Introduction to Simulation Methods for Polymer Systems

Outline: Today we will introduce some basic stochastic simulation methods for polymer systems. We begin by looking at methods to generate random numbers. Then we introduce the simple sampling method with a side look at percolation theory (important for the description of gels etc). Next we develop methods to generate states with a predescribed distribution. Finally, we introduce the  importance sampling method.
The development of simulation methods and algorithms over the last couple of years has been matched only by the equally rapid advances that have taken place in the field of computer technology. The many-fold increases in the speed, memory size and the flexibility of computers has opened-up a vast number of new possibilities for studying science and engineering problems, and enabled both new insights and new lines of inquiry. However, the complexity of the problems studied and the accuracy of the results required are such that the computing resources available are hardly able to keep up with the demand.

But before proceeding to look at various methods of simulation we need to say some remarks about the importance of simulation for computational science. The computer simulation of a physical system is in essence a numerical experiment that differs from its laboratory counterpart in a number of crucial ways. In a computer simulation we have the freedom to choose both the type of model we wish to study and the conditions (i.e., temperature, inter-particle couplings, etc.) under which we wish to study it, often in ways that are not possible in a laboratory experiment. This gives us an enormous range of phenomena that can be investigated and a tremendous flexibility in investigating even the basic assumptions of our understanding of such systems, although we must recognize that there are also limitations to be encountered. These limitations, which arise from the finite simulation time, finite system size and so forth, are reflected in the accuracy of the results that we obtain.

In this chapter we also have the opportunity to expose approaches that complement each other. An example is the simulation if polymer liquids. The deterministic simulation of a polymer liquid has the advantage of being equipped with a mapping to a time constant. We thus have an immediate and intuitive interpretation for time-dependent phenomena, as for example diffusion. For a probabilistic simulation the time-dependent phenomena are little more intricate and perhaps a little less intuitive. On the other hand, when it comes to simulating a polymer liquid at a constant temperature the probabilistic simulation method, yielding the correct statistical mechanical ensemble, complement a simulation at constant energy using a deterministic method. Often it depends on the authors taste which method, deterministic or probabilistic he chooses. The model not always dictates the method. For some lattice models the choice is obvious, especially if the degrees of freedom are discrete. For models allowing both approaches, it is a matter, as said, of taste whether putting more emphasis on the force, as done in deterministic method or on the potential, as done in the probabilistic methods.

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