Boltzmann equation with double-well potentials

We study the dynamics of an interacting classical gas trapped in a double-well potential at finite temperature. Two model potentials are considered: a cubic box with a square barrier in the middle, and a harmonic trap with a Gaussian barrier along one direction. The study is performed using the Boltzmann equation, solved numerically via the test-particle method. We introduce and discuss a simple analytical model that allows one to provide estimates of the relaxation time, which are compared with numerical results. Finally, we use our findings to make numerical and analytical predictions for the case of a fermionic mixture in the normal-fluid phase in a realistic double-well potential relevant for experiments with cold atoms.

Figure 1

Plot of the SDW (top) and HGDW (bottom) potentials. The figures represent $V(x,y,z=0)$, where $V$ is the potential energy. Particles are also pictorially shown, with their vertical coordinate representing their energy.

Figure 2

Rates at equilibrium. Panel (a): equilibrium rates (lines) of the processes $A, B$ (dashed), $C, D$ (dot-dashed), $E$, and $F$ (double-dashed), defined in (20), and their sum vs $\beta V_0$. The rates are equal two by two and correspond to $h_1,h_2,h_3/2$. The simulations (points) were performed for a square single-well potential, with parameters $N=5000, L=0.5$, and $T=10$. Results with (full symbols) and without (open symbols) a barrier are shown. Panel (b): the adimensional functions $h_1, h_2$, and $h_3$ (with linear scale in vertical axis) vs $\beta V_0$.

Figure 3

Dynamics in the SDW potential. Time evolution of the well population for two barrier heights. Parameters: $N=5000, L=0.5, T=10$, and $\sigma /\pi =0.{001}^2$. The barrier height is $V_0=1$ for the lower curve and $V_0=7.5$ for the upper one. The simulation curves are obtained averaging over 40 runs with different microscopic initial conditions and smoothing over small time intervals. The fits (dotted lines) are done with the function $f_1\left(t\right)=a+b{e}^{-t/\tau }$.

Figure 4

SDW potential relaxation times. The inverse relaxation time $1/\tau$, in units of $1/{\tau }_{\mathrm{trap}}$, as a function of $\beta V_0$ for different values of the cross section: $\sigma =\pi {(0.001,0.004,0.008,0.016)}^2$ with $x_L(t=0)=0.6$. The system is a box of size $2L=1$, with $N=5000$ particles, at $T=10$ and different barriers $V_0$. The numerical results are denoted by points. Each point is obtained from an average over 40 runs [64 for the largest value of the cross section $\sigma /\pi =0.{016}^2$] and smoothing over small time intervals. The fits are done in the whole time interval, using one of the three functions $f_1\left(t\right)$ (full circles), $f_2\left(t\right)$ (stars), and $f_3\left(t\right)$ (full squares) defined in the text. Different gray scales indicate different values of the cross section. For comparison, we show (lines) the results obtained in an approximated analytical solution presented in Sec. 3, namely ${\tau }_1$ obtained using Eqs. (40), (27), (41), and (42). The diffusive limit ${\tau }_{1,\mathrm{diff}}$ is also shown (short-dashed line). Inset: inverse of $\tau$ rescaled in units of collisional time ${\tau }_{\mathrm{coll}}$. Gray scales correspond to the same cross section as the main figure.

Figure 5

SDW model potential. Inverse relaxation time $1/\tau$, in units of $1/{\tau }_{\mathrm{trap}}$, as a function of $d_{\mathrm{int}}$ for a given barrier height, $\beta V_0=0.75$. Parameters and the meaning of symbols and lines are the same of Fig. 4. The numerical results are denoted by points and the model prediction by lines.

Figure 6

Dynamics in the HGDW. Evolution of $x_L$ with time, for three values of the temperature. The gas contains $N=5000$ particles, with an initial imbalance of $x_L(t=0)=0.6$. The well parameters, in trap units, are $\tilde{V}=10, d_{\mathrm{int}}=0.06, w=0.8$ (therefore, $V_0\simeq 7.6$), and the temperatures $T=5$ [panel (a)], 7 [panel (b)], and 20 [panel (c)]. To express them in dimensional units, reported in the top $x$ axis, we need to specify the trap frequency ${\omega }_0$ and the particle mass: for ${}^6\mathrm{Li}$ atoms and with ${\omega }_0=2\pi \times 300$ Hz one has $\tilde{V}/k_B=144\text{nK}, d_{\mathrm{int}}=142\text{nm}=2684a_0, w=1.89\mu \text{m}$, and $T=72,101,288\text{nK}$. These values are realistic for ${}^6\mathrm{Li}$ atoms in a double well potential as in the recent experiment [16] (which is anyway done below $T_c$). The fits (dotted lines) are done with the function $f_2\left(t\right)=a+b{e}^{-t/\tau }+ccos(\omega t+d)e^{-t/{\tau }_2}$. The values of $\tau$ and $\omega$ fitted from the data for figures (a),(b),(c) in trap units are respectively $\tau =(37.2\pm 0.2,22.8\pm 0.2,18.6\pm 0.4)$ and $\omega =(0.99\pm 0.03,0.962\pm 0.002,0.9859\pm 0.0001)$.

Figure 7

HGDW relaxation time. The inverse relaxation time $1/\tau$ as a function of $\beta V_0$, obtained varying the temperature (the trap is fixed as in Fig. 6 with a barrier height $V_0\simeq 7.6$). The numerical results are denoted by points. Different gray scales correspond to different values of the cross section: $\sigma =\pi {(0.03,0.06,0.12)}^2=\pi {(71,142,284)}^2{\text{nm}}^2$. Each point is obtained from an average over 40 runs. The fits are done in the whole time interval, using either $f_2\left(t\right)$ (full circles) or $f_3\left(t\right)$ (stars). The dimensional quantities reported in the figure are obtained as in Fig. 6. For comparison, we show as lines the results obtained in an approximated analytical solution presented in Sec. 3, namely ${\tau }_1$ obtained using Eqs. (40), (27), (43), and (44). Inset: inverse of $\tau$ rescaled in units of collisional time ${\tau }_{\mathrm{coll}}$. Gray scales correspond to the same cross section as the main figure.

Figure 8

HGDW potential. Inverse relaxation time $1/\tau$ as a function of $d_{\mathrm{int}}$ for the same trap of Fig. 6 and a given temperature ($T=10$, then $\beta V_0=0.76$). The numerical results are denoted by points and the model prediction by lines. The meaning of symbols and lines are the same of Fig. 7. The dimensional units are obtained as explained in the caption of Fig. 6.

Figure 9

HGDW potential. Inverse relaxation time $1/\tau$ as a function of $T/T_F$, for a fixed barrier $V_0\simeq 7.6$. The results for $\tau$ are the same of Fig. 7, here shown in a different representation. The Fermi temperature has been computed as $T_F={\left(6N_{\uparrow }\right)}^{1/3}$: that is, the Fermi temperature for a balanced mixture having $N_{\uparrow }=N_{\downarrow }=N$ in a harmonic well. For details on the parameters and the dimensional units, see the caption of Fig. 6.