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We establish a rigorous time-dependent density functional theory of classical fluids for a wide class of microscopic dynamics. We obtain a stationary action principle for the density. We further introduce an exact practical scheme, to obtain hydrodynamical effects in density evolution, that is analogous to the Kohn-Sham theory of quantum systems. Finally, we show how the current theory recovers existing phenomenological theories in an adiabatic limit.

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... For classical systems, Chan and Finken [2] asserted uniqueness following the idea of Runge and Gross [26]. However, because higher-body correlations due to interparticle interactions were omitted, the argument so far holds only under the adiabatic approximation. ...

... Similar to the idea of Runge and Gross (or Chan and Finken) [26,2], we assume analytic potentials and can thus reduce the uniqueness of the mapping to the uniqueness of a solution to a (semi-)elliptic PDE. In contradistinction to the available proofs in quantum mechanics [26,12,13,24], we explicitly have to take the hierarchy of reduced Smoluchowski equations into account. ...

... An essential step in the proof is to reduce the uniqueness of the mapping to the uniqueness of a solution to a (semi-)elliptic PDE. As discussed in the introduction, this approach is, in parts, similar to the argument by Runge and Gross (or Chan and Finken) [26,2], but it differs in that we have to take the hierarchy of reduced Smoluchowski equations from Theorem 3.1 into account. We, of course, pay close attention to a rigorous treatment of the boundary terms. ...

When can we uniquely map the dynamic evolution of a classical density to a time-dependent 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).

... Equation (4) is the exact drift-diffusion equation for overdamped motion of a mutually noninteracting system, i.e., the ideal gas. Besides Evans' original proposal [4] based on the continuity equation and undoubtedly his physical intuition, derivations of the DDFT (1) were founded much more recently on Dean's equation of motion for the density operator [5], the Smoluchowski equation [24], a stationary action principle for the density [25], the projection operator formalism [26], a phase-space approach [27], the meanfield approximation [28], a local equilibrium assumption Page 2 of 21 AUTHOR SUBMITTED MANUSCRIPT -JPCM-121672.R1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 A c c e p t e d M a n u s c r i p t [29], and a non-equilibrium free energy [30]. The question of the well-posedness of the DDFT was addressed [31] and several extensions beyond overdamped Brownian dynamics were formulated, such as e.g. for dynamics including inertia [32][33][34][35] and for particles that experience hydrodynamic interactions [35,36] or undergo chemical reactions [37,38]. ...

... The fact that both the static limit for the fully interacting system (2) as well as the full dynamics of the noninteracting system (4) are exact, taken together with the heft of the DDFT literature, appears to give much credibility to the equation of motion (1). However, despite the range of theoretical techniques employed [5,[24][25][26][27][28][29][30] neither of these approaches has provided us with a concrete way of going beyond Eq. (1). ...

... The proof can either be based on the fact that Eq. (25) is merely Eq. (24) for the special case of an equilibrium system, from which then Eq. (26) follows from the force splitting (11). Alternatively and starting from a very fundamental point of view, the global translational invariance of the excess free energy functional F exc [ρ] and of the superadiabatic free power functional P exc t [ρ, v], here considered instantaneously at time t, lead directly to Eqs. (25) and (26), see Refs. [45,46] for detailed derivations. ...

We argue in favour of developing a comprehensive dynamical theory for rationalizing, predicting, designing, and machine learning nonequilibrium phenomena that occur in soft matter. To give guidance for navigating the theoretical and practical challenges that lie ahead, we discuss and exemplify the limitations of dynamical density functional theory. Instead of the implied adiabatic sequence of equilibrium states that this approach provides as a makeshift for the true time evolution, we posit that the pending theoretical tasks lie in developing a systematic understanding of the dynamical functional relationships that govern the genuine nonequilibrium physics. While static density functional theory gives a comprehensive account of the equilibrium properties of many-body systems, we argue that power functional theory is the only present contender to shed similar insights into nonequilibrium dynamics, including the recognition and implementation of exact sum rules that result from the Noether theorem. As~a~demonstration of the power functional point of view, we consider an idealized steady sedimentation flow of the three-dimensional Lennard-Jones fluid and machine-learn the kinematic map from the mean motion to the internal force field. The trained model is capable of both predicting and designing the steady state dynamics universally for various target density modulations. This demonstrates the significant potential of using such techniques in nonequilibrium many-body physics and overcomes both the conceptual constraints of dynamical density functional theory as well as the limited availability of its analytical functional approximations.

... Under the assumption that collisions are binary and well-separated, the result should also be appropriate for systems of N particles. We do not go into any more explicit detail here, but refer the reader to the derivations in [29,57] When considering the associated Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy [5] (in weak form), the additional term integrates to the well-known Boltzmann collision operator for α = 1, and when α > 1 we can derive its inelastic counterpart: (11) where p i is the pre-collisional velocity associated to p i , determined using Eq. (4). ...

... To derive a computationally efficient model for use in simulation, we can rely on the result of [11], which relates the N -particle distribution function and the one-body position density as follows: ...

... where L coll ( f (2) ) incorporates binary collisions via a collision operator. We consider the inelastic collision operator Eq. (11). Long range interactions are included in ...

We construct a new mesoscopic model for granular media using Dynamical Density Functional Theory (DDFT). The model includes both a collision operator to incorporate inelasticity and the Helmholtz free energy functional to account for external potentials, interparticle interactions and volume exclusion. We use statistical data from event-driven microscopic simulations to determine the parameters not given analytically by the closure relations used to derive the DDFT. We numerically demonstrate the crucial effects of each term and approximations in the DDFT, and the importance of including an accurately parametrised pair correlation function.

... In equilibrium DFT, the one-body density gives, as discussed in Section 2.2, all relevant information about the system. For a system initially out of equilibrium, however, this is not the case due to an additional dependence on the initial condition [60,130]. Consequently, it is possible that a nonequilibrium system has an equilibrium density profile that minimizes the free energy functional, but a nonequilibrium correlation. ...

... By inserting this expansion into Eq. (130) and converting the functional derivative with respect to ( r, u, t) into derivatives with respect to the fields ρ( r, t), P ( r, t), and Q( r, t), one obtains a set of coupled dynamic equations for these fields. A detailed calculation following this procedure can be found in Ref. [556]. ...

... A DDFT for atomic fluids, where, unlike in colloidal fluids, inertia always plays a role, has been derived by Archer [131] (general fluids were also treated by Chan and Finken [130], see Section 5.4.2). Starting from Newton's equation of motion and performing an ensemble average, one finds for the one-body density ρ( r, t) the exact equation of motion ...

Classical dynamical density functional theory (DDFT) is one of the cornerstones of modern statistical mechanics. It is an extension of the highly successful method of classical density functional theory (DFT) to nonequilibrium systems. Originally developed for the treatment of simple and complex fluids, DDFT is now applied in fields as diverse as hydrodynamics, materials science, chemistry, biology, and plasma physics. In this review, we give a broad overview over classical DDFT. We explain its theoretical foundations and the ways in which it can be derived. The relations between the different forms of deterministic and stochastic DDFT as well as between DDFT and related theories, such as quantum-mechanical time-dependent DFT, mode coupling theory, and phase field crystal models, are clarified. Moreover, we discuss the wide spectrum of extensions of DDFT, which covers methods with additional order parameters (like extended DDFT), exact approaches (like power functional theory), and systems with more complex dynamics (like active matter). Finally, the large variety of applications, ranging from fluid mechanics and polymer physics to solidification, pattern formation, biophysics, and electrochemistry, is presented.

... For phase field models in particular, the free energy functionals of DFT directly follow from the statistical mechanical considerations of molecular-level interactions, which can be viewed as a firstprinciples generalization of the otherwise phenomenological phase field models based on the Ginzburg-Landau functionals. The square gradient approximation, in particular, allows to retrieve the Ginzburg-Landau-type functional within DFT. 36 While most of work on DDFT focus on colloidal systems, [37][38][39][40] the underlying assumptions of DDFT allow for a generalization to molecular fluids, as was initially suggested by Evans 14 and further developed by Chan and Finken 41 and Archer. 42 Thiele et al. 43 applied these DDFT concepts to open systems by investigating the de-wetting of colloidal suspensions, whereby the conservation equations of DDFT for colloid particles were coupled with the dissipative equation for the solvent. ...

... Equation (19) was first introduced by Evans 14 and later justified by more rigorous arguments by Dietrich et al., 67 Marconi and Tarazona 68 and Chan and Frinken. 41 In the present study we assume that both the dissipative and conservative mechanisms contribute to the evolution of the fluid density, so that the dynamic equation takes the from ...

... Thus, the integrating operator for the discrete version of Eq. (43) is essentially the sum of the three integration matrices constructed for each of its three constitutive terms, by using Eq. (41). With these considerations, we complete the presentation of our discretization scheme, since in a similar manner, any 1D convolution-type integral may be conveniently cast as a matrix operating on the discrete values of ρ(z). ...

Classical Density Functional Theory (DFT) is a statistical-mechanical framework to analyze fluids, which accounts for nanoscale fluid inhomogeneities and non-local intermolecular interactions. DFT can be applied to a wide range of interfacial phenomena, as well as problems in adsorption, colloidal science and phase transitions in fluids. Typical DFT equations are highly non-linear, stiff and contain several convolution terms. We propose a novel, efficient pseudo-spectral collocation scheme for computing the non-local terms in real space with the help of a specialized Gauss quadrature. Due to the exponential accuracy of the quadrature and a convenient choice of collocation points near interfaces, we can use grids with a significantly lower number of nodes than most other reported methods. We demonstrate the capabilities of our numerical methodology by studying equilibrium and dynamic two-dimensional test cases with single- and multispecies hard-sphere and hard-disc particles modelled with fundamental measure theory, with and without van der Waals attractive forces, in bounded and unbounded physical domains. We show that our results satisfy statistical mechanical sum rules.

... It is convenient to calculate the relaxation spectrum using the response functions [31,48,49]. Let us formulate the basis of this formalism. ...

... Consider the system of the atomic particles, which can interact with each other as well as the environment such as liquid, solid, or surface. Let us show, following [49], that many-particles distribution functions of such a system are the density functionals. Assume that each particle at the coordinate r interacts with time-dependent external field V(r, ). ...

... Following [49], let us write the action for the Liouville equation in the form ...

The paper is devoted to the analysis of the correlation effects and manifestations of general
properties of 1D systems (such as spatial heterogeneity that is associated with strong density fluctuations, the lack of phase transitions, the presence of frozen disorder, confinement, and blocked movement of nuclear particle by its neighbours) in nonequilibrium phenomena by considering the four examples. The anomalous transport in zeolite channels is considered. The mechanism of the transport may appear in carbon nanotubes and MOF structures, relaxation, mechanical properties, and stability of nonequilibrium states of free chains of metal atoms, non-Einstein atomic mobility in 1D atomic systems. Also we discuss atomic transport and separation of two-component mixture of atoms in a 1D system—a zeolite membrane with subnanometer channels. We discuss the atomic transport and separation of two-component mixture of atoms in a 1D system—zeolite membrane with subnanometer channels. These phenomena are described by the response function method for nonequilibrium systems of arbitrary density that allows us to calculate the dynamic response function and the spectrum of relaxation of density fluctuations 1D atomic system.

... This is independent of the number of particles N, allowing arbitrarily large numbers of particles to be studied for constant computational cost. Whilst it has been shown rigorously that the full N-body distribution is a functional of the one-body distribution [13], the functional itself is unknown. For computations this functional, or a good approximation to it, needs to be known explicitly. ...

... density functional theory) systems is to average over the degrees of freedom of all but a few particles, leading to a lower-dimensional problem. In both the classical [13] and quantum [47] cases it is known rigorously that the full N-particle probability distribution is a functional of the one-body position distribution ...

... Going beyond this approximation introduces additional coupling between the momentum moment equations, analogous to the term in (19) containing f neq , which must then also be approximated. The functional dependence of f (2) on ρ can be rigorously justified from the fact [13] that the full time-dependent N-body distribution is a functional of ρ [13], and hence so are all the reduced distributions. However, as with the excess over the ideal gas term, the exact form of this functional is unknown. ...

Starting from the Kramers equation for the phase-space dynamics of the N-body probability distribution, we derive a dynamical density functional theory (DDFT) for colloidal fluids including the effects of inertia and hydrodynamic interactions (HI). We compare the resulting theory to extensive Langevin dynamics simulations for both hard rod systems and three-dimensional hard sphere systems with radially symmetric external potentials. As well as demonstrating the accuracy of the new DDFT, by comparing with previous DDFTs which neglect inertia, HI, or both, we also scrutinize the significance of including these effects. Close to local equilibrium we derive a continuum equation from the microscopic dynamics which is a generalized Navier-Stokes-like equation with additional non-local terms governing the effects of HI. For the overdamped limit we recover analogues of existing configuration-space DDFTs but with a novel diffusion tensor.

... Equation (4) is the exact drift-diffusion equation for overdamped motion of a mutually noninteracting system, i.e., the ideal gas. Besides Evans' original proposal [4] based on the continuity equation and undoubtedly his physical intuition, derivations of the DDFT (1) were founded much more recently on Dean's equation of motion for the density operator [22], the Smoluchowski equation [23], a stationary action principle for the density [24], the projection operator formalism [25], a phase-space approach [26], the mean-field approximation [27], a local equilibrium assumption [28], and a non-equilibrium free energy [29]. The question of the well-posedness of the DDFT was addressed [30] and several extensions beyond overdamped Brownian dynamics were formulated, such as e.g. for dynamics including inertia [31][32][33][34] and for particles that experience hydrodynamic interactions [34,35] or undergo chemical reactions [36,37]. ...

... The fact that both the static limit for the fully interacting system (2) as well as the full dynamics of the noninteracting system (4) are exact, taken together with the heft of the DDFT literature, appears to give much credibility to the equation of motion (1). However, despite the range of theoretical techniques employed [22][23][24][25][26][27][28][29] neither of these approaches has provided us with a concrete way of going beyond Eq. (1). ...

We argue in favour of developing a comprehensive dynamical theory for rationalizing, predicting, and machine learning nonequilibrium phenomena that occur in soft matter. To give guidance for navigating the theoretical and practical challenges that lie ahead, we discuss and exemplify the limitations of dynamical density functional theory. Instead of the implied adiabatic sequence of equilibrium states that this approach provides as a makeshift for the true time evolution, we posit that the pending theoretical tasks lie in developing a systematic understanding of the dynamical functional relationships that govern the genuine nonequilibrium physics. While static density functional theory gives a comprehensive account of the equilibrium properties of many-body systems, we argue that power functional theory is the only present contender to shed similar insights into nonequilibrium dynamics, including the recognition and implementation of exact sum rules that result from the Noether theorem. As a demonstration of the power functional point of view, we consider an idealized steady sedimentation flow of the three-dimensional Lennard-Jones fluid and machine-learn the kinematic map from the mean motion to the internal force field. This proof of concept demonstrates the significant potential of machine learning the inherent functional relationships that govern nonequilibrium many-body physics.

... Here, we aim to provide a link between such PDE-constrained optimization problems and state-of-the-art methods in statistical mechanics (known as Dynamic Density Functional Theory, or DDFT) [27,34,52,55,68,79,86,87], before devising numerical methods for such problems using a pseudospectral method in space and time, allowing highly efficient and accurate solution of both the forward and optimization problems [14,40,60,81]. Having derived first-order optimality conditions using the formal Lagrange method, we modify existing 'sweeping', or fixed-point, algorithms [4,17] to reliably solve such systems, and apply a recently developed Newton-Krylov method [41,42] to tackle non-linear optimization problems to higher order. ...

... However, an extremely efficient and accurate example of coarse-graining which captures such effects is Dynamic Density Functional Theory (DDFT) [27,55]. The crucial observation here is that the full N -body information in a system is a functional of the 1-body density, ρ( x, t) (i.e., the probability of finding any one particle at a given position at a given time). ...

We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we investigate problems where the control acts as an advection ‘flow’ vector or a source term of the partial differential equation, and the constraint is equipped with boundary conditions of Dirichlet or no-flux type. After deriving continuous first-order optimality conditions for such problems, we solve the resulting systems by developing a link with computational methods for statistical mechanics, deriving pseudospectral methods in space and time variables, and utilizing variants of existing fixed-point methods as well as a recently developed Newton–Krylov scheme. Numerical experiments indicate the effectiveness of our approach for a range of problem set-ups, boundary conditions, as well as regularization and model parameters, in both two and three dimensions. A key contribution is the provision of software which allows the discretization and solution of a range of optimization problems constrained by differential equations describing particle dynamics.

... In timedependent quantum mechanical systems, the Runge-Gross theorem [32] ensures the existence of a unique mapping between the density distribution and a time-dependent external potential. A classical analog of the Runge-Gross theorem was proposed by Chan and Finken [33]. The existence of a unique mapping between the kinematic fields and the external force field plays a central role in power functional theory, an exact variational principle for nonequilibrium classical many-body overdamped Brownian [34] and Hamiltonian systems [35] as well as for many-body quantum systems [36]. ...

... The existence of a unique mapping between the density distribution and a time-dependent external potential is at the core of time-dependent density functional theory [33]. Such mapping is not completely general but is restricted to the occurrence of gradientlike forces only. ...

Driving an inertial many-body system out of equilibrium generates complex dynamics due to memory effects and the intricate relationships between the external driving force, internal forces, and transport effects. Understanding the underlying physics is challenging and often requires carrying out case-by-case analysis. To systematically study the interplay between all types of forces that contribute to the dynamics, a method to generate prescribed flow patterns could be of great help. We develop a custom flow method to numerically construct the external force field required to obtain the desired time evolution of an inertial many-body system, as prescribed by its one-body current and density profiles. We validate the custom flow method in a Newtonian system of purely repulsive particles by creating a slow-motion dynamics of an out-of-equilibrium process and by prescribing the full time evolution between two distinct equilibrium states. The method can also be used with thermostat algorithms to control the temperature.

... In timedependent quantum mechanical systems, the Runge-Gross theorem [32] ensures the existence of a unique mapping between the density distribution and a time-dependent external potential. A classical analog of the Runge-Gross theorem was proposed by Chan and Finken [33]. The existence of a unique mapping between the kinematic fields and the external force field plays a central role in power functional theory, an exact variational principle for nonequilibrium classical many-body overdamped Brownian [34] and Hamiltonian systems [35] as well as for many-body quantum systems [36]. ...

... The existence of a unique mapping between the density distribution and a time-dependent external potential is at the core of time-dependent density functional theory [33]. Such mapping is not completely general but is restricted to the occurrence of gradientlike forces only. ...

Driving an inertial many-body system out of equilibrium generates complex dynamics due to memory effects and the intricate relationships between the external driving force, internal forces, and transport effects. Understanding the underlying physics is challenging and often requires carrying out case-by-case analysis. To systematically study the interplay between all types of forces that contribute to the dynamics, a method to generate prescribed flow patterns could be of great help. We develop a custom flow method to numerically construct the external force field required to obtain the desired time evolution of an inertial many-body system, as prescribed by its one-body current and density profiles. We validate the custom flow method in a Newtonian system of purely repulsive particles by creating a slow-motion dynamics of an out-of-equilibrium process and by prescribing the full time evolution between two distinct equilibrium states. The method can also be used with thermostat algorithms to control the temperature.

... In time-dependent quantum mechanical systems, the Runge-Gross theorem [31] ensures the existence of a unique mapping between the density distribution and a time-dependent external potential. A classical analogue of the Runge-Gross theorem was proposed by Chan and Finken [32]. The existence of a unique mapping between the kinematic fields and the external force field plays a central role in power functional theory, an exact variational principle for nonequilibrium classical many-body overdamped Brownian [33] and Hamiltonian systems [34] as well as for many-body quantum systems [35]. ...

... The existence of a unique mapping between the density distribution and a time-dependent external potential is at the core of time-dependent density functional theory [32]. Such mapping is not completely general but restricted to the occurrence of gradient-like forces only. ...

Driving an inertial many-body system out of equilibrium generates complex dynamics due to memory effects and the intricate relationships between the external driving force, internal forces, and transport effects. Understanding the underlying physics is challenging and often requires carrying out case-by-case analysis. To systematically study the interplay between all types of forces that contribute to the dynamics, a method to generate prescribed flow patterns could be of great help. We develop a custom flow method to numerically construct the external force field required to obtain the desired time evolution of an inertial many-body system, as prescribed by its one-body current and density profiles. We validate the custom flow method in a Newtonian system of purely repulsive particles by creating a slow motion dynamics of an out-of-equilibrium process and by prescribing the full time evolution between two distinct equilibrium states. The method can also be used with thermostat algorithms to control the temperature.

... Our methods are widely applicable to the optimization of many systems described by non-local, non-linear partial differential equations (PDEs), some cases of which have recently received attention in the literature [2,3,5,12,14,25]. The principal novelties are a link to state-of-the-art methods in statistical mechanics (known as Dynamic Density Functional Theory, or DDFT) [22,29,45,48,58,68,69], the implementation of a pseudospectral method, both in space and time, allowing highly efficient and accurate solution of both the forward and optimization problems [13,65], and a modification of existing 'sweeping', or fixed point, algorithms [3,16] to increase the stability for the problems studied here. We also demonstrate how to efficiently implement Neumann (no-flux) boundary conditions, provide a number of exact and validation test cases, and accompany the paper with an open source software implementation [1], based on 2DChebClass [35,51]. ...

... However, an extremely efficient and accurate example of coarse-graining which captures such effects is Dynamic Density Functional Theory (DDFT) [22,48]. The crucial observation here is that the full N -body information in a system is a functional of the 1-body density, ρ( x, t) (i.e., the probability of finding any one particle at a given position at a given time). ...

We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we investigate problems where the control acts as an advection 'flow' vector or a source term of the partial differential equation, and the constraint is equipped with boundary conditions of Dirichlet or no-flux type. After deriving continuous first-order optimality conditions for such problems, we solve the resulting systems by developing a link with computational methods for statistical mechanics, deriving pseudospectral methods in both space and time variables, and utilizing variants of existing fixed point methods. Numerical experiments indicate the effectiveness of our approach for a range of problem set-ups, boundary conditions, as well as regularization and model parameters.

... Equation 17, combined with Equation 9, shows that the Lagrangian multipliers can be identified with the electrochemical potentials of each species λ i = μ i (r). [21] This indicates that the electrochemical potential of every species must be uniform throughout the system (even if the concentration profiles are not) at equilibrium. From Equation 18, we find that the shape of the electric field can be determined from the principle that it always adjusts itself to minimize the free energy. ...

... We note that this phenomenological expression for the flux can also be motivated from more fundamental statistical mechanical arguments. The one-body density (and hence the free energy) is a unique function of a time-dependent external field, 21 and under the assumption that two-particle correlations are identical in equilibrium and non-equilibrium fluids, the dynamics of a species i in the system can be approximated by 22 ...

In this study we develop a general framework for describing reaction-diffusion processes in a multi-component electrolyte in which multiple reactions of different types may occur. Our motivation for this is the need to understand how the interactions between species and processes occurring in a complex electrochemical system. We use the framework to develop a modified Poisson-Nernst-Planck model which accounts for the excluded volume interaction (EVI) and incorporates both electrochemical and chemical reactions. Using this model, we investigate how the EVI influences the reactions and how the reactions influence each other in the contexts of the equilibrium state of a system and of a simple electrochemical device under load. Complex behavior quickly emerges even in relatively simple systems, and deviations from the predictions of ideal solution theory, together with how they may influence the behavior of more complex system, are discussed.

... Many of them treat the solvent as an incompressible liquid. There have been several proposals to incorporate hydrodynamic interactions into DDFT [259,271,278,279]. Their role is minor in the scope of this thesis, because we discuss single particles in the chapters 3, 4, and 5 and soft particles in chapters 6 and 7. ...

... However, correlations are still hard to incorporate into a DDFT because of the lack of suitable non-equilibrium free energy functionals. There are ideas [278,293] how to use such a functional to describe the motion but to my knowledge there are no proposals for actual non-equilibrium free energy functionals. However, the deviations from the adiabatic approximation could be quantified [268]. ...

In dieser Arbeit untersuchen wir den Transport von Kolloiden in einer räumlichen Dimension mit theoretischen und numerischen Methoden. Wir betrachten die Brownsche Bewegung der Kolloide unter dem Einfluss eines starken externen räumlich moduliertes Feldes. Wir modellieren den externen Einfluss mit einem periodischen Potential und einer konstanten treibenden Kraft. Wir interessieren uns für Prozesse außerhalb des thermischen Gleichgewichts. Insbesondere studieren wir die Bewegung auf Längen- und Zeitskalen, die zwischen der Bewegung in den Tälern des Potentials und der Bewegung zwischen den Tälern liegen. Für die theoretische Beschreibung der Brownschen Bewegung setzen wir die Langevin Gleichung und die zugehörige Fokker-Planck Gleichung, die Smoluchowski Gleichung, ein. Weiterhin benutzen wir die Dynamische Dichtefunktionaltheorie um die Teilchenwechselwirkungen auf großen Längen- und Zeitskalen berechnen zu können. In den ersten beiden Kapiteln betrachten wir die Bewegung eines einzelnen Brownschen Teilchens im gekippten Waschbrettpotential. Wir betrachten die Kurzzeitdiffusion, die durch die mittlere quadratische Verschiebung des Teilchens quantifiziert ist. Wir präsentieren ein einfaches Modell, das analytische Voraussagen für die zeitabhängigen Diffusionseigenschaften liefert. Weiterhin studieren wir die Zeit, die ein Brownsches Teilchen für die Überquerung einer Energiebarriere benötigt. Dazu schlagen wir eine Verallgemeinerung der Wartezeitverteilung vor, die zuvor nur für diskrete System verfügbar war. Mit dieser Wartezeitverteilung können Systeme, die nahezu diskrete Dynamik besitzen, nun mit hoher Genauigkeit charakterisiert werden. Nach diesen Grundlagenuntersuchungen wenden wir uns einer Rückkoppelsteuerung zu. Solch eine Steuerung optimiert das Verhalten eines System unter Verwendung von Informationen aus dem System selbst. Unsere Rückkoppelsteuerung steuert ein Brownsches Teilchen in einem asymmetrischen periodischen Potential auf der Basis von zeitverzögerter Bereitstellung der Information aus dem System. Wir zeigen, dass dadurch ein gerichteter Strom in dem System entsteht. Der Vergleich mit dem zugehörigen System ohne rückgekoppelte Steuerung zeigt, dass der Strom durch rückgekoppelte Steuerung verstärkt werden kann. In den letzten beiden Kapiteln behandeln wir Diffusion und Transport von wechselwirkenden Teilchen. Wir konzentrieren uns wieder auf die mittlere quadratische Verschiebung und zeigen, dass die Diffusion durch repulsive Teilchenwechselwirkung verstärkt wird. Danach schlagen wir eine weitere Rückkoppelsteuerung für den kollektiven Transport vor. Die Steuerung arbeitet wie eine optische Falle, die mit den Teilchen mitbewegt wird. Wir zeigen, dass die Mobilität der Teilchengruppe durch das Zusammenspiel von Falle und Teilchenwechselwirkung um mehrere Größenordnungen erhöht wird.

... While such an approximation certainly reduces the numerical complexity [since determining A is now O(N ) rather than O(N 3 )], it physically corresponds to ignoring HIs, thus neglecting crucial physical information. Another approach to tackle the problem, which is the one we adopt, is to derive a coarse-grained/mean-field model by averaging over the degrees of freedom of all but a few particles [14,33,35,81]. This method yields a lower-dimensional system but requires knowledge of the functional relation between f (N ) and the reduced distribution function. ...

... The integral of Eq. (27) depends only on f (2) (r 1 , r 2 , p 1 , p 2 ; t) and we now need a closure relation between the one-and two-body distributions. From a previous rigorous derivation of DDFT [14], it is known that the one-body density determines the N -body distribution and, therefore, all the time-dependent properties of the system can be expressed as functionals of ρ(r; t). Thus, we make the following approximation ...

Over the last few decades, classical density-functional theory (DFT) and its dynamic extensions (DDFTs) have become powerful tools in the study of colloidal fluids. Recently, previous DDFTs for spherically-symmetric particles have been generalised to take into account both inertia and hydrodynamic interactions, two effects which strongly influence non-equilibrium properties. The present work further generalises this framework to systems of anisotropic particles. Starting from the Liouville equation and utilising Zwanzig’s projection-operator techniques, we derive the kinetic equation for the Brownian particle distribution function, and by averaging over all but one particle, a DDFT equation is obtained. Whilst this equation has some similarities with DDFTs for spherically-symmetric colloids, it involves a translational-rotational coupling which affects the diffusivity of the (asymmetric) particles. We further show that, in the overdamped (high friction) limit, the DDFT is considerably simplified and is in agreement with a previous DDFT for colloids with arbitrary-shape particles.

... DDFT has been tested successfully in a variety of situations [10,11], including non-equilibrium sedimentation of hard spheres under gravity, where excellent agreement with results from Brownian Dynamics computer simulations and from experiments using confocal microscopy of colloidal dispersions was found [12]. Nevertheless the DDFT is approximative [13,14] and it is hence of practical interest to develop the framework further. Figure 1. ...

... is the Boltzmann factor of the free energy cost of an identity (species) exchange of a particle at position x, and is hence intimately related to the hopping rate in the TASEP, equation (14), because moving a particle at site x corresponds to choosing a particle at rest (species 1) and transforming it into a particle that moves (species 2), see the Lower panel of figure 1. We note further that using equation (12) the right-hand side of equation (10) becomes ρ(x)/(1 − ρ(x)) which is known as the 'auxiliary field' in the mean-field theory for the TASEP, which we recover here as the local fugacity z 1 (x) in the binary mixture. ...

We consider the totally asymmetric exclusion process (TASEP) of particles on a one-dimensional lattice that interact with site exclusion and are driven into one direction only. The mean-field approximation of the dynamical equation for the one-particle density of this model is shown to be equivalent to the exact Euler–Lagrange equations for the equilibrium density profiles of a binary mixture. In this mixture particles occupy one (two) lattice sites and correspond to resting (moving) particles in the TASEP. Despite the strict absence of bulk phase transitions in the equilibrium mixture, the influence of density-dependent external potentials is shown to induce abrupt changes in the one-body density that are equivalent to the exact out-of-equilibrium phase transitions between steady states in the TASEP with open boundaries.

... Such approaches can also be used when spin interactions are relevant (such as in quantum ferrofluids [285]), since Wigner functions can also model spins [284,286]. A further possible connection between classical and quantum nonequilibrium DFT is the Runge-Gross theorem [287], which forms the basis of quantum mechanical time-dependent DFT, but which was also extended to classical mechanics by Chan and Finken [288]. ...

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.

... This is exploited in equilibrium density functional theory (Evans [1979], p. 148). However, there is an additional dependence on the initial condition for out-of-equilibrium systems (Chan and Finken [2005], pp. 1-2). ...

Thermodynamics has a clear arrow of time, characterized by the irreversible approach to equilibrium. This stands in contrast to the laws of microscopic theories, which are invariant under time-reversal. In this article, I show that the difficulty in solving this "problem of irreversibility" partly arises from the fact that it actually consists of five different sub-problems. These concern the source of irreversibility in thermodynamics, the definition of equilibrium and entropy, the justification of coarse-graining, the approach to equilibrium and the arrow of time. Not distinguishing them can lead and has led to terminological confusion, apparent contradictions between positions that are actually compatible, and difficulties in connecting the debates in physics and philosophy of physics.

... There are a number of directions in which the formal frame- work suggested here can be extended, paralleling developments from the traditional density-functional theory literature. Exten- sions to time-dependent DFT methods (TDDFT) 36,37 would enable the prediction of situations in which crowds gather and disperse in response to changes in the environment. This approach would also apply to situations in which the center of mass of the entire group is moving as whole, such as in herd migration and bacterial and insect swarming. ...

A primary goal of collective population behavior studies is to determine the rules governing crowd distributions in order to predict future behaviors in new environments. Current top-down modeling approaches describe, instead of predict, specific emergent behaviors, whereas bottom-up approaches must postulate, instead of directly determine, rules for individual behaviors. Here, we employ classical density functional theory (DFT) to quantify, directly from observations of local crowd density, the rules that predict mass behaviors under new circumstances. To demonstrate our theory-based, data-driven approach, we use a model crowd consisting of walking fruit flies and extract two functions that separately describe spatial and social preferences. The resulting theory accurately predicts experimental fly distributions in new environments and provides quantification of the crowd "mood". Should this approach generalize beyond milling crowds, it may find powerful applications in fields ranging from spatial ecology and active matter to demography and economics.

... There are TD external potentials and densities that are non-analytic functions around the initial time. Hence, the application of TDDFT to non-analytic fields requires formal support that might assist in setting new rules and conditions for the development of improved approximations and computational methods within TDDFTand other research areas such as classical fluids, where Chan and Finken 7 showed that the Runge-Gross (RG) theorem is applicable. ...

Provided the initial state, the Runge-Gross theorem establishes that the time-dependent (TD) external potential of a system of non-relativistic electrons determines uniquely their TD electronic density, and vice versa (up to a constant in the potential). This theorem requires the TD external potential and density to be Taylor-expandable around the initial time of the propagation. This paper presents an extension without this restriction. Given the initial state of the system and evolution of the density due to some TD scalar potential, we show that a perturbative (not necessarily weak) TD potential that induces a non-zero divergence of the external force-density, inside a small spatial subset and immediately after the initial propagation time, will cause a change in the density within that subset, implying that the TD potential uniquely determines the TD density. In this proof, we assume unitary evolution of wavefunctions and first-order differentiability (which does not imply analyticity) in time of the internal and external force-densities, electronic density, current density, and their spatial derivatives over the small spatial subset and short time interval.

... Note that this transformation does not reduce the numerical complexity, as taking M discretization points for each spatial/momentum dimension would require (2M ) 3N points in total; the only way to solve the Kramers equation in high-dimensions is by using Monte Carlo methods, i.e. by solving (1). However, it is known rigorously [22] that f (N ) is a functional of the one-body position distribution ρ(r 1 , t) = N dpdr f (N ) (r, p, t), where dr denotes integration over all but r 1 , the position of the first particle. Hence, for any number of particles at fixed time, the system is, at least in principle, completely described by a function of only a single three-dimensional position variable (this is analogous to time-dependent density functional theory in quantum mechanics [23]). ...

We study the dynamics of a colloidal fluid including inertia and hydrodynamic interactions, two effects which strongly influence the nonequilibrium properties of the system. We derive a general dynamical density functional theory which shows very good agreement with full Langevin dynamics. In suitable limits, we recover existing dynamical density functional theories and a Navier-Stokes-like equation with additional nonlocal terms.

The rich and diverse dynamics of particle-based systems ultimately originates from the coupling of their degrees of freedom via internal interactions. To arrive at a tractable approximation of such many-body problems, coarse graining is often an essential step. Power functional theory provides a unique and microscopically sharp formulation of this concept. The approach is based on an exact one-body variational principle to describe the dynamics of both overdamped and inertial classical and quantum many-body systems. In equilibrium, density-functional theory is recovered, and hence spatially inhomogeneous systems are described correctly. The dynamical theory operates on the level of time-dependent one-body correlation functions. Two- and higher-body correlation functions are accessible via the dynamical test-particle limit and the nonequilibrium Ornstein-Zernike route. The structure of this functional approach to many-body dynamics is described, including much background as well as applications to a broad range of dynamical situations, such as the van Hove function in liquids, flow in nonequilibrium steady states, motility-induced phase separation of active Brownian particles, lane formation in binary colloidal mixtures, and both steady and transient shear phenomena.

The rich and diverse dynamics of particle-based systems ultimately originates from the coupling of their degrees of freedom via internal interactions. To arrive at a tractable approximation of such many-body problems, coarse-graining is often an essential step. Power functional theory provides a unique and microscopically sharp formulation of this concept. The approach is based on an exact one-body variational principle to describe the dynamics of both overdamped and inertial classical and quantum many-body systems. In equilibrium, density functional theory is recovered, and hence spatially inhomogeneous systems are described correctly. The dynamical theory operates on the level of time-dependent one-body correlation functions. Two- and higher-body correlation functions are accessible via the dynamical test particle limit and the nonequilibrium Ornstein-Zernike route. We describe the structure of this functional approach to many-body dynamics, including much background as well as applications to a broad range of dynamical situations, such as the van Hove function in liquids, flow in nonequilibrium steady states, motility-induced phase separation of active Brownian particles, lane formation in binary colloidal mixtures, and both steady and transient shear phenomena.

Dynamic density functional theory (DDFT) allows the description of microscopic dynamical processes on the molecular scale extending classical DFT to non-equilibrium situations. Since DDFT and DFT use the same Helmholtz energy functionals, both predict the same density profiles in thermodynamic equilibrium. We propose a molecular DDFT model, in this work also referred to as hydrodynamic DFT, for mixtures based on a variational principle that accounts for viscous forces as well as diffusive molecular transport via the generalized Maxwell–Stefan diffusion. Our work identifies a suitable expression for driving forces for molecular diffusion of inhomogeneous systems. These driving forces contain a contribution due to the interfacial tension. The hydrodynamic DFT model simplifies to the isothermal multicomponent Navier–Stokes equation in continuum situations when Helmholtz energies can be used instead of Helmholtz energy functionals, closing the gap between micro- and macroscopic scales. We show that the hydrodynamic DFT model, although not formulated in conservative form, globally satisfies the first and second law of thermodynamics. Shear viscosities and Maxwell–Stefan diffusion coefficients are predicted using an entropy scaling approach. As an example, we apply the hydrodynamic DFT model with a Helmholtz energy density functional based on the perturbed-chain statistical associating fluid theory equation of state to droplet and bubble coalescence in one dimension and analyze the influence of additional components on coalescence phenomena.

Thermodynamics has a clear arrow of time, characterized by the irreversible approach to equilibrium. This stands in contrast to the laws of microscopic theories, which are invariant under time-reversal. Foundational discussions of this "problem of irreversibility" often focus on historical considerations, and do therefore not take results of modern physical research on this topic into account. In this article, I will close this gap by studying the implications of dynamical density functional theory (DDFT), a central method of modern nonequilibrium statistical mechanics not previously considered in philosophy of physics, for this debate. For this purpose, the philosophical discussion of irreversibility is structured into five problems, concerned with the source of irreversibility in thermodynamics, the definition of equilibrium and entropy, the justification of coarse-graining, the approach to equilibrium and the arrow of time. For each of these problems, it is shown that DDFT provides novel insights that are of importance for both physicists and philosophers of physics.

Classical dynamical density functional theory (DDFT) is one of the cornerstones of modern statistical mechanics. It is an extension of the highly successful method of classical density functional theory (DFT) to nonequilibrium systems. Originally developed for the treatment of simple and complex fluids, DDFT is now applied in fields as diverse as hydrodynamics, materials science, chemistry, biology, and plasma physics. In this review, we give a broad overview over classical DDFT. We explain its theoretical foundations and the ways in which it can be derived. The relations between the different forms of deterministic and stochastic DDFT as well as between DDFT and related theories, such as quantum-mechanical time-dependent DFT, mode coupling theory, and phase field crystal models, are clarified. Moreover, we discuss the wide spectrum of extensions of DDFT, which covers methods with additional order parameters (like extended DDFT), exact approaches (like power functional theory), and systems with more complex dynamics (like active matter). Finally, the large variety of applications, ranging from fluid mechanics and polymer physics to solidification, pattern formation, biophysics, and electrochemistry, is presented.

While solvation dynamics plays an important role in many processes, most available studies focus on the solvent response to the variations of charge distribution on dissolved solute, and the solvent response to the structural and geometrical variations of solute is rarely explored. Herein, by using dynamical density functional theory, we study the solvation dynamics of simple solvent in response to the variations of solute size and potential strength. We find that increasing solute size causes opposite effect of increasing its potential strength on solvation time. On the solvent structure, varying solute's potential strength brings a long-range yet weak influence while changing solute size generates a short-range yet strong effect, and therefore the penalty in solvent free energy is generally dominated by the change in solute's potential strength while the variations of coordination number and solvation shell radius are dominated by the change in solute size. This work unravels the nonpolar solvation dynamics in response to the variation of solute geometry, and casts helpful insights on the design and control of solvation-dynamics related applications in molecular engineering.

The mechanism of reaction-diffusion (RD) processes has great scientific and practical importance in many fields. However, theoretical studies of RD systems is limited by their multiscale and non-equilibrium nature. In this work, we propose an RD model on the basis of the classical density functional theory (CDFT) and reaction-diffusion equation (RDE), and applied it to an electrode system as an example. The model shows both long- and short-range behaviors in the RD system, which are different from the corresponding equilibrium systems. A γ-shaped polarization curve consistent with experimental data is predicted. The model also indicates that the microscopic/mesoscopic RD process is correlated with macroscopic convection through the width of the diffusion region.

The coalescence of nanodroplets is an important interfacial phenomenon and is the key to many real‐world applications. However, the microscopic mechanism for this process remains unclear. In this work, we propose a time‐dependent density functional theory (TDDFT) to understand and predict this process. The formation of a “peanut” nanodroplet seems key to the coalescence process, before and after which the system is dominated by a nucleation mechanism and an ordinary diffusion mechanism, respectively. It appears that molecular attraction is not only the driving force but also the resistance of droplet coalescence. The velocity distribution indicates that there is significant mass transfer on the vapor–liquid interphase. During coalescence, there is a clear linear correlation between the free energy and the surface area of the vapor–liquid interface, which means surface tension is the dominant contribution to the free energy. This article is protected by copyright. All rights reserved.

Microswimmers typically operate in complex environments. In biological systems, often diverse species are simultaneously present and interact with each other. Here, we derive a (time-dependent) particle-scale statistical description, namely, a dynamical density functional theory, for such multispecies systems, extending existing works on one-component microswimmer suspensions. In particular, our theory incorporates not only the effect of external potentials but also steric and hydrodynamic interactions between swimmers. For the latter, a previously introduced force-dipole-based minimal (pusher or puller) microswimmer model is used. As a limiting case of our theory, mixtures of hydrodynamically interacting active and passive particles are captured as well. After deriving the theory, we apply it to different planar swimmer configurations. First, these are binary pusher–puller mixtures in external traps. In the considered situations, we find that the majority species imposes its behavior on the minority species. Second, for unconfined binary pusher–puller mixtures, the linear stability of an orientationally disordered state against the emergence of global polar orientational order (and thus emergent collective motion) is tested analytically. Our statistical approach predicts, qualitatively in line with previous particle-based computer simulations, a threshold for the fraction of pullers and for their propulsion strength that lets overall collective motion arise. Third, we let driven passive colloidal particles form the boundaries of a shear cell, with confined active microswimmers on their inside. Driving the passive particles then effectively imposes shear flows, which persistently acts on the inside microswimmers. Their resulting behavior reminds of the one of circle swimmers although with varying swimming radii.

Based on statistical mechanics for classical fluids, general expressions for hydrodynamic stress in inhomoge-neous colloidal suspension are derived on a molecular level. The result is exactly an extension of the Iving-Kirkwood stress for atom fluids to colloidal suspensions where dynamic correlation emerges. It is found that besides the inter-particle distance, the obtained hydrodynamic stress depends closely on the velocity of the colloidal particles in the suspension, which is responsible for the appearance of the solvent-mediated hydrodynamic force. Compared to Brady's stresslets for the bulk stress, our results are applicable to inhomogeneous suspension, where the inhomogeneity and anisotropy of the dynamic correlation should be taken into account. In the near-field regime where the packing fraction of colloidal particles is high, our results can reduce to those of Brady. Therefore, our results are applicable to the suspensions with low, moderate, or even high packing fraction of colloidal particles.

We propose a three-dimensional time-dependent density functional theory (TDDFT) to investigate the freezing/melting of Lennard-Jones fluids in interfacial systems including flat interfaces and nano-droplets. The theory is based on a modified fundamental measure theory (MFMT) for the hard-sphere reference system plus mean field theory (MFT) for the attractive contributions plus an additional weighted density approximation (WDA) contribution to enforce that bulk liquid equation of state is that of modified Benedict, Webb and Rubin (MBWR). By using different initial states, our theory generated a series of equilibrium structures including crystals and polyhedral particles. The non-linear effect and the irreversibility of the freezing/melting process were captured. The profiles for free energy and the order parameter indicate that the free energy barrier existing in the freezing/melting process is caused by the break in symmetry. Additionally, the time-dependent properties were also examined, and provided insight into the mechanism of the freezing process.

Previous particle-based computer simulations have revealed a significantly more pronounced tendency of spontaneous global polar ordering in puller (contractile) microswimmer suspensions than in pusher (extensile) suspensions. We here evaluate a microscopic statistical theory to investigate the emergence of such an order through a linear instability of the disordered state. For this purpose, input concerning the orientation-dependent pair-distribution function is needed, and we discuss the corresponding approaches, particularly a heuristic variant of the Percus test-particle method applied to active systems. Our theory identifies an inherent evolution of polar order in planar systems of puller microswimmers, if mutual alignment due to hydrodynamic interactions overcomes the thermal dealignment by rotational diffusion. In our theory, the cause of orientational ordering can be traced back to the actively induced hydrodynamic rotation-translation coupling between the swimmers. Conversely, disordered pusher suspensions remain linearly stable against homogeneous polar orientational ordering. We expect that our results can be confirmed in experiments on (semi-)dilute active microswimmer suspensions, based, for instance, on biological pusher- and puller-type swimmers.

We study the dynamics of colloidal fluids in both unconfined geometries and when confined by a hard wall. Under minimal assumptions, we derive a dynamical density functional theory (DDFT) which includes hydrodynamic interactions (HI; bath-mediated forces). By using an efficient numerical scheme based on pseudospectral methods for integro-differential equations, we demonstrate its excellent agreement with the full underlying Langevin equations for systems of hard disks in partial confinement. We further use the derived DDFT formalism to elucidate the crucial effects of HI in confined systems.

Supercapacitors, or more specifically electric double-layer capacitors, (EDLCs) store electrical energy by adsorbing ionic species to the inner surfaces of porous electrodes. Porous carbons are the most commonly used electrodes, while ionic liquids, organic electrolytes, and aqueous electrolytes are used as charge carriers. Approximating the porous carbon as a slit pore and ions as charged hard spheres, we address the differential capacitance and layering of the ionic-liquid/carbon interface and the dependence of the capacitance on the pore size from a classical density functional theory (CDFT) perspective. We further introduce the solvent into the electrolyte to model the organic electrolyte EDLC. We demonstrate that the CDFT is uniquely amenable to the investigations of the electrochemical behavior of confined electrolytes given its applicability to electrodes with a wide distribution of pore sizes, ranging from ionic dimensionality to mesoscopic scales.

A dynamics density functional theory approach was presented to investigate the polymer-mediated nanoparticle deposits on a solid surface. The equilibrium and nonequilibrium behaviors of nanoparticles in the flexible linear, flexible star, semiflexible linear, and semiflexible star polymer solutions were investigated to evaluate the polymer-induced entropic effects and solvent-mediated hydrodynamic interactions. The theoretical results are in remarkable agreement with the Brownian dynamic simulation data, providing the quantitative verification of particle agglomeration and polymer depletion. The description at the microscopic level reveals new insight into the structure–function relationship of semidilute polymer–particle suspensions under confinement.

The ability of the phase-field-crystal (PFC) model to quantitatively predict atomistic defect structures in crystalline solids is addressed. First, general aspects of the PFC model are discussed within the context of obtaining quantitative results in solid materials. Then a specific example is used to illustrate major points. Specifically, accelerated molecular dynamics is used to compute the one-particle probability density rho((1))(r) in a complex atomistic defect consisting of a Lomer dislocation with an equilibrium distribution of vacancies in the core, and the results are considered within the general framework of the PFC model. As expected,.(1)(r) shows numerous spatially localized peaks with integrated densities smaller than unity, as would arise in a PFC computation. However, the rho((1))(r) actually corresponds to a time-averaged superposition of a few well-defined atomic configurations each having a well-defined energy. The deconvolution of rho((1))(r) to obtain the actual distinct atomic configurations is not feasible. Using a potential energy functional that accurately computes the energies of distinct configurations, the potential energy computed using rho((1))(r) differs from the actual average atomistic energy by similar to 50 eV divided among approximately 46 atoms in the core of the defect. Attempts to rectify this deviation by introducing correlations cannot significantly reduce this error. The simulations show energy barriers between distinct configurations varying by up to 0.5 eV, indicating that the simple kinetic evolution law used in PFC cannot accurately capture the true time evolution in this problem. Overall, these results demonstrate, in one nontrivial case, that the PFC model is probably unable to predict atomistic defect structures, energies, or kinetic barriers at the quantitative levels needed for application to problems in materials science.

We study the dynamics of a multi-species colloidal fluid in the full position-momentum phase space. We include both inertia and hydrodynamic interactions, which strongly influence the non-equilibrium properties of the system. Under minimal assumptions, we derive a dynamical density functional theory (DDFT), and, using an efficient numerical scheme based on spectral methods for integro-differential equations, demonstrate its excellent agreement with the full underlying Langevin equations. We utilise the DDFT formalism to elucidate the crucial effects of hydrodynamic interactions in multi-species systems.

The dynamics of nano-droplets on structured substrates differs qualitatively from the macroscopically expected behavior. The long-range van der Waals interactions acting in such systems induce lateral forces on nano-droplets in the vicinity of chemical and topographical steps even if the three-phase contact line of the droplets does not touch the step. Moreover, macroscopic considerations based on differences or gradients of local equilibrium contact angles are invalid because nano-droplets do not sample the long-range tails of the effective interface potential. As one of the consequences, nano-droplets spanning chemical steps can move towards the less wettable side.

We employ dynamical density functional theory (DDFT) and Brownian Dynamics (BD) simulations to examine the fully developed dynamics of ultrasoft colloids interacting via a Gaussian pair potential in time-dependent external fields. The DDFT formalism employed is that of Marconi and Tarazona [J. Chem. Phys., 110, 8032 (1999)], which allows for determination of the time-dependent density profile based on knowledge of the static, equilibrium density functional. Three different dynamical situations are examined: firstly, the behaviour of Gaussian particles in a spherical cavity of oscillating size, including both sudden and continuous changes in the size of the cavity. Secondly, a spherical cavity with a fixed size but varying sharpness. Finally, to investigate a strong inhomogeneity in the density profile we study the diffusion of one layer of particles which is initially strongly confined and separated from the remaining system via an external potential. In all cases, DDFT is in excellent agreement with BD results, demonstrating the applicability of the theory to dynamical problems involving overdamped interacting particles in a solvent.

Classical density functional theory (DFT) provides an exact variational framework for determining the equilibrium properties of inhomogeneous fluids. We report a generalization of DFT to treat the non-equilibrium dynamics of classical many-body systems subject to Brownian dynamics. Our approach is based upon a dynamical functional consisting of reversible free energy changes and irreversible power dissipation. Minimization of this "free power" functional with respect to the microscopic one-body current yields a closed equation of motion. In the equilibrium limit the theory recovers the standard variational principle of DFT. The adiabatic dynamical density functional theory is obtained when approximating the power dissipation functional by that of an ideal gas. Approximations to the excess (over ideal) power dissipation yield numerically tractable equations of motion beyond the adiabatic approximation, opening the door to the systematic study of systems far from equilibrium.

The aim of this paper is to review and to preview some selected topics of crystal nucleation in colloidal suspensions. First we discuss how the structure of critical nuclei can be calculated by computer simulations, in particular how linear shear flow affects the size and shape of the critical nuclei. Second, we preview the possibilities to access heterogeneous crystal nucleation and dynamics of a crystal by using the recently developed formalism of dynamical density functional theory. In particular, data for global crystal heating are presented.

We present a generalization of the Density Functional Theory to distributions in μ-space rather than in configuration space. This equilibrium theory is the basic ingredient for constructing a dynamic theory with projection operators. The reversible part of the dynamics is computed exactly while the irreversible part is approximated with a fast momentum relaxation assumption. As a result the irreversible operator is given in terms of a viscosity tensor. We show that the kinetic equation has an H-theorem.

We propose a numerical scheme based on the Chebyshev pseudo-spectral collocation method for solving the integral and integro-differential equations of the density-functional theory and its dynamic extension. We demonstrate the exponential convergence of our scheme, which typically requires much fewer discretization points to achieve the same accuracy compared to conventional methods. This discretization scheme can also incorporate the asymptotic behavior of the density, which can be of interest in the investigation of open systems. Our scheme is complemented with a numerical continuation algorithm and an appropriate time stepping algorithm, thus constituting a complete tool for an efficient and accurate calculation of phase diagrams and dynamic phenomena. To illustrate the numerical methodology, we consider an argon-like fluid adsorbed on a Lennard-Jones planar wall. First, we obtain a set of phase diagrams corresponding to the equilibrium adsorption and compare our results obtained from different approximations to the hard sphere part of the free energy functional. Using principles from the theory of sub-critical dynamic phase field models, we formulate the time-dependent equations which describe the evolution of the adsorbed film. Through dynamic considerations we interpret the phase diagrams in terms of their stability. Simulations of various wetting and drying scenarios allow us to rationalize the dynamic behavior of the system and its relation to the equilibrium properties of wetting and drying.

We examine the out-of-equilibrium dynamical evolution of density profiles of
ultrasoft particles under time-varying external confining potentials in three
spatial dimensions. The theoretical formalism employed is the dynamical
density functional theory (DDFT) of Marini, Bettolo, Marconi and Tarazona
(1999 J. Chem. Phys. 110 8032), supplied by an equilibrium excess free
energy functional that is essentially exact. We complement our theoretical
analysis by carrying out extensive Brownian dynamics simulations. We find
excellent agreement between theory and simulations for the whole time
evolution of density profiles, demonstrating thereby the validity of the
DDFT when an accurate equilibrium free energy functional is employed.

We resolve an existing paradox regarding the causality and symmetry properties of response functions within time-dependent density-functional theory. We do this by defining a new action functional within the Keldysh formalism. By functional differentiation the new functional leads to response functions which are symmetric in the Keldysh time contour parameter, but which become causal when a transition to physical time is made. The new functional is further used to derive the equations of the time-dependent optimized potential method.

We give an overview of the underlying concepts of time-dependent density-functional theory. The basic relations between densities, potentials and initial states, for time-dependent many-body systems are discussed. We obtain some new results concerning the invertability of response functions. Some fundamental difficulties associated with the time-dependent action principle are discussed and we show how these difficulties can be resolved by means of the Keldysh formalism.

We present a new time-dependent Density Functional approach to study the relaxational dynamics of an assembly of interacting particles subject to thermal noise. Starting from the Langevin stochastic equations of motion for the velocities of the particles we are able by means of an approximated closure to derive a self-consistent deterministic equation for the temporal evolution of the average particle density. The closure is equivalent to assuming that the equal-time two-point correlation function out of equilibrium has the same properties as its equilibrium version. The changes in time of the density depend on the functional derivatives of the grand canonical free energy functional $F[\rho]$ of the system. In particular the static solutions of the equation for the density correspond to the exact equilibrium profiles provided one is able to determine the exact form of $F[\rho]$. In order to assess the validity of our approach we performed a comparison between the Langevin dynamics and the dynamic density functional method for a one-dimensional hard-rod system in three relevant cases and found remarkable agreement, with some interesting exceptions, which are discussed and explained. In addition, we consider the case where one is forced to use an approximate form of $F[\rho]$. Finally we compare the present method with the stochastic equation for the density proposed by other authors [Kawasaki,Kirkpatrick etc.] and discuss the role of the thermal fluctuations. Comment: 14 pages, 6 figures, accepted by Journal of Chemical Physics

We present a simple derivation of the stochastic equation obeyed by the density function for a system of Langevin processes interacting via a pairwise potential. The resulting equation is considerably different from the phenomenological equations usually used to describe the dynamics of non-conserved (model A) and conserved (model B) particle systems. The major feature is that the spatial white noise for this system appears not additively but multiplicatively. This simply expresses the fact that the density cannot fluctuate in regions devoid of particles. The steady state for the density function may, however, still be recovered formally as a functional integral over the coursed grained free energy of the system as in models A and B.

This paper deals with the ground state of an interacting electron gas in an external potential v(r). It is proved that there exists a universal functional of the density, Fn(r), independent of v(r), such that the expression Ev(r)n(r)dr+Fn(r) has as its minimum value the correct ground-state energy associated with v(r). The functional Fn(r) is then discussed for two situations: (1) n(r)=n0+n(r), n/n01, and (2) n(r)= (r/r0) with arbitrary and r0. In both cases F can be expressed entirely in terms of the correlation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented.

The full text of this article is available as a PDF file.

A density-functional theory that treats all states of an electronic system on the same footing is introduced. The corresponding Kohn-Sham formalism can be applied to ground and excited states alike, does not suffer from a v-representability problem, and represents a rigorous formal basis for the common, but so far unjustified practice to treat excited states by Kohn-Sham methods. The presented density-functional theory emerges from a generalization of the constrained-search procedure. The new Kohn-Sham formalism is based on generalized adiabatic connections introduced here. The possible topologies of those generalized adiabatic connections are discussed. A density-based stationarity principle and a density theorem that represents a more general counterpart of the Hohenberg-Kohn theorem are presented. A method to take into account exactly exchange interactions in the presented Kohn-Sham formalism is introduced, implemented, and applied to atoms.

A phenomenological dynamic extension of the classical density functional theory is proposed. Our evolution equation for the density has the form of a generalized Smoluchowski equation which involves a correlation potential to be derived from the microscopic theory. By means of an expansion in terms of generalized Hermite polynomials we also present numerical results for simple cases of nonlinear diffusion.

The theory of differential forms and time-dependent vector fields on manifolds is applied to formulate response theory for non-Hamiltonian systems. This approach is manifestly coordinate-free, and provides a transparent derivation of the response of a thermostatted system to a time-dependent perturbation.

A variational property of the ground-state energy of an electron gas in an external potential v(r), derived by Hohenberg and Kohn, is extended to nonzero temperatures. It is first shown that in the grand canonical ensemble at a given temperature and chemical potential, no two v(r) lead to the same equilibrium density. This fact enables one to define a functional of the density F[n(r)] independent of v(r), such that the quantity Ω=∫v(r)n(r)dr+F[n(r)] is at a minimum and equal to the grand potential when n(r) is the equilibrium density in the grand ensemble in the presence of v(r).

A density-functional formalism comparable to the Hohenberg-Kohn-Sham theory of the ground state is developed for arbitrary time-dependent systems. It is proven that the single-particle potential $v(\stackrel{$\rightarrow${}}{\mathrm{r}}t)$ leading to a given $v$-representable density $n(\stackrel{$\rightarrow${}}{\mathrm{r}}t)$ is uniquely determined so that the corresponding map $v$\rightarrow${}n$ is invertible. On the basis of this theorem, three schemes are derived to calculate the density: a set of hydrodynamical equations, a stationary action principle, and an effective single-particle Schr\"odinger equation.

From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of 23.) Electronic systems at finite temperatures and in magnetic fields are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations.

The time-dependent density-functional theory of Runge and Gross [Phys.
Rev. Lett. 52, 997 (1984)] is reexamined on the basis of its
limitations, and the criticisms raised by Xu and Rajagopal [Phys. Rev. A
31, 2682 (1985)] are addressed, within the imposition of natural
boundary conditions of vanishing density and potential at infinity.
Also, for a single-particle system characterized by an arbitrary
time-dependent potential, the uniqueness of the density-to-potential
mapping is established explicitly for both bound and scattering states.

Universal variational functionals of densities, first-order density matrices, and natural spin-orbitals are explicitly displayed for variational calculations of ground states of interacting electrons in atoms, molecules, and solids. In all cases, the functionals search for constrained minima. In particular, following Percus [Formula: see text] is identified as the universal functional of Hohenberg and Kohn for the sum of the kinetic and electron-electron repulsion energies of an N-representable trial electron density rho. Q[rho] searches all antisymmetric wavefunctions Psi(rho) which yield the fixed. rho. Q[rho] then delivers that expectation value which is a minimum. Similarly, [Formula: see text] is shown to be the universal functional for the electron-electron repulsion energy of an N-representable trial first-order density matrix gamma, where the actual external potential may be nonlocal as well as local. These universal functions do not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus, the v-representability problem, which is especially severe for trial first-order density matrices, has been solved. Universal variational functionals in Hartree-Fock and other restricted wavefunction theories are also presented. Finally, natural spin-orbital functional theory is compared with traditional orbital formulations in density functional theory.

This paper develops a quantitatively accurate first-principles description for the frequency and the linewidth of collective electronic excitations in inhomogeneous weakly disordered systems. A finite linewidth in general has intrinsic and extrinsic sources. At low temperatures and outside the region where electron-phonon interaction occurs, the only intrinsic damping mechanism is provided by electron-electron interaction. This kind of intrinsic damping can be described within time-dependent density-functional theory (TDFT), but one needs to go beyond the adiabatic approximation and include retardation effects. It was shown previously that a density-functional response theory that is local in space but nonlocal in time has to be constructed in terms of the currents, rather than the density. This theory will be reviewed in the first part of this paper. For quantitatively accurate linewidths, extrinsic dissipation mechanisms, such as impurities or disorder, have to be included. In the second part of this paper, we discuss how extrinsic dissipation can be described within the memory function formalism. We first review this formalism for homogeneous systems, and then present a synthesis of TDFT with the memory function formalism for inhomogeneous systems, to account simultaneously for intrinsic and extrinsic damping of collective excitations. As example, we calculate frequencies and linewidths of intersubband plasmons in a 40 nm wide GaAs/AlGaAs quantum well. Comment: 20 pages, 3 figures