Cynosure对《Statistical Mechanics》的笔记(1)

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This book is aimed primarily at graduate students in chemistry or computationalbiology and graduate or advanced undergraduate students in physics or engineering.
for this reason, many books with the words“statistical mechanics” in their titles can differ considerably. Here, I have attemptedto bring together topics that reflect what I see as the modern landscape of statistical mechanics.
it aims to show how the development of computational algorithms derives from the underlying theory with the hopeof enabling readers to understand the methodology-oriented literature 
the focus is on the molecular dynamics and MonteCarlo techniques
. I assume that readers have an understanding ofcalculus (through calculus of several variables), linear algebra, and ordinary differentialequations. 
reasons to employ an importance function h(x) in a Monte Carlocalculation. First, the function h(x) might be easier to sample than f(x). If h(x) retainssome of the most important features of f(x), then h(x) will be a good choice for animportance function. In this sense, employing importance sampling is akin to usinga reference potential in molecular dynamics, which we discussed in Section 3.11. Asecond reason concerns the behavior of the integrand φ(x) itself. If φ(x) is a highlyoscillatory function, then positive and negative contributions will tend to cancel in theMonte Carlo evaluation of eqn. (7.3.12), rendering the convergence of the samplingalgorithm extremely slow and inefficient because of the large variance. A judiciouslychosen importance function can help tame such oscillatory behavior, leading to asmaller variance and better convergence.
The reason for the failure of this importance function is that 1 − x is only a goodrepresentation of exp(−x) for x very close to 0,
The lesson from this example is that importance functions must be chosencarefully.
better convergence can oftenbe achieved if the vectors are generated sequentially x1 → x2 → · · · → xM with a rulethat specifies how to generate xi+1 given xi. Such a sequence of vectors, in which xi+1is generated based only knowledge of xi is called a Markov chain. Markov chains arethe core of many Monte Carlo algorithms.
The detailed balance condition ensuresthat the Markov process is microscopically reversible and hence guarantees unbiased sampling of the state space. It has been argued that the detailed balance condition isa sufficient but not strictly necessary condition to ensure proper sampling of the statespace。
M(RT)2 algorithm (M(RT)2stands in for the last names of the five authors), belongs to a class of Monte Carloschemes known as rejection methods. The M(RT)2 method starts with a rule for generating trial。
MC methods are comparable in their efficiency when molecular dynamics。
An advantage of moleculardynamics over Monte Carlo is that it is straightforward to couple and uncouple thermostats and barostats in order to switch between sampling and dynamics calculations,which makes writing an elegant, object-oriented code that encompasses both types ofcalculations conceptually seamless. Monte Carlo, as described here, is only useful asa sampling technique and therefore, a separate molecular dynamics module would beneeded to study the dynamics of a system. Because molecular dynamics moves all particles simultaneously, it is also easier to devise and implement algorithms suitable forparallel computing architectures in order to tackle very large-scale applications. Onthe other hand, Monte Carlo allows for considerable flexibility to invent new types ofmoves since one needs to worry only about satisfying detailed balance. It is, of course,likewise possible to devise clever molecular dynamics methods, as we have seen inChapters 4–5. However, in molecular dynamics, “cleverness” appears in the equationsof motion and the demonstration that the algorithm achieves its objective. Finally,due to inherent randomness, Monte Carlo calculations are, by construction, ergodic,even if a large number of Monte Carlo passes is required to achieve converged results.In molecular dynamics, because of its deterministic nature, achieving ergodicity is asignificant challenge
a key difference between Monte Carlo and moleculardynamics calculations is the ability of the latter to generate moves of the entire system(global moves) with acceptance probability 1.
The Wang–Landau algorithm is asimple yet elegant approach that can generate Ω(E) with impressive efficiency.
The key step in the Wang–Landau algorithm is that the densityof states Ω(E) is modified after each move according to Ω(E) → Ω(E)f, where fis a scaling factor with f > 1。
An interesting point concerning the Wang–Landau algorithm is that it does notsatisfy detailed balance due to the application of the scaling factor to the densityof states Ω(E),
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