Sensitivity to the initial conditions of the Time-Dependent Density Functional Theory

TL;DR

This paper investigates the sensitivity of Time-Dependent Density Functional Theory (TDDFT) solutions to initial conditions, finding that for large many-body systems, the theory's dynamics are effectively predictable with negligible chaos.

Contribution

The study demonstrates that, despite the non-linear nature of TDDFT equations, the maximum Lyapunov exponents are negligible in large systems, confirming the predictability of TDDFT for complex many-fermion dynamics.

Findings

Lyapunov exponents are negligible in large systems

TDDFT solutions are effectively predictable

No practical butterfly effect in TDDFT

Abstract

Time-Dependent Density Functional Theory is mathematically formulated through non-linear coupled time-dependent 3-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 Schr\"odinger 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 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 the study of non-equilibrium 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 ${\cal O}(10^{10})$ and larger, suggests that its maximum Lyapunov exponents are negligible for all practical purposes.