Gradient of a potential

From classical physics, the negative gradient of a potential energy is a force - AVTr) = F(r). It follows that a force can also be associated with a process described by a potential energy VTRC) ... 

Consider, then, a fluid containing ions that are nonuniformly distributed, producing a position-dependent electric field E. Since an electric field is conservative, it is given by the gradient of a potential, E = — VtJ/. Negative ions tend to collect at locations where this potential is positive relative to some datum, and positive ions collect where it is negative. At equilibrium, the number density n, of ionic species i at each location is given by the Boltzmann distribution ... 

In classical mechanics, we are accustomed to relating the gradient of a potential to the direction and magnitude of the force P producing the motion of a point that is,... 

The same result is obtained from the theory of the Brownian motion taking into account the external force field. When the external force field F can be derived from the gradient of a potential... 

In Eq. 12, FB is the boundary force at r0, F(r0 — rT) is the force of interaction between a particle at rT in RR and a particle at r0 in RZ, and dTTpTg( o rr) is the probability of the pair (0, T) having a separation r<> — tT. The boundary force may be written as the gradient of a potential, the boundary potential. The boundary potential for the oxygen atom of ST2 water12 in an 11 A reaction zone is plotted in Fig. 9. In the calculation of this potential only the van der Waals part of the ST2-ST2 interaction was included in Eq. 12. A methodology that consistently incorporates electrostatic forces into the boundary potential is under development.110 In its present simplified form, the model has proven successful in the simulation of localized regions of pure... 

The type of forces depends on the MD model exploited. A simulation is realistic — that is, it mimics the behavior of the real system — only to the extent that interatomic forces are similar to those that real atoms (or, more exactly, nuclei) would experience when arranged in the same configuration. Forces are usually obtained as the gradient of a potential energy function d>(rj), depending on the positions of the particles. The realism of the simulation therefore depends on the ability of the potential chosen to reproduce the behavior of the material under the conditions at which the simulation is mn. The forces can be defined as follows ... 

The force acting on the ith molecule can be written as the gradient of a potential as... 

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