Forces nonlocality

Leikin S, Kornyshev AA. Theory of hydration forces. Nonlocal electrostatic interaction of neutral surfaces. J. Chem. Phys. 1990 92 6890-6898. 

The function f contains all the constitutive (material-dependent) properties and describes how the internal forces depend on the deformation. A standard assumption is that for a given material, f = 0 for all u when jr -rj > S for some > 0 called the horizon. This characteristic distance represents a length scale for the nonlocality of force. Nonlocal models allow to incorporate directly the description of atomistic effects and long-range interactions into a continuum theory. In contrast, classical elasticity theory does not have a length scale - the horizon consists of a point because of the assumption of a contact force. 

In LN, the bonded interactions are treated by the approximate linearization, and the local nonbonded interactions, as well as the nonlocal interactions, are treated by constant extrapolation over longer intervals Atm and At, respectively). We define the integers fci,fc2 > 1 by their relation to the different timesteps as Atm — At and At = 2 Atm- This extrapolation as used in LN contrasts the modern impulse MTS methods which only add the contribution of the slow forces at the time of their evaluation. The impulse treatment makes the methods symplectic, but limits the outermost timestep due to resonance (see figures comparing LN to impulse-MTS behavior as the outer timestep is increased in [88]). In fact, the early versions of MTS methods for MD relied on extrapolation and were abandoned because of a notable energy drift. This drift is avoided by the phenomenological, stochastic terms in LN. 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

In our approach to membrane breakdown we have only taken preliminary steps. Among the phenomena still to be understood is the combined effect of electrical and mechanical stress. From the undulational point of view it is not clear how mechanical tension, which suppresses the undulations, can enhance the approach to membrane instability. Notice that pore formation models, where the release of mechanical and electrical energy is considered a driving force for the transition, provide a natural explanation for these effects [70]. The linear approach requires some modification to describe such phenomena. One suggestion is that membrane moduli should depend on both electrical and mechanical stress, which would cause an additional mode softening [111]. We hope that combining this effect with nonlocality will be illuminating. 

This formulation results very insightful according to Equation 8.30, particles move under the action of an effective force — We , i.e., the nonlocal action of the quantum potential here is seen as the effect of a (nonlocal) quantum force. From a computational viewpoint, these formulation results are very interesting in connection to quantum hydrodynamics [21,27]. Thus, Equations 8.27 can be reexpressed in terms of a continuity equation and a generalized Euler equation. As happens with classical fluids, here also two important concepts that come into play the quantum pressure and the quantum vortices [28], which occur at nodal regions where the velocity field is rotational. 

The (nonlocal) polarizabilities are important DFT reactivity descriptors. But, how are polarizabilities related to chemistry As stated above, an essential ingredient of the free energy surface is the potential energy surface and, in particular, its gradients. In a classical description of the nuclei, they determine the many possible atomic trajectories. Thanks to Feynman, one knows a very elegant and exact formulation of the force between the atoms namely [22,23]... 

For larger displacement UA-, the variation of i(r, r ) relative to UK can be computed by using the nonlinear polarizability kernels defined below [26] (see Section 24.4). Forces and nonlocal polarizabilities are thus intimately related. 

The symmetry changes of the vortex lattice in borocarbide superconductors affect their pinning properties as was shown for YNi2B2C (Silhanek et al. 2001). For the field orientation // c, the reorientation transition of the vortex lattice mentioned above was found to be associated with a significant kink in the volume pinning force Fp, whereas in the basal plane (for H c) the signature of nonlocal effects is a fourfold periodicity of Fp. 

In this book we summarize the state of the art in the study of peculiarities of chemical processes in dense condensed media its aim is to present the unique formalism for a description of self-organization phenomena in spatially extended systems whose structure elements are coupled via both matter diffusion and nonlocal interactions (chemical reactions and/or Coulomb and elastic forces). It will be shown that these systems could be described in terms of nonlinear partial differential equations and therefore are complex enough for the manifestation of wave processes. Their spatial and temporal characteristics could either depend on the initial conditions or be independent on the initial as well as the boundary conditions (the so-called autowave processes). 

The expectation value of the property A at the space-time point (r, t) depends in general on the perturbing force F at all earlier times t — t and at all other points r in the system. This dependence springs from the fact that it takes the system a certain time to respond to the perturbation that is, there can be a time lag between the imposition of the perturbation and the response of the system. The spatial dependence arises from the fact that if a force is applied at one point of the system it will induce certain properties at this point which will perturb other parts of the system. For example, when a molecule is excited by a weak field its dipole moment may change, thereby changing the electrical polarization at other points in the system. Another simple example of these nonlocal changes is that of a neutron which when introduced into a system produces a density fluctuation. This density fluctuation propagates to other points in the medium in the form of sound waves. 

It was recently shown via molecular dynamics simulations14 that, in the close vicinity of a surface, water molecules exhibit an anomalous dielectric response, in which the local polarization is not proportional to the local electric field. The recent findings are also in agreement with earlier molecular dynamics simulations, which showed that the polarization of water oscillates in the vicinity of a dipolar surface,11,14 leading therefore to a nonmonotonic hydration force.15 Previous models for oscillatory hydration forces, based either on volume-excluded effects,18,19 or on a nonlocal dielectric constant,f4 predicted many oscillations with a periodicity of 2 A, which is inconsistent with these molecular dynamics simulations,11,18,14 in which the polarization exhibits only a few oscillations in the vicinity of the surface, with a larger periodicity. 

To calculate the quantum potential it is first necessary to find R, which by equation 9 is clearly a function of all particle coordinates, because of nonlocal interaction in terms of the molecular quantum entanglement inferred before. Even when all particle coordinates and velocities (v = 0) are identical in the two states, the state S must therefore still be at a lower energy. The difference can only be attributed to the quantum potential which differs for the two states. The force that holds the molecule together is therefore seen to be a special case of non-local interaction within the molecule. 

---