Sensitivity of time-dependent density functional theory to initial conditions

Time-dependent density-functional theory is mathematically formulated through nonlinear coupled time-dependent three-dimensional partial differential equations, and it is natural to expect a strong sensitivity of its solutions to variations of the initial conditions, akin to the butterfly effect ubiquitous in classical dynamics. Since the Schrödinger equation for an interacting many-body system is, however, linear and mathematically the exact equations of the density-functional theory reproduce the corresponding one-body properties, it would follow that the Lyapunov exponents are also vanishing within a density-functional theory framework. Whether for realistic implementations of the time-dependent density-functional theory the question of the absence of the butterfly effect and whether the dynamics provided is indeed a predictable theory was never discussed. At the same time, since the time-dependent density-functional theory is a unique tool allowing us to study the nonequilibrium dynamics of strongly interacting many-fermion systems, the question of predictability of this theoretical framework is of paramount importance. Our analysis, for a number of quantum superfluid many-body systems (unitary Fermi gas, nuclear fission, and heavy-ion collisions) with a classical equivalent number of degrees of freedom $O\left({10}^{10}\right)$ and larger, suggests that its maximum Lyapunov exponents are negligible for all practical purposes.

Figure 1

Several consecutive frames demonstrating the evolution of a system with 12 quantized vortices, perturbed by a spherical ball stirring them in the time interval $\mathrm{\Delta }t=170{\varepsilon }_F^{-1}$, after which the UFG is left to evolve undisturbed. The system remains superfluid during the entire evolution. Time is in units of $1/{\varepsilon }_F$ and in the case of UFG one also typically chooses $\hslash =1$.

Figure 2

The time evolution of $\lambda \left(t\right)$ for different methods of introducing the noise: perturbation of quasiparticle wave functions ($\mathrm{\Psi }+\delta \mathrm{\Psi }$), and initial Hamiltonian through density ($n+\delta n$) or the order parameter ($\mathrm{\Delta }+\delta \mathrm{\Delta }$). The value after label ($0.5\%$ or $1.0\%$) indicates strength of the noise $\epsilon$. On top the upper bound by Maldacena et al. [20] is depicted.

Figure 3

In the upper panel we display the evolution of the neutron (upper half of each frame) and proton (lower half of each frame) number densities of fissioning ${}^{236}\mathrm{U}$ in a simulation box ${30}^3\times 60{\mathrm{fm}}^3$ into a heavy (left) and light (right) fragments. The neutron and proton numbers of the heavy (left) and of the light (right) fragments are (83.6, 52.2) and (60.4, 39.8), respectively The shapes of the distributions and the neutron and proton numbers of the fission fragments are hardly affected by the level of noise level. In the lower panel we show the time evolution of $\lambda \left(t\right)$ as obtained from Eq. (41). After scission, which occurs at $t\approx 2400$ $\mathrm{fm}/c$, the excited fission fragments emerge, after thermalization, with temperatures $T\approx 1$ MeV [41, 63], which results in ${\lambda }_{\text{MSS}}^{\text{nuclear}}\approx 0.016$ $c/\mathrm{fm}$.

Figure 4

In the upper panel we display the evolution of the neutron (upper half of each frame) and proton (lower half of each frame) number densities of a fissioning ${}^{238}\mathrm{U}+{}^{238}\mathrm{U}$ collision in a simulation box ${30}^3\times 64{\mathrm{fm}}^3$ into two highly excited and not yet equilibrated fragments. The shapes of the neutron and proton distributions are hardly affected by the noise level. In the lower panel, the time evolution of $\lambda \left(t\right)$, Eq. (41), in the case collision of two heavy ions ${}^{238}\mathrm{U}+{}^{238}\mathrm{U}$ at a center-of-mass energy of 1200 MeV is shown. After separation the excited fragments emerge, after thermalization, with temperatures $T=3.55$ MeV, which results in a ${\lambda }_{\text{MSS}}^{\text{nuclear}}\approx 0.057$ $c/\mathrm{fm}$.