External potential dynamics

Maurits, N.M., Fraaije, J.G.E.M. Mesoscopic dynamics of copolymer melts from density dynamics to external potential dynamics using nonlocal kinetic coupling. J. Chem. Phys. 107 (1997) 5879-5889. 

Keywords Polymer blends Self-consistent field theory External potential dynamics Field-theoretic polymer simulations Polymeric microemulsion Polymer dynamics... 

The approach is commonly referred to as external potential dynamics (EPD). A related approach was originally introduced by Maurits and Fraaije [31]. However, these authors do not determine U W) exactly, but only approximately by solving separate Langevin equations for real fields Wa and Wb. This amounts to introducing a separate Langevin equation for a real field iU (i.e., an imaginary U in our notation) in addition to Eq. 97. 

Here, is the volume fraction of A block in diblock copolymer. To study the dynamics of phase separation, the polymeric external potential dynamics (EPD) method can be employed, which was proposed by Maurits and Fraaije [23] in dynamic density functional theory (DDFT) method (bead-string model). In EPD, the monomer concentration is a conserved quantity, and the polymer dynamics is inherently of Rouse type. The external dynamical equation in terms of the potential field m,- is expressed as... 

The thermodynamic integration scheme can be appUed to different models including coarse-grained, partide-based models of amphiphihc systems and membranes [133, 134] (e.g., soft DPD-models [135-137], Lennard-Jones models [138,139], or solvent-free models [140-142] of membranes) as well as field-theoretic representations [28]. It can be implemented in Monte Carlo or molecular dynamic simulations, as well as SCMF simulations [40-42, 86], field-theoretic simulations [28], and external potential dynamics [27, 63, 64] or dynamic density functional theory [143, 144]. 

Many of the electrochemical techniques described in this book fulfill all of these criteria. By using an external potential to drive a charge transfer process (electron or ion transfer), mass transport (typically by diffusion) is well-defined and calculable, and the current provides a direct measurement of the interfacial reaction rate [8]. However, there is a whole class of spontaneous reactions, which do not involve net interfacial charge transfer, where these criteria are more difficult to implement. For this type of process, hydro-dynamic techniques become important, where mass transport is controlled by convection as well as diffusion. 

The specific form of the short-time transition probability depends on the type of dynamics one uses to describe the time evolution of the system. For instance, consider a single, one-dimensional particle with mass m evolving in an external potential energy V(q) according to a Langevin equation in the high-friction limit... 

Once the diabatic potential energy surfaces relevant to describing a process, the integration of the sources of external potential (nuclear dynamics) can be done in real space using numerical integration methods. 

We recall here that, in quantum mechanics, each dynamic variable used in classical mechanics is associated with a linear hermitian operator. As a consequence, energy and momentum of a particle placed in an external potential , = Ep/q, will respectively be associated to the following operators ... 

In the case of dynamical interaction the pair potentials U r), (7BB(r) ar d Uab(t) should be incorporated into equation (2.3.45). It could be done using the Smoluchowski equation [27, 83, 84] for a particle drift in the external potential W (r) and expressed in terms of single particle DF (or concentration of such non-interacting particles)... 

Brownian transport processes and the related relaxation dynamics in the presence and absence of an external potential are most conveniently described in terms of partial differential equations of the Fokker-Planck (Smo-luchowski) [13, 14, 17-19], Rayleigh [13, 20], and Klein-Kramers [13, 14,... 

Sources of external potential can be produced in a number of ways in which there is no need for special massive nuclei. As they identify external sources they are classical variables, namely, position coordinates for the sources. There is no quantum dynamics related to them yet. Symmetry constraints can be naturally defined. We formally write He(p a) to distinguish this situation from the standard approach. Since the primacy is given to the electronic wave function, and no Schrodinger equation is available at this point, its existence has to be taken as workinghypotheses. 

SA = 0 subject to the energy constraint restates the principle of least action. When the external potential function is constant, the definition of ds as a path element implies that the system trajectory is a geodesic in the Riemann space defined by the mass tensor m . This anticipates the profound geometrization of dynamics introduced by Einstein in the general theory of relativity. 

NMR spectroscopy is a very useful tool for determining the local chemical surroundings of various atoms. Komin et al studied theoretically this for the adenine molecule of Fig. 20 both in vacuum and in an aqueous solution using different computational approaches. In all cases, density-functional calculations were used for the adenine molecule, but as basis functions they used either a set of localized functions (marked loc in Table 45) or plane waves (marked pw). Furthermore, in order to include the effects of the solvent they used either the polarizable continuum approach (marked PCM) or an explicit QM/MM model with a force field for the solvent and a molecular-dynamics approach for optimizing the structure (marked MD). In that case, the chemical shifts were calculated as averages over 40 snapshots from the molecular-dynamics simulations. Finally, in one case, an extra external potential from the solvent acting on the solute was included, too, marked by the asterisk in the table. 

In the method proposed by Solmajer and Mehler for modeling the effects of bulk solvent in protein simulations, the functional form for the configurational energy of the system is obtained by adding an external potential Vext for restraining the dynamics of solvent molecules to the standard potential energy function in Equation 1 to give... 

In most cases, as in the Born-Oppenheimer Molecular Dynamics scheme discussed earlier (see Sect. 2.3) what is given to us is the external potential vq (r ) instead of the ground state density. In its turn, the ground state density no for the external potential vq can be shown to minimize the energy functional [124] ... 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

A convenient way to formulate a dynamical equation for a Levy flight in an external potential is the space-fractional Fokker-Planck equation. Let us quickly review how this is established from the continuous time random walk. We will see below, how that equation also emerges from the alternative Langevin picture with Levy stable noise. Consider a homogeneous diffusion process, obeying relation (16). In the limit k — 0 and u > 0, we have X(k) 1 — CTa fe and /(w) 1 — uz, whence [52-55]... 

Provided the time tc is short enough for the system to remain near the bottom of these two bands, the dynamics can be interpreted in terms of the appropriate effective masses the "heavy" diatoms with mass (28) move much slower than the "light" single atoms with mass (15b). Thus, after changing the external potential from harmonic to linear repulsive, the unpaired atoms will be ejected out of the lattice and separated from the diatoms as glumes from grains. 

The equilibrium is dynamic with metal ions being discharged and metal atoms being ionized, but these two effects cancel each other and there is no net change in the system. For the realization of metal deposition at the cathode and metal dissolution at the anode, the system must be moved away from the equilibrium condition. An external potential must be provided for the useful electrode reactions to take place at a practical rate this external potential may have several causes. 

The present review has been very selective, stressing the rationale behind density-functional methods above their applications and excluding many important topics (both theoretical and computational). The interested reader may refer to anyone of the many books [91-93] or review articles [94-101] on density-functional theory for more details. Of special importance is the extension of density-functional theory to time-dependent external potentials [102-105], as this enables the dynamical behavior of molecules, including electronic excitation, to be addressed in the context of DFT [106-108]. As they are particularly relevant to the present discussion, we cite several articles related to the formal foundations of density-functional theory [85,100,109-111], linear-scaling methods [63,112-116], exchange-correlation energy functionals [25, 117-122], and qualitative tools for describing chemical reactions [123-126,126-132]. 

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