Particle infinite systems

There are two difficulties that arise in trying to model an (essentially) infinite system in terms of a representative box of particles such as the one above. First of all, particles near the walls of the box experience very different forces compared to molecules in the middle of the box. Secondly, as the simulation progresses, molecules can leave the box and so the density can change. 

Because only the B-particle is charged (we shall suppose the B-particle is denoted by the index 1 = a thus eZx = ea = 0 eZt = 0 for j 7 1), it is clear that, in the limit of an infinite system, the velocity distribution of the fluid particles will not be affected. This result may be proved rigorously32 but we shall not demonstrate it here. We thus have ... 

This equation is readily transformed to an integral equation for different from i and in <— k,- Y(z] — k )) never appear in two successive collision operators because otherwise we would get a negligible contribution in the limit of an infinite system moreover as these dummy particles have zero wave vectors in the initial state, they have a Maxwellian distribution of velocities (see Eq. (418)). This allows us to write Eq. (A.74) in the compact form ... 

The investigation of the behaviour of Z(r, t) in a finite system is a difficult problem due to the fact that the boundary conditions for (7.3.3) are unknown for fractal systems although one can use some scaling arguments based on the knowledge of the properties of the infinite system. In the case of the independent production of different particles we obtain... 

In contrast to fhe static methods discussed in the previous section, molecular dynamics (MD) includes thermal energies exphcitly. The method is conceptually simple an ensemble of particles represents fhe system simulated and periodic boundary conditions (PBC) are normally apphed to generate an infinite system. The particles are given positions and velocities, fhe latter being assigned in accordance with a... 

Still, as discussed in Section 2,8.1, normalization is in some sense still a useful concept even for such processes. As we saw in Section 2.8.1, we may think of an infinite system as a Q oo limit of a finite system of volume Q. Intuition suggests that a scattering process characterized by a short range potential should not depend on system size. On the other hand the normalization condition dx fl/(x)p = 1 implies that scattering wavefunctions will vanish everywhere like as Q CX3. We have noted (Section 2.8) that physically meaningful results are associated either with products such as A V<(x) 2 or yo V (x)p, where jV, the total number of particles, and p, the density of states, are both proportional to Q. Thus, for physical observables the volume factor cancels. 

Finite size effects on the critical temperature for Bose-Einstein condensation of a noninteracting Bose gas conhned in a harmonic trap manifests the reduction of the condensate fraction and the lowering of the transition temperature, as compared to the infinite system [14, 127]. Eor an N particle condensate, the shift of the critical temperature Tc, relative to that for the N by the cluster size scaling relation [14, 127]... 

The propagator methods originally developed in quantum-fields theory9 were rather simple in principle, but they became extremely complicated when applied in practice to many-particle systems, e.g., to the very large or infinite systems in quantum statistics,10 or to atomic nuclei. In... 

MRF results for dipolar hard spheres at p = 0.8 have been reported by Adams and by Levesque, et al. Adams has carried out a 500-particle = 3.85d) MRF calculation at =2.75 assuming that c=50 in the re-action-field expression (3.47). The MRF results (Figs. 9 and 10) lie below the LHNC and QHNC results for an infinite system, but well above the SC... 

Computer simulation results for S2 are somewhat sparse and involve the usual uncertainties involved in extrapolating results for a truncated T(r) used in a periodic box to untruncated T(r) in an infinite system." Nevertheless for polarizable hard-sphere and Lennard-Jones particles, it is probably safe to say that the estimates currently available from the combined use of analytic and simulation input are enough to provide a reliable guide to the p and dependence of Sj over the full fluid range of those variables. The most comprehensive studies of have been made by Stell and Rushbrooke" and by Graben, Rushbrooke, and Stell," for the hard-sphere and Lennard-Jones cases, respectively. Both these works utilize the simulation results of Alder, Weis, and Strauss," as well as exact density-expansion results, and numerical results of the Kirkwood superposition approximation... 

Consider an infinite system divided into macroscopic cubic cells of volume Q = l that are open to particle flux. Assume periodic boundary conditions on ea h cell and expand the concentration fluctuation Sc(t, t) in a Fourier series... 

The topic of flow through an effectively infinite system of particles belongs to the more general domain of flow through porous media (C12, C13, PIO, R5, S3, S4). In the absence of physicochemical interaction between the particles and fluid, the slow quasi-static Newtonian flow of incompressible fluids through such media is governed by Darcy s law. This linear phenomenological law has the form... 

To simulate an infinite system as closely as possible, a small number of particles was enclosed in a cell of volume V and periodic boundary conditions were applied to the system. These overcome the problem intrinsic to the use of a small number of particles which is the escape of particles from the cell. The periodic boundary conditions are implemented as follows. Each particle inside the cell has a ghost particle outside that mimics its position and motion exactly. As a result when one particle moves out of the cell, another compensatory particle enters from the other side (Fig. 13.8). 

Real systems contain a practically infinite number of particles, while the ensembles that are used in simulations contain typically a few hundred, at best a few thousand, particles. In order to mimic an infinite system, cychc boundary conditions can be imposed. In electrochemical investigations they are usually imposed in the two directions parallel to the electrode surface. 

Cluster calculations The conceptually simplest method consists in replacing the infinite system by a large cluster of particles. Such calculations can be performed with several packages, from both commercial and public domain, and have therefore become quite popular. However, even in relatively big clusters a large fraction of the atoms lie at the surface. Therefore the results of such calculations have to be interpreted with care - particularly, if the system is charged or has an uneven distribution of charges, since charges tend to accumulate on surfaces. In addition, the results obtained often depend on the size of the cluster, so that one has to focus on relative rather than on absolute values. 

For certain choices of trial wave functions further simplifications are possible, and this has been exploited in an interesting way in a series of variational calculations on the ground states of He and He, begun by McMillan and continued by the Orsay group. The model is again that of N particles with pair interactions, confined to a box, and with periodic boundary conditions to mimic an infinite system. The wave function has to be able to prevent strong overlaps of the particles, and a popular form for the boson case is therefore simply a product of pair functions. 

In order to simulate more closely the behavior of an infinite system, periodic boundary conditions are imposed on the solution to the equations of motion. If a molecule labeled i is located at position (xj, yt, z,) at time t, we imagine that there are 26 additional images of i located at (x, L, 0 yj L, 0 L, 0). The particle and its 26 images have the same orientation and velocity. Another molecule j may interact with any i within its interaction range. If molecule i should cross a face of the box, it is reinserted at the opposite face. Constant density is thus maintained. These periodic boundary conditions avoid the strong surface effects that would result from a box with reflecting walls. 

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