (t) - \frac{1}{4} \big ( 1 + G_2(t) G_2''(t) - G_2'(t) \big )^2 \\= & {} \frac{2 \pi ^2 \delta ^2 \Gamma ^3}{\beta ^2} G_2(t)^2 \left( t \left( \int \limits _0^t G_2(s) \, \mathrm{d}s \right) ^2 - \left( \int \limits _0^t G_2(s) \, \mathrm{d}s \right) \left( \int \limits _0^t s G_2(s) \, \mathrm{d}s \right) \right) \\&- \frac{1}{4} \big ( 1 + G_2(t) G_2''(t) - G_2'(t) \big )^2 , \end{aligned}$$

which results in a seventh degree polynomial whose coefficients depend upon the physical parameters $\delta$, $\beta$, $\Omega$ and $\Gamma$ in the following way:

$$\begin{aligned} {\hbox {det}}(\Psi _t)=e_7 t^7 + e_6 t^6 + e_5 t^5 + e_4 t^4 + e_3 t^3 + e_2 t^2 + e_1 t + e_0 , \end{aligned}$$

with

$$\begin{aligned}&e_7 = \frac{\pi ^2 \delta ^2 \Gamma ^3}{6 \beta ^2} , \quad e_6 = \frac{7 \sqrt{\pi \delta } \Gamma ^{\frac{7}{2}}}{36 \sqrt{3} \beta ^2} + \frac{7 \pi \delta \Gamma ^3}{3 \beta ^2} , \quad e_5= \frac{17 \Gamma ^{\frac{7}{2}}}{9 \sqrt{3 \pi \delta } \beta ^2} + \frac{34 \Gamma ^3}{3 \beta ^2} ,\\&e_4 = \frac{17 \Gamma ^{\frac{7}{2}}}{3 \sqrt{3 \pi ^3 \delta ^3} \beta ^2 } + \frac{\sqrt{\pi \delta } \Gamma ^{\frac{7}{2}}}{36 \sqrt{3}} +\frac{68 \Gamma ^3}{3 \pi \delta \beta ^2} +\frac{\pi \delta \Gamma ^3}{6} + \frac{\sqrt{\pi \delta } \Omega ^2 \Gamma ^{\frac{3}{2}}}{6 \sqrt{3}} ,\\&e_3 = \frac{16 \Gamma ^{\frac{7}{2}}}{3 \sqrt{3 \pi ^5 \delta ^5} \beta ^2} +\frac{23 \Gamma ^{\frac{7}{2}}}{144 \sqrt{3 \pi \delta }} +\frac{16 \Gamma ^3}{\pi ^2 \delta ^2 \beta ^2} +\frac{3 \Gamma ^3}{4} +\frac{\Omega ^2 \Gamma ^{\frac{3}{2}}}{4 \sqrt{3 \pi \delta } } , \\&e_2 = \frac{127 \Gamma ^{\frac{7}{2}}}{576\sqrt{3 \pi ^3 \delta ^3}} + \frac{1331 \Gamma ^3}{1728 \pi \delta } - \frac{\Gamma ^{\frac{5}{2}}}{24 \sqrt{3 \pi \delta }} -\frac{\Gamma ^2}{4} + \frac{29 \Omega ^2 \Gamma ^{\frac{3}{2}}}{12 \sqrt{3 \pi ^3 \delta ^3}} -\frac{19 \Omega ^2 \Gamma }{72 \pi \delta } - \frac{\Omega ^2 \Gamma ^{\frac{1}{2}}}{2 \sqrt{3 \pi \delta }} , \\&e_1 = \frac{37 \Gamma ^{\frac{7}{2}}}{5184 \sqrt{3 \pi ^5 \delta ^5}} -\frac{\Gamma ^3}{27 \pi ^2 \delta ^2} + \frac{\Gamma ^{\frac{5}{2}}}{12 \sqrt{3 \pi ^3 \delta ^3}} - \frac{31 \Omega ^2 \Gamma ^{\frac{3}{2}}}{12 \sqrt{\pi ^5 \delta ^5}} , \\&e_0 = \frac{17 \Gamma ^{\frac{7}{2}}}{2592 \sqrt{3 \pi ^7 \delta ^7}} -\frac{\Gamma ^3}{432 \pi ^3 \delta ^3} + \frac{\Omega ^2 \Gamma ^{\frac{3}{2}}}{216 \sqrt{3 \pi ^7 \delta ^7} } . \end{aligned}$$

Requiring its positivity for positive times provides us with a wide range of choices within our approximation. Indeed, by choosing the same relations among the parameters as before, the positivity of the determinant holds for (i) $0.02< \delta < 0.66$, (ii) $0.02< \delta < 0.26$, and (iii) $0.02< \delta < 0.22$, with the same positivity criterium as for the trace (Fig. 2).

Fig. 2

Graphical visualization of the functions ${\hbox {tr}}(\Psi _t)$ (left-hand side) and ${\hbox {det}}(\Psi _t)$ (right-hand side) for the following choices of the physical parameters in the HPZ equation: $\beta = 0.1 \delta$, $\Gamma = \delta ^{\frac{3}{4}}$, $\delta =0.15$ and for different time intervals. The dashed/full/dotted lines correspond to the cases $\Omega = 0.3 \Gamma$, $\Omega = \Gamma$ and $\Omega = 1.3 \Gamma$, respectively