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The European Physical Journal B

On the inclusion of dissipation on top of mean-field approaches

Phuong Mai Dinh, Lionel Lacombe, P.‐G. Reinhard, E. Suraud, Marc Vincendon

Published Oct 1, 2018• 19 citations
Publisher

Abstract

We discuss extensions of time-dependent mean-field theories such as time-dependent local density approximation (TDLDA) in order to include incoherent dynamical correlations, which are known to play a key role in far-off equilibrium dynamics. We focus here on the case of irradiation dynamics in clusters and molecules. The field, still largely unexplored, requires quantum approaches which represents a major formal and computational effort. We present several approaches we have investigated to address such an issue. We start with time-dependent current-density functional theory (TDCDFT), known to provide damping in the linear regime and explore its capability far-off equilibrium. We observe difficulties with the scaling of relaxation times with deposited energy. We next briefly discuss semi-classical approaches which deliver kinetic equations applicable at sufficiently large excitation energies. We then consider a first quantum kinetic equation at the level of a simplified, though rather elaborate in its content, relaxation time approximation (RTA). Thanks to its sophistication, the method allows us to address numerous realistic irradiation scenarios beyond the usual domain of reliability of such theories. We demonstrate in particular the key role played by dense spectral regions in the impact of dissipation in the response of the irradiated system. RTA nevertheless remains a phenomenological approach which calls for more fundamental descriptions. This is achieved by a stochastic extension of mean field theory, coined stochastic time dependent Hartree–Fock (STDHF), which provides an ensemble description of far-off equilibrium dynamics. The method is equivalent to a quantum kinetic equation complemented by a stochastic collision term. STDHF clearly leads to proper thermalization behaviors in 1D test systems considered here. It remains limited by its ensemble nature which requires possibly huge ensembles to properly sample small transition rates. An alternative approach, coined average STDHF (ASTDHF), consists in overlooking mean field fluctuations of STDHF. ASTDHF provides a robust tool, properly matching STDHF when possible and allowing extension to realistic dynamical scenarios in full 3D. It can also be used in open systems to explore, as done in RTA, the competition between ionization and dissipation.

DOI
10.1140/epjb/e2018-90147-0

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We discuss extensions of time-dependent mean-field theories such as time-dependent local density approximation (TDLDA) in order to include incoherent dynamical correlations, which are known to play a key role in far-off equilibrium dynamics. We focus here on the case of irradiation dynamics in clusters and molecules. The field, still largely unexplored, requires quantum approaches which represents a major formal and computational effort. We present several approaches we have investigated to address such an issue. We start with time-dependent current-density functional theory (TDCDFT), known to provide damping in the linear regime and explore its capability far-off equilibrium. We observe difficulties with the scaling of relaxation times with deposited energy. We next briefly discuss semi-classical approaches which deliver kinetic equations applicable at sufficiently large excitation energies. We then consider a first quantum kinetic equation at the level of a simplified, though rather elaborate in its content, relaxation time approximation (RTA). Thanks to its sophistication, the method allows us to address numerous realistic irradiation scenarios beyond the usual domain of reliability of such theories. We demonstrate in particular the key role played by dense spectral regions in the impact of dissipation in the response of the irradiated system. RTA nevertheless remains a phenomenological approach which calls for more fundamental descriptions. This is achieved by a stochastic extension of mean field theory, coined stochastic time dependent Hartree–Fock (STDHF), which provides an ensemble description of far-off equilibrium dynamics. The method is equivalent to a quantum kinetic equation complemented by a stochastic collision term. STDHF clearly leads to proper thermalization behaviors in 1D test systems considered here. It remains limited by its ensemble nature which requires possibly huge ensembles to properly sample small transition rates. An alternative approach, coined average STDHF (ASTDHF), consists in overlooking mean field fluctuations of STDHF. ASTDHF provides a robust tool, properly matching STDHF when possible and allowing extension to realistic dynamical scenarios in full 3D. It can also be used in open systems to explore, as done in RTA, the competition between ionization and dissipation.
Abstract We propose a collisional extension of time-dependent mean-field theories on the basis of a recently proposed stochastic extension of mean-field dynamics (stochastic time-dependent Hartree–Fock, STDHF). The latter theory is unfortunately too involved to envision practical applications in realistic systems in the near future and is thus bound to model systems. It is also hard to explore moderate to low energies with STDHF, because of vanishing transition probabilities that are impossible to sample properly. For such moderately excited situations covering small fluctuations, we compactify sampling by employing the same average mean field for all STDHF trajectories. The new approach, coined average STDHF (ASTDHF), ignores the fluctuations of the mean field but still accounts correctly for the collisional correlations responsible for dissipative features on top of mean-field dynamics. We detail the main features of the new approach in relation to existing equations, in particular quantum kinetic theories. The new theory is directly connected to STDHF, both formally and practically. We thus discuss in detail how the two approaches are related to each other. We apply the new scheme to illustrative examples taking as benchmark STDHF dynamics in 1D. ASTDHF provides results that are in remarkable agreement with the more elaborate STDHF. It makes it a promising approach to deal with dissipative dynamics in finite quantum systems, because of its moderate cost allowing applications in realistic systems and the possibility of exploring any excitation energy range where collisional correlations are expected to play a role.
We present a formalism to describe collisional correlations responsible for thermalization effects in finite quantum systems . The approach consists in a stochastic extension of time dependent mean field theory. Correlations are treated in time dependent perturbation theory and loss of coherence is assumed at some time intervals allowing a stochastic reduction of the correlated dynamics in terms of a stochastic ensemble of time dependent mean-fields. This theory was formulated long ago in terms of density matrices but never applied in practical cases because of its complexity. We propose here a reformulation of the theory in terms of wave functions and use a simplified 1D model of cluster and molecules allowing to test the theory in a schematic but realistic manner. We illustrate the performance in terms of several observables, in particular global moments of the density matrix and single particle entropy built on occupation numbers. The occupation numbers remain fixed in time dependent mean-field propagation and change when evaluating the correlations, then taking fractional values. They converge asymptotically towards Fermi distributions which is a clear indication of thermalization.
The time-dependent Hartree-Fock (TDHF) and density functional theory (DFT) are among the most useful approaches within mean-field theories for studying static and dynamic properties of complex many-body systems in different branches of physics. Despite the fact that they provide a good approximation for the average properties of one-body degrees of freedoms, they are known to fail to include quantal fluctuations of collective observables and they do not provide sufficient dissipation of collective motion. In order to incorporate these missing effects the stochastic mean-field (SMF) approach was proposed (Ayik 2008). In the SMF approach a set of stochastic initial one-body densities are evolved. Each stochastic one-body density matrix consists of a set of stochastic Gaussian random numbers that satisfy the first and second moments of collective one-body observables. Recent works indicate that the SMF approach provides a good description of the dynamics of the nuclear systems (Yilmaz et al. 2018; Ayik et al. 2019). In this work, the one-dimensional Fermi-Hubbard model is simulated with the SMF approach by using different distributions such as Gaussian, uniform, bimodal and two-point distributions. The dissipative dynamics are discussed and the predictive power of the SMF approach with different probability distributions are compared with each other and the exact dynamics. As a result it is shown that by considering different distributions, the predictive power of the SMF approach can be improved.
In the present note, a summary of selected aspects of timedependent mean-field theory is first recalled. This approach is optimized to describe one-body degrees of freedom. A special focus is made on how this microscopic theory can be reduced to a macroscopic dynamic for a selected set of collective variables. Important physical phenomena like adiabaticity/diabaticity, one-body dissipation or memory effect are discussed. Special aspects related to the use of a time-dependent density functional instead of a time-dependent Hartree-Fock theory based on a bare hamiltonian are underlined. The absence of proper description of complex internal correlations however strongly impacts the predictive power of mean-field. A brief overview of theories going beyond the independent particles/quasi-particles theory is given. Then, a special attention is paid for finite fermionic systems at low internal excitation. In that case, quantum fluctuations in collective space that are poorly treated at the meanfield level, are important. Several approaches going beyond mean-field, that are anticipated to improve the description of quantum fluctuations, are discussed: the Balian-V´en´eroni variational principle, the Time-Dependent Random Phase Approximation and the recently proposed Stochastic Mean-Field theory. Relations between these theories are underlined as well as their advantages and shortcomings.
Standard methods used for computing the dynamics of a quantum many-body system are the mean-field (MF) approximations such as the time-dependent Hartree-Fock (TDHF) approach. Even though MF approaches are quite successful, they suffer some well-known shortcomings, one of which is insufficient dissipation of collective motion. The stochastic mean-field approach (SMF), where a set of MF trajectories with random initial conditions are considered, is a good candidate to include dissipative effects beyond mean field. In this approach, the one-body density matrix elements are treated initially as a set of stochastic Gaussian c numbers that are adjusted to reproduce first and second moments of collective one-body observables. It is shown that the predictive power of the SMF approach can be further improved by relaxing the Gaussian assumption for the initial probabilities. More precisely, using Gaussian or uniform distributions for the matrix elements generally leads to overdamping for long times, whereas distributions with smaller kurtosis lead to much better reproduction of the long time evolution.
The dynamics of a many-body system coupled to an external environment represents a fundamentally important problem. To this class of open quantum systems pertains the study of energy transport and dissipation, dephasing, quantum measurement and quantum information theory, phase transitions driven by dissipative effects, etc. Here, we discuss in detail an extension of time-dependent current-density-functional theory (TDCDFT), we named stochastic TDCDFT [Phys. Rev. Lett. 98, 226403 (2007)], which allows the description of such problems from a microscopic point of view. We discuss the assumptions of the theory, its relation to a density-matrix formalism, and the limitations of the latter in the present context. In addition, we describe a numerically convenient way to solve the corresponding equations of motion and apply this theory to the dynamics of a one-dimensional gas of excited bosons confined in a harmonic potential and in contact with an external bath.