The Box

The Feldberg approach to digital simulation [1] uses a somewhat different method of discretisation, and the method is alive and well [2]. It begins with Pick s first diffusion equation, using fluxes between boxes or finite volumes, rather than concentrations at points in the discretisation process (see below). 

Strutwolf, Digital Simulation in Electrochemistry, Monographs 

This is the formula in [4], seen again in Rudolph s chapter in [5]. Also, the box 

This expanding box strategy is mathematically equivalent to the transformation from X into Y as described for point positions in Chap. 7, Eq.(7.3), as is shown in Appendix C. Its implementation in the discretisation process is however different. 

The D coefficients can be worked out from (9.8), substituting for the x terms using (9.3). The denominators in the two terms in brackets on the right-hand side of (9.8) can be simplified. As an example, consider the first of these. It is simplified in the following manner. From (9.3), 

Dieter Britz Digital Simulation in Electrochemistry, Lect. Notes Phys. 666, 145—187 (2005) www.springerlink.ccxD c Springer-Verlag Berlin Heidelberg 2005 

There are several commercial packages that realise the above strategy for molecularly realistic systems. It is useful to discuss some of the limitations. Ideally, one would like to do simulations on macroscopic systems. However, it is impossible to use a computer to deal with numbers of degrees of freedom on the order of /Vav. In lipid systems, where the computations of all the interactions in the system are expensive, a typical system can contain of the order of tens of thousands of particles. Recently, massive systems with up to a million particles have been considered [33], Even for these large simulations, this still means that the system size is limited to the order of 10 nm. Because of this small size, one refers to this volume as a box, although the system boundaries are typically not box-like. Usually the box has periodic boundary conditions. This implies that molecules that move out of the box on one side will enter the box on the opposite side. In such a way, finite size effects are minimised. In sophisticated simulations, i.e. (N, p, y, Tj-ensembles, there are rules defined which allow the box size and shape to vary in such a way that the intensive parameters (p, y) can assume a preset value. 

The finite size of the box has several important consequences. One of them is that the area of the membrane piece is only of the order of 100 nm2. It is expected that the membrane is, on this length scale, roughly flat, i.e. the area is small as compared with the persistence length for the bilayer. Interestingly, however, in recent simulations the first signs of fluctuations away from the flat bilayer structure (undulations) have reportedly been found by MD simulations [33], 

The periodic boundary conditions applied in the system have the consequence that one bilayer leaflet can interact in the normal direction with the other leaflet, not only through the contact region in the core of the bilayer, but also through the water phase. To minimise artifacts, one should systematically increase the size of the water phase. However, this is expensive, especially if the main interest is in the behaviour of the lipids. Another solution is to cut off the 

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Another design option that can be considered if a column will not fit is use of an intermediate reboiler or condenser. An intermediate condenser is illustrated in Fig. 14.5. The shape of the box is now altered because the intermediate condenser changes the heat flow through the column. The particular design shown in Fig. 14.5 would require that at least part of the heat rejected from the intermediate condenser be passed to the process. An analogous approach can be used to evaluate the possibilities for use of intermediate reboilers. Flower and Jackson," Kayihan, and Dhole and Linnhofl have presented procedures for the location of intermediate reboilers and condensers. 

From an overall economic viewpoint, any investment proposal may be considered as an activity which initially absorbs funds and later generates money. The funds may be raised from loan capital or from shareholders capital, and the net (after tax and costs) money generated may be used to repay interest on loans and loan capital, with the balance being due to the shareholders. The shareholders profit can either be paid out as dividends, or reinvested in the company to fund the existing venture or new ventures. The following diagram indicates the overall flow of funds for a proposed project. The detailed cash movements are contained within the box labelled the project . 

One procedure makes use of a box on whose silk screen bottom powdered desiccant has been placed, usually lithium chloride. The box is positioned 1-2 mm above the surface, and the rate of gain in weight is measured for the film-free and the film-covered surface. The rate of water uptake is reported as u = m/fA, or in g/sec cm. This is taken to be proportional to - Cd)/R, where Ch, and Cd are the concentrations of water vapor in equilibrium with water and with the desiccant, respectively, and R is the diffusional resistance across the gap between the surface and the screen. Qualitatively, R can be regarded as actually being the sum of a series of resistances corresponding to the various diffusion gradients present ... 

The second model is a quantum mechanical one where free electrons are contained in a box whose sides correspond to the surfaces of the metal. The wave functions for the standing waves inside the box yield permissible states essentially independent of the lattice type. The kinetic energy corresponding to the rejected states leads to the surface energy in fair agreement with experimental estimates [86, 87],... 

Figure Bl.16.5. An example of the CIDNP net effect for a radical pair with one hyperfme interaction. Initial conditions g > g2, negative and the RP is initially singlet. Polarized nuclear spin states and schematic NMR spectra are shown for the recombination and scavenging products in the boxes.

This gives the total energy, which is also the kinetic energy in this case because the potential energy is zero within the box , m tenns of the electron density p x,y,z) = (NIL ). It therefore may be plausible to express kinetic energies in tenns of electron densities p(r), but it is by no means clear how to do so for real atoms and molecules with electron-nuclear and electron-electron interactions operative. 

Here we have introduced scaled coordinates where L is the box length (assumed cubic). 

For a creation attempt, a position is chosen unifomily at random within the box, and an attempt made to create a new particle there. The probability ratio for creation is ... 

Thus, this multi-level process produces a finer and finer covering of rsE -r) B)-Up to now, the parameters 6Ek are adapted to the size of the boxes according to some heuristics. Recall that an approximation of the energy surface Eo(E) would only be possible in the limit r —+ 0 which implies 5E t) —+ 0. 

The invariant measure corresponding to Aj = 1 has already been shown in Fig. 6. Next, we discuss the information provided by the eigenmeasure U2 corresponding to A2. The box coverings in the two parts of Fig. 7 approximate two sets Bi and B2, where the discrete density of 1 2 is positive resp. negative. We observe, that for 7 > 4.5 in (15) the energy E = 4.5 of the system would not be sufficient to move from Bi to B2 or vice versa. That is, in this case Bi and B2 would be invariant sets. Thus, we are exactly in the situation illustrated in our Gedankenexperiment in Section 3.1. 

The third eigenmeasure 1 3 corresponding to A3 provides information about three additional almost invariant sets on the left hand side in Fig. 8 we have the set corresponding to the oscillation C D, whereas on the fight hand side the two almost invariant sets around the equilibria A and B are identified. Again the boxes shown in the two parts of Fig. 8 approximate two sets where the diserete density of 1/3 is positive resp. negative. In this case we can use Proposition 2 and the fact that A and B are symmetrically lelated to conclude that for all these almost invariant sets 5 > A3 = 0.9891. 

To separate the non-bonded forces into near, medium, and far zones, pair distance separations are used for the van der Waals forces, and box separations are used for the electrostatic forces in the Fast Multipole Method,[24] since the box separation is a more convenient breakup in the Fast Multipole Method (FMM). Using these subdivisions of the force, the propagator can be factorized according to the different intrinsic time scales of the various components of the force. This approach can be used for other complex systems involving long range forces. 

Fig. 1. Periodic boundary conditions protect the inner simulation cell from disturbing effects of having all its particles close to the surface. With PBCs in force, as a particle moves out of the box on one side, one of its images will move back into the box on the opposite side.

The problems already mentioned at the solvent/vacuum boundary, which always exists regardless of the size of the box of water molecules, led to the definition of so-called periodic boundaries. They can be compared with the unit cell definition of a crystalline system. The unit cell also forms an "endless system without boundaries" when repeated in the three directions of space. Unfortunately, when simulating hquids the situation is not as simple as for a regular crystal, because molecules can diffuse and are in principle able to leave the unit cell. 

However, it is common practice to sample an isothermal isobaric ensemble NPT, constant pressure and constant temperature), which normally reflects standard laboratory conditions well. Similarly to temperature control, the system is coupled to an external bath with the desired target pressure Pq. By rescaling the dimensions of the periodic box and the atomic coordinates by the factor // at each integration step At according to Eq. (46), the volume of the box and the forces of the solvent molecules acting on the box walls are adjusted. 

Also in this case, Tp corresponds to a relaxation time which determines the coupling of the modulated variable to the external bath. The pressure scaling can be applied isotropically, whidi means that the factor is the same in all three spatial directions. More realistic is an anisotropic pressure scaling, because the box dimensions also change independently during the course of the simulation. 

Sin ce the Molecu lar Dynamics Results window con lain ing plots is a true window, an image of it alone can be captured into the clipboard or a file using Top-level in the File/Preferences/Setiip Image dialog box. Th is captured image, in addition to shtnving the molecular dynamics plots, shows the Restart and Done buttons, etc. If you on ly want the plots, you can erase the details of the box with a paint program, such as Microsoft Windows Pain thru sh which comes with Microsoft Windows. 

For some simulations it is inappropriate to use standard periodic boundary conditions in all directions. For example, when studying the adsorption of molecules onto a surface, it is clearly inappropriate to use the usual periodic boundary conditions for motion perpendicular to the surface. Rather, the surface is modelled as a true boundary, for example by e, plicitly including the atoms in the surface. The opposite side of the box must still be treated when a molecule strays out of the top side of the box it is reflected back into the simulation cell, as indicated in Figure 6.6. Usual periodic boundary conditions apply to motion parallel to the surface. 

The most commonly used method for applying constraints, particularly in molecula dynamics, is the SHAKE procedure of Ryckaert, Ciccotti and Berendsen [Ryckaert et a 1977]. In constraint dynamics the equations of motion are solved while simultaneous satisfying the imposed constraints. Constrained systems have been much studied in classics mechanics we shall illustrate the general principles using a simple system comprising a bo sliding down a frictionless slope in two dimensions (Figure 7.8). The box is constrained t remain on the slope and so the box s x and y coordinates must always satisfy the equatio of the slope (which we shall write as y = + c). If the slope were not present then the bo... 

CE uses holonomic constraints. In a constrained system the coordinates of the particles 5t independent and the equations of motion in each of the coordinate directions are cted. A second difficulty is that the magnitude of the constraint forces is unknown, in the case of the box on the slope, the gravitational force acting on the box is in the ction whereas the motion is down the slope. The motion is thus not in the same direc-s the gravitational force. As such, the total force on the box can be considered to arise wo sources one due to gravity and the other a constraint force that is perpendicular to otion of the box (Figure 7.8). As there is no motion perpendicular to the surface of the the constraint force does no work. 

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