Appendix 1. Pseudo-thermal density matrix
Let
$$\begin{aligned} |{{\psi _{\theta }}(t)}\rangle = \int \limits _{-L/2}^{L/2}\,\textrm{d}{y}\;\int \limits _{-\infty }^{\infty }\,\textrm{d}{q}\;\gamma (y,q;\theta )\;|{\psi _{}^{\mathrm{(G)}}(t;y,q)}\rangle , \end{aligned}$$
(28)
where \(\gamma\) and \(\psi _{}^{\mathrm{(G)}}\) are defined in Eqs. (15) and (10), respectively, and L is a length. Furthermore let
$$\begin{aligned} {\rho _{\theta }}\left( t;{q^\prime },q\right)= & {} \langle {\phi (0;{q^\prime })}|{{\psi _{\theta }}(t)}\rangle \langle {{\psi _{\theta }}(t)}|{\phi (0;q)}\rangle \nonumber \\= & {} \int \limits _{-L/2}^{L/2}\,\textrm{d}{{\tilde{y}}}\;\int \limits _{-\infty }^{\infty }\,\textrm{d}{{\tilde{q}}}\;\gamma ^*\left( {\tilde{y}},{\tilde{q}};\theta \right) \; \int \limits _{-L/2}^{L/2}\,\textrm{d}{{\tilde{\tilde{y}}}}\;\int \limits _{-\infty }^{\infty }\,\textrm{d}{{\tilde{\tilde{q}}}}\;\gamma \left( {\tilde{\tilde{y}}},{\tilde{\tilde{q}}};\theta \right) \; \nonumber \\{} & {} \quad \times \langle {\phi \left( 0;{q^\prime }\right) }|{\psi _{}^{\mathrm{(G)}}\left( t;{\tilde{y}},{\tilde{q}}\right) }\rangle \langle {\psi _{}^{\mathrm{(G)}}\left( t;{\tilde{\tilde{y}}},{\tilde{\tilde{q}}}\right) }|{\phi (0;q)}\rangle , \end{aligned}$$
(29)
be the density matrix element of state \({\psi _{\theta }}(t)\) in the basis of eigenstates of the system’s Hamiltonian. Averaging over random phases yields (see Eq. (17))
$$\begin{aligned}{} \left\langle {\rho _{\theta }}\left( t;{q^\prime },q\right) \right\rangle _{\theta } {} & =\;\textrm{e}^{\displaystyle \textrm{i}\,\left( {q^\prime }^2-q^2\right) \,D_{\textrm{q}}\,t}\nonumber\\ {}&\quad\times \int \limits _{-L/2}^{L/2}\,\textrm{d}{{\tilde{y}}}\; \int \limits _{-\infty }^{\infty }\,\textrm{d}{{\tilde{q}}}\;\;|\gamma \left( {\tilde{y}},{\tilde{q}};0\right) |^2\; f^*\left( {q^\prime };{\tilde{y}},{\tilde{q}}\right) \;f\left( q;{\tilde{y}},{\tilde{q}}\right) \nonumber \\{} & {} =\;\textrm{e}^{\displaystyle \textrm{i}\,\left( {q^\prime }^2-q^2\right) \,D_{\textrm{q}}\,t}\;\;\frac{2\,d\,\ell }{\pi \,L}\; \nonumber \\{} & {} \quad \times \underbrace{ \int \limits _{-\infty }^{\infty }\,\textrm{d}{{\tilde{q}}}\; \textrm{e}^{\displaystyle -2\,\ell ^2\,{\tilde{q}}^2-d^2\,\left[ \left( {q^\prime }-{\tilde{q}}\right) ^2+\left( q-{\tilde{q}}\right) ^2\right] } }_{\displaystyle \sqrt{\frac{\pi }{2\,\left( \ell ^2+d^2\right) }}\; \textrm{e}^{\displaystyle -\frac{\ell ^2 d^2}{\ell ^2+d^2}\,\left( {q^\prime }^2+q^2\right) }\; \textrm{e}^{\displaystyle -\frac{d^4}{2\,\left( \ell ^2+d^2\right) }\,} } \nonumber \\[5mm]{} & {} \quad \left( {q^\prime }-q\right) ^2 \times \underbrace{ \int \limits _{-L/2}^{L/2}\,\textrm{d}{{\tilde{y}}}\;\textrm{e}^{\displaystyle \textrm{i}\,\left( {q^\prime }-q\right) \,{\tilde{y}}} }_{\displaystyle \frac{2\,\sin \left( \left[ {q^\prime }-q\right] \,L/2\right) }{{q^\prime }-q} } \nonumber \\[5mm]{} & {} =\;\textrm{e}^{\displaystyle \textrm{i}\,\left( {q^\prime }^2-q^2\right) \,D_{\textrm{q}}\,t}\;\sqrt{\frac{2}{\pi }}\;s\; \; \textrm{e}^{\displaystyle -s^2\,\left( {q^\prime }^2+q^2\right) }\; \nonumber \\{} & {} \quad \times \; \textrm{e}^{\displaystyle -\frac{d^2}{2\ell ^2}\,s^2\;\left( {q^\prime }-q\right) ^2} \; \frac{\sin \left( \left[ {q^\prime }-q\right] \,L/2\right) }{[{q^\prime }-q]\,L/2}, \end{aligned}$$
(30)
where \(s\equiv \ell \,d\,/\,\sqrt{\ell ^2+d^2}\).
The diagonal element \(\left\langle {\rho _{\theta }}(t;q,q)\right\rangle _{\theta }=\sqrt{2/\pi }\;s\; \textrm{e}^{\displaystyle -2 s^2 q^2}\) has the form of a Boltzmann distribution. Indeed, by setting \(s = \lambda _{\textrm{th}}/ (2\sqrt{2\pi })\), where \(\lambda _{\textrm{th}}\) is the thermal de Broglie wave length of the particle, these diagonal elements yield the Boltzmann distribution for the population of eigenstates [2].
Strictly, the density matrix pertaining to state \({\psi _{\theta }}\) becomes diagonal only in the limit \(L\rightarrow \infty\), i.e. when the Gaussian wave packets sample the entire position space of the particle.
Appendix 2. Matrix elements of \({\hat{x}}\)
The matrix element of the position operator between two time dependent Gaussian wave packets is given by
$$\begin{aligned}{} & {} x_{}^{\mathrm{(G)}}\left( t;{y^\prime },{q^\prime },y,q\right) \equiv \langle {\psi _{}^{\mathrm{(G)}}\left( t;{y^\prime },{q^\prime }\right) }|\,\hat{x}\,|{\psi _{}^{\mathrm{(G)}}(t;y,q)}\rangle \nonumber \\{} & {} \quad = \frac{1}{\sqrt{2\pi }\,d^{}(t)}\,\int \limits _{-\infty }^{\infty }\,\textrm{d}{x}\,x\, \textrm{e}^{\displaystyle -\frac{\left( x\,-\,{y^\prime }(t)\right) ^2+\left( x\,-\,y(t)\right) ^2}{4\,d^{2}(t)}} \nonumber \\{} & {} \qquad \times \textrm{e}^{\displaystyle \textrm{i}\, \frac{\left( x\,-\,y(t)\right) ^2-\left( x\,-\,{y^\prime }(t)\right) ^2}{4\,d^{2}(t)}\,\frac{D_{\textrm{q}}\,t}{d^2} }\nonumber \\{} & {} \qquad \times \textrm{e}^{\displaystyle \textrm{i}\,\left( q\,\left( x\,-\,D_{\textrm{q}}\,t\,q\right) -{q^\prime }\,\left( x\,-\,D_{\textrm{q}}\,t\,{q^\prime }\right) \right) } \end{aligned}$$
(31)
with \({y^\prime }(t)\equiv {y^\prime }(t,{q^\prime }) = {y^\prime }+ 2 D_{\textrm{q}}{q^\prime }t\) and \(y(t)\equiv y(t,q) = y + 2 D_{\textrm{q}}q t\). Let
$$\begin{aligned} u= & {} \frac{{q^\prime }+q}{\sqrt{2}} \end{aligned}$$
(32)
$$\begin{aligned} v= & {} \frac{{q^\prime }-q}{\sqrt{2}} \end{aligned}$$
(33)
$$\begin{aligned} s(t)= & {} \frac{{y^\prime }(t)+y(t)}{\sqrt{2}} = s(0) + 2\,u\,D_{\textrm{q}}\,t \end{aligned}$$
(34)
$$\begin{aligned} r(t)= & {} \frac{{y^\prime }(t)-y(t)}{\sqrt{2}} = r(0) + 2\,v\,D_{\textrm{q}}\,t \end{aligned}$$
(35)
Then
$$\begin{aligned}{} & {} \left( x\,-\,{y^\prime }(t)\right) ^2+\left( x\,-\,y(t)\right) ^2\nonumber \\{} & {} \quad = 2 x^2 - 2 x \left( {y^\prime }(t)+y(t)\right) + {y^\prime }^2(t)+y^2(t) \nonumber \\{} & {} \quad = 2 x^2 - 2\sqrt{2} x s(t) + s^2(t) + r^2(t) \nonumber \\{} & {} \quad = 2\,\left( x-\frac{s(t)}{\sqrt{2}}\right) ^2 + r^2(t) \end{aligned}$$
(36)
$$\begin{aligned}{} & {} \left( x\,-\,y(t)\right) ^2-\left( x\,-\,{y^\prime }(t)\right) ^2\nonumber \\{} & {} \quad = - 2 x \left( y(t)-{y^\prime }(t)\right) + y^2(t)-{y^\prime }^2(t) \nonumber \\{} & {} \quad = 2\sqrt{2}\,x\,r(t) - 2\,s(t)\,r(t) \nonumber \\{} & {} \quad = 2\sqrt{2}\,r(t)\,\left( x-\frac{s(t)}{\sqrt{2}}\right) \end{aligned}$$
(37)
Given that
$$\begin{aligned}{} & {} q\,\left( x\,-\,D_{\textrm{q}}\,t\,q\right) -{q^\prime }\,\left( x\,-\,D_{\textrm{q}}\,t\,{q^\prime }\right) \nonumber \\{} & {} \quad = -\left( {q^\prime }-q\right) \;x +\left( {q^\prime }^2-q^2\right) \;D_{\textrm{q}}\,t \nonumber \\{} & {} \quad = -\sqrt{2}\,v\,x + 2\,v\,u\,D_{\textrm{q}}\,t \nonumber \\{} & {} \quad = -\sqrt{2}\,v\,\left( x - \sqrt{2}\,u\,D_{\textrm{q}}\,t\right) \nonumber \\{} & {} \quad = -\sqrt{2}\,v\,\left( x - \frac{s(t)}{\sqrt{2}} + \underbrace{ \frac{s(t)}{\sqrt{2}} - \sqrt{2}\,u\,D_{\textrm{q}}\,t }_{\displaystyle s(0)/\sqrt{2}} \right) \nonumber \\{} & {} \quad = -\sqrt{2}\,v\,\left( x - \frac{s(t)}{\sqrt{2}}\right) - v\,s(0) \end{aligned}$$
(38)
the matrix element becomes, after substitution \(x^{\prime }=x-s(t)/\sqrt{2}\),
$$\begin{aligned}{} & {} x_{}^{\mathrm{(G)}}(t;{y^\prime },{q^\prime },y,q) = \frac{1}{\sqrt{2\pi }\,d^{}(t)}\; \textrm{e}^{\displaystyle -\frac{r^2(t)}{4d^{2}(t)} -\textrm{i}\left( v\,s(0)\right) } \; \nonumber \\{} & {} \qquad \times \; \int \limits _{-\infty }^{\infty }\,\textrm{d}{x^{\prime }}\,\left( x^{\prime }+\frac{s(t)}{\sqrt{2}}\right) \, \textrm{e}^{\displaystyle -\frac{{{x^{\prime }}^{2}}}{2\,d^{2}(t)} +\textrm{i}\,\left( \frac{r(t)}{\sqrt{2}\,d^{2}(t)} \,\frac{D_{\textrm{q}}\,t}{d^2} -\sqrt{2}\,v \right) \,x^{\prime }} \nonumber \\{} & {} \quad = \frac{1}{\sqrt{2\pi }\,d^{}(t)}\; \textrm{e}^{\displaystyle -\frac{r^2(t)}{4d^{2}(t)} -\textrm{i}\left( v\,s(0)\right) } \, \times \, \left\{ I_1\left( t\right) \;\frac{s(t)}{\sqrt{2}} + I_2\left( t\right) \right\} \nonumber \\ \end{aligned}$$
(39)
where
$$\begin{aligned} I_1\left( t\right) = \int \limits _{-\infty }^{\infty }\,\textrm{d}{x}\, \textrm{e}^{\displaystyle -b\,x^2 +\textrm{i}\,a\,x }= & {} \sqrt{\frac{\pi }{b}}\,\textrm{e}^{\displaystyle -\frac{a^2}{4\,b}} \end{aligned}$$
(40)
$$\begin{aligned} I_2\left( t\right) = \int \limits _{-\infty }^{\infty }\,\textrm{d}{x}\,x\, \textrm{e}^{\displaystyle -b\,x^2 +\textrm{i}\,a\,x }= & {} \textrm{i}\frac{a}{2\,b}\,\sqrt{\frac{\pi }{b}}\,\textrm{e}^{\displaystyle -\frac{a^2}{4\,b}} \end{aligned}$$
(41)
with
$$\begin{aligned} a \left( = a(t)\right)= & {} \frac{r(t)}{\sqrt{2}\,d^{2}(t)} \,\frac{D_{\textrm{q}}\,t}{d^2} -\sqrt{2}\,v \end{aligned}$$
(42)
$$\begin{aligned} b \big (= b(t)\big )= & {} \frac{1}{2d^{2}(t)} \end{aligned}$$
(43)
To simplify this equation, consider
$$\begin{aligned} \frac{r^2(t)}{4d^{2}(t)} + \frac{a^2}{4\,b}= & {} \frac{r^2(t)}{4d^{2}(t)} + \frac{d^{2}(t)}{2}\, \left( \frac{r(t)}{\sqrt{2}\,d^{2}(t)} \,\frac{D_{\textrm{q}}\,t}{d^2} -\sqrt{2}\,v \right) ^2 \nonumber \\= & {} r^2(t)\, \underbrace{ \frac{d^4+D_{\textrm{q}}^2t^2}{4\,d^4\,d^{2}(t)} }_{\displaystyle 1/(4\,d^2)} \,+\, d^{2}(t)\,v^2 \,-\, \frac{v\,r(t)\,D_{\textrm{q}}\,t}{d^2} \nonumber \\= & {} \frac{1}{4\,d^2} \, \left( r^2(t) +4\,v\, \left( d^2\,d^{2}(t)\,v - r(t)\,D_{\textrm{q}}\,t\right) \right) \nonumber \\= & {} \frac{1}{4\,d^2} \, \big ( r^2(0) + 4\,r(0)\,v\,D_{\textrm{q}}\,t + 4\,v^2\,\left( D_{\textrm{q}}\,t\right) ^2 \nonumber \\{} & {} \quad + 4\,d^4\,v^2 + 4\,\left( D_{\textrm{q}}\,t\right) ^2\,v^2 - 4\,v\,r(0)\,D_{\textrm{q}}\,t\nonumber \\{} & {} \quad - 8\,v^2\,\left( D_{\textrm{q}}\,t\right) ^2 \big ) \nonumber \\= & {} d^2\,v^2 + \frac{r^2(0)}{4\,d^2} \end{aligned}$$
(44)
$$\begin{aligned} \frac{a}{2\,b}= & {} \frac{r(0)\,D_{\textrm{q}}\,t}{\sqrt{2}\,d^2}-d^2\,\sqrt{2}\,v \end{aligned}$$
(45)
One then obtains
$$\begin{aligned}{} & {} x_{}^{{\mathrm{(G)}}}\left( t;{y^\prime },{q^\prime },y,q\right) =\textrm{e}^{\displaystyle - d^2\,v^2 - \frac{r^2(0)}{4\,d^2} -\textrm{i}\,v\,r(0) } \\{} & {}\qquad \times \;\left\{ \frac{s(0)}{\sqrt{2}}+\sqrt{2}\,u\,D_{{\textrm{q}}}\,t +\textrm{i}\,\frac{r(0)}{\sqrt{2}}\,\frac{D_{{\textrm{q}}}\,t}{d^2} -\textrm{i}\,d^2\,\sqrt{2}\,v \right\} \\{} & {} \quad =\textrm{e}^{\displaystyle - d^2\,\frac{\left( {q^\prime }-q\right) ^2}{2} - \frac{\left( {y^\prime }-y\right) ^2}{8\,d^2} -\textrm{i}\left( \frac{\left( {q^\prime }-q\right) \,\left( {y^\prime }+y\right) }{2} \right) } \\{} & {} \qquad \times \;\left\{ \frac{{y^\prime }+y}{2}+\left( {q^\prime }+q\right) \,D_{\textrm{q}}\,t\right. \\{} & {} \qquad\quad\; \left. +\,\textrm{i}\,\frac{\left( {y^\prime }-y\right) }{2}\,\frac{D_{{\textrm{q}}}\,t}{d^2} - \textrm{i}\,d^2\,\left( {q^\prime }-q\right) \right\} \\ \end{aligned}$$
(46)
We will focus on differences
$$\begin{aligned}{} & \Delta x_{}^{\mathrm{(G)}}\left( t;{y^\prime },{q^\prime },y,q\right) \nonumber\\{} &\quad\equiv x_{}^{\mathrm{(G)}}\left( t;{y^\prime },{q^\prime },y,q\right) -x_{}^{\mathrm{(G)}}\left( 0;{y^\prime },{q^\prime },y,q\right) \nonumber \\[3mm]{} & {} \quad= \textrm{e}^{\displaystyle - d^2\,\frac{\left( {q^\prime }-q\right) ^2}{2} - \frac{\left( {y^\prime }-y\right) ^2}{8\,d^2} -\textrm{i}\left( \frac{\left( {q^\prime }-q\right) \,\left( {y^\prime }+y\right) }{2} \right) } \nonumber \\{} & {} \qquad \times \;\left[ \left( {q^\prime }+q\right) +\textrm{i}\,\frac{\left( {y^\prime }-y\right) }{2\,d^2} \right] \,\,D_{\textrm{q}}\,t. \end{aligned}$$
(47)
Diagonal elements are simplified to
$$\begin{aligned} \Delta x_{}^{\mathrm{(G)}}\left( t;y,q,y,q\right)= & {} 2\,q\,D_{\textrm{q}}\,t. \end{aligned}$$
(48)
Note that, in the limit \(d\rightarrow \infty\), the differential matrix elements become diagonal in q and independent of both y and \({y^\prime }\):
$$\begin{aligned} \lim \limits _{d\rightarrow \infty } \Delta x_{}^{\mathrm{(G)}}\left( t;{y^\prime },{q^\prime },y,q\right)= & {} \left\{ \begin{array}{lr} 0&\quad {}{q^\prime }\ne q\\ 2\,q \,D_{\textrm{q}}\,t&\quad {}{q^\prime }=q. \end{array} \right. \end{aligned}$$
(49)
Appendix 3. Route I calculation of the MSD
Following the definition in Eq. (1), the quantum mechanical MSD of the particle is
$$\begin{aligned} \delta _x^2(t)= & {} \int \limits _{-L/2}^{L/2}\;\textrm{d}{{y^{\prime \prime \prime }}}\; \int \limits _{-L/2}^{L/2}\;\textrm{d}{{y^{\prime \prime }}}\; \int \limits _{-L/2}^{L/2}\;\textrm{d}{{y^\prime }}\; \int \limits _{-L/2}^{L/2}\;\textrm{d}{y}\;\nonumber \\{} & {} \quad \int \limits _{-\infty }^{\infty }\;\textrm{d}{{q^{\prime \prime \prime }}}\; \int \limits _{-\infty }^{\infty }\;\textrm{d}{{q^{\prime \prime }}}\; \int \limits _{-\infty }^{\infty }\;\textrm{d}{{q^\prime }}\; \int \limits _{-\infty }^{\infty }\;\textrm{d}{q}\; \nonumber \\{} & {} \; \left\langle \gamma ^*\left( {y^{\prime \prime \prime }},{q^{\prime \prime \prime }};\theta \right) \,\gamma \left( {y^\prime },{q^\prime };\theta \right) \,\gamma ^* \left( {y^{\prime \prime }},{q^{\prime \prime }};\theta \right) \,\gamma \left( y,q;\theta \right) \right\rangle _{\theta } \nonumber \\{} & {} \;\;\times \; \Delta x_{}^{\mathrm{(G)}}\left( t;{y^{\prime \prime }},{q^{\prime \prime }},y,q\right) \; \Delta x_{}^{\mathrm{(G)}}\left( t;{y^{\prime \prime \prime }},{q^{\prime \prime \prime }},{y^\prime },{q^\prime }\right) . \end{aligned}$$
(50)
Here
$$\begin{aligned}{} & {} \left\langle \gamma ^*\left( {y^{\prime \prime \prime }},{q^{\prime \prime \prime }};\theta \right) \,\gamma \left( {y^\prime },{q^\prime };\theta \right) \,\gamma ^* \left( {y^{\prime \prime }},{q^{\prime \prime }};\theta \right) \,\gamma (y,q;\theta )\right\rangle _{\theta } \nonumber \\{} & {} \quad = \frac{\displaystyle 2\,\ell ^2}{\displaystyle \pi \,L^2} \, \textrm{e}^{\displaystyle -\ell ^2\, \left( q^2 + {q^\prime }^2 + {q^{\prime \prime }}^2 + {q^{\prime \prime \prime }}^2 \right) }\; \;\nonumber \\{} & {} \quad \left\langle \textrm{e}^{\displaystyle \textrm{i}\,\left( \theta (y,q)-\theta ({y^{\prime \prime }},{q^{\prime \prime }})+\theta ({y^\prime },{q^\prime })-\theta ({y^{\prime \prime \prime }},{q^{\prime \prime \prime }})\right) }\right\rangle _{\theta }. \end{aligned}$$
(51)
The averaged random phases yield (see Eq. (17))
$$\begin{aligned}{} & {} \left\langle \textrm{e}^{\displaystyle \textrm{i}\,\left( \theta (y,q)-\theta \left( {y^{\prime \prime }},{q^{\prime \prime }}\right) +\theta \left( {y^\prime },{q^\prime }\right) -\theta \left( {y^{\prime \prime \prime }},{q^{\prime \prime \prime }}\right) \right) }\right\rangle _{\theta } \nonumber \\{} & \quad = \delta \left( {y^{\prime \prime \prime }}-{y^{\prime }}\right) \delta \left( {q^{\prime \prime \prime }}-{q^{\prime }}\right) \delta \left( y^{\prime\prime }-y\right) \delta \left( q^{\prime\prime }-q\right) \nonumber \\{} & {} \qquad + \delta \left( {y^{\prime \prime \prime }}-y\right)\delta \left( {q^{\prime \prime \prime }}-q\right) \delta \left( {y^{\prime \prime }}-{y^\prime }\right) \delta \left( {q^{\prime \prime }}-{q^\prime }\right) . \end{aligned}$$
(52)
Hence
$$\begin{aligned} \delta _x^2(t)= & {} \left( D_{\textrm{q}}\,t\right) ^2 \; \frac{2\,\ell ^2}{\pi \,L^2} \; \times \left( C_1\;+\;C_2\right) \end{aligned}$$
(53)
where
$$\begin{aligned} C_1= & {} \left( 2\; \int \limits _{-L/2}^{L/2}\;\textrm{d}{y}\; \int \limits _{-\infty }^{\infty }\;\textrm{d}{q}\; q\,\textrm{e}^{\displaystyle -2\,\ell ^2\,q^2} \right) ^2 \;=\;0 \end{aligned}$$
(54)
$$\begin{aligned} C_2= & {} \int \limits _{-L/2}^{L/2}\;\textrm{d}{{y^\prime }}\; \int \limits _{-L/2}^{L/2}\;\textrm{d}{y}\; \int \limits _{-\infty }^{\infty }\;\textrm{d}{{q^\prime }}\; \int \limits _{-\infty }^{\infty }\;\textrm{d}{q}\; \nonumber \\{} & {} \qquad \textrm{e}^{\displaystyle -2\ell ^2\left( q^2+{q^\prime }^2\right) -d^2\left( {q^\prime }-q\right) ^2-\left( {y^\prime }-y\right) ^2/(4d^2)} \;\nonumber \\{} & {} \qquad \left( \left( q+{q^\prime }\right) ^2+\frac{\left( y-{y^\prime }\right) ^2}{4\,d^4}\right) \nonumber \\= & {} 4d^2\; \int \limits _{-L/4d}^{L/4d}\textrm{d}{{\tilde{\tilde{y}}}}\; \int \limits _{-L/4d}^{L/4d}\textrm{d}{{\tilde{y}}}\; \int \limits _{-\infty }^{\infty }\textrm{d}{u}\;\nonumber \\{} & {} \qquad \int \limits _{-\infty }^{\infty }\textrm{d}{v}\; \textrm{e}^{\displaystyle -2\ell ^2\left( u^2+v^2\right) -2d^2v^2-\left( {\tilde{\tilde{y}}}-{\tilde{y}}\right) ^2} \nonumber \\{} & {} \qquad \times \; \left( 2u^2+\frac{\left( {\tilde{y}}-{\tilde{\tilde{y}}}\right) ^2}{d^2}\right) \nonumber \\= & {} 4d^2\; \int \limits _{-L/4d}^{L/4d}\textrm{d}{{\tilde{\tilde{y}}}}\; \int \limits _{-L/4d}^{L/4d}\textrm{d}{{\tilde{y}}}\; \textrm{e}^{\displaystyle -\left( {\tilde{\tilde{y}}}-{\tilde{y}}\right) ^2} \; \underbrace{ \int \limits _{-\infty }^{\infty }\textrm{d}{v}\; \textrm{e}^{\displaystyle -2\left( \ell ^2+d^2\right) v^2} }_{\displaystyle =\sqrt{\frac{\pi }{2}}\;\frac{1}{\sqrt{\ell ^2+d^2}}\; } \nonumber \\{} & {} \qquad \times \left( 2\, \underbrace{ \int \limits _{-\infty }^{\infty }\textrm{d}{u}\;\textrm{e}^{\displaystyle -2\ell ^2u^2}\,u^2 }_{\displaystyle = \frac{\sqrt{2\pi }}{8\,\ell ^3} } \; + \; \frac{\left( {\tilde{\tilde{y}}}-{\tilde{y}}\right) ^2}{d^2} \; \underbrace{ \int \limits _{-\infty }^{\infty }\textrm{d}{u}\;\textrm{e}^{\displaystyle -2\ell ^2u^2} }_{\displaystyle = \frac{\sqrt{\pi }}{\sqrt{2}\,\ell } } \right) \nonumber \\= & {} \pi \; \frac{d^2}{\ell \sqrt{\ell ^2+d^2}}\; \left( \frac{1}{\ell ^2}\;J_1\left( \frac{L}{4d}\right) \; + \; \frac{2}{d^2}\;J_2\left( \frac{L}{4d}\right) \right) \end{aligned}$$
(55)
The double integrals in the last equation are given by
$$\begin{aligned} J_1(x)= & {} \int \limits _{-x}^{x}\textrm{d}{{\tilde{\tilde{y}}}}\; \int \limits _{-x}^{x}\textrm{d}{{\tilde{y}}}\; \textrm{e}^{\displaystyle -\left( {\tilde{\tilde{y}}}-{\tilde{y}}\right) ^2} \nonumber \\{} & {} = 2\,\sqrt{\pi }\,\textrm{erf}\left( 2 x\right) \,x\,-\,1\,+\,\textrm{e}^{\displaystyle -4 x^2} \end{aligned}$$
(56)
$$\begin{aligned} J_2(x)= & {} \int \limits _{-x}^{x}\textrm{d}{{\tilde{\tilde{y}}}}\; \int \limits _{-x}^{x}\textrm{d}{{\tilde{y}}}\; \textrm{e}^{\displaystyle -\left( {\tilde{\tilde{y}}}-{\tilde{y}}\right) ^2}\,\left( {\tilde{\tilde{y}}}-{\tilde{y}}\right) ^2 \nonumber \\= & {} \sqrt{\pi }\,\textrm{erf}\left( 2 x\right) \,x\,-\,1\,+\,\textrm{e}^{\displaystyle -4 x^2} \end{aligned}$$
(57)
where \(\textrm{erf}(x)\) is the error function \(\textrm{erf}(x)=1/\sqrt{\pi }\,\int \nolimits _{-x}^x\,\textrm{e}^{\displaystyle -y^2}\textrm{d}{y}\)
\(\equiv 2/\sqrt{\pi }\,\int \nolimits _{-\infty }^x\,\textrm{e}^{\displaystyle -y^2}\textrm{d}{y} -1\), so that
$$\begin{aligned} \delta _x^2(t)= & {} \left( D_{\textrm{q}}\,t\right) ^2 \; 2\,\frac{d^2\ell }{L^2\,\sqrt{\ell ^2+d^2}} \; \left( \frac{1}{\ell ^2}\,J_1\left( \frac{L}{4d}\right) \;\right. \nonumber \\{} & {} \left. +\;\frac{2}{d^2}\,J_2\left( \frac{L}{4d}\right) \right) \nonumber \\= & {} F(d,L)\;\;v_{\textrm{th}}^2\,t^2 \end{aligned}$$
(58)
In the last equation use was made of Eq. (16) and of the identity \(D_{\textrm{q}}^2=\lambda _{\textrm{th}}^2\,v_{\textrm{th}}^2/(8\pi )\). The function F(d, L) is given by
$$\begin{aligned} F(d,L)= & {} \frac{1}{4\pi \sqrt{8\pi }}\;\frac{\lambda _{\textrm{th}}}{L}\;\Big\{ 4\pi \sqrt{\pi }\;\textrm{erf}\left( \frac{L}{2d}\right) \nonumber \\{} & {} \quad+\;\left( 8\pi \,\frac{d}{L}+\frac{\lambda _{\textrm{th}}^2}{d\,L}\right) \,\left( \textrm{e}^{\displaystyle -\frac{L^2}{4\,d^2}}-1\right)\Big\} \nonumber \\ \end{aligned}$$
(59)
Appendix 4. Route II calculation of the MSD
Let \(\bar{n}= (n_x,n_q)\) be a set of indices such as used in Eq. (18). From Eq. (9) we may set \(f_{k,\bar{n}}\equiv f_k(x_{n_x},q_{n_q})\) and then write
$$\begin{aligned} {x_{\theta }}(t)= & {} \sum \limits _{k\,j}\,\sum \limits _{\bar{n}\,\bar{m}}\; g_{\bar{n}}(\theta )\,g_{\bar{m}}^*(\theta )\, f_{k,\bar{n}}\,f_{j,\bar{m}}^*\; \textrm{e}^{\displaystyle -\textrm{i}\,\left( q_k^2-q_j^2\right) \,D_{\textrm{q}}\,t}\;x_{kj} \nonumber \\= & {} \sum \limits _{k\,j}\; \textrm{e}^{\displaystyle -\textrm{i}\,\left( q_k^2-q_j^2\right) \,D_{\textrm{q}}\,t}\;x_{kj}\; \underbrace{ \sum \limits _{\bar{n}\,\bar{m}}\; g_{\bar{n}}(\theta )\,g_{\bar{m}}^*(\theta )\, f_{k,\bar{n}}\,f_{j,\bar{m}}^*\ }_{\displaystyle \equiv a_{kj}(\theta )}\nonumber \\ \end{aligned}$$
(60)
where all indices run from \(-N\) to N and \(\sum \nolimits _{\bar{n}}\) stands for \(\sum \nolimits _{n_x\,n_q}\).
Following Eqs. (9) and (18), the expansion matrix elements can be written as
$$\begin{aligned}{} & {} a_{kj}(\theta ) = \sum \limits _{\bar{n}\,\bar{m}}\; g_{\bar{n}}(\theta )\,g_{\bar{m}}^*(\theta )\, f_{k,\bar{n}}\,f_{j,\bar{m}}^*\ \nonumber \\{} & {} \quad = \frac{4\pi \,d\,\ell }{N\,L^2}\; \sum \limits _{\bar{n}\,\bar{m}}\; \textrm{e}^{\displaystyle \textrm{i}\,\left( \theta _{\bar{n}}-\theta _{\bar{m}}\right) }\nonumber \\{} & {} \qquad \times\textrm{e}^{\displaystyle -\ell ^2\,\left( q_{n_q}^2+q_{m_q}^2\right) }\; \textrm{e}^{\displaystyle -d^2\,\left[ \left( q_k+q_{n_q}\right) ^2+\left( q_j+q_{m_q}\right) ^2\right] }\; \nonumber \\{} & {} \qquad \times \textrm{e}^{\displaystyle \textrm{i}\,\left[ x_{n_x}\,\left( q_k+q_{n_q}\right) -x_{m_x}\,\left( q_j+q_{m_q}\right) \right] }\;\nonumber \\{} & {} \end{aligned}$$
(61)
where \(\theta _{\bar{n}}=\theta (x_{n_x},q_{n_q})\).
Following the definition in Eq. (1), the quantum mechanical MSD of the particle is then
$$\begin{aligned} \delta _x^2(t)= & {} \sum \limits _{k} \sum \limits _{j} \sum \limits _{k^\prime } \sum \limits _{j^\prime } \; x_{kj}\,x_{k^\prime j^\prime }\; \left\langle a_{kj}(\theta )\;a_{k^\prime j^\prime }(\theta )\right\rangle _{\theta } \nonumber \\{} & {} \quad \times \left( \textrm{e}^{\displaystyle -\textrm{i}\,\left( q_k^2-q_j^2\right) \,D_{\textrm{q}}\,t}-1\right) \;\nonumber \\{} & {} \quad \times \left( \textrm{e}^{\displaystyle -\textrm{i}\,\left( q_{k^\prime }^2-q_{j^\prime }^2\right) \,D_{\textrm{q}}\,t}-1\right) \; \end{aligned}$$
(62)
Here
$$\begin{aligned}{} & {} \left\langle a_{kj}(\theta )\;a_{k^\prime j^\prime }(\theta )\right\rangle _{\theta } = \left( \frac{4\pi \,d\,\ell }{N\,L^2}\right) ^2\;\nonumber \\{} & {} \quad \sum \limits _{\bar{n}} \sum \limits _{\bar{m}} \sum \limits _{\bar{n}^\prime } \sum \limits _{\bar{m}^\prime } \; \textrm{e}^{\displaystyle -\ell ^2\,\left( q_{n_q}^2+q_{n^\prime _q}^2+q_{m_q}^2+q_{m^\prime _q}^2\right) } \nonumber \\{} & {} \quad \times \, \textrm{e}^{\displaystyle -d^2\,\left( q_k+q_{n_q}\right) ^2+\textrm{i}\,x_{n_x}\,\left( q_k+q_{n_q}\right) } \nonumber \\{} & {} \quad \times \, \textrm{e}^{\displaystyle -d^2\,\left( q_j+q_{m_q}\right) ^2-\textrm{i}\,x_{m_x}\,\left( q_j+q_{m_q}\right) } \nonumber \\{} & {} \quad \times \, \textrm{e}^{\displaystyle -d^2\,\left( q_{k^\prime }+q_{n^\prime _q}\right) ^2+\textrm{i}\,x_{n^\prime _x}\,\left( q_{k^\prime }+q_{n^\prime _q}\right) } \nonumber \\{} & {} \quad \times \, \textrm{e}^{\displaystyle -d^2\,\left( q_{j^\prime }+q_{m^\prime _q}\right) ^2-\textrm{i}\,x_{m^\prime _x}\,\left( q_{j^\prime }+q_{m^\prime _q}\right) } \nonumber \\{} & {} \quad \times \, \left\langle \textrm{e}^{\displaystyle \textrm{i}\,\left( \theta _{\bar{n}}-\theta _{\bar{m}}+\theta _{\bar{n}^\prime }-\theta _{\bar{m}^\prime }\right) }\right\rangle _{\theta } \end{aligned}$$
(63)
The averaged quantities \(\left\langle \textrm{e}^{\displaystyle \textrm{i}\,\left( \theta _{\bar{n}}-\theta _{\bar{m}}+\theta _{\bar{n}^\prime }-\theta _{\bar{m}^\prime }\right) }\right\rangle _{\theta }\) yield zero, unless \(\bar{n}=\bar{m}\) and \(\bar{n}^\prime =\bar{m}^\prime\), or \(\bar{n}=\bar{m}^\prime\) and \(\bar{m}=\bar{n}^\prime\), when they take the value one (see Eq. (19)).
Hence
$$\begin{aligned} \left\langle a_{kj}(\theta )\;a_{k^\prime j^\prime }(\theta )\right\rangle _{\theta }= & {} A^{(1)}_{k j k^\prime j^\prime } + A^{(2)}_{k j k^\prime j^\prime } \end{aligned}$$
(64)
where
$$\begin{aligned} A^{(1)}_{k j k^\prime j^\prime }= & {} \left( \frac{4\pi \,d\,\ell }{N\,L^2}\right) ^2\; \sum \limits _{\bar{n}} \sum \limits _{\bar{n}^\prime } \; \textrm{e}^{\displaystyle -2\ell ^2\,\left( q_{n_q}^2+q_{n^\prime _q}^2\right) } \nonumber \\{} & {} \quad \times \, \textrm{e}^{\displaystyle -d^2\,\left( q_k+q_{n_q}\right) ^2+\textrm{i}\,x_{n_x}\,\left( q_k+q_{n_q}\right) } \nonumber \\{} & {} \quad \times \, \textrm{e}^{\displaystyle -d^2\,\left( q_j+q_{n_q}\right) ^2-\textrm{i}\,x_{n_x}\,\left( q_j+q_{n_q}\right) } \nonumber \\{} & {} \quad \times \, \textrm{e}^{\displaystyle -d^2\,\left( q_{k^\prime }+q_{n^\prime _q}\right) ^2+\textrm{i}\,x_{n^\prime _x}\,\left( q_{k^\prime }+q_{n^\prime _q}\right) } \nonumber \\{} & {} \quad \times \, \textrm{e}^{\displaystyle -d^2\,\left( q_{j^\prime }+q_{n^\prime _q}\right) ^2-\textrm{i}\,x_{n^\prime _x}\,\left( q_{j^\prime }+q_{n^\prime _q}\right) } \end{aligned}$$
(65)
We shall consider the limit \(N\rightarrow \infty\). In this limit [8] Footnote 4
$$\begin{aligned} \sum _{n=-N}^N\;\textrm{e}^{\displaystyle \textrm{i}\,x_n\,(q_k-q_j)}\approx & {} 2N\,\delta _{jk} \end{aligned}$$
(66)
Therefore we can write, in this limit,
$$\begin{aligned} A^{(1)}_{k j k^\prime j^\prime }\approx & {} \delta _{kj}\,\delta _{k^\prime j^\prime }\; \left( \frac{8\pi \,d\,\ell }{L^2}\right) ^2\; \sum \limits _{n_q} \sum \limits _{n^\prime _q} \; \textrm{e}^{\displaystyle -2\ell ^2\,\left( q_{n_q}^2+q_{n^\prime _q}^2\right) }\; \nonumber \\{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\times \, \textrm{e}^{\displaystyle -2d^2\,\left[ \left( q_k+q_{n_q}\right) ^2+\left( q_{k^\prime }+q_{n^\prime _q}\right) ^2\right] } \end{aligned}$$
(67)
Because of the Kronecker-\(\delta\) factors, the contribution of these terms to the MSD in Eq. (62) is effectively nil.
Similarly, in the limit \(N\rightarrow \infty\),
$$\begin{aligned}{} & {} A^{(2)}_{k j k^\prime j^\prime } = \left( \frac{4\pi \,d\,\ell }{N\,L^2}\right) ^2\; \sum \limits _{\bar{n}} \sum \limits _{\bar{m}} \; \textrm{e}^{\displaystyle -2\ell ^2\,\left( q_{n_q}^2+q_{m_q}^2\right) } \nonumber \\{} & {} \qquad \times \, \textrm{e}^{\displaystyle -d^2\,\left( q_k+q_{n_q}\right) ^2+\textrm{i}\,x_{n_x}\,\left( q_k+q_{n_q}\right) } \nonumber \\{} & {} \qquad \times \, \textrm{e}^{\displaystyle -d^2\,\left( q_j+q_{m_q}\right) ^2-\textrm{i}\,x_{m_x}\,\left( q_j+q_{m_q}\right) } \nonumber \\{} & {} \qquad \times \, \textrm{e}^{\displaystyle -d^2\,\left( q_{k^\prime }+q_{m_q}\right) ^2+\textrm{i}\,x_{m_x}\,\left( q_{k^\prime }+q_{m_q}\right) } \nonumber \\{} & {} \qquad \times \, \textrm{e}^{\displaystyle -d^2\,\left( q_{j^\prime }+q_{n_q}\right) ^2-\textrm{i}\,x_{n_x}\,\left( q_{j^\prime }+q_{n_q}\right) } \nonumber \\{} & {} \quad \approx \delta _{kj^\prime }\,\delta _{j k^\prime }\; \left( \frac{8\pi \,d\,\ell }{L^2}\right) ^2\; \sum \limits _{n_q} \sum \limits _{m_q} \; \textrm{e}^{\displaystyle -2\ell ^2\,\left( q_{n_q}^2+q_{m_q}^2\right) }\; \nonumber \\{} & {} \qquad \times \, \textrm{e}^{\displaystyle -2d^2\,\left[ \left( q_k+q_{n_q}\right) ^2+\left( q_j+q_{m_q}\right) ^2\right] } \nonumber \\{} & {} \quad = \delta _{kj^\prime }\,\delta _{j k^\prime }\; \left( \frac{8\pi \,d\,\ell }{L^2}\right) ^2\; \left( \sum \limits _{n}\,\textrm{e}^{\displaystyle -2\ell ^2\,q_n^2-2d^2\,\left( q_k+q_n\right) ^2}\right) \nonumber \\{} & {} \qquad \times \, \; \left( \sum \limits _{n}\,\textrm{e}^{\displaystyle -2\ell ^2\,q_n^2-2d^2\,\left( q_j+q_n\right) ^2}\right) \nonumber \\{} & {} \quad \approx \delta _{kj^\prime }\,\delta _{jk^\prime }\; \underbrace{ \left( \frac{8\pi \,d\,\ell }{L^2}\right) ^2\;\left( \frac{L}{2}\right) ^2\; \frac{1}{2\pi \,\left( \ell ^2\!+\!d^2\right) }\,\textrm{e}^{\displaystyle -2\frac{\ell ^2d^2}{\ell ^2\!+\!d^2}\,\left( q_k^2\!+\!q_j^2\right) } }_{\displaystyle \frac{8\pi s^2}{L^2}\,\textrm{e}^{\displaystyle -2\,s^2\,\left( q_k^2+q_j^2\right) }\;\;\;\text{(see } \text{ Appendix } \text{1) }} \nonumber \\ \end{aligned}$$
(68)
where \(s=\ell d\,/\,\sqrt{\ell ^2+d^2}\equiv \lambda _{\textrm{th}}/\sqrt{8\pi }\) and the approximation in the last equation holds the better, the larger L (see Eq. (3.2) in Ref. [8]).
Hence
$$\begin{aligned} \delta _x^2(t)\approx & {} \frac{8\pi \,s^2}{L^2}\sum \limits _{k}\,\sum \limits _{j}\,|x_{kj}|^2 \, |\textrm{e}^{\displaystyle -\textrm{i}\,\left( q_k^2\!-\!q_j^2\right) \,D_{\textrm{q}}\,t}\!-\!1|^2 \;\textrm{e}^{\displaystyle -2\,s^2\, \left( q_k^2\!+\!q_j^2 \right) } \nonumber \\ \end{aligned}$$
(69)
