Simulations continuous-deterministic

A computational method was developed by Gillespie in the 1970s [381, 388] from premises that take explicit account of the fact that the time evolution of a spatially homogeneous process is a discrete, stochastic process instead of a continuous, deterministic process. This computational method, which is referred to as the stochastic simulation algorithm, offers an alternative to the Kolmogorov differential equations that is free of the difficulties mentioned above. The simulation algorithm is based on the reaction probability density function defined below. 

Over the last several years much effort has gone into the extension of MD to describe other than microcanonical systems (Nose and references therein). Nose, in particular, has provided a method that is capable in principle of producing canonical averages from time averages of static physical properties along continuous, deterministic trajectories. This method can also be used for the calculation of dynamic propierties that are inaccessible from MC simulations. However, the interpretation of time-dejjendent properties calculated from this and similar methods is not straightforward, particularly for very small systems. ... 

A better alternative for tissue growth modehng is the hybrid discrete-continuous approach. Hybrid discrete-continuous models employ a discrete algorithm to simulate the dynamics of a cell population, while processes such as diffusion and consumption of GFs or nutrients are described with a continuous, deterministic component usually based on partial differential equations. 

Figure 18.1 Regimes of the problem space for multiscale stochastic simulations of chemical reaction kinetics. The r-axis represents the number of molecules of reacting species, x, and the ) -axis measures the frequency of reaction events, A. The threshold variables demarcate the partitions of modeling formalisms. In area I, the number of molecules is so small and the reaction events are so infrequent that a discrete-stochastic simulation algorithm, like the SSA, is needed. In contrast, in area V, which extends to infinity, the thermodynamic limit assumption becomes vahd and a continuous-deterministic modehng formalism becomes valid. Other areas admit different modehng formalisms, such as ones based on chemical Langevin equations, or probabilistic steady-state assumptions.

A final comment on the interpretation of stochastic simulations We are so accustomed to writing continuous functions—differential and integrated rate equations, commonly called deterministic rate equations—that our first impulse on viewing these stochastic calculations is to interpret them as approximations to the familiar continuous functions. However, we have got this the wrong way around. On a molecular level, events are discrete, not continuous. The continuous functions work so well for us only because we do experiments on veiy large numbers of molecules (typically 10 -10 ). If we could experiment with very much smaller numbers of molecules, we would find that it is the continuous functions that are approximations to the stochastic results. Gillespie has developed the stochastic theory of chemical kinetics without dependence on the deterministic rate equations. 

Hydrate nucleation is the process during which small clusters of water and gas (hydrate nuclei) grow and disperse in an attempt to achieve critical size for continued growth. The nucleation step is a microscopic phenomenon involving tens to thousands of molecules (Mullin, 1993, p. 173) and is difficult to observe experimentally. Current hypotheses for hydrate nucleation are based upon the better-known phenomena of water freezing, the dissolution of hydrocarbons in water, and computer simulations of both phenomena. Evidence from experiments shows that nucleation is a statistically probable (not deterministically certain see Section 3.1.3) process. 

Fig. 7.3 Deterministic (a) and noisy (b) computer simulations of the time course of affective disorders showing the intervals between successive disease episodes (interval duration) as a function of a disease variable S and examples of episode generation from different disease states (figure modified after [2]). In deterministic simulations (a), there is a progression from steady state (S = 18) to subthreshold oscillations (S = 22) with immediate onset of periodic event generation at a certain value of S (slightly below S = 60). With further increase of S, the intervals between successive episodes are continuously...

This chapter provides an overview of the most frequently applied numerical methods for the simulation of polymerization processes, that is, die calculation of the polymer microstructure as a function of monomer conversion and process conditions such as the temperature and initial concentrations. It is important to note that such simulations allow one to optimize the macroscopic polymer properties and to influence the polymer processability and final polymer product application range. Both deterministic and stochastic modeling techniques are discussed. In deterministic modeling techniques, time variation is seen as a continuous and predictable process, whereas in stochastic modeling techniques, a random-walk process is assumed instead. 

Non-continuous approach can be deterministic or stochastic. In deterministic approaches, such as the molecular dynamics (MD) method and the lattice Boltzmann method (LBM), the particle or molecule s trajectory, velocity and intermolecular collision are calculated or simulated in a deterministic manner. In the stochastic approaches, such as the direct simulation Monte Carlo (DSMC) method, randomness is introduced in to the solution variables. 

It is an accurate method of time integration and is explicit in nature. Table 3.15 summarises this method. This method of higher order is a robust algorithm used to solve non-linear equations but may have problems which have discontinuous coefficients which take place spatially. Some coefficients change discontinu-ously as the load increases. Strain localisation will be difficult to produce in numerical simulations. Non-linear equations have bifurcation. The non-linear equation of stochastic elasto-plasticity is more suitable for finding the most unstable solution compared with the non-linear equation of deterministic elasto-plasty since the coefficients change continuously. 

Several codes are available for carrying out a direct Monte Carlo simulation of a reactor problem using detailed geometrical models and continuous energy (or very fine group) representation of nuclear data. These can be used to provide reference values and investigate the effects of approximations in deterministic methods. Some widely used Monte Carlo codes are MCNP [4.39], MORSE [4.66] and KENO [4.67], amongst others. 

In a continuous-flow chemical reactor, the concern is not only with probabilistic transitions among chemical species but also with probabilistic liansitions of each chemical species between the interior and exterior of the reactor. Pippel and Philipp [8] used Markov chains for simulating the dynamics of a chemical system. In their approach, the kinetics of a chemical reaction are treated deterministically and the flow through the system are treated stochastically by means of a Markov chain. Shinnar et al. [9] superimposed the kinetics of the first order chemical reactions on a stochastically modeled mixing process to characterize the performance of a continuous-flow reactor and compared it with that of the corresponding batch reactor. Most stochastic approaches to analysis and modeling of chemical reactions in a flow system have combined deterministic chemical kinetics and stochastic flows. 

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