Method exact enumeration

Figure 9.30. Kinetics of the photoinduced back ET reaction for (NH3)5Fe"(CN)(Ru) "(CN)5 in water at BOOK with F = 2500cm , — AG = 3900cm and , = 3800cm . Ohmic spectral density with exponential cutoff at 220 cm is assumed for the solvent. Exact enumeration method (dashed) and the transfer matrix path integral approach (solid) are compared with the Golden Rule prediction (dash-dotted). (Reproduced from [132] with permission. Copyright (1998) by the American Institute of Physics.)...

What makes this problem so difficult—and indeed most copolymer problems difficult— is the necessity to take the average over all possible 2" distributions of monomers on the polymer. Nevertheless, some exact enumeration studies of this problem have been conducted. Martin [16] estimated the location of the transition using exact enumeration methods. An open question is the value of the crossover exponent, (j>, which describes the shape of the free-energy near the adsorption critical temperature. The conclusion from the above study, and others, is that the difference between this exponent and its homopolymer counterpart, if it exists, is too small to be detectable by any current numerical studies. 

Golding and Kantor [19] and Kantor and Kardar [20] stndied a version of this problem by both exact enumeration and Monte Carlo methods. In the former situation described above, they find a collapse transition much like the transition discnssed above in the... 

At d = 1 one has a completely stretched chain with ly = 1. At d = 2 the exact result v = 3/4) [13] is obtained. The upper critical dimension is d = 4, above which the polymer behaves as a random walker. The values of the universal exponents for SAWs on d - dimensional regular lattices have also been calculated by the methods of exact enumerations and Monte Carlo simulations. In particular, at the space dimension d = 3 in the frames of field-theoretical renormalization group approach one has (v = 0.5882 0.0011 [11]) and Monte Carlo simulation gives (i/ = 0.592 0.003 [12]), both values being in a good agreement. 

This Chapter is organized as follows Section 2 revisits general definitions of quantities of interest, for both deterministic and random fractals, and introduces the notation used. Section 3 contains the numerical techniques such as Monte Carlo (MC) methods and the exact enumeration (EE) technique as well as data analysis schemes. Section 4 provides a discussion of SAWs on deterministic fractals, specifically the Sierpinski square and... 

Because of the short chain limitation of the exact enumeration method, Monte Carlo (MC) techniques have been widely applied to the excluded volume problem, notably by Wall and co-workers. One recent paper has investigated the mean square end-to-end distance of chains as a function of concentration and... 

In an exact-enumeration study, one first generates a complete list of all SAWs up to a certain length (usually N w 15-35 steps), keeping track of the properties of interest such as the number of such walks or their squared end-to-end distances. One then performs an extrapolation to the limit N - oo, using techniques such as the ratio method. Fade approximants or differential approximants. ° Inherent in any such extrapolation is an assumption about the behavior of the coefficients beyond those actually computed. Sometimes this assumption is fairly explicit other times it is hidden in the details of the extrapolation method. In either case, the assumptions made have a profound effect on the numerical results obtained. For this reason, the claimed error bars in exact-enumeration/extrapolation studies should be viewed with a healthy skepticism. 

Finally, within the ideality hypothesis oj k) can always be obtained for any model from computer simulation of an isolated chain, which is now feasible for fairly long chains. For the rotational isomeric state model, a hybrid approach has been shown to be quite accurate. In this method the distribution function is calculated from exact enumeration for sites separated by five bonds or less and approximately using the Koyama distribution for sites separated by more than five bonds. 

For the case of fully developed excluded volume limit, the exact enumeration method " consists of carrying out enumerations of the number, C , of all possible nonintersecting random walks of n steps on a lattice and the number, f (/ )A/ , of those walks whose end points lie between R and R -h A/ from the origin. Since all possible configurations are a priori equally possible... 

There is an alternative method, called the Monte Carlo method, to obtain v where nonintersecting random walks are simulated on various lattices and statistical data describing the chain are accumulated. This method is restricted in accuracy by statistical fluctuations in contrast to the exact enumeration method. But long walks can be studied here. This method has been applied to understand a variety of phenomena involving polymers as reviewed in refs. 73-75. 

A method is outlined by which it is possible to calculate exactly the behavior of several hundred interacting classical particles. The study of this many-body problem is carried out by an electronic computer which solves numerically the simultaneous equations of motion. The limitations of this numerical scheme are enumerated and the important steps in making the program efficient on the computer are indicated. The applicability of this method to the solution of many problems in both equilibrium and nonequilibrium statistical mechanics is discussed. 

For sufficiently short chains it is possible to calculate C , u , and all other features of interest exactly. Such enumerations were initiated independently because of their application to the statistical mechanics of interacting systems on crystal lattices,10 and a variety of analytical and computational methods (including the use of digital computers) has been employed to extend the enumerations to as large a value of n as practicable. These exact results are then used to conjecture the pattern of asymptotic behavior,... 

The overlap between small libraries such as these could be determined by enumerating each and comparing the specific structures using the existing methods for exact match. Enumeration is prohibitive for interactive comparisons of larger libraries, so the generic structures themselves must be compared with each other [5],... 

Exact results of combinatorial optimization of PWCs structure are presented in the Table. The data were obtained by different methods. For small and middle-size clusters (N < 28), we used a complete enumeration method. For large clusters (N = 36, 60), the data were obtained using symbolic calculations and max-plus algebra. Characteristics of two lowest energy levels for the SWEB model are presented. For water fullerene N = 60) on the base of SWEB model, the data obtained only for ground state level without isomorphism analysis. Values in italics correspond to ground sates of the SWB model. For example, the... 

---