Einstein equation simulations

Self-diffusion coefficients are dynamic properties that can be easily obtained by molecular dynamics simulation. The properties are obtained from mean-square displacement by the Einstein equation ... 

Steric requirements, hydrogen and deuterium, 299 Stem-Volmer plot, 181 Stiff differential equations, 109 Stochastic simulation, 109 Stoichiometric coefficients, 11 Stokes-Einstein equation, 135 Stopped flow, 179 Stmetured water, 395 Structure-reactivity relationships, 311 Sublimation energy, 403 Substituent, 313 Substituent constant, 323 alkyl group, 341 electrophilic, 322 Hammett, 316 inductive, 325, 338 normal. 324 polar, 339 primary, 324 resonance, 325... 

The particle trajectories from the MD simulations can be used to determine the self-diffusion coefficient using the Einstein equation ... 

The relation between the means-square displacement (MSD) to the diffusion constant is valid in the physical limit of the observation time being larger compared with the mean-collision time (Haile 1991). We used Equation 10.6 to compute diffusivities through MD simulations. We computed diffusivity, D, along each direction using the projected Einstein equations ... 

Consider the simplest dynamic property one can compute, the self-diffusivity, Dj. The standard approach for computing Ds is to conduct an equilibrium MD simulation and accumulate the mean-square displacement as a function of time. The self-diffusivity is then computed using the Einstein equation ... 

The molar conductivity may be determined, in addition to its direct determination from the specific conductivity k and the molar volume T (asA = kV), also from the ionic self-diffusion coefficients, obtained fi om NMR measurements as mentioned above, according to the Nemst-Einstein equation A = F (D+ +D-)/RT, with results in fair agreement with the directly determined values according to Tokuda et al. [80, 352, 374]. Ionic self-diffusion coefficients can be estimated from molecular d5mamics simulations and can then be transformed into the molar conductivity, as applied in [365, 387, 388], showing fair agreement with experimental values.. 

Xi, y>i and Z are direct results from any MD simulation, and D is thus proportional to the slope of the mean-square displacement (MSD) function s time-development (see Fig. 8.6). From the diffusion coefficient, the molar conductivity (A°m) can be calculated by the Nemst-Einstein equation ... 

The experimental and simulation results presented here indicate that the system viscosity has an important effect on the overall rate of the photosensitization of diary liodonium salts by anthracene. These studies reveal that as the viscosity of the solvent is increased from 1 to 1000 cP, the overall rate of the photosensitization reaction decreases by an order of magnitude. This decrease in reaction rate is qualitatively explained using the Smoluchowski-Stokes-Einstein model for the rate constants of the bimolecular, diffusion-controlled elementary reactions in the numerical solution of the kinetic photophysical equations. A more quantitative fit between the experimental data and the simulation results was obtained by scaling the bimolecular rate constants by rj"07 rather than the rf1 as suggested by the Smoluchowski-Stokes-Einstein analysis. These simulation results provide a semi-empirical correlation which may be used to estimate the effective photosensitization rate constant for viscosities ranging from 1 to 1000 cP. 

P 61] The numerical simulations were based on the solution of the incompressible Navier-Stokes equation and a convection-diffusion equation for a concentration field by means of the finite-volume method [152], The Einstein convention of summation over repeated indices was used. For pressure-velocity coupling, the SIMPLEC algorithm and for discretization of the species concentration equation the QUICK differencing scheme were applied. Hybrid and the central differencing schemes referred to velocities and pressure, respectively (commercial flow solvers CFX4 and CFX5). 

Clearly, a general theory able to naturally include other solvent modes in order to simulate a dissipative solute dynamics is still lacking. Our aim is not so ambitious, and we believe that an effective working theory, based on a self-consistent set of hypotheses of microscopic nature is still far off. Nevertheless, a mesoscopic approach in which one is not limited to the one-body model, can be very fruitful in providing a fairly accurate description of the experimental data, provided that a clever choice of the reduced set of coordinates is made, and careful analytical and computational treatments of the improved model are attained. In this paper, it is our purpose to consider a description of rotational relaxation in the formal context of a many-body Fokker-Planck-Kramers equation (MFPKE). We shall devote Section I to the analysis of the formal properties of multivariate FPK operators, with particular emphasis on systematic procedures to eliminate the non-essential parts of the collective modes in order to obtain manageable models. Detailed computation of correlation functions is reserved for Section II. A preliminary account of our approach has recently been presented in two Letters which address the specific questions of (1) the Hubbard-Einstein relation in a mesoscopic context [39] and (2) bifurcations in the rotational relaxation of viscous liquids [40]. 

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