Sampling importance

Consider the free energy difference, AA ( ), obtained from sampling the system with the non-Boltzmann probability density p C), given by  

Substituting this expression into (3.20), we obtain the important result 

Since uj is always positive this can be done without loss of generality. We also define 

Despite this deficiency, importance sampling is a very powerful and versatile technique. It can be used with different types of order parameters, including those which describe an actual Hamiltonian coordinate of the system and those which 

The connection between importance sampling and FEP can be easily established by noting that free energy difference between the reference and the target system in (2.8) can be rewritten as 

Because MC is a numerical technique to calculate multidimensional integrals like the one over configuration space in Eq. (5.1), we begin by discretizing configuration space and rewrite the integral as 

To appreciate this latter point, consider the trajectory of a single molecule in space. Let us discretize this trajectory such that we represent the trajectory of the molecule by a succession of regularly spaced points. Thus, these points may be viewed as the p nodes of a cubic lattice. Clearly, in each spatial dimension, 2 otit of the total number of the p nodes in that direction lie on the surface of the cube. Extending these considerations to N instead of just a single molecule, it is immediately clear that we need to replace the original cube by an Af-dimensionaJ hypercube such that in eacli dimension the fraction p — 2/p represents the ratio of nodes not on the surface of the hypercube relative to the total number of nodes. To estimate this fraction 

Moreover, to represent the continuous trajectory of each molecule by a succession of regularly spaced nodes, the lattice constant of the hypercube should be sufficiently small that is, p should be large enough so that x = 2/p C 1. In this case, we can approximate the logarithm in Ekj. (5.3) through ln(l — x) = -X -h O (j ) —.T such that 

However, there is a twofold problem with this naive approadi. Firet, because the amount of computer time is ineAdtably finite, we need to limit the summation in Eq. (5.2) to some maximum value M - A/max- It tmns out that in most cases A / ,ax = C (10 — 10 ) is sufficient from a practical perspective so that this problem can be surmounted. 

The second and by far more serious problem with an application of Eq. (5.2) involves the probability density p r X) which is a priori unknown. An inspection of Ek. (2.117) reveals that p (r ,x) depends on the (classic) partition function, which involves the configuration integral as Eq. (2.118) shows. However, the partition function itself is unknown such that the probability with which O (r ) needs to be sampled at points remains undetermined. 

However, there is a twofold problem with this naive appruadi. First, because the amount of computer time is inevitably finite, we need to limit 

In statistical mechanics the goal is to generate a chain of microscopic states that sample the limiting, equilibrium probability distribution in phase space. Of course, the transition matrix is immense. For example, for NVE ensembles the transition matrix is an matrix. This is incomprehensibly large. In addition, the matrix elements are unknowable, since it is not possible to determine the transition probability between any two arbitrary points in phase space. 

Proving this is beyond the scope of this text but we can explain it by focusing on the ATT ensemble. The goal of Metropolis Monte Carlo 

p is the limiting, equilibrium distribution, with elements 

There are two major steps in computational implementation of importance sampling  

generation of a point in phase space using information of the previously generated phase point  

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There are a few variations on this procedure called importance sampling or biased sampling. These are designed to reduce the number of iterations required to obtain the given accuracy of results. They involve changes in the details of how steps 3 and 5 are performed. For more information, see the book by Allen and Tildesly cited in the end-of-chapter references. 

Recent Uses of Solid-Surface Luminescence Analysis in Environmental Analysis. Vo-Dinh and coworkers have shown very effectively how solid-surface luminescence techniques can be used for environmentally important samples (17-22). RTF has been used for the screening of ambient air particulate samples (17,18). In addition, RTF has been employed in conjunction with a ranking index to characterize polynuclear aromatic pollutants in environmental samples (19). A unique application of RTF reported recently is a personal dosimeter badge based on molecular diffusion and direct detection by RTF of polynuclear aromatic pollutants (20). The dosimeter is a pen-size device that does not require sample extraction prior to analysis. 

Canadian samples analyzed, 1 sample (carrots) contained a methyl parathion residue at a level of <0.05 ppm. Of the imported samples, 14 contained methyl parathion residues. Levels of methyl parathion ranged from <0.05 ppm (in pears and snowpeas), to 0.10 ppm (in apples, oranges, pears, and tomatoes), to a maximum of 0.50 ppm (in grapes, apples, oranges, and pears, plus another unspecified sample). 

Domestic and imported pears and tomatoes were collected and analyzed for pesticide residues from July 1992-July 1993 (Roy et al 1995). Endosulfan (both isomers) was found in 471 of 1,219 domestic tomato samples at a maximum concentration of 0.2 mg/kg and in 80 of 144 imported tomato samples at a maximum concentration of 0.55 mg/kg. In pears, endosulfan was found in 144 of 710 domestic samples at a maximum concentration of 1.1 mg/kg, and in 4 of 949 imported samples at a maximum concentration of 0.13 mg/kg. 

The attractive features of spllti ess injection techniques are that they allow the analysis of dilute samples without preconcentration (trace analysis) and the analysis of dirty samples, since the injector is easily dismantled for cleaning. Success with individual samples, however, depends on the selection of experimental variables of which the most Important sample... 

Mineral acid dissolution is an important sample preparation process for instrumental analysis, as it liberates element ions into a solution that can be directly introduced into an analytical instrument. For quantitative analysis, most instruments require a solution. 

The direct calculation of the reduced partition function ((e-AU kT))o,h=i is expected to be limited by the variance of the function e-AU/kT [ Iowever, representing the averaged quantity as a ratio opens possibilities for importance sampling, evaluating both numerator and denominator on the basis of a single sample designed to reduce the variance of e AU/kTm... 

The basic idea of importance sampling can be illustrated simply in the example of the transformation from 0 to 1 along A, as described above. In lieu of sampling from the true probability distribution, P A), we design simulations in which A is sampled according to P A). The latter probability should be chosen so that it is more uniform than P A). The relation between the two probabilities may then be expressed as follows ... 

The fundamental issue in implementing importance sampling in simulations is the proper choice of the biased distribution, or, equivalently, the weighting factor, q. A variety of ingenious techniques that lead to great improvement in the efficiency and accuracy of free energy calculations have been developed for this purpose. They will be mentioned frequently throughout this book. 

Srinivasan, R., Importance Sampling, Springer Berlin, Heidelberg, New York, 2002... 

Fig. 2.4. Schematic representation of the different relationships between the important regions in phase space for the reference (0) and the target (1) systems, and their possible interpretation in terms of probability distributions - it should be clarified that because AU can be distributed in a number of different ways, there is no obvious one-to-one relation between P0(AU), or Pi (AU), and the actual level of overlap of the ensembles [14]. (a) The two important regions do not overlap, (b) The important region of the target system is a subset of the important region of the reference system, (c) The important region of the reference system overlaps with only a part of the important region of the target state. Then enhanced sampling techniques of stratification or importance sampling that require the introduction of an intermediate ensemble should be employed (d)...

ABF shares some similarities with the technique of Laio et al. [30-34], in which potential energy terms in the form of Gaussian functions are added to the system in order to escape from energy minima and accelerate the sampling of the system. However, this approach is not based on an analytical expression for the derivative of the free energy but rather on importance sampling. 

Since the potential of mean force is a statistical property, it is insufficient to calculate it directly by importance sampling which, by design, emphasizes potential minima... 

Zuckerman, D.M. Woolf, T.B., Efficient dynamic importance sampling of rare events in one dimension, Phys. Rev. E 2001, 6302, 016702... 

In addition, and most importantly, samples of typical B2 materials always seem to exhibit minority domains which behave quite differently from the stripe domains. Specifically, at zero field these minority domains have the lower An green birefringence color and show a smooth SmA-like focal conic... 

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