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csx-home>vol-01>issue-02>Foundation of classical dynamical density functional theory: uniqueness of time-dependent density–potential mappings
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Submitted 10 March 2023
Revised 12 April 2023
Accepted 23 April 2023
Foundation of classical dynamical density functional theory: uniqueness of time-dependent density–potential mappings
Michael Andreas Klatt
Hartmut L¨owen
Ren´e Wittmann
Canadian Science Letters X
2023 ° 23(04) ° 01-02
https://www.wikipt.org/csx-home
DOI: 10.1490/698502.365csx
Funding Agent Details
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), through the SPP 2265, under Grant Nos. WI 5527/1-1 and LO 418/25-1.
Abstract
When can we uniquely map the dynamic evolution of a classical density to a timedependent potential? In equilibrium, without time dependence, the one-body density uniquely specifies the external potential that is applied to the system. This mapping from a density to the potential is the cornerstone of classical density functional theory (DFT). Here, we derive rigorous and explicit conditions for such a unique mapping between a nonequilibrium density profile and a time-dependent external potential. We thus prove the underlying assertion of dynamical density functional theory (DDFT) — with or without the so-called adiabatic approximation often used in applications. We also illustrate loopholes when our conditions are violated so that two distinct external potentials result in the same density profiles but different currents — as suggested by the framework of power functional theory (PFT).
Introduction
The foundation of classical density functional theory (DFT) [21, 9, 10] rests on the fact that the one-body density uniquely determines the external potential and hence the underlying Hamiltonian if the interaction potential is known. In essentially all relevant cases, there exists a unique mapping from the one-body density ρ(x) to an external potential V (x) for x ∈ R d in d dimensions and for a given interaction potential, temperature, and number of particles (or chemical potential). Remarkably, because of this unique mapping, the one-body density specifies a many-body system in equilibrium and hence all higher-body correlations. The existence of such a unique density–potential mapping was first proven in the context of quantum mechanics by Hohenberg and Kohn [15], Kohn and Sham [16], and Mermin [21]. Mermin’s generalized arguments can be directly applied to classical many-body systems as elaborated by Evans [9] and later rigorously confirmed by Chayes, Chayes, and Lieb [3]. The unique mapping exists under mild and natural conditions on the density and interparticle interactions that essentially assume finite energies. Among others, this result implies a formal equivalence of Mermin-Evans DFT to the alternative framework [7] based on Levy constrained search [17] (which does not a priori restrict to density profiles that are realizable by an external potential).
Conclusion
So far, we have assumed the existence of a well-behaved solution P(x N , t), but a proof of existence can be constructed similarly to our proof of uniqueness. Analogously, van Leeuwen [32] generalized the argument by Runge and Gross [26] in quantum mechanics. We, therefore, expect that our proof can also be generalized, but an additional difficulty arises. The existence of a suitable potential requires the solution to an inhomogeneous PDE analogous to (4.9). The resulting conditions on the density and interaction potential should include, as a special case, the known conditions for systems in equilibrium [3]. Similar questions have recently been discussed in quantum mechanics [35]. Another open problem is to drop the condition of analytic potentials. As mentioned above, a fixed-point approach as in [23, 24] could avoid this restriction. A useful generalization would also be to include unnormalizable densities to rigorously treat periodic boundary conditions. Finally, we can generalize the pairwise-interacting passive particles to (i) many-body interactions and marked particles, as well as to (ii) non-conservative forces, such as for active particles. (i) Higher-body interactions lead to more complex average interaction forces but do not change the structure of the hierarchy, so our method of proof should apply. Similarly, our proof should be generalizable to marked particles, where the marks may represent different particle shapes or orientations [22]. (ii) If a known non-conservative force field is added to (2.3), we expect that the corresponding terms drop out similar to (4.2) and (4.4). Thus, the uniqueness of the density-potential mapping equally holds for intrinsically nonequilibrium systems, such as active particles [34].
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