Continuous space

In some cases, the ventilation process in the room can be simplified and mechanisms of air and contaminant movement under the influence of each of the above factors can be described using simplified theoretical principles of fluid mechanics, empirical data, and observations from numerous research studies. In general, the ventilation process in a room is complex and different factors have a joint effect on airflow patterns and characteristics, in continued spaces and in industrial buildings particularly. 

Lattice models have the advantage that a number of very clever Monte Carlo moves have been developed for lattice polymers, which do not always carry over to continuum models very easily. For example, Nelson et al. use an algorithm which attempts to move vacancies rather than monomers [120], and thus allows one to simulate the dense cores of micelles very efficiently. This concept cannot be applied to off-lattice models in a straightforward way. On the other hand, a number of problems cannot be treated adequately on a lattice, especially those related to molecular orientations and nematic order. For this reason, chain models in continuous space are attracting growing interest. 

The simplest atomistic model for the formation of a crystal in continuous space requires the definition of some effective attractive potential between any two atoms, which is defined independently of the other atoms in the cluster or crystal. The most frequently studied potential is the Lennard-Jones potential... 

This is possible within the framework of the self-consistent field (SCF) approach to polymer configurations, described more completely elsewhere [18, 19, 51, 52]. Implementation of this method in its full form invariably requires numerical computations which are done in one of two equivalent ways (1) as solutions to diffusion- or Schrodinger-type equations for the polymer configuration subject to the SCF (in which solutions to the continuous-space formulation of the equations are obtained by discretization) or (2) as solutions to matrix equations resulting from a discrete-space formulation of the problem on a lattice. 

A suitable approach to the equilibration of an amorphous polymer system at bulk density becomes much more likely when the fully atomistic model in continuous space is replaced by an equivalent coarse-grained model on a lattice with sufficient conformational flexibility. Different strategies, which seek results at different levels of detail, can be employed to create an appropriate coarse-grained model. Section 4 (Doruker, Mattice) describes an approach which attempts to retain a connection with the covalent bonds in the polymer. The rotational isomeric state (RIS) [35,36] model for the chain is mapped into... 

The most austere representation of a polymer backbone considers continuous space curves with a persistence in their tangent direction. The Porod-Kratky model [99,100] for a chain molecule incorporates the concept of constant curvature c0 everywhere on the chain skeleton c0 being dependent on the chemical structure of the polymer. It is frequently referred to as the wormlike chain, and detailed studies of this model have already appeared in the literature [101-103], In his model, Santos accounts for the polymer-like behavior of stream lines by enforcing this property of constant curvature. 

Fig. 4.1. Schematic representation of three numbered steps in a MC simulation on a high coordination lattice (solid arrows) that replace a simulation of the fully atomistic system in continuous space (single dashed line)...

MD simulations of melts of C44H90, based on classic techniques in continuous space, have been reported recently using united atom [146] and fully atomistic [145] representations of the chain. Time in the conventional MD simulations is expressed in seconds, whereas time in the simulation of the coarse-grained chains on the 2nnd lattice is expressed in MC steps. Nevertheless, a few comparisons are possible via the longest relaxation time, rr, deduced from the decorrelation of the end-to-end vector ... 

Export processes are often more complicated than the expression given in Equation 7, for many chemicals can escape across the air/water interface (volatilize) or, in rapidly depositing environments, be buried for indeterminate periods in deep sediment beds. Still, the majority of environmental models are simply variations on the mass-balance theme expressed by Equation 7. Some codes solve Equation 7 directly for relatively large control volumes, that is, they operate on "compartment" or "box" models of the environment. Models of aquatic systems can also be phrased in terms of continuous space, as opposed to the "compartment" approach of discrete spatial zones. In this case, the partial differential equations (which arise, for example, by taking the limit of Equation 7 as the control volume goes to zero) can be solved by finite difference or finite element numerical integration techniques. 

This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

As the dynamics of the system are removed from the model, it is no longer necessary to allow the molecules to live in a continuous space. Instead, the use of lattices - discrete sets of coordinates on to which the molecules are restricted - is popular. Digital computers are of course much more efficient with discrete space than with continuum space. The use of a lattice implies that one removes all properties that occur on shorter length scales than the lattice spacing from the model. This is no problem if the main interest is in phenomena that are larger than this length scale. 

Discrete vs. continuous space is related to the possible values of the decision variables deciding production quantities for instance is reflected by a continuous decision variable while deciding to make a change-over or not is a binary decision requiring a discrete decision reflected by integer variables in this case 0 or 1. 

In continuous space the thermodynamical probability W(x) of the realization of state in... 

As same as for the lattice space, general number L of SARW trajectories in continuous space let us determine in the form (10), that is L (2d)N c(x). That is why... 

The milieu of effect can be represented by a hypothetical compartment, as if this were a continuous space or a composite of similar spaces the concentration within the effect compartment can then be related to the observed effect ... 

Population transfer between sites in a lattice is a ubiquitous phenomenon in condensed matter physics. Because a lattice space has a different symmetry than does continuous space, protocols to alter the evolution of population in a lattice system with an external field must differ slightly from the protocols used for the same purpose in continuous space. With this observation in mind, Masuda and Rice extended the fast-forward protocol to apply to lattice systems and applied then-formalism to enhancing site-to-site particle transfer in a BEC [ 18]. We now briefly review that formalism. 

---