Unitary evolution with fluctuations and dissipation

We outline an extension of the classical Langevin equation to a quantum formulation of the treatment of dissipation and fluctuations of all collective degrees of freedom with unitary evolution of a many-fermion system within an extension of the time-dependent density functional theory. We illustrate the method by computing the distribution of fission fragment yields for ${}^{258}\mathrm{Fm}$ in a quantum hydrodynamic approach and a typical trajectory with full unrestricted density functional theory augmented with dissipation and fluctuations.

Figure 1

The expectation value of the energy of the 1D harmonic oscillator as a function of time.

Figure 2

The initial, final $\rho \left(x\right)=\frac{1}{\tau }{\int }_0^{\tau }dt{\left|\psi (x,t)\right|}^2$, and expected theoretical density distribution, using the temperature estimated from Eq. (15). By increasing the simulation time one can improve on the tails of the calculated density distribution.

Figure 3

The mass yields obtained solving the quantum hydrodynamics equations [82] including dissipation and fluctuations as in Eq. (16) at an excitation energy $E^{*}\approx 14$ MeV, obtained using different strengths $\mathrm{\Gamma }$ = 0.1 (solid), 0.05 (dash), and 0.02 (dots) MeV of the fluctuating field $u_0(\mathbf{r},t)$ (keeping $\mathrm{\Gamma }/\gamma$ fixed), compared to experimental data (thick solid) [85] for spontaneous fission of ${}^{258}\mathrm{Fm}$. The variance of the mass (24.9, 18.4, and 12.9 amu vs experiment 15.2 amu) and of the TKE (18.3, 14.9, and 8.5 MeV vs experiment 19.3 MeV) distributions are approximately proportional to $\sqrt{\mathrm{\Gamma }}$. In the insets we show the TKE (left inset) distributions and typical nuclear shapes at scission (right inset) with and without fluctuations and dissipation. The arrow points to the TKE for symmetric splitting in the absence of dissipation and fluctuations.

Figure 4

In the main panel we show two typical full TDDFT trajectories for ${}^{240}\mathrm{Pu}$ projected into the $Q_{20}=\langle 2z^2-x^2-y^2\rangle$ and $Q_{30}=\langle (5z^2-3r^2)z\rangle$ plane obtained by evolving in time the TDDFT equations [14, 66] without (dashed line) and with (full line) dissipation and fluctuations included, using the NEDF SeaLL1 [14, 83]. In the insets we show the fluctuations of the moments $Q_{2m}=\langle z^{2-m}{(x+iy)}^m\rangle$, for $m=1,2$, which vanish in the absence of fluctuations and otherwise break axial symmetry.