Montecarlo Techniques

                     These are the techniques that uses random numbers to describe the physical or mathematical systems. These techniques follows some algorithms.
                      Montecarlo techniques are useful when the system has highest degrees of freedom. In a raw statement one can define that any technique that uses random numbers is known as montecarlo techniques. We have to use suitable computational algorithms to solve these type of problems.


Random numbers
Random numbers are generated by some process/algorithm whose outcome is not predictable. To use statistical mechanics or montecarlo techniques, random numbers should be uniformly distributed. There are several methods to create random numbers. Of course, from nature one can obtain random numbers. For example, counting cosmic particles and calculating modulo, calculating decay or throwing a die/coin etc. But these natural processes are not fast. In fact these doesn't satisfy some tests which are necessary to use statistical mechanics. Hence we used to generate random numbers using computer. These random numbers are known as pseudo random numbers, since these are not completely random. These follow some algorithm.
There are different methods to generate random numbers.

1. Linear Congruential method  

                     X(n+1) = (AX(n)+B) modulo M
         Here X(0) is the seed and M is the modulo. A and B are constants. Using this recurrence relation we can get random numbers. Accuracy of randomness depends on values of A and B. This is the simplest and most efficient method.

Tests for checking randomness

• Uniform test
• Frequency test
• Correlation test
• Run up - run down test 

These are the basics tests. In addition to above tests there are variety of advanced tests like ks test, chi - square test etc depending on the model.
Montecarlo Techniques

• Acceptance rejection techniques
• Inversion technique

There are different types of montecarlo techniques.

• Classical montecarlo
• Quantum montecarlo
• Path integral montecarlo
• Simulation montecarlo  etc...

Library:

1. Random number generation and montecarlo methods by James E. Gentle
2. Montecarlo methods in statistical physics by K.P.N. Murthy