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Modelling of many real-world processes, such as drug delivery, wastewater treatment, and pharmaceutical production, requires accurate descriptions of the dynamics of hard particles confined in complicated domains. In particular, when modelling sedimentation processes or systems with driven flows, it is important to accurately capture volume exclusion effects. This work applies Dynamic Density Functional Theory to the evolution of a particle density under diffusion, external forces, particle–particle interaction, and volume exclusion. Using a spectral element framework, for the first time it is possible to include all of these effects in dynamic simulations on complex domains. Moreover, this allows one to apply complicated no-flux, and other non-local, non-linear, boundary conditions. The methodology is also extended to control problems, addressing questions of how to enhance production set-up in industrially-motivated processes. In this work the relevant models are introduced, numerical methods are discussed, and several example problems are solved to demonstrate the methods' versatility. It is shown that incorporating volume exclusion is crucial for simulation accuracy and we illustrate that the choice of boundary conditions significantly impacts the dynamics. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Classical Density Functional Theory (DFT) is a statistical–mechanical framework to analyse 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 specialised 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate the transition of a quantum wave-packet through a one-dimensional avoided crossing of molecular energy levels when the energy levels at the crossing point are tilted. Using superadiabatic representations, and an approximation of the dynamics near the crossing region, we obtain an explicit formula for the transition wave function. Our results agree extremely well with high precision ab-initio calculations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Schrödinger equation for atoms and ions with N = 1 to 10 electrons. In the asymptotic limit of large nuclear charge Z, we determine explicitly the low-lying energy levels and eigenstates. The asymptotic energies and wavefunctions are in good quantitative agreement with experimental data for positive ions, and in excellent qualitative agreement even for neutral atoms (Z = N). In particular, the predicted ground state spin and angular momentum quantum numbers (¹S for He, Be, Ne, ²S for H and Li, 4S for N, ²P for B and F, and ³P for C and O) agree with experiment in every case. The asymptotic Schrödinger ground states agree, up to small corrections, with the semiempirical hydrogen orbital configurations developed by Bohr, Hund, and Slater to explain the periodic table. In rare cases where our results deviate from this picture, such as the ordering of the lowest ¹Do and ³So states of the carbon isoelectronic sequence, experiment confirms our predictions and not Hund's. [ABSTRACT FROM AUTHOR]

The rate of convergence of a CI calculation on the ground state of the helium atom is rigorously derived under suitable assumptions on the regularity of the exact wavefunction. For bases consisting of all partial waves with angular momentum less than or equal to L, the large L asymptotic energies are found to obey the well-known formula EL - E = CL-3 + o(L-3), where C is an explicit constant defined in terms of the exact wavefunction. [ABSTRACT FROM AUTHOR]

Configuration-interaction (CI) models are approximations to the electronic Schrödinger equation which are widely used for numerical electronic structure calculations in quantum chemistry. Based on our recent closed-form asymptotic results for the full atomic Schrödinger equation in the limit of fixed electron number and large nuclear charge [SIAM J. Math. Anal., 41 (2009), pp. 631-664], we introduce a class of CI models for atoms which reproduce, at fixed finite model dimension, the correct Schrödinger eigenvalues and eigenstates in this limit. We solve exactly the ensuing minimal model for the second period atoms, Li to Ne, except for optimization of eigenvalues with respect to orbital dilation parameters, which is carried out numerically. The energy levels and eigenstates are in remarkably good agreement with experimental data (comparable to that of much larger scale numerical simulations in the literature) and facilitate a mathematical understanding of various spectral, chemical, and physical properties of small atoms. [ABSTRACT FROM AUTHOR]

Developments in automated code generation have allowed extremely compact representations of numerical models and for associated adjoint models to be derived automatically via high level algorithmic differentiation. In this article these principles are extended to enable the calculation of higher order derivative information. The higher order derivative information is computed through the automated derivation of tangent-linear equations, which are then treated as new forward equations, and from which higher order tangent-linear and adjoint information can be derived. The principal emphasis is on the calculation of partial differential equation constrained Hessian actions, but the approach generalizes for derivative information at arbitrary order. The derivative calculations are further combined with an advanced data checkpointing strategy. Applications which make use of partial differential equation constrained Hessian actions are presented. [ABSTRACT FROM AUTHOR]

We introduce a versatile bottom-up derivation of a formal theoretical framework to describe (passive) soft-matter systems out of equilibrium subject to fluctuations.We provide a unique connection between the constituent-particle dynamics of real systems and the time evolution equation of their measurable (coarse-grained) quantities, such as local density and velocity. The starting point is the full Hamiltonian description of a system of colloidal particles immersed in a fluid of identical bath particles. Then,we average out the bath viaZwanzig’s projection-operator techniques and obtain the stochastic Langevin equations governing the colloidal-particle dynamics. Introducing the appropriate definition of the local number and momentum density fields yields a generalisation of the Dean–Kawasaki (DK) model, which resembles the stochastic Navier–Stokes description of a fluid.Nevertheless, the DK equation still contains all the microscopic information and, for that reason, does not represent the dynamical law of observable quantities.We address this controversial feature of the DK description by carrying out a nonequilibrium ensemble average. Adopting a natural decomposition into local-equilibrium and nonequilibrium contribution,where the former is related to a generalised version of the canonical distribution,we finally obtain the fluctuating-hydrodynamic equation governing the time-evolution of themesoscopic density and momentum fields.Along the way,we outline the connection between the ad hoc energy functional introduced in previous DK derivations and the free-energy functional from classical density-functional theory.The resultant equation has the structure of a dynamical density functional theory (DDFT) with an additional fluctuating force coming from the random interactions with the bath.We show that our fluctuating DDFT formalism corresponds to a particular version of the fluctuating Navier–Stokes equations, originally derived by Landau and Lifshitz. Our framework thus provides the formal apparatus for ab initio derivations of fluctuating DDFT equations capable of describing the dynamics of soft-matter systems in and out of equilibrium. [ABSTRACT FROM AUTHOR]

We examine the nanoscale behavior of an equilibrium three-phase contact line in the presence of long-ranged intermolecular forces by employing a statistical mechanics of fluids approach, namely, density functional theory (DFT) together with fundamental measure theory (FMT). This enables us to evaluate the predictive quality of effective Hamiltonian models in the vicinity of the contact line. In particular, we compare the results for mean field effective Hamiltonians with disjoining pressures defined through (i) the adsorption isotherm for a planar liquid film, and (ii) the normal force balance at the contact line. We find that the height profile obtained using (i) shows good agreement with the adsorption film thickness of the DFT-FMT equilibrium density profile in terms of maximal curvature and the behavior at large film heights. In contrast, we observe that while the height profile obtained by using (ii) satisfies basic sum rules, it shows little agreement with the adsorption film thickness of the DFT results. The results are verified for contact angles of 20°, 40°, and 60°. [ABSTRACT FROM AUTHOR]

Transcription elongation rates influence RNA processing, but sequence-specific regulation is poorly understood. We addressed this in vivo , analyzing RNAPI in S. cerevisiae. Mapping RNAPI by Miller chromatin spreads or UV crosslinking revealed 5′ enrichment and strikingly uneven local polymerase occupancy along the rDNA, indicating substantial variation in transcription speed. Two features of the nascent transcript correlated with RNAPI distribution: folding energy and GC content in the transcription bubble. In vitro experiments confirmed that strong RNA structures close to the polymerase promote forward translocation and limit backtracking, whereas high GC in the transcription bubble slows elongation. A mathematical model for RNAPI elongation confirmed the importance of nascent RNA folding in transcription. RNAPI from S. pombe was similarly sensitive to transcript folding, as were S. cerevisiae RNAPII and RNAPIII. For RNAPII, unstructured RNA, which favors slowed elongation, was associated with faster cotranscriptional splicing and proximal splice site use, indicating regulatory significance for transcript folding. • Structures in the nascent RNA correlate with rapid elongation by RNAPI in vivo • Stable RNA structures limit RNAPI backtracking in vitro • GC content in the transcription bubble tunes transcription elongation rate • Nascent transcript folding modulates dynamics of all three RNAPs in vivo Transcription elongation is a stochastic process that can go backward as well as forward. Here Turowski et al. report that folding of the nascent transcript immediately behind the polymerase resists backtracking and promotes forward movement. This results in faster elongation relative to unstructured sequences and is potentially important for cotranscriptional processing. [ABSTRACT FROM AUTHOR]

The paper by Nold et al. [Phys. Fluids 26 (7), 072001 (2014)] examined density profiles and the micro-scale structure of an equilibrium three-phase (liquid-vapour-solid) contact line in the immediate vicinity of the wall using elements from the statistical mechanics of classical fluids, namely density-functional theory. The present research note, building on the above work, further contributes to our understanding of the nanoscale structure of a contact line by quantifying the strong dependence of the liquid-vapour density profile on the normal distance to the interface, when compared to the dependence on the vertical distance to the substrate. A recent study by Benet et al. [J. Phys. Chem. C 118 (38), 22079 (2014)] has shown that this could explain the emergence of a film-height-dependent surface tension close to the wall, with implications for the Frumkin-Derjaguin theory. [ABSTRACT FROM AUTHOR]