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Differential force law

Within a Lagrangian formulation of the mechanics of a field as used here one may define an energy-momentum tensor whose components summarize the principal properties of the field (Morse and Feshbach 1953 Landau and Lifshitz 1975). We show that the divergence equations satisfied by the spatial components of this tensor for the Schrodinger field yield the differential form of the atomic force law, eqn (8.175) (Bader 1980). [Pg.396]

Equation (8.178) has been previously obtained by Pauli (1958) and by Epstein (1975), who termed it the differential force law. The integration of this expression over the coordinates of all electrons but one by the usual recipe yields (Bader 1980) an equation governing a force density [Pg.397]

Equation (8.179) is the differential form of the integrated atomic force law given in cqn (8.175), [Pg.397]

As previously discussed in Chapter 6 in eqn (6.17) and following for a stationary state, the force density F(r, t) is the total instantaneous force exerted on the electron at Tj by the nuclei and the remaining electrons [Pg.397]

The variational derivation of the integral atomic force law, eqn (8.175), is applicable only to a region of space bounded by a zero-flux surface in Vp(r), i.e. to an open system whose Lagrangian integral vanishes at the point of variation. Thus the variational derivation of the atomic force. [Pg.397]


The general time-dependent virial theorem for an atom in a molecule is derived from the atomic variational principle. We shall find a close connection between the expressions so obtained for the virial and those derived in the previous section for the force. In particular, the differential force law leads directly to a corresponding local expression for the virial theorem. [Pg.398]

The average of the atomic virial V,(12), eqn (8.192), as defined by the atomic variational principle, is the virial of the quantum mechanical force density as defined in the differential force law, averaged over the atomic volume. By taking the virial of F(r, t) in eqn (8.179), one obtains... [Pg.401]

Epstein ST (1975) Coordinate invariance, the differential force law, and the divergtaice of the stress-enCTgy tensor. J Chem Phys 63 3573-3574... [Pg.122]

The basic principles are described in many textbooks [24, 26]. They are thus only sketchily presented here. In a conventional classical molecular dynamics calculation, a system of particles is placed within a cell of fixed volume, most frequently cubic in size. A set of velocities is also assigned, usually drawn from a Maxwell-Boltzmann distribution appropriate to the temperature of interest and selected in a way so as to make the net linear momentum zero. The subsequent trajectories of the particles are then calculated using the Newton equations of motion. Employing the finite difference method, this set of differential equations is transformed into a set of algebraic equations, which are solved by computer. The particles are assumed to interact through some prescribed force law. The dispersion, dipole-dipole, and polarization forces are typically included whenever possible, they are taken from the literature. [Pg.271]

For charged particles on a lattice interacting through screened Coulomb forces, differentiation of the force law with respect to interparticle distance gives... [Pg.209]

Maxwell s equations, which were first presented in 1864 and published in 1865 [40], completely describe the classical behavior of electric and magnetic fields and — supported by the Lorentz force law — their interaction with charged particles and currents. In Gaussian units their differential form is given by... [Pg.36]

This kinetic description can be extended to the case where the tagged and fluid particles are assumed to interact via the same force law that holds for fluid particles. The mass m of the tagged particle is assumed to be large compared to that of a fluid particle mj ). Therefore, the mass ratio e = nijr/m is a small parameter in terms of which the Boltzmann- Lorentz collision operator may be expanded. If this expansion is carried out to the leading order, the Boltzmann-Lorentz operator reduces to a differential operator yielding a kinetic Fokker Planck equation for the tagged particle distribution F... [Pg.107]

Israelachvili (2011) describes how this definition for a force law arose historically. Equation 10.2 arises from Equation 10.1 by using Leibnitz s rule of differentiation of integrals. Negative F and V imply attraction while positive F and V mean repulsion. [Pg.212]

This is a general rule we differentiate U by the coordinates of the particle in question in order to recover the force on that particle, from the expression for the mutual potential energy. Knowing the force F, we can use Newton s second law... [Pg.62]

In Chapter 2, I gave you a brief introduction to molecular dynamics. The idea is quite simple we study the time evolution of our system according to classical mechanics. To do this, we calculate the force on each particle (by differentiating the potential) and then numerically solve Newton s second law... [Pg.252]

The Navier-Stokes equations have a complex form due to the necessity of treating many of the terms as vector quantities. To understand these equations, however, one need only recognize that they are not mass balances but an elaboration of Newton s second law of motion for a flowing fluid. Recall that Newton s second law states that the vector sum of all the forces acting on an object ( F) will be equal to the product of the object s mass (m) and its acceleration (a), or XF = ma. Now consider the first of the three Navier-Stokes equations listed above, Eq. (10). The object in this case is a differential fluid element, that is, a small cube of fluid with volume dx dy dz and mass p(dx dy dz). The left-hand side of the equation is essentially the product of mass and acceleration for this fluid element (ma), while the right-hand side represents the sum of the forces... [Pg.28]

Navier-Stokes equations A series of differential equations derived from Newton s second law of motion (XF = raa) that describe the relationship between fluid velocity and applied forces in a moving fluid. See Eqs. (10)—(12). [Pg.37]

Therefore, any result that follows from considerations of the form of Fick s second law applies to evolution of heat as well as concentration. However, the thermal and mass diffusion equations differ physically. The mass diffusion equation, dc/dt = V DVc, is a partial-differential equation for the density of an extensive quantity, and in the thermal case, dT/dt = V kVT is a partial-differential equation for an intensive quantity. The difference arises because for mass diffusion, the driving force is converted from a gradient in a potential V/u to a gradient in concentration Vc, which is easier to measure. For thermal diffusion, the time-dependent temperature arises because the enthalpy density is inferred from a temperature measurement. [Pg.79]

A plot of 2 vs. -t2 for symmetrical systems (i.e., ii vo) is shown in Fig. 1 for a series of values of the heat lerm, It shows how the partial vapor pressure of a component of a binary solution deviates positively from Raoult s law more and mure as the components become more unlike in their molecular attractive forces. Second, the place of T in die equation shows that tlic deviation is less die higher the temperature. Third, when the heat term becomes sufficiently large, there are three values of U2 for the same value of ay. This is like the three roots of the van der Waals equation, and corresponds to two liquid phases in equilibrium with each other. The criterion is diat at the critical point the first and second partial differentials of a-i and a are all zero. [Pg.1522]

The GMC control response can be designed via the tuning parameters K and K2 based on the tuning curve given by [24], It should be noted that the GMC approach is a special case of the global input output linearizing control technique in which a transformed control action is chosen properly with the external PI controller. The use of Eq. (16) forces y toward its set point, ysp, with zero offset. If Eq. (15) is differentiated, and the Eq. (16) is substituted into the resulting equation, the GMC control law is... [Pg.107]

This change in the wall s momentum in a very short time (the duration of the collision AL0ii) implies the wall exerts a large force (F = Ap/Atco ). If it is not obvious to you that F = Ap/ At, review Section 3.1 F = dp/dt is actually Newton s Second Law. We use Ap and At instead of the differentials dp and dt because the collisions, while brief, are not infinitesimal. If gas molecules and container walls really were incompressible, they would be in contact with the wall for an infinitesimal time, and the force would have to be infinite. [Pg.154]


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Force law

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