key: cord-0482748-qx8ugflf
authors: Vrugt, Michael te; Wittkowski, Raphael
title: Perspective: New directions in dynamical density functional theory
date: 2022-04-24
journal: nan
DOI: nan
sha: db26ebb28309a5af7a9f57621f694515bef0d160
doc_id: 482748
cord_uid: qx8ugflf

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.

DDFT is an extension of classical density functional theory (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 Refs. [16, 27] . The microscopic description of a classical manybody 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 

The free energy F can be split into three parts:

The first term in Eq. (3) is the exactly known ideal gas free energy F id (T, [ρ]) = k B Tˆd 3 r ρ( r)(ln(Λ 3 ρ( r)) − 1). (4)

Here, k B is the Boltzmann constant and Λ is the (irrelevant) thermal de Broglie wavelength. The third term in Eq.

(3) is the external free energy F ext ([ρ]) =ˆd 3 r ρ( r)U 1 ( r)

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.) 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

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 ρ( r 1 , t) = Nˆd 3 r 2 · · ·ˆd 3 r N Ψ({ r k }, t).

Integrating Eq. (6) over the coordinates of all particles except for one and using Eq. (7) gives ∂ ∂t ρ( r, t) = D ∇ 2 ρ( r, t) + Γ ∇ · (ρ( r, t) ∇U 1 ( r, t)) + Γ ∇ ·ˆd 3 r ρ (2) ( r, r , t) ∇U 2 ( r, r )

with the diffusion constant D = Γk B T , where we write r for r 1 and r for r 2 . Since Eq. (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 ρ( r) ∇ δF exc [ρ] δρ( r) =ˆd 3 r ρ (2) ( r, r ) ∇U 2 ( r, r )

into Eq. (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 Ref. [16] . Phase field crystal (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

(with a mobility M ) and can be derived from Eq. (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 Eq. (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 Refs. [16, 18, 29 ]. An extension of DDFT that has gained some popularity is power functional theory (PFT), which was developed by Schmidt and Brader [19] (see Refs. [16, 20, 30] 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". This functional can be split into an "ideal part" (the part that is already present in DDFT) and an "excess part" P exc . This leads to the governing equation

of PFT, which reduces to Eq. (10) for P exc = 0. The relations between DFT, DDFT, PFC models, and PFT are visualized in Fig. 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 [31, 32]).

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 Fig. 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. 

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-infectedrecovered (SIR) model [33] with DDFT), repulsive particle interactions are used to represent social distancing measures [34] . The SIR-DDFT model has been extended to model governmental intervention strategies that can lead to multiple waves of a pandemic [35] . 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 [36] . Yi et al. [37] 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 Ref. [38] . A brief overview was given in Ref. [39] .

The SIR-DDFT model has the mathematical structure of a reaction-diffusion DDFT (RDDFT) [40, 41] , i.e., a reaction-diffusion equation [42] with the diffusion terms replaced by the right-hand side of Eq. (10). RDDFT has recently been used also to study actively switching Brownian particles [43] [44] [45] . Here, it is assumed that the particles can switch between two states with different sizes at a certain rate. This work also explicitly compares RDDFT to Brownian dynamics simulations and reports good agreement [44] . Another active matter model based on RDDFT has been developed by Alston et al. [46] . Finally, RDDFT has also been applied in actual chemistry to study catalytic oxidation [47] , crystal nucleation [48] , metal corrosion [49] , reactions on catalytic substrates [50] , and reactions on electrode surfaces [51] .

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 [ [67] . Finally, DDFT can be used to study nanoparticle separation [68] , the release of molecules from nanoparticles [69] or porous surfaces [70] , and wound healing [71] .

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 [72] [73] [74] . Further work considered the influence of correlations on polymer dynamics [75] , memory effects [76, 77] , micelle relaxation [78] , and morphological phase transitions [79] . The relation to other relaxation models was briefly discussed in Ref. [80] . Finally, the MesoDyn software [81] , which allows to simulate polymer systems based on DDFT, remains an important tool in the study of polymer dynamics [82] [83] [84] [85] [86] .

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 [87] [88] [89] [90] and at extending DDFT towards the canonical case in a formalism known as "particle-conserving dynamics" (PCD) [91] . Schindler et al. [92] , 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. This problem is solved in "order-preserving dynamics" (OPD) [25], a variant of PCD based on an asymmetric interaction potential.

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 [93] . 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 [94] [95] [96] , and to ana-lyze the problem of scientific reduction [97] . In particular, Ref. [94] had the specific aim of developing a philosophy of DDFT.

Some of these discussions [94, 95] discuss DDFT in relation to the Mori-Zwanzig formalism [98] [99] [100] [101] [102] , which allows to derive irreversible transport equations from reversible microdynamics and thus to understand irreversibility [95, 103] . This formalism has also been used to derive [6, 7] and extend [104] [105] [106] [107] DDFT. It is also important for more recent work. In particular, it plays a prominent role in Fang's development of a DDFT for ferrofluids [108] [109] [110] [111] , in analyzing memory effects in polymers [76, 77] , and in the study of crystal elasticity [112] [113] [114] . A stochastic theory related to stochastic DDFT was derived using the Mori-Zwanzig formalism in Ref. [115] . Since the Mori-Zwanzig formalism continues to be improved [116] [117] [118] [119] , it is likely to play an important role also in future work on DDFT.

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 [120] have used DDFT to study colloids with Casimir and magnetic interactions. Jia and Kusaka [121] used an extended form of DDFT [106] to model nonisothermal hard spheres. The transport of soft Brownian particles was analyzed by Antonov et al. [122] . Montañez-Rodríguez et al. [123] studied diffusion on spherical surfaces. The nonequilibrium self-consistent generalized Langevin equation [124] , an extension of DDFT, was applied to arrested density fluctuations by Lira-Escobedo et al. [125, 126] . Density fluctuations were also modeled using an adiabatic approximation by Szamel [127] . Finally, Sharma et al. [128] have studied the local softness parameter (which is useful for the description of caging) using DDFT.

In addition, stochastic DDFT [9, 10], commonly referred to as "Dean-Kawasaki equation" (a name that is somewhat unfortunate as it fails to acknowledge the differences between Dean's and Kawasaki's approaches [16]), has remained an important tool in the study of interacting particles with stochastic dynamics. Recent examples include active matter [129, 130] , chemotaxis [131, 132] , electrolytes [59, 61, 62], densely packed spheres [133] , and proteins [134] . Satin [135] suggested a link between stochastic DDFT and theories of gravity.

Moreover, there have been several extensions of DDFT towards physical systems not previously considered in the context of DDFT. Building up on earlier work [108, 109] , Fang [110, 111, 136] has recently derived a DDFT for ferrofluids. Another example is the development of a DDFT for granular media [137, 138] . Stanton et al. [139] have modeled cellular membranes in DDFT. These can be described as a mixture of lipids and proteins. Finally, Wittmann et al. [140] have derived a DDFT that allows to describe mechano-sensing in growing bacteria colonies.

DDFT is also frequently studied in relation to hydrodynamics. Several works have investigated the relation between DDFT and the Navier-Stokes equation [141] [142] [143] . Stierle and Gross [144] have derived a "hydrodynamic DFT", which describes underdamped mixtures. An inertial DDFT with hydrodynamic interactions was studied by Goddard et al. [145] . Moreover, DDFT allows to model drying colloidal films [146] , droplets [147] , flow in nanopores [64], hydrodynamics of ferrofluids [136] , and polymer mixtures [148] .

DDFT is of interest not only in physics and chemistry, but also for applied mathematics and software development. Recent work has considered stochastic DDFT (the Dean-Kawasaki equation) [149] [150] [151] [152] [153] [154] [155] and the McKean-Vlasov equation (a DDFT-type model) [156] [157] [158] [159] from a mathematical perspective. Moreover, numerical methods were developed for DDFT [160] [161] [162] [163] [164] [165] and PFC models [166] [167] [168] [169] . A particularly rapidly growing subfield is the application of machine learning, which can be used to learn static free energy functionals [170] [171] [172] [173] that can be used also in DDFT. However, machine learning is also used in the dynamical case [174] [175] [176] [177] [178] [179] . An example of the latter type is multiscale modeling of proteins based on machine learning in Ref. [176] , where the DDFT from Ref. [139] is used as a macroscale model.

Recent studies of PFC models have focused on active matter with [180] [181] [182] and without [183] [184] [185] [186] [187] [188] inertia, bifurcation diagrams [187] [188] [189] [190] , colored noise [191] , cubic terms [192] , crystals [193] , dislocation lines [194] , electromigration [195] , grain boundaries [196, 197] , mixtures [169, 188, [198] [199] [200] , nucleation [201] , solidification [202, 203] , and stress tensors [204] .

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 [205] , active matter [206] , the dynamics of the van Hove function [207, 208] , shear flow [209] , and superadiabatic demixing [210] . On the other hand, there have been more theoretical developments such as the derivation of Noether's theorem for statistical mechanics [211] [212] [213] (which also served as the basis for a "force-based DFT" [214] ), a classification of nonequilibrium forces [215] , a custom flow method [216] , philosophical investigations of PFT [93, 94] , and a reassessment of the original derivation of PFT [217] .

Finally, further overview articles covering (also) DDFT have been published; in particular an extensive review of PFT by Schmidt [20], a tutorial on active DDFT by Löwen [218] , and several reviews on biology and medicine [219] [220] [221] [222] [223] [224] , coarse-graining [30, 225] , electrochemistry [226] [227] [228] [229] , multiscale modeling [230] , PFC models [231] , and polymers [232] [233] [234] [235] [236] [237] in which DDFT is mentioned. In our view, this large number of overview articles published in two years further highlights how timely the topic is.

A. Phase field crystal 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 [238] , 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 [29] .

DFT functionals can, at least for hard particles, be derived pretty much "ab initio". Fundamental measure theory (FMT) [239] 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. In contrast, the free energy in PFC models is typically just assumed to have a very simple Swift-Hohenberg-type form [18, 240] , and the parameters of this free energy can then be adjusted to fit a wide class of materials [29] . 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 [241] .

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 [242, 243] , which are also connected to DDFT [244] ), and in general a better understanding of scientific reduction [97] . Recently, some work has been done in this direction. This includes a microscopic extension of the active PFC model towards mixtures [188] , the development of a framework for obtaining gradient-based free energies from more general expressions [245] , and in particular a systematic assessment of the derivation of PFC models from DDFT by Archer et al. [29] , 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.

Since, as explained in Section II, 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 Eq. (12). This would allow to obtain a model that combines the simplicity of the PFC approach with the ability of PFT to model far-fromequilibrium processes, and would allow, e.g., to study memory in active matter within the PFC framework (as done phenomenologically in Ref. [182] ). Such a theory would fit in the currently empty spot at the top right of Fig. 1 . A further interesting idea, suggested in Ref. [246] , would be to combine PFT with RDDFT (see Section III) 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 [217] 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 something like an FMT for the excess power functional, which provides an accurate expression obtained from first principles. Moreover, as discussed in Refs. [25, 93] , 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.

Active matter physics [31, 32], 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 II) is still the same. Active DDFT has a number of interesting applications, in particular in the study of microswimmers [15, [247] [248] [249] (see Ref. [218] for an overview). Moreover, active DDFT serves as the basis for the derivation of active PFC models [23, 24, 188] .

A conceptual challenge in modeling active particles using DDFT is that, as explained in Section II, 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 Refs. [250, 251] , inherited by active PFC models. Dhont et al. [252] have argued that active DDFT is inappropriate for particles with steep and short-ranged interactions. Moreover, DDFT models for microswimmers [15, 247] can become inaccurate at higher densities [253] 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-toequilibrium 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 [206] . Microscopically derived active matter models generally require as an input knowledge (or assumptions) about the correlations in the system [254] , 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 Refs. [255] [256] [257] .

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 [258] . Given that topological defects are of central importance for the understanding of active matter systems [259] , 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 [260, 261] , 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 [262] .

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 [263, 264] , 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 [139] . Going to larger scales, we arrive at DDFT models of entire cells as used in applications to cancer growth [265, 266] , microswimmers [15, 218, [247] [248] [249] , and bacteria [140] . In the SIR-DDFT model [34, 35] , the considered "particles" are humans. It even does not have to stop there, since a (quantumbased) DFT has been applied to entire ecosystems [267] . 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.

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 [34-37, 39, 43-51] and the many works on electrochemistry [52-64, [226] [227] [228] [229] . 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 reac-tants 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 [268] . 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.

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. 

Containing papers of a

Effects of social distancing and isolation on epidemic spreading modeled via dynamical density functional theory

Containing a pandemic: nonpharmaceutical interventions and the "second wave

sir ddft -a Rust implementation of the SIR-DDFT model with Python and JavaScript bindings

The case for small-scale, mobile-enhanced COVID-19 epidemiology

More than a year after the onset of the CoVid-19 pandemic in the UK: lessons learned from a minimalistic model capturing essential features including social awareness and policy making

Abstand halten! Physikalische Modelle aus der Theorie der weichen Materie liefern neue Einblicke in die Ausbreitung von Infektionskrankheiten

Mechanism for the stabilization of protein clusters above the solubility curve: the role of non-ideal chemical reactions

Mechanism for the stabilization of protein clusters above the solubility curve

The chemical basis of morphogenesis

Controlling the microstructure and phase behavior of confined soft colloids by active interaction switching

Active binary switching of soft colloids: stability and structural properties

Active interaction switching controls the dynamic heterogeneity of soft colloidal dispersions

Intermittent attractive interactions lead to microphase separation in non-motile active matter

Development of reaction-diffusion DFT and its application to catalytic oxidation of NO in porous materials

Formation and stabilization mechanism of mesoscale clusters in solution

Development of a BV-TDDFT model for metal corrosion in aqueous solution

Molecular theory of solvation and solvation dynamics in a binary dipolar liquid

Permeability and selectivity analysis for affinity-based nanoparticle separation through nanochannels

Scaling laws in the diffusive release of neutral cargo from hollow hydrogel nanoparticles: paclitaxel-loaded poly

Flow-driven release of molecules from a porous surface explored using dynamical density functional theory

Antimicrobial rubber nanocapsule-based iodophor promotes wound healing

Dynamic coarsegraining of polymer systems using mobility functions

Bottom-up construction of dynamic density functional theories for inhomogeneous polymer systems from microscopic simulations

Dynamic self-consistent field approach for studying kinetic processes in multiblock copolymer melts

Influence of small-scale correlation on the interface evolution of semiflexible homopolymer blends

Kinetic pathways of block copolymer directed self-assembly: insights from efficient continuum modeling

Memory in the relaxation of a polymer density modulation

Universal scaling for the exit dynamics of block copolymers from micelles at short and long time scales

3D pattern formation from coupled Cahn-Hilliard and Swift-Hohenberg equations: morphological phases transitions of polymers, bock and diblock copolymers

Kudryavtsev, Block copolymers in high-frequency electric field: mean-field approximation

The MesoDyn project: software for mesoscale chemical engineering

The interpenetration polymer network in a cement paste-waterborne epoxy system

High-performance multiblock pEMs containing a highly acidic fluorinatedhydrophilic domain for water electrolysis

Mesoscale modeling and experimental study of quercetin organization as nanoparticles in the polylactic-co-glycolic acid/water system under different conditions

Reinforced polymer blend membranes with liposome-like morphology for polymer electrolyte membrane fuel cells operating under low-humidity conditions

The effects of temperature on surfactant solution: a molecules dynamics simulation

Classical density functional theory in the canonical ensemble

Full canonical information from grand-potential density-functional theory

The Ornstein-Zernike equation in the canonical ensemble

The extended variable space approach to density functional theory in the canonical ensemble

Particle conservation in dynamical density functional theory

Particleconserving dynamics on the single-particle level

How to distinguish between indistinguishable particles

The five problems of irreversibility

Understanding probability and irreversibility in the Mori-Zwanzig projection operator formalism

Master equations for Wigner functions with spontaneous collapse and their relation to thermodynamic irreversibility

Is thermodynamics fundamental?

Transport, collective motion, and Brownian motion

On quantum theory of transport phenomena: steady diffusion

Ensemble method in the theory of irreversibility

Projection Operator Techniques in Nonequilibrium Statistical Mechanics

Projection operators in statistical mechanics: a pedagogical approach

The physical basis of the direction of time

Extended dynamical density functional theory for colloidal mixtures with temperature gradients

Microscopic approach to entropy production

Functional thermo-dynamics: a generalization of dynamic density functional theory to non-isothermal situations

Nanoscale hydrodynamics near solids

First-principles magnetization relaxation equation of interacting ferrofluids with applications to magnetoviscous effects

Generic theory of the dynamic magnetic response of ferrofluids

Dynamical effective field model for interacting ferrofluids: I. Derivations for homogeneous, inhomogeneous, and polydisperse cases

Dynamical effective field model for interacting ferrofluids: II. The proper relaxation time and effects of dynamic correlations

Elasticity of disordered binary crystals

Microscopic density-functional approach to nonlinear elasticity theory

Elasticity in crystals with a high density of local defects: insights from ultra-soft colloids

Application of projection operator method to coarse-grained dynamics with transient potential

Comments on the validity of the non-stationary generalized Langevin equation as a coarse-grained evolution equation for microscopic stochastic dynamics

On the dynamics of reaction coordinates in classical, timedependent, many-body processes

Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians

Mori-Zwanzig formalism for general relativity: a new approach to the averaging problem

Statics and dynamics of a finite two-dimensional colloidal system with competing attractive critical Casimir and repulsive magnetic dipole interactions

Density functional study of nonisothermal hard sphere fluids

Driven transport of soft Brownian particles through pore-like structures: effective size method

Spectral analysis of the collective diffusion of Brownian particles confined to a spherical surface

General nonequilibrium theory of colloid dynamics

Spatially heterogeneous dynamics and locally arrested density fluctuations from firstprinciples

Ultra-slow and arrested densityfluctuations as precursor of spatial heterogeneity

An alternative, dynamic density functionallike theory for time-dependent density fluctuations in glass-forming fluids

Identifying structural signature of dynamical heterogeneity via the local softness parameter

Statistical mechanics of active Ornstein-Uhlenbeck particles

Spatial organization of active particles with field-mediated interactions

Nonequilibrium polarityinduced chemotaxis: emergent Galilean symmetry and exact scaling exponents

Stochastic dynamics of chemotactic colonies with logistic growth

Non-hyperuniform metastable states around a disordered hyperuniform state of densely packed spheres: stochastic density functional theory at strong coupling

Binding of thermalized and active membrane curvature-inducing proteins

Correspondences of matter field fluctuations in semiclassical and classical gravity in the decoherence limit

Consistent hydrodynamics of ferrofluids

Granular media at multiple scales: mathematical analysis, modelling and computation

Modelling inelastic granular media using dynamical density functional theory

Dynamic density functional theory of multicomponent cellular membranes

Mechano-sensing in growing bacteria colonies with dynamical density functional theory

On the relation between dynamical density functional theory and Navier-Stokes equation

Analysis and applications of dynamic density functional theory

Self-consistent equations governing the dynamics of non-equilibrium binary colloidal systems

Hydrodynamic density functional theory for mixtures from a variational principle and its application to droplet coalescence

The singular hydrodynamic interactions between two spheres in Stokes flow

Dynamical density functional theory for the drying and stratification of binary colloidal dispersions

Changing the flow profile and resulting drying pattern of dispersion droplets via contact angle modification

Stratification of polymer mixtures in drying droplets: Hydrodynamics and diffusion

From weakly interacting particles to a regularised Dean-Kawasaki model

Wellposedness for a regularised inertial Dean-Kawasaki model for slender particles in several space dimensions

Ranked diffusion, delta Bose gas, and Burgers equation

Improved field theoretical approach to noninteracting Brownian particles in a quenched random potential

From interacting agents to density-based modeling with stochastic PDEs

The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles

Well-posedness of the Dean-Kawasaki and the nonlinear Dawson-Watanabe equation with correlated noise

Mean field limits for interacting diffusions with colored noise: phase transitions and spectral numerical methods

A law of large numbers for interacting diffusions via a mild formulation

On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions, Archive for Rational Mechanics and Analysis 241

Minimum action method for nonequilibrium phase transitions

High-order well-balanced finite-volume schemes for hydrodynamic equations with nonlocal free energy

A finite volume method for continuum limit equations of nonlocally interacting active chiral particles

A finite-volume scheme for gradient-flow equations with non-homogeneous diffusion

Numerical approximation of singular-degenerate parabolic stochastic PDEs

PDE-constrained optimization models and pseudospectral methods for multiscale particle dynamics

Pavliotis, Noisy bounded confidence models for opinion dynamics: the effect of boundary conditions on phase transitions

A second order accurate scalar auxiliary variable (SAV) numerical method for the square phase field crystal equation

Error estimates for second-order SAV finite element method to phase field crystal model

Numerical scheme for solving the nonuniformly forced cubic and quintic Swift-Hohenberg equations strictly respecting the Lyapunov functional

Numerical approximation of the twocomponent PFC models for binary colloidal crystals: efficient, decoupled, and second-order unconditionally energy stable schemes

A classical density functional from machine learning and a convolutional neural network

Analytical classical density functionals from an equation learning network

Machine-learning free-energy functionals using density profiles from simulations

Physicsconstrained Bayesian inference of state functions in classical density-functional theory

Flux: Overcoming scheduling challenges for exascale workflows

Bremer, Machine-learning-based dynamic-importance sampling for adaptive multiscale simulations

Streitz, Machine learning-driven multiscale modeling reveals lipid-dependent dynamics of RAS signaling proteins

Image inversion and uncertainty quantification for constitutive laws of pattern formation

Learning the physics of pattern formation from images

Lightstone, and H. I. Ingolfsson, Adaptable coordination of large multiscale ensembles: Challenges and learnings at scale

Active phase field crystal systems with inertial delay and underdamped dynamics

Mean field approach of dynamical pattern formation in underdamped active matter with short-ranged alignment and distant antialignment interactions

Jerky active matter: a phase field crystal model with translational and orientational memory

Phase-field-crystal description of active crystallites: elastic and inelastic collisions

Two-dimensional localized states in an active phasefield-crystal model

Dynamical crystallites of active chiral particles

Deformable active nematic particles and emerging edge currents in circular confinements

Localized states in passive and active phase-field-crystal models

Derivation and analysis of a phase field crystal model for a mixture of active and passive particles

Efficient calculation of phase coexistence and phase diagrams: application to a binary phase-field crystal model

Dissipative systems, in Emerging Frontiers in Nonlinear Science

Correlated noise effect on the structure formation in the phase-field crystal model

Phase-field crystal method for multiscale microstructures with cubic term

A comparison of different approaches to enforce lattice symmetry in two-dimensional crystals

A phase field crystal theory of the kinematics and dynamics of dislocation lines

Connecting the phase-field-crystal model of electromigration with electronic and continuum theories

Evaluation of grain boundary energy, structure and stiffness from phase field crystal simulations, Modelling and Simulation in Materials Science and Engineering

Microscopic patterns in the 2D phase-fieldcrystal model

Structure diagram and dynamics of formation of hexagonal boron nitride in phase-field crystal model

Atomic-scale study of compositional and structural evolution of early-stage grain boundary precipitation in Al-Cu alloys through phase-field crystal simulation

Mesoscale defect motion in binary systems: effects of compositional strain and cottrell atmospheres

Nucleation and postnucleation growth in diffusion-controlled and hydrodynamic theory of solidification

Traveling waves of the solidification and melting of cubic crystal lattices

A thermodynamicallyconsistent phase field crystal model of solidification with heat flux

Stress in ordered systems: Ginzburg-Landau-type density field theory

Shear and bulk acceleration viscosities in simple fluids

Phase separation of active Brownian particles in two dimensions: anything for a quiet life

Universality in driven and equilibrium hard sphere liquid dynamics

Dynamic decay and superadiabatic forces in the van Hove dynamics of bulk hard sphere fluids

Shear-induced deconfinement of hard disks

Superadiabatic demixing in nonequilibrium colloids

Noether's theorem in statistical mechanics

Why Noether's theorem applies to statistical mechanics

Variance of fluctuations from Noether invariance

Force density functional theory inand out-of-equilibrium

Flow and structure in nonequilibrium Brownian many-body systems

Custom flow in molecular dynamics

Reconsidering power functional theory

Dynamical density functional theory for "dry" and "wet" active matter

Polymeric nanocomposites for cancertargeted drug delivery

Not only in silico drug discovery: molecular modeling towards in silico drug delivery formulations

Modelling collective cell migration: neural crest as a model paradigm

Understanding the interaction of polyelectrolyte architectures with proteins and biosystems

Strategies for the treatment of breast cancer: from classical drugs to mathematical models

Protein-mimetic self-assembly with synthetic macromolecules

Introducing memory in coarse-grained molecular simulations

Advances in in-situ characterizations of electrode materials for better supercapacitors

Nanospace-confinement synthesis: Designing highenergy anode materials toward ultrastable lithium-ion batteries

Multiscale modeling of electrolytes in porous electrode: from equilibrium structure to non-equilibrium transport

Microscopic simulations of electrochemical doublelayer capacitors

Multiscale modeling for the science and engineering of materials

A review of continuous modeling of periodic pattern formation with modified phase-field crystal models

Theory and Modeling of Polymer Nanocomposites

Review on the computer simulation tools for polymeric membrane researches

Polymeric hybrid nanocomposites processing and finite element modeling: an overview

Process-directed self-assembly of copolymers: results of and challenges for simulation studies

A decade of innovation and progress in understanding the morphology and structure of heterogeneous polymers into rigid confinement

A review on multiscale modelling and simulation for polymer nanocomposites

Polymorphism, crystal nucleation and growth in the phase-field crystal model in 2D and 3D

Fundamental measure theory for hard-sphere mixtures: a review

Hydrodynamic fluctuations at the convective instability

Thermodynamics of bcc metals in phase-fieldcrystal models

Advances of and by phase-field modeling in condensed-matter physics

A phase field concept for multiphase systems

Non-local phase field revisited

A unified theory of free energy functionals and applications to diffusion

Modulating internal transition kinetics of responsive macromolecules by collective crowding

Dynamical density functional theory for circle swimmers

Particle-scale statistical theory for hydrodynamically induced polar ordering in microswimmer suspensions

Multi-species dynamical density functional theory for microswimmers: derivation, orientational ordering, trapping potentials, and shear cells

Collective dynamics of active Brownian particles in three spatial dimensions: a predictive field theory

From a microscopic inertial active matter model to the Schrödinger equation

Motilityinduced inter-particle correlations and dynamics: a microscopic approach for active Brownian particles

Active Brownian particles in external force fields: fieldtheoretical models, generalized barometric law, and programmable density patterns

Pairdistribution function of active Brownian spheres in two spatial dimensions: simulation results and analytic representation

Active Brownian particles at interfaces: an effective equilibrium approach

Active Brownian particles in two-dimensional traps

Effective interactions in active Brownian suspensions

Particle-resolved topological defects of smectic colloidal liquid crystals in extreme confinement

Topological active matter

Topology of orientational defects in confined smectic liquid crystals

Topological fine structure of smectic grain boundaries and tetratic disclination lines within three-dimensional smectic liquid crystals

Defect dynamics in active smectics steered by extreme confinement

Energetics of divalent selectivity in a calcium channel: the ryanodine receptor case study

De)constructing the ryanodine receptor: modeling ion permeation and selectivity of the calcium release channel

Dynamic density functional theory of solid tumor growth: preliminary models

Dynamical density-functional-theory-based modeling of tissue dynamics: application to tumor growth

A mechanistic density functional theory for ecology across scales

The influence of macromolecular crowding and macromolecular confinement on biochemical reactions in physiological media