Interaction multiparticle

Friction interactions, multiparticle collision dynamics, single-particle friction and diffusion, 114—118 Friction kernel, transition state trajectory, colored noise, 209... 

Multiparticle collision dynamics can be generalized to treat systems with different species. While there are many different ways to introduce multiparticle collisions that distinguish between the different species [16, 17], all such rules should conserve mass, momentum, and energy. We suppose that the A-particle system contains particles of different species a=A,B,... with masses ma. Different multiparticle collisions can be used to distinguish the interactions among the species. For this purpose we let V 1 denote the center of mass velocity of particles of species a in the cell i ,3... 

Multiparticle collision dynamics can be combined with full molecular dynamics in order to describe the behavior of solute molecules in solution. Such hybrid MPC-MD schemes are especially useful for treating polymer and colloid dynamics since they incorporate hydrodynamic interactions. They are also useful for describing reactive systems where diffusive coupling among solute species is important. 

Hybrid MPC-MD schemes may be constructed where the mesoscopic dynamics of the bath is coupled to the molecular dynamics of solute species without introducing explicit solute-bath intermolecular forces. In such a hybrid scheme, between multiparticle collision events at times x, solute particles propagate by Newton s equations of motion in the absence of solvent forces. In order to couple solute and bath particles, the solute particles are included in the multiparticle collision step [40]. The above equations describe the dynamics provided the interaction potential is replaced by Vj(rJVs) and interactions between solute and bath particles are neglected. This type of hybrid MD-MPC dynamics also satisfies the conservation laws and preserves phase space volumes. Since bath particles can penetrate solute particles, specific structural solute-bath effects cannot be treated by this rule. However, simulations may be more efficient since the solute-solvent forces do not have to be computed. 

Most descriptions of the dynamics of molecular or particle motion in solution require a knowledge of the frictional properties of the system. This is especially true for polymer solutions, colloidal suspensions, molecular transport processes, and biomolecular conformational changes. Particle friction also plays an important role in the calculation of diffusion-influenced reaction rates, which will be discussed later. Solvent multiparticle collision dynamics, in conjunction with molecular dynamics of solute particles, provides a means to study such systems. In this section we show how the frictional properties and hydrodynamic interactions among solute or colloidal particles can be studied using hybrid MPC-MD schemes. 

The nature of the solvent influences both the structure of the polymer in solution and its dynamics. In good solvents the polymer adopts an expanded configuration and in poor solvents it takes on a compact form. If the polymer solution is suddenly changed from good to poor solvent conditions, polymer collapse from the expanded to compact forms will occur [78], A number of models have been suggested for the mechanism of the collapse [79-82], Hydrodynamic interactions are expected to play an important part in the dynamics of the collapse and we show how MPC simulations have been used to investigate this problem. Hybrid MD-MPC simulations of the collapse dynamics have been carried out for systems where bead-solvent interactions are either explicitly included [83] or accounted for implicitly in the multiparticle collision events [84, 85]. 

The effect of hydrodynamic interactions on polymer collapse has also been studied using MPC dynamics, where the polymer beads are included in the multiparticle collision step [28, 84]. Hydrodynamic interactions can be turned off by replacing multiparticle collisions in the cells by sampling of the particle velocities from a Boltzmann distribution. Collapse occurs more rapidly in the... 

Multiparticle collision dynamics provides an ideal way to simulate the motion of small self-propelled objects since the interaction between the solvent and the motor can be specified and hydrodynamic effects are taken into account automatically. It has been used to investigate the self-propelled motion of swimmers composed of linked beads that undergo non-time-reversible cyclic motion [116] and chemically powered nanodimers [117]. The chemically powered nanodimers can serve as models for the motions of the bimetallic nanodimers discussed earlier. The nanodimers are made from two spheres separated by a fixed distance R dissolved in a solvent of A and B molecules. One dimer sphere (C) catalyzes the irreversible reaction A + C B I C, while nonreactive interactions occur with the noncatalytic sphere (N). The nanodimer and reactive events are shown in Fig. 22. The A and B species interact with the nanodimer spheres through repulsive Lennard-Jones (LJ) potentials in Eq. (76). The MPC simulations assume that the potentials satisfy Vca = Vcb = Vna, with c.,t and Vnb with 3- The A molecules react to form B molecules when they approach the catalytic sphere within the interaction distance r < rc. The B molecules produced in the reaction interact differently with the catalytic and noncatalytic spheres. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

Brownian motion, multiparticle collision dynamics, hydrodynamic interactions, 118—121... 

Generalized relativstic effective core potentials (GRECP), ab initio calculations, P,T-odd interactions, 253-259 Gene transcription, multiparticle collisions, reactive dynamics, 108-111 Geometric transition state theory, 195-201 Gillespie s algorithm, multiparticle collisions, reactive dynamics, 109-111 Golden Rule approximation, two-pathway... 

Monte Carlo heat flow simulation, nonequilibrium molecular dynamics, 79—81 multiparticle collision dynamics hydrodynamic interactions, 118-121 single-particle friction and diffusion, 114-118... 

Multiparticle collision dynamics (continued) hydrodynamic equations, 104—107 flow simulation, 107 friction interactions, 118-121 immiscible fluids, 138-139 macroscopic laws and transport coefficients, 99-104... 

The magnetic interaction energy increases if the interaction works between multiparticles. The multi-interaction becomes important in a concentrated dispersion (3). 

C. Multiparticle Interactions, Deformable Interfaces, and Protein-Surface Interactions... 

There seems to be much going for this proposition. Before acceding to it completely, it is necessary to recall what has been written earlier about the use of pair potentials in place of multiparticle interactions. The quantities A and 5 in the Lennard-... 

A semirigorous multiple-scattering theory of Shaqfeh and Fredrickson (1990) that accounts for multiparticle hydrodynamic interactions for slender bodies has verified Batchelor s theory and has given a slightly improved formula for str ... 

The above results illustrate the utility of multiparticle Brownian dynamics for the analysis of diffusion controlled polymerizations. The results presented here are, however, qualitative because of the assumption of a two-dimensional system, neglect of polymer-polymer interactions and the infinitely fast kinetics in which every collision results in reaction. While the first two assumptions may be easily relaxed, incorporation of slower reaction kinetics by which only a small fraction of the collisions result in reaction may be computationally difficult. A more computationally efficient scheme may be to use Brownian dynamics to extract the rate constants as a function of polymer difflisivities, and to incorporate these in population balance models to predict the molecular weight distribution [48-50]. We discuss such a Brownian dynamics method in the next section. 

If a multiparticle system is considered and the election interaction is introduced, we may use the Dirac-Coulomb-Breit (DCB) Hamiltonian which is given by a sum of one-particle Dirac operators coupled by the Coulomb interaction 1 /r,7 and the Breit interaction Bij. Applying the Douglas-Kroll transformation to the DCB Hamiltonian, we arrive at the following operator (Hess 1997 Samzow and Hess 1991 Samzow et al. 1992), where an obvious shorthand notation for the indices pi has been used ... 

Taking anharmonicity of the lattice vibrations into account leads to interaction of the phonons with one another. When this interaction turns out to be sufficiently strong, the formation of bound states of quasiparticles becomes possible in addition to the above-mentioned multiparticle states. Such states are... 

It is an essential fact that the above-mentioned gaps in the polariton spectrum, if they arise, as well as the corresponding interaction between the photon and phonon, are nonzero within the framework of linear theory and, in general, do not require that anharmonicity be taken into account. Therefore, it makes sense to denote as a polariton Fermi resonance only such situations where vibrations of overtone or combination tone frequencies resonate with the polariton. We now turn our attention to an analysis of such rather complex situations, requiring that multiparticle excited states of the crystal be taken into consideration. Shown schematically in Fig. 6.6 is a typical polariton spectrum, as well as a band of two-particle states of B phonons. If, under the effect of anharmonicity, biphonons with energy E = E are formed, these states also resonate with the polariton, influencing its spectrum. 

It is perhaps also Important to emphasize that the oscillator strength /< assumes a much more complicated form than normally realized, even for the first set of assumptions. In condensed systems it is formidable because simultaneous multiparticle interactions must be considered. 

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