Average force

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

The thennodynamic properties are calculated from the ion-ion pair correlation fimctions by generalizing the expressions derived earlier for one-component systems to multicomponent ionic mixtures. For ionic solutions it is also necessary to note that the interionic potentials are solvent averaged ionic potentials of average force ... 

The time-averaged force, equation (Cl.4.3), consists of two tenns tire first tenn is proportional to tire gradient of tire electric field amplitude tire second tenn is proportional to tire gradient of tire phase. Substituting equation (Cl.4.4) and equation (Cl.4.5) into equation (Cl.4.3), we have for tire two tenns. 

If there are no reactions, the conservation of the total quantity of each species dictates that the time dependence of is given by minus the divergence of the flux ps vs), where (vs) is the drift velocity of the species s. The latter is proportional to the average force acting locally on species s, which is the thermodynamic force, equal to minus the gradient of the thermodynamic potential. In the local coupling approximation the mobility appears as a proportionality constant M. For spontaneous processes near equilibrium it is important that a noise term T] t) is retained [146]. Thus dynamic equations of the form... 

Stiff Spring For a stiff spring, satisfying K (fiUjdx, under the overdamped condition assumed in (3) the average force measured by the spring can be expressed as... 

Induction heating is thought to cause vigorous convection because of the spatially varying average force field imposed along the melt surface. 

The function W(X) is called the potential of mean force (PMF). The fundamental concept of the PMF was first introduced by Kirkwood [4] to describe the average structure of liquids. It is a simple matter to show that the gradient of W(X) in Cartesian coordinates is related to the average force. 

As shown by Eq. (6) the PME is the reversible work done by the average force. It is possible to express relative values of the PME between different solute configurations X i and X2 using Eq. (6) and the reversible work theorem [4] ... 

A chemical will be a solvent for another material if the molecules of the two materials are compatible, i.e. they can co-exist on the molecular scale and there is no tendency to separate. This statement does not indicate the speed at which solution may take place since this will depend on additional considerations such as the molecular size of the potential solvent and the temperature. Molecules of two different species will be able to co-exist if the force of attraction between different molecules is not less than the forces of attraction between two like molecules of either species. If the average force of attraction between dissimilar molecules A and B is and that between similar molecules of type B Fbb and between similar molecules of type A F a then for compatibility Fab - bb and AB - P/KA- This is shown schematically in Figure 5.5 (a). 

Maxwell found that he could represent the distribution of velocities statistically by a function, known as the Maxwellian distribution. The collisions of the molecules with their container gives rise to the pressure of the gas. By considering the average force exerted by the molecular collisions on the wall, Boltzmann was able to show that the average kinetic energy of the molecules was... 

A 1,000 kg (metric ton) car accelerating from 0—30 m/s (68 mph) in 6 seconds, thus requires an average force of 5,000 neivtoiis. This is a force approximatel y half the tveight of the car. It is likel y that the car would burn rubber as the drive wheels spun on the road during this acceleration. To accelerate to the same speed in 12 seconds would require a force of onl y 2,500 newtons. 

Ea = Arrhenius activation energy Es = excess stress energy AEr = potential barrier for bond rotation Eel = molecular elastic energy F = mean force potential f = average force on the chain fb = bond breaking force H0 = Hookean spring constant kB = Boltzmann constant... 

The average force f(r) in the chain when the ends are held a distance r apart could then be obtained from Eq. (10) providing the appropriate configuration distribution function p(r) is known. In the limit of a small extension ratio, p(r) is approximately proportional to peq(r) ... 

Likewise the even part of the probability can be tested by taking the steady-state average force. Using similar linearization arguments to the earlier ones, it may be shown that... 

Flere, A (T) is a function of temperature alone. Equation (6) represents the elastic free energy of a Gaussian chain with ends fixed at a separation of r. The average force required to keep the two ends at this separation is obtained from the thermodynamic expression [28]... 

Darve, E. Pohorille, A., Calculating free energies using average force, J. Chem. Phys. 2001,775,9169-9183... 

The second approach, ABF, consists of estimating the average force at different and removing it from the system. This leads to a diffusive-like motion along and a much better convergence of the calculation. We will describe ABF in Sect. 4.6. 

We previously showed that the free energy can be computed using (4.33). This is done by binning the force and computing the average force along the interval. 

Binning is used to calculate statistics for the average force. /... 

In some cases, one may not be able to collect enough trajectories for a histogram analysis. Instead, one can use moments [15]. In particular, the derivative of the free energy is approximately given by the average force (weighted by the Boltzmann factor of the accumulated work)... 

Figure 10.3 Carbide hardnesses vs. characteristic vibrational energy densities derived from average force constants (entropic specific heat). After Grimvall and Theissen (1986). The crystal structures are of the NaCi type. The hardness data are fromTeter (1998).

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