Perspective Perspective: New directions in dynamical density functional theory

Classical dynamical density functional theory (DDFT) has become one of the central modeling approaches in nonequilibrium soft matter physics. Recent years have seen the emergence of novel and interesting fields of application for DDFT. In particular, there has been a remarkable growth in the amount of work related to chemistry. Moreover, DDFT has stimulated research on other theories such as phase field crystal models and power functional theory. In this perspective, we summarize the latest developments in the field of DDFT and discuss a variety of possible directions for future research.


Introduction
Classical dynamical density functional theory (DDFT) is a theory for the time evolution of the one-body density ρ of a fluid, which is based on extending results from equilibrium density functional theory (DFT) [1,2] toward the nonequilibrium case. DDFT exists in deterministic and stochastic variants. Deterministic DDFT was first introduced phenomenologically by Evans [3] and later derived from microscopic particle dynamics by Marini Bettolo Marconi and Tarazona [4], Archer and Evans [5], Yoshimori [6], and Español and Löwen [7]. Stochastic DDFT was pioneered by Munakata [8], Kawasaki [9], and Dean [10]. Finally, Fraaije and coworkers [11,12] have developed a DDFT for polymers. Nowadays, DDFT has been applied in a large and diverse number of fields ranging from simple [13] and colloidal [4] fluids to plasmas [14] and microswimmers [15]. A recent review can be found in [16].
Apart from its 'older brother' DFT, DDFT has two younger 'siblings'-namely phase field crystal (PFC) models [17,18] and power functional theory (PFT) [19,20]. PFC models are simpler than DDFT (and can be derived from it [21]), whereas PFT is more complex and contains DDFT as a limiting case. The development of these two other theories is intimately connected to that of DDFT, and progress in DDFT has stimulated progress in these other theories. For example, the development of active DDFT [22] allowed to derive an active PFC model [23,24], and improvements in DDFT allow also for more accurate PFT equations [25]. * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. This perspective article complements our review [16] in two ways. First, we present articles on DDFT from the past two years, thereby also covering articles not included in the review because they are too recent. Second, we discuss perspectives for future work, thereby providing also a more speculative outlook on the new directions the field is developing towards.
This article is structured as follows: In section 2, we briefly review the derivation of DDFT and its relations to some other methods. Recent developments are explained in section 3. In section 4, we discuss possible future directions. We conclude in section 5.

Density functional theory
DDFT is an extension of classical DFT, which describes the equilibrium state of a classical fluid. Classical DFT, in turn, originates from the more widely known quantum DFT developed by Hohenberg and Kohn [26], which allows to model the ground state of a many-electron system.
We start by briefly introducing DFT following [16,27]. The microscopic description of a classical many-body system requires, in principle, knowledge of the exact phase-space distribution function. Classical DFT makes use of the fact that the state of an equilibrium fluid is completely determined once the one-body density ρ is known. The equilibrium density ρ eq can be calculated from the grand-canonical free energy functional Ω (depending on the temperature T and the chemical potential µ) via the minimization principle (DFT equation) From the grand-canonical functional Ω, the canonical free energy functional F can be obtained via the Legendre transformation The free energy F can be split into three parts: The first term in equation (3) is the exactly known ideal gas free energy Here, k B is the Boltzmann constant and Λ is the (irrelevant) thermal de Broglie wavelength. The third term in equation (3) is the external free energy depending on the external potential U 1 . Finally, the excess free energy F exc describes interactions of the particles in the system and is not known exactly.
(Parametric dependencies are suppressed from here on.)

Dynamical density functional theory
Now, we turn to the nonequilibrium case and present the derivation of DDFT following Archer and Evans [5]. The starting point is the Smoluchowski equation (6) describing the dynamics of the distribution function Ψ depending on the positions ⃗ r k of the N particles (we consider spherical overdamped particles with two-body interactions only) and on time t. Here, Γ is the mobility of a particle and U = U 1 + U 2 (with the pair-interaction potential U 2 ) the total potential. The one-body density is defined as Integrating equation (6) over the coordinates of all particles except for one and using equation (7) gives with the diffusion constant D = Γk B T, where we write ⃗ r for ⃗ r 1 and ⃗ r ′ for ⃗ r 2 . Since equation (8) depends also on the unknown two-body density ρ (2) , we require a closure. For this purpose, one uses the adiabatic approximation, which corresponds to the assumption that the correlations in the system are the same as in an equilibrium system. This allows to insert the equilibrium relation into equation (8) to obtain the DDFT equation Important alternative derivation routes start from the Langevin equations [4,28] that describe the motion of the particles in the system or use the Mori-Zwanzig formalism [6,7]. A complete overview is given in [16]. It is also worth mentioning here the most important limitations of DDFT (discussed in more detail in [16]): (a) The adiabatic approximation required for the derivation of (deterministic) DDFT breaks down in a variety of contexts. For example, it does not capture shear flow 1 , which can bring a system arbitrarily far out of equilibrium without creating density gradients and therefore without creating adiabatic forces [20]. Colloidal many-body systems are governed by viscous and structural nonequilibrium forces not captured within the adiabatic approximation [20,30]. Moreover, the adiabatic approximation leads to an underestimation of relaxation times [4,31]. (b) DDFT usually employs grand-canonical free energy functionals, which leads to inaccuracies for systems with a small number of particles [32,33].
(c) Free energy functionals used in practice are typically approximate. A possible consequence of this is that the system gets stuck in a local minimum of the free energy, leading to an arrested state that would not be predicted by a better functional [4]. (d) Due to an underlying ergodicity assumption, DDFT predicts that hard particles in one dimension can pass through each other (which is unphysical) [25,33].

Related approaches
PFC models [17,18] are a closely related approach. They are based on an order parameter ψ that is related to the density ρ by ρ = ρ 0 (1 + ψ), where ρ 0 is a spatially and temporally constant reference density. The governing equation of PFC models is given by δψ(⃗ r, t) (11) (with a mobility M) and can be derived from equation (10) by making the approximation of a constant mobility. The free energy F in PFC models is also considerably simpler than that of (D)DFT and can be derived by performing a Taylor expansion for the logarithm in equation (4) and a functional Taylor expansion combined with a gradient expansion for the excess free energy F exc . A detailed discussion of this derivation can be found in [16,18,34]. An extension of DDFT that has gained some popularity is PFT, which was developed by Schmidt and Brader [19] (see [16,20,35] for a review). PFT describes the nonequilibrium dynamics of many-body systems and is, like DFT, a formally exact variational theory. The variational principle is formulated here not for the density ρ, but for the current ⃗ J that minimizes the so-called 'power functional'. One can split the dissipative intrinsic part of this functional into an 'ideal part' (the part that is already present in DDFT) and an 'excess part' P exc . This leads to the governing equation [19] ⃗ J(⃗ r, t) δρ(⃗ r, t) (12) of PFT, which reduces to equation (10) for P exc = 0. (In PFT for Newtonian or quantum mechanics, one considers the power rate instead, which is minimized with respect to⃗ J [36,37]). Given that PFT does not rely on the adiabatic approximation, it allows to avoid the problems related to limitation 1. For instance, PFT is capable of describing sheared systems and all relevant viscous and structural forces [20,30,38,39]. The excess power functional P exc contains contributions that counteract the excessively fast relaxation predicted by the adiabatic part of the dynamics [20]. Notably, the functional P exc is non-local in space and time, and therefore allows to incorporate memory effects not present in DDFT. The presence of memory can give rise to interesting dynamics [40]. A further advantage of PFT is that it, in analogy to equilibrium DFT, is based on a formally exact variational principle [20]. PFT is also more accurate than PFC models (as a direct consequence of the fact that PFT is more accurate than DDFT and DDFT is more accurate than PFC models).
The relations between DFT, DDFT, PFC models, and PFT are visualized in figure 1. DFT is an exact theory (apart from approximations required for the free energy functional) for an equilibrium fluid. PFC models also allow to describe equilibrium systems, but with a more approximate free energy functional. If we go to the nonequilibrium case and use DDFT or PFC models, we are making an approximation (namely the adiabatic approximation), such that the theory is not exact. PFT, finally, provides an exact nonequilibrium theory. For the three dynamical theories DDFT, PFC models, and PFT, we include also an active variant which is based on the same sorts of approximations, but will usually be applied to systems further away from equilibrium (namely active ones [41,42]).
A connection can also be established between DDFT and the path-integral approach. While equation (10) is deterministic, a very similar equation with a noise term (a stochastic DDFT) governs the dynamics of the microscopic density operator [10]. This theory, where F exc takes a simple mean-field form, is an exact reformulation of the underlying Brownian dynamics (BD), and also the starting point for the derivation of deterministic DDFT in [4]. As demonstrated in [43], stochastic DDFT can be re-written in path-integral form. Action functionals can be obtained within the Martin-Siggia-Rose (MSR) path-integral formalism [44] (reviewed in [45]). In the context of stochastic DDFT, the MSR formalism has been used to study mode-coupling theory and the glass transition [46,47]. Recent applications can be found not only in works addressing this problem [48,49], but also in active matter physics [50]. Such methods have been used to study the ergodic-nonergodic transition [47], whereas deterministic DDFT faces general problems with nonergodic systems [33]. However, path-integral formulations are also considerably more complex than the rather simple equation (10).
Another approach that is worth mentioning here is the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) developed by Grmela and Öttinger [51,52], which aims to provide a general structure for the dynamics of nonequilibrium systems. In a recent article based on earlier work [53], Haussmann [54] has established a connection between DDFT and the GENERIC formalism. It was also briefly remarked in [55] that an extension of DDFT derived there is consistent with the GENERIC approach.
In theoretical models of fluids, the main alternative to field-theoretical approaches such as DDFT or PFC models are microscopic particle-based approaches such as BD or molecular dynamics (MD) simulations [56,57]. Here, the equations of motion (Langevin equations or Newton's equations) governing the individual constituents of a fluid are solved numerically. Such approaches are more accurate than DDFT, but also computationally far more expensive for a many-particle system. Consequently, BD and MD simulations can be used if the microscopic details are important, whereas DDFT has advantages if one wishes to model larger length or time scales that are not accessible by particle-based simulations. Also, the more compact form of field-theoretical approaches makes it easier to gain qualitative insights. In practice, BD and MD simulations and DDFT can complement each other. For example, a comparison to BD or MD simulation results allows to test DDFT [16], and parameters of field theories can be obtained by fits to BD or MD simulation results [40].

Recent developments in DDFT
Since its original development, DDFT has found a very remarkable number of applications. A detailed overview has been given in our recent review article [16]. Since the purpose of the present manuscript is to highlight more recent developments and, in particular, future perspectives, we now briefly discuss the work on DDFT from the past two years, thereby covering (though not exclusively) articles not yet discussed in our review.
The amount of articles published on certain selected topics is visualized in figure 2. While for some topics the number of existing articles mainly originates from the work of a single author or research group, it is possible to identify certain trends. Perhaps the most interesting one is chemistry, which is addressed by a variety of authors in a variety of ways.

Epidemiology
Already in 2020, the worldwide outbreak of the coronavirus disease COVID-19 has motivated the application of DDFT to disease spreading. In the SIR-DDFT model (a combination of the susceptible-infected-recovered (SIR) model [58] with DDFT), repulsive particle interactions are used to represent social distancing measures [59]. The SIR-DDFT model has been extended to model governmental intervention strategies that can lead to multiple waves of a pandemic [60]. This extension represents the first DDFT with a time-dependent interaction potential. Moreover, a software package has been developed to simulate epidemic outbreaks in the SIR-DDFT model [61]. Yi et al [62] have proposed a way of combining the SIR-DDFT model with WiFi data in order to get estimates for values of the model's parameters. Some further extensions were suggested in [63]. A brief overview was given in [64].

Chemistry
The SIR-DDFT model has the mathematical structure of a reaction-diffusion DFT (RDDFT) [65,66], i.e. a reaction-diffusion equation [67] with the diffusion terms replaced by the right-hand side of equation (10). RDDFT has recently been used also to study actively switching Brownian particles [68][69][70]. Another active matter model based on RDDFT has been developed by Alston et al [71]. Finally, RDDFT has also been applied in actual chemistry to study catalytic oxidation [72], crystal nucleation [73], metal corrosion [74], reactions on catalytic substrates [75], and reactions on electrode surfaces [76].
As an example for such a theory, we discuss here the RDDFT for actively switching colloids by Moncho-Jordá and Dzubiella [68]. The model assumes soft Gaussian particles, which are well described by a simple mean-field free energy functional [77] (a fact that is also exploited in the SIR-DDFT model [59]). These particles can switch their size, which allows for a description as a two-component mixture of big and small particles with a diffusive contribution from DDFT and reaction terms for the switching process. By changing the switching rate, the phase behavior of the liquid can be controlled. Bley et al [69] report good agreement of the RDDFT to BD simulations. Also apart from RDDFT, quite a number of recent applications of DDFT come from chemistry and chemical physics, broadly construed. In particular, DDFT has been used in electrochemistry to model systems and processes such as charging of electric double layers [78,79] and supercapacitors [80,81], counterions [82], dielectricity [83,84], electrolytes [85][86][87][88][89][90], impedance response [91], and ion adsorption [92]. A further example is solvation dynamics [93,94], which has a long tradition as an application of DDFT [95]. Finally, DDFT can be used to study nanoparticle separation [96], the release of molecules from nanoparticles [97] or porous surfaces [98], and wound healing [99].
DDFT for polymer systems [11,12], which is already well established in chemical physics, should of course also be mentioned here. On the theoretical side, the microscopic construction of mobility functions was studied [100][101][102]. Further work considered Bayesian model calibration [103], the influence of correlations on polymer dynamics [104], memory effects [105,106], micelle relaxation [107], and morphological phase transitions [108]. The relation to other relaxation models was briefly discussed in [109]. Finally, the MesoDyn software [110], which allows to simulate polymer systems based on DDFT, remains an important tool in the study of polymer dynamics [111][112][113][114][115].

Theoretical developments
While the majority of DDFT-related works from the past two years focuses on applications rather than on theoretical foundations, there has also been some progress on the theoretical side, which we discuss here. (See section 3.6 for a discussion of theoretical progress in PFC models and PFT.) One line of work is concerned with the fact that classical DFT is formulated in the grand-canonical ensemble, which is inappropriate for very small closed systems and somewhat inconsistent with the fact that DDFT has the form of a (particle-conserving) continuity equation. Work to address this issue has been directed at formulating a canonical DFT [116][117][118][119] and at extending DDFT towards the canonical case in a formalism known as 'particle-conserving dynamics' (PCD) [32]. Such approaches allow to address limitation 2 of DDFT mentioned in section 2.2.
Schindler et al [33], who developed a PCD for mixtures, have noted that this theory makes the unphysical prediction of allowing hard rods in one dimension to pass through each other (limitation 4). This problem is solved in 'order-preserving dynamics' (OPD) [25], a variant of PCD based on an asymmetric interaction potential. Here, the ensemble averages are constructed in such a way that 'unphysical' particle configurations have probability zero. This theory employs a different equilibrium framework than standard DDFT, which turns out to affect the effects of the adiabatic approximation [25]. OPD has also been of interest for philosophers of physics. Since it treats observationally indistinguishable particle configurations in different ways, it is of relevance for the long-standing philosophical debate concerned with whether such distinctions are possible [120]. Wittmann et al [25] and te Vrugt [120] also discuss OPD in relation to PFT, where the form of the superadiabatic contribution might be affected by the fact that the adiabatic one has a different form in OPD.
More generally, DDFT has been discussed in philosophy in relation to the problem of thermodynamic arrow of time, i.e. the question how the irreversibility of macroscopic thermodynamics is compatible with the reversibility of the microscopic laws of physics [121][122][123], and to analyze the problem of scientific reduction [124]. In particular, [121] has the specific aim of developing a philosophy of DDFT.

Applications in new physical contexts
Simple and colloidal fluids remain a central field of application for DDFT, although more recent work on these systems has gone beyond 'standard' DDFT in several ways. For example, Marolt and Roth [145] have used DDFT to study colloids with Casimir and magnetic interactions. Jia and Kusaka [146] used an extended form of DDFT [132] to model nonisothermal hard spheres. The transport of soft Brownian particles was analyzed by Antonov et al [147]. Zimmermann et al [148] investigated the effects of barriers on the transport of colloids through microchannels. Archer et al [149,150] modeled quasicrystals. Montañez-Rodríguez et al [151] studied diffusion on spherical surfaces. The nonequilibrium self-consistent generalized Langevin equation [152], an extension of DDFT, was applied to arrested density fluctuations by Lira-Escobedo et al [153,154]. Density fluctuations were also modeled using an adiabatic approximation by Szamel [155]. Finally, Sharma et al [156] have studied the local softness parameter (which is useful for the description of caging) using DDFT.
Moreover, there have been several extensions of DDFT toward physical systems not previously considered in the context of DDFT. Building up on earlier work [134,135], Fang [136,137,164] has recently derived a DDFT for ferrofluids. Another example is the development of a DDFT for granular media [165,166]. Stanton et al [167] have modeled cellular membranes in DDFT. These can be described as a mixture of lipids and proteins. Finally, Wittmann et al [168] have derived a DDFT that allows to describe mechano-sensing in growing bacteria colonies.
As an example, we discuss here two derivations of active PFC models from DDFT. First, Arold and Schmiedeberg [212,213] have derived a theory for underdamped active particles by extending previous work on inertial DDFT [240]. Here, the free energy functional is chosen based on active PFC models. Second, te Vrugt et al [220] studied the construction of free energy functionals by deriving a PFC model for a mixture of active and passive particles from DDFT. Here, it was investigated in detail which approximations lead to which types of nonlinearities and coupling terms in the free energy functional.
The past two years have seen a significant amount of work on PFT and superadiabatic forces (forces that are not captured within the adiabatic approximation). On the one hand, the formalism has found several applications in the study of acceleration viscosities [241], active matter [242], the dynamics of the van Hove function [243,244], shear flow [38], and superadiabatic demixing [245]. On the other hand, there have been more theoretical developments such as the derivation of Noether's theorem for statistical mechanics [246][247][248][249] (which also served as the basis for a 'force-based DFT' [250]), a classification of nonequilibrium forces [30], a custom flow method [251], philosophical investigations of PFT [120,121], and a reassessment of the original derivation of PFT [252]. A method for calculating superadiabatic forces from first principles was recently proposed by Tschopp and Brader [253].
We here present one of these approaches in more detail, namely the extension of Noether's theorem [254] to statistical mechanics by Hermann and Schmidt [246]. Noether's theorem is commonly applied to a variational theory based on action functionals, where it yields conservation laws based on invariances (such as momentum conservation based on translational invariance). Hermann and Schmidt [246] exploited the fact that statistical mechanics in the form of DFT (equilibrium) and PFT (nonequilibrium) can also be based on variational principles. Making use of various invariances allows to derive hierarchies of sum rules, ranging from simple mechanical laws to complex nonequilibrium relations including memory. Hermann and Schmidt [246] also presents applications to active phase separation and sedimentation.

PFC models
The relation of PFC models to DDFT is a very complex one whose understanding demands further work. Actually, it is not even clear how to draw the boundary between them. Some authors see PFC models simply as a special case of DDFT [280], some see the difference in the fact that PFC models use a gradient expansion for the excess free energy [21], while others reserve the name 'PFC' for models that also have a constant mobility approximation and an expanded logarithm in the ideal gas free energy [34].
DFT functionals can, at least for hard particles, be derived pretty much 'ab initio'. Fundamental measure theory (FMT) [281] provides highly accurate expressions for the free energy functional in hard-particle systems, such that, if we know the particle shapes (and other basic parameter such as the temperature), we can construct the DDFT equation (10) without having to adjust any free parameters. A comparison of several hard-sphere functionals can be found in [282]. In contrast, the free energy in PFC models is typically just assumed to have a very simple Swift-Hohenberg-type form [18,283], and the parameters of this free energy can then be adjusted to fit a wide class of materials [34]. Nevertheless, a derivation of PFC models from DDFT does give microscopic expressions for all these parameters, and so in principle, assuming the free energy to be known for a certain interaction, PFC models also do not contain any free parameters. However, this option is almost never used in practice. This has to do with the fact that the predictions of DFT for the PFC parameters can turn out to be quite inaccurate as a consequence of the fact that the approximations made in the derivation of PFC models from DDFT (ψ is assumed to be small and slowly varying in space) are not well justified [284].
Thus, more work is required in understanding the microscopic origins of PFC models from DDFT. This might allow for more accurate predictions of model parameters, the development of more accurate PFC models (and perhaps also phase field models [285,286], which are also connected to DDFT [287]), and in general a better understanding of scientific reduction [124]. Recently, some work has been done in this direction. This includes a microscopic extension of the active PFC model toward mixtures [220], the development of a framework for obtaining gradient-based free energies from more general expressions [288], and in particular a systematic assessment of the derivation of PFC models from DDFT by Archer et al [34], who argued that the order parameter ψ of PFC models should be interpreted not as the dimensionless deviation of the density from a reference value, but as the logarithm of the density.

Power functional theory
Since, as explained in section 2, PFT contains all of DDFT, but also adds additional structure, it can be quite complex. If one is interested in a model that allows to describe far-from-equilibrium processes but that is also easy to handle, one could also use PFC approaches to approximate the DDFT terms in equation (12). This would allow to obtain a model that combines the simplicity of the PFC approach with the ability of PFT to model far-from-equilibrium processes, and would allow, e.g. to study memory in active matter within the PFC framework (as done phenomenologically in [214]). Such a theory would fit in the currently empty spot at the top right of figure 1. A further interesting idea, suggested in [289], would be to combine PFT with RDDFT (see section 3) in order to model far-from-equilibrium effects in chemical reactions.
On the other hand, also the theoretical foundations of PFT merit further investigation. In particular, a recent article by Lutsko and Oettel [252] has highlighted certain issues in the original derivation of PFT by Schmidt and Brader [19]. More generally, the usefulness of PFT in practice strongly depends on the availability of a good approximation for the excess power functional. Something that would significantly increase the power of PFT would be the development of an analogon of FMT for the excess power functional, which provides an accurate expression obtained from first principles. Moreover, as discussed in [25,120], the question whether a particular effect is to be classified as superadiabatic or not can strongly depend on the choice of the underlying equilibrium framework (e.g. on whether or not one uses OPD in one dimension), since this framework affects the effects of the adiabatic approximation.

Quantum mechanics
Although it is a classical theory, approaches based on DDFT can also be relevant for quantum systems. For example, DDFT can be used to close equations of motion in quantum hydrodynamics [290], and PFT has been extended to quantum systems [36]. However, quantum PFT has not found many applications up to now (an exception is [291]). Given the many interesting results of classical PFT, quantum PFT could allow for further insights into the dynamics of many-electron systems.
Moreover, generalized frameworks connecting classical and quantum descriptions can be developed based on Wigner functions [292]. These provide a phase-space description of quantum systems that (with certain caveats [293,294]) reduces to the classical Liouville equation in the classical limit. This fact has been exploited to develop DDFT-like frameworks for hybrid quantum-classical systems [295,296]. Such approaches represent a promising direction of research, in particular given that in the Wigner framework order parameters can be derived in a very similar way as in the classical case [297] and that the Wigner equations of motion for stochastic variants of quantum mechanics closely resemble classical Fokker-Planck equations [123]. Such approaches can also be used when spin interactions are relevant (such as in quantum ferrofluids [298]), since Wigner functions can also model spins [297,299]. A further possible connection between classical and quantum nonequilibrium DFT is the Runge-Gross theorem [300], which forms the basis of quantum mechanical time-dependent DFT, but which was also extended to classical mechanics by Chan and Finken [301].

Active matter
Active matter physics [41,42], the study of systems that contain self-propelled particles, continues to be a rapidly growing subfield of soft matter physics in which a number of interesting effects are presumably still to be discovered. Active particles can be described in DDFT using a one-body density that depends also on the orientation of the particles. Apart from this, the general idea behind the derivation (see section 2) is still the same. Active DDFT has a number of interesting applications, in particular in the study of microswimmers [15,[302][303][304] (see [255] for an overview). Moreover, active DDFT serves as the basis for the derivation of active PFC models [23,24,220].
A conceptual challenge in modeling active particles using DDFT is that, as explained in section 2, DDFT is based on the assumption that the two-body correlations are the same as in an equilibrium system. Therefore, DDFT is based on a close-to-equilibrium assumption, which is problematic since active systems are far from equilibrium. This problem is, as mentioned in [305,306], inherited by active PFC models. Dhont et al [307] have argued that active DDFT is inappropriate for particles with steep and short-ranged interactions. Moreover, DDFT models for microswimmers [15,302] can become inaccurate at higher densities [308] since hydrodynamic interactions are modeled using a far-field approximation. PFT allows, in principle, to overcome the low-activity limitation as it does not require a close-to-equilibrium assumption, although the governing equation (12) of PFT in practice typically takes the form 'DDFT equation + correction term'. Consequently, PFT has been successfully applied to active phase separation [242]. Microscopically derived active matter models generally require as an input knowledge (or assumptions) about the correlations in the system [309], and it is among the main virtues of DDFT that it provides such an input. Therefore, a promising direction would be to develop a DDFT-like theory based on correlations from a nonequilibrium steady state. Ideas of this form have been used in [310][311][312].
From a more 'applied' perspective, an interesting project could be the study of topological defects in active matter using DDFT. For equilibrium systems, it has been found that DFT provides a quantitatively accurate description (as compared to experiments) of the topology of confined smectics [313]. Given that topological defects are of central importance for the understanding of active matter systems [314], this suggests the investigation of defect dynamics in active matter systems as a further application of active DDFT. Since even the topology of equilibrium smectics remains a topic of active research [315,316], the nonequilibrium case (that can be accessed by DDFT) promises even more interesting discoveries. A first step in this direction is the application of an active PFC model to this problem [317].

Biology
Closely related to active matter are biological applications of DDFT, which have a remarkable diversity. DDFT allows to understand biological systems across all scales. Ion channels [318,319], which can be found in cell membranes, are a small-scale biological system that can be modeled in DDFT. Moreover, DDFT has been used to model the membranes themselves [167]. Going to larger scales, we arrive at DDFT models of entire cells as used in applications to cancer growth [320,321], microswimmers [15,255,[302][303][304], and bacteria [168]. In the SIR-DDFT model [59,60], the considered 'particles' are humans. It even does not have to stop there, since a (quantum-based) DFT has been applied to entire ecosystems [322]. This brief list should make clear the particular advantage DDFT has in biology-the same concept can be applied across all length scales, making DDFT an ideal tool for multiscale modeling.

Chemistry
When taking a look at the publications on DDFT from the past two years, it is notable that quite a number of them are in some way related to chemistry. Examples are the numerous applications of RDDFT [59-62, 64, 68-76] and the many works on electrochemistry [78-88, 91, 92, 263-266]. This is an interesting observation given that DDFT was developed as and is generally thought of as a theory for simple and colloidal fluids.
Since this trend is a rather recent development, DDFT has a lot of unexplored potential for chemistry. Essentially, any system in which chemical reactions occur in combination with other interactions-among the reactants or with other molecules in the environment-could get an improved description from DDFT. This includes, in particular, many biochemical reactions which take place in crowded environments [323]. Moreover, DDFT for ions can be used to improve the design of capacitors and batteries and in medical applications for studying ion channels. In the future, DDFT can therefore be expected to be relevant not only for basic research in statistical mechanics, but also for applications in biotechnology, nanotechnology, and chemical engineering.

Conclusions
In this article, we have summarized recent progress in the field of classical DDFT and outlined perspectives for the future. Interesting work remains to be done at the interface between DDFT and other closely related theories, namely PFC models and PFT. Moreover, DDFT has recently found quite a number of applications that are related to chemistry, which strongly suggests that this is a promising area for future work. Finally, DDFT is a powerful tool for the multiscale modeling of active and biological matter.

Data availability statement
No new data were created or analysed in this study.