Time scale equation

The time-scaling method used and why it was selected (include the rationale for the value of n in the time-scaling equation). 

The characteristics of the groundwater and the porous medium that appear in Darcy s equation vary independently on a geological time scale. Equation 1.9 should be used when dealing with large temporal and spatial scales. 

The nickel(I) complex [CpNiL2] (L = PR3), disproportionates rapidly on the cyclic voltammetry time scale [equation (23)]/ The [CpNiL2] complexes are suggested to be likely intermediates in the reaction of nickelocene with phosphorous ligands (L) to give [NiL4] and CioHio. 

The foregoing equations apply to any arbitrary reference temperature To. Of particular interest, however, is the use of Tg as the reference temperature, with the reservation of course that the latter depends somewhat on the experimental time scale. Equation 38 is unchanged since it is valid for any reference temperature, as may be shown by combining equations 24 and 25. Equation 37 becomes... 

Macromixing, produced by turbulence, involves different space and time scales. We have already discussed in Chapter 8 the large scale of turbulence, characterized by the integral scale It and the turbulent velocity Wnns. With that is associated the time scale (equation [8.12]) which is referred to as the eddy turn-over time of turbulence as it gives an order of magnitude of the revolution time of an eddy at the large scale of turbulence ... 

The increase in effective diffusivity, due to rotation will lead to reduction in Taylor time scale, (equation (4.63)). 

Due to the conservation law, the diffiision field 5 j/ relaxes in a time much shorter than tlie time taken by significant interface motion. If the domain size is R(x), the difhision field relaxes over a time scale R Flowever a typical interface velocity is shown below to be R. Thus in time Tq, interfaces move a distanc of about one, much smaller compared to R. This implies that the difhision field 6vj is essentially always in equilibrium with tlie interfaces and, thus, obeys Laplace s equation... 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

With M = He, experimeuts were carried out between 255 K aud 273 K with a few millibar NO2 at total pressures between 300 mbar aud 200 bar. Temperature jumps on the order of 1 K were effected by pulsed irradiation (< 1 pS) with a CO2 laser at 9.2- 9.6pm aud with SiF or perfluorocyclobutaue as primary IR absorbers (< 1 mbar). Under these conditions, the dissociation of N2O4 occurs within the irradiated volume on a time scale of a few hundred microseconds. NO2 aud N2O4 were monitored simultaneously by recording the time-dependent UV absorption signal at 420 run aud 253 run, respectively. The recombination rate constant can be obtained from the effective first-order relaxation time, A derivation analogous to (equation (B2.5.9). equation (B2.5.10). equation (B2.5.11) and equation (B2.5.12)) yield... 

The method of molecular dynamics (MD), described earlier in this book, is a powerful approach for simulating the dynamics and predicting the rates of chemical reactions. In the MD approach most commonly used, the potential of interaction is specified between atoms participating in the reaction, and the time evolution of their positions is obtained by solving Hamilton s equations for the classical motions of the nuclei. Because MD simulations of etching reactions must include a significant number of atoms from the substrate as well as the gaseous etchant species, the calculations become computationally intensive, and the time scale of the simulation is limited to the... 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

Molecular dynamics simulations ([McCammon and Harvey 1987]) propagate an atomistic system by iteratively solving Newton s equation of motion for each atomic particle. Due to computational constraints, simulations can only be extended to a typical time scale of 1 ns currently, and conformational transitions such as protein domains movements are unlikely to be observed. 

Extending time scales of Molecular Dynamics simulations is therefore one of the prime challenges of computational biophysics and attracted considerable attention [2-5]. Most efforts focus on improving algorithms for solving the initial value differential equations, which are in many cases, the Newton s equations of motion. 

The limit equation governing limj -,o qc can be motivated by referring to the quantum adiabatic theorem which originates from work of Born and FOCK [4, 20] The classical position g influences the Hamiltonian very slowly compared to the time scale of oscillations of in fact, infinitely slowly in the limit e — 0. Thus, in analogy to the quantum adiabatic theorem, one would expect that the population of the energy levels remain invariant during the evolution ... 

When the friction coefficient is set to zero, HyperChem performs regular molecular dynamics, and one should use a time step that is appropriate for a molecular dynamics run. With larger values of the friction coefficient, larger time steps can be used. This is because the solution to the Langevin equation in effect separates the motions of the atoms into two time scales the short-time (fast) motions, like bond stretches, which are approximated, and longtime (slow) motions, such as torsional motions, which are accurately evaluated. As one increases the friction coefficient, the short-time motions become more approximate, and thus it is less important to have a small timestep. 

The generalized transport equation, equation 17, can be dissected into terms describing bulk flow (term 2), turbulent diffusion (term 3) and other processes, eg, sources or chemical reactions (term 4), each having an impact on the time evolution of the transported property. In many systems, such as urban smog, the processes have very different time scales and can be viewed as being relatively independent over a short time period, allowing the equation to be "spht" into separate operators. This greatly shortens solution times (74). The solution sequence is... 

In a simulation it is not convenient to work with fluctuating time intervals. The real-variable formulation is therefore recommended. Hoover [26] showed that the equations derived by Nose can be further simplified. He derived a slightly different set of equations that dispense with the time-scaling parameter s. To simplify the equations, we can introduce the thermodynamic friction coefficient, = pJQ. The equations of motion then become... 

A variety of techniques have been introduced to increase the time step in molecular dynamics simulations in an attempt to surmount the strict time step limits in MD simulations so that long time scale simulations can be routinely undertaken. One such technique is to solve the equations of motion in the internal degree of freedom, so that bond stretching and angle bending can be treated as rigid. This technique is discussed in Chapter 6 of this book. Herein, a brief overview is presented of two approaches, constrained dynamics and multiple time step dynamics. 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

In deciding whether to write an elementary reaction as either a reversible or an irreversible reaction, we take the practical view that if the reverse reaction is negligibly slow on the exp>erimental time scale, the reaction is essentially irreversible. Consider the alkaline hydrolysis of an ester, for which the rate equation is... 

Substituting this into the second equation of (6-185), and making some intermediate calculations (and change of time scale) (Reference 6, p. 457) one obtains ultimately the differential equation ... 

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