QMC 量子蒙卡 （1） 经典蒙特卡罗算法

最近准备系统学习一下蒙卡算法。使用的教材是”Monte Carlo Simulation ain Statistical Physics - An Introduction, Sixth Edition, Kurt Binder, Dieter W. Heermann”, 是比较基本的教材，学校有买电子书。
另有一本 “Quantum Monte Carlo Methods: Algorithms for Lattice Models, J. Gubernatis, et. al., 2016” 这本书写也还行，但是似乎到处都没找到电子版，好在校图书馆有一本。
量子系统和静电系统的区别就在于量子系统中存在非对异性，以及根据粒子统计性质的不同需要对称或者反对称的波函数。上述量子性体现在蒙卡算法中，还会出现符号问题（sign problem）。
首先，需要学习经典的蒙特卡罗算法。因为量子蒙卡是利用蒙卡算法计算量子系统。

0. Concepts in probability

• A Sample space is s set of possible outcomes .

• An event is a set of outcomes that satisfies some criterion.
  The probability of an event , , must be:
  (i) ;
  (ii) if the event include all outcomes in the sample space, then ;
  (iii) if breaks into and that have no common outcome, then .

• A random variable maps an outcome to a real number.
  Events can thus be described by random variables, e.g. . In the following we will work on the level of random variables. We write the probability as as or .

• The cumulative distribution function of a random variable is .
  If is everywhere differentiable, then the random variable is continuous and we can define the probability density: .
  If the random variable is discrete, we may only write .

• If we have two random variables and , the joint probability is . The relation to sigle-variable probability is:
  
  If , then the two variables are said to be statistically independent.

• The conditional probability is the probability of given that occurs.
  
  We can find the Bayes’ theorem:
  。

• For multiple random variables, the conditional probability distribution:
  $f(x1,\dots,x_k|x{k+1},\dots,xn) = \frac{f(x_1,\dots , x_n)}{f(x{k+1},\dots,x_n)}$.

• From this we can derive the Chapman-KOlmogotoff equation:
  
• And we can also derive the conditional probability chain rule:
  $f(x1,\dots,x_n) = f(x_n|x{n-1},\dots,x_1)\dots f(x_2|x_1)f(x_1)$.