Monte Carlo methods are essentially computational algorithms that involve repeated random sampling to compute results.  This random sampling occurs through pseudo-random numbers generated by computer applications.  The name Monte Carlo comes from an administrative region of Monaco famous for its casinos and gambling.  The methods include a variety of approaches though most common are these steps: 

  -Defining a domain of possible inputs

-Generating random inputs from the domain

 -Performing a deterministic computation on those inputs

-Combine different individual computations into a final result

  A most simple application of Monte Carlo applications is a hit-and-miss computation.  One of course involves determining the value of pi.  In this calculation, there is a circle inscribed in a square or a simpler method is a quarter-circle inscribed in a square. 

From simple geometry we can show that:

 Number of darts in circle/Number of darts in square = ¼*pi*r^2/r^2 = ¼*pi

  Thus an estimate of 1/4 pi can be obtained from this simulation. 

 Another simulation involves the use of a full circle inscribed in a square.  From here, random or in better terms pseudo-random numbers can be generated for x and y and the distance from the origin can be calculated.  If the distance is less than 1, the generation is within the circle.  From here, an estimate for pi can be computed.  Code provided for this as below is:

 x=(random#)

 y=(random#)

 dist=sqrt(x^2 + y^2)

     if dist.from.origin (less.than.or.equal.to) 1.0

          let hits=hits+1.0

 A simple application of this is offered here: http://polymer.bu.edu/java/java/montepi/montepiapplet.html

These simulations thus do provide an alternate route to finding pi.  A method covered in the text was the determination of pi through numerical integration. 

 Monte Carlo simulations have been used in various branches of the physical sciences.  One area is polymer science.  Polymers are molecules made up of smaller molecule subunits known as monomers that are chemical bonded together.   Plastics and fabrics such as polyethylene, polyurethanes, and polystyrene are examples of polymers.  There are numerous examples of polymers in biology.  Examples are DNA, proteins, and complex sugars. 

 An simple example of Monte Carlo methods involving polymers applies to determining the distance between two endpoints of a polymer., r  One assumption that can be made is that the bonds in each polymer are fixed at certain angles.  For carbon-carbon single bonds, the bonds are fixed at 109.5 degrees.  For a smaller molecule, there are a small number of states that the molecule can assume.  A molecule with four subunits can assume two positions here:

However with a larger molecule the number of states increases.  A molecule with 5 atoms has 4 states: 

A molecule with 8 atoms has 32 states:

However, it becomes impossible to review every type of state especially as polymers generally have thousands of subunits.  Yet with Monte Carlo simulations, a randomly generated states from an input domain can be used to determine the average size or average squared size of a polymer.  Monte Carlo simulations can be used to make a probability distribution of the various values of r^2.  This of course may require simpler assumptions such as allowing for only 90 or 180-degree angles.  Or it could require complications such as electric dipole considerations. 

  These steps usually involve calculating energy of the various states.  From the energy and Boltzmann factor, a probability of the polymer being in a given state can be determined. After that, an ensemble can be constructed which builds a probability distribution.  Of course variations in the energy of the system need to be considered as well.   The link http://www.chem.uci.edu/undergrad/applets/conf/conf_explain.htm provides a good explanation of a simple application of this. 

  While the above is a simple application, Monte Carlo methods have been used to study diffusion in polymeric systems, effects of shear and strain, branching of polymers, lattice structures, star polymers, and so forth.  This is only a simple overview for the laymen. 

References:   

  http://www.chem.uci.edu/undergrad/applets/conf/conf_explain.htm

 http://en.wikipedia.org/wiki/Monte_Carlo_method

http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html

http://polymer.bu.edu/java/java/montepi/montepiapplet.html