Power functional theory for Newtonian many-body dynamics

Schmidt, Matthias

doi:10.1063/1.5008608

Cited by 25 publications

(26 citation statements)

References 160 publications

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“…DDFT is based on equilibrium density functional theory (DFT) [25][26][27] and for an equilibrium system, DDFT is equivalent to DFT. DDFT was originally developed as a theory for Brownian particles with over-damped stochastic equations of motion [28][29][30][31], but it has also been extended to describe the dynamics of under-damped systems and atomic or molecular systems where the particle dynamics is governed by Newton's equations of motion [32][33][34][35][36][37]. This body of work shows that when such systems are not too far from equilibrium, then the dynamics predicted by the original DDFT is still often correct in the long-time limit where the particle dynamics is dominated by diffusive processes.…”

Section: Introductionmentioning

confidence: 99%

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

et al. 2019

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Phase field crystal (PFC) theory is extensively used for modelling the phase behaviour, structure, thermodynamics and other related properties of solids. PFC theory can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Here, we carefully identify all of these approximations and explain the consequences of each. One approximation that is made in standard derivations is to neglect a term of form ∇ · [n∇Ln], where n is the scaled density profile and L is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and that dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in n. However, the consequences of doing this are: (i) the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and (ii) other periodic structures, such as stripes, are erroneously predicted to be thermodynamic equilibrium structures. In general, L consists of a nonlocal convolution involving the pair direct correlation function. A second approximation sometimes made in deriving PFC theory is to replace L by a gradient expansion involving derivatives. We show that this leads to the possibility of the density going to zero, with its logarithm going to −∞ whilst being balanced by the fourth derivative of the density going to +∞. This subtle singularity leads to solutions failing to exist above a certain value of the average density. We illustrate all of these conclusions with results for a particularly simple model two-dimensional fluid, the generalised exponential model of index 4 (GEM-4), chosen because a DDFT is known to be accurate for this model. The consequences of the subsequent PFC approximations can then be examined. These include the phase diagram being both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal phase. Thus, although PFC models are very successful as phenomenological models of crystallisation, we find it impossible to derive the PFC as a theory for the (scaled) density distribution when starting from an accurate DDFT, without introducing spurious artefacts. However, we find that making a simple one-mode approximation for the logarithm of the density distribution log ρ(x) (rather than for ρ(x)), is surprisingly accurate. This approach gives a tantalising hint that accurate PFC-type theories may instead be derived as theories for the field log ρ(x), rather than for the density profile itself.

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Section: Introductionmentioning

confidence: 99%

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

et al. 2019

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“…For the case of sedimentation of the active ideal gas, an analytical solution for these fields could be constructed [55]. PFT was extended for the description of inertial classical [51] and of quantum many-body dynamics [52,53]. The fundamental differences between canonical and grand canonical schemes have been addressed in equilibrium [56] and for the dynamics [57].…”

Section: Introductionmentioning

confidence: 99%

Shear-induced deconfinement of hard disks

Jahreis

Schmidt

2020

Colloid Polym Sci

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Using Brownian dynamics simulations, we investigate the response to shear of a two-dimensional system of quasi-hard disks that are confined in the velocity gradient direction by a smooth external potential. Shearing the confined system leads to a homogenization of the one-body density profile. In order to rationalize this deconfinement effect, we split the internal onebody force field into adiabatic and superadiabatic contributions. We demonstrate that the superadiabatic force field consists of viscous and of structural contributions. We give an empirical scaling law that yields results for the superadiabatic force profiles both in the flow and in the gradient direction, in excellent agreement with the simulation data.

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“…PFT provides a formally exact method for including such effects and for calculating the full current in a non-equilibrium system [52]; see Ref. [66] for a pedagogical introduction to the framework. Both adiabatic forces, which give rise to the adiabatic current J ad (r, t) via equation ( 21), but also superadiabatic forces, which characteristically depend functionally on both the density profile and on the current distribution, are included.…”

Section: Power Functional Theorymentioning

confidence: 99%

Dynamic Decay and Superadiabatic Forces in the van Hove Dynamics of Bulk Hard Sphere Fluids

Treffenstädt,

Schindler,

Schmidt

2022

Preprint

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We study the dynamical decay of the van Hove function of Brownian hard spheres using event-driven Brownian dynamics simulations and dynamic test particle theory. Relevant decays mechanisms include deconfinement of the self particle, decay of correlation shells, and shell drift. Comparison to results for the Lennard-Jones system indicates the generality of these mechanisms for dense overdamped liquids. We use dynamical density functional theory on the basis of the Rosenfeld functional with self interaction correction. Superadiabatic forces are analysed using a recent power functional approximation. The power functional yields a modified Einstein long-time self diffusion coefficient in good agreement with simulation data.

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Power functional theory for Newtonian many-body dynamics

Cited by 25 publications

References 160 publications

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

Shear-induced deconfinement of hard disks

Dynamic Decay and Superadiabatic Forces in the van Hove Dynamics of Bulk Hard Sphere Fluids

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