## 4 Dynamics
### Estimate of the Droplet Lifetime  
Before analyzing the detailed dynamical behavior of our system it is convenient to discuss the role of three-body losses on the droplet lifetime. To this aim, we shall consider a droplet in free space, at equilibrium (at \(t=0\)). Then, _assuming_ that the shape of the droplet is weakly affected by the atom losses (which is justified at the early stages at the evolution, at least), we can write  
\[n_{i}(r,t)\simeq N_{i}(t)\rho(r), \tag{14}\]  
where \(N_{1}(0)/N_{2}(0)=\sqrt{g_{22}/g_{11}}\equiv\gamma\), and with \(\rho(r)\) representing the droplet density profile normalized to unity [see Equations (4) and (7)]. With this in mind, the evolution of \(N_{i}(t)\) can be obtained from the previous Equation (13) by neglecting the kinetic and chemical potential terms [which concur in determining the shape \(\rho(r)\)], left multiplying each equation by \(\psi_{i}^{*}\), and integrating over the volume (see also Ref. [36]), yielding  
\[\frac{d\widetilde{N}_{1}}{d\tau}=-\widetilde{N}_{1}\widetilde{N}_{2}^{2} \tag{15}\]  
Figure 1: Heat map of \((\widetilde{N}-\widetilde{N}_{c})^{1/4}\) as a function of \(a_{12}\) and \(N_{2}\), with \(\widetilde{N}_{c}=18.65\) being the critical value for the existence of a droplet [1]. The dotted-dashed lines, corresponding to \(\widetilde{N}=933.7\), 94.2, 18.65 (from left to right), represent the boundaries between the different regimes (see text). The vertical red dashed-dotted line at \(a_{12}=-73.6\) corresponds to the onset of the MF collapse. The two circles at \(a_{12}=-82a_{0}\) refer to the parameter configurations considered in Section 4.  
\[\frac{d\bar{N}_{2}}{d\tau}=-2\gamma\bar{N}_{1}\bar{N}_{2}^{2}, \tag{16}\]  
where we have defined \(\bar{N}_{i}(\tau)\equiv N_{i}(\tau)/N_{i}(0)\) and \(\tau\equiv tK_{3}N_{2}^{2}(0)\int\rho^{3}d^{3}\tau\). Notice that the factor of 2 in Equation (16) corresponds to the fact that here we are considering the dominant recombination channel K-Rb-Rb (two atoms of Rb are lost for each atom of K).  
The above equations have an approximate solution of the form  
\[\bar{N}_{1}(\tau) =\frac{1-\beta}{1+(\tau/\tau_{1})^{a_{1}}}+\beta, \tag{17}\] \[\bar{N}_{2}(\tau) =\frac{1}{1+(\tau/\tau_{2})^{a_{2}}}, \tag{18}\]  
which is very accurate, indeed. In the present case (\(\gamma\simeq 0.8736\)), from a fit of the exact numerical solution of Equations (15) and (16) we find \(\tau_{1}\simeq 0.70\), \(a_{1}\simeq 0.88\), \(\beta\simeq 0.43\), \(\tau_{2}\simeq 0.71\), \(\alpha_{2}\simeq 0.86\). In particular, \(\tau_{2}\) corresponds to the _half-life_ of the \(i=2\) component (regardless of the value of \(a_{2}\)), in which losses are dominant. Therefore, it represents a characteristic time through which we can measure the impact of three-body losses on the lifetime of the droplet. In our simple model the half-life is therefore \(t_{1/2}\simeq\tau_{2}/\left[K_{3}N_{2}^{2}(0)\int\rho^{3}d^{3}\tau\right]\) that, besides the explicit dependence on \(N_{2}(0)\), also depends implicitly on \(a_{12}\) through the density distribution \(\rho(r)\). The behavior of \(t_{1/2}\) as a function of \(a_{12}\) and \(N_{2}(0)\) is shown in Figure 2. There we compare the prediction of the above analytical model with the actual values obtained from the solution of the GP equations in (13). The qualitative agreement is remarkable.