Monte carlo Simulation in java

Monte carlo Simulation in java

This section under major construction.In 1953 Enrico Fermi, John Pasta, and Stanslaw Ulam created the first "computer experiment" to study a vibrarting atomic lattice. Nonlinear system couldn't be analyzed by classical mathematics.

Simulation = analytic method that imitates a physical system. Monte Carlo simulation = use randomly generated values for uncertain variables. Named after famous casino in Monaco. At essentially each step in the evolution of the calculation, Repeat several times to generate range of possible scenarios, and average results. Widely applicable brute force solution. Computationally intensive, so use when other techniques fail. Typically, accuracy is proportional to square root of number of repetitions. Such techniques are widely applied in various domains including: designing nuclear reactors, predicting the evolution of stars, forecasting the stock market, etc.

Generating random numbers.The math library functionMath.randomgenerate a pseudo-random number greater than or equal to 0.0 and less than 1.0. If you want to generate random integers or booleans, the best way is to use the libraryRandom. ProgramRandomDemo.javaillustrates how to use it.

Random random = new Random();

boolean a = random.nextBoolean(); // true or false

int b = random.nextInt(); // between -2^31 and 2^31 - 1

int c = random.nextInt(100); // between 0 and 99

double d = random.nextDouble(); // between 0.0 and 1.0

double e = random.nextGaussian(); // Gaussian with mean 0 and stddev = 1

Note that you should only create a newRandomobject once per program. You will not get more "random" results by creating more than one. For debugging, you may wish to produce the same sequence of pseudo-random number each time your program executes. To do this, invoke the constructor with alongargument.

Random random = new Random(1234567L);

The pseudo-random number generator will use 1234567 as the seed. UseSecureRandomfor cryptographically secure pseudo-random numbers, e.g., for cryptography or slot machines.

Linear congruential random number generator.With integer types we must be cognizant of overflow. Consider a * b (mod m) as an example (either in context of a^b (mod m) or linear congruential random number generator: Given constants a, c, m, and a seed x[0], iterate: x = (a * x + c) mod m. Park and Miller suggest a = 16807, m = 2147483647, c = 0 for 32-bit signed integers. To avoid overflow, use Schrage's method.

Precompute: q = m / a, r = m % a

Iterate: x = a * (x - x/ q) * q) - r * (x / q)

Exercise: compute cycle length.

Library of probability functions.OR-Objects contains many classicprobability distributions and random number generators, including Normal, F, Chi Square, Gamma, Binomial, Poisson. You can download thejar filehere. ProgramProbDemo.javaillustrates how to use it. It generate one random value from the gamma distribution and 5 from the binomial distribution. Note that the method is calledgetRandomScalerand notgetRandomScalar.

GammaDistribution x = new GammaDistribution(2, 3);

System.out.println(x.getRandomScaler());

BinomialDistribution y = new BinomialDistribution(0.1, 100);

System.out.println(y.getRandomVector(5));

Queuing models.M/M/1, etc. A manufacturing facility has M identical machines. Each machine fails after a time that is exponentially distributed with mean 1 / . A single repair person is responsible for maintaining all the machines, and the time to fix a machine is exponentially distributed with mean 1 / . Simulate the fraction of time in which no machines are operational.

Diffusion-limited aggregation.

Diffuse = undergo random walk. The physical processdiffusion-limited aggregation(DLA) models the formation of an aggregate on a surface, including lichen growth, the generation of polymers out of solutions, carbon deposits on the walls of a cylinder of a Diesel engine, path of electric discharge, and urban settlement.The modeled aggregate forms when particles are released one at a time into a volume of space and, influenced by random thermal motion, they diffuse throughout the volume. There is a finite probability that the short-range attraction between particles will influence the motion. Two particles which come into contact with each other will stick together and form a larger unit. The probability of sticking increases as clusters of occupied sites form in the aggregate, stimulating further growth. Simulate this process in 2D using Monte Carlo methods: Create a 2D grid and introduce particles to the lattice through a launching zone one at a time. After a particle is launched, it wanders throughout with a random walk until it either sticks to the aggregate or wanders off the lattice into thekill zone. If a wandering particle enters an empty site next to an occupied site, then the particle's current location automatically becomes part of the aggregate. Otherwise, the random walk continues. Repeat this process until the aggregate contains some pre-determined number of particles.Reference:Wong, Samuel, Computational Methods in Physics and Engineering, 1992.

ProgramDLA.javasimulates the growth of a DLA with the following properties. It uses the helper data typePicture.java. Set the initial aggregate to be the bottom row of the N-by-N lattice. Launch the particles from a random cell in top row. Assume that the particle goes up with probability 0.15, down with probability 0.35, and left or right with probability 1/4 each. Continue until the particles stick to a neighboring cell (above, below, left, right, or one of the four diagonals) or leaves the N-by-N lattice. The preferred downward direction is analogous to the effect of a temperature gradient on Brownian motion, or like how when a crystal is formed, the bottom of the aggregate is cooled more than the top; or like the influence of a gravitational force. For effect, we color the particles in the order they are released according to the rainbow from red to violet. Below are three simulations with N = 176; here is an image withN = 600.

Brownian motion.Brownian motion is a random process used to model a wide variety of physical phenomenon including the dispersion of ink flowing in water, and the behavior of atomic particles predicted by quantum physics. (more applications). Fundamental random process in the universe. It is the limit of a discrete random walk and the stochastic analog of the Gaussian distribution. It is now widely used in computational finance, economics, queuing theory, engineering, robotics, medical imaging, biology, and flexible manufacturing systems. First studied by a Scottish botanist Robert Brown in 1828 and analyzed mathematically by Albert Einstein in 1905. Jean-Baptiste Perrin performed experiments to confirm Einstein's predictions and won a Nobel Prize for his work.Anappletto illustrate physical process that may govern cause of Brownian motion.

Simulating a Brownian motion.Since Brownian motion is a continuous and stochastic process, we can only hope to plot one path on a finite interval, sampled at a finite number of points. We can interpolate linearly between these points (i.e., connect the dots). For simplicitly, we'll assume the interval is from 0 to 1 and the sample points t0, t1, ..., tNare equally spaced in this interval. To simulate a standard Brownian motion, repeatedly generate independent Gaussian random variables with mean 0 and standard deviation sqrt(1/N). The value of the Brownian motion at time i is the sum of the first i increments.

Geometric Brownian motion.A variant of Brownian motion is widely used to model stock prices, and the Nobel-prize winning Black-Scholes model is centered on this stochastic process.A geometric Brownian motion with drift and volatility is a stochastic process that can model the price of a stock. The parameter models the percentage drift. If = 0.10, then we expect the stock to increase by 10% each year. The parameter models the percentage volatility. If = 0.20, then the standard deviation of the stock price over one year is roughly 20% of the current stock price. To simulate a geometric Brownian motion from time t = 0 to t = T, we follow the same procedure for standard Brownian motion, but multiply the increments, instead of adding them, and incorporate the drift and volatility parameters. Specifically, we multiply the current price by by (1 + t + sqrt(t)Z), where Z is a standard Gaussian and t = T/N Start with X(0) = 100, = 0.04.

construction of BM.

Black-Scholes formula.Move to here?

Ising model.The motions of electrons around a nucleus produce a magnetic field associated with the atom. Theseatomic magnetsact much like conventional magnets. Typically, the magnets point in random directions, and all of the forces cancel out leaving no overall magnetic field in a macroscopic clump of matter. However, in some materials (e.g., iron), the magnets can line up producing a measurable magnetic field. A major achievement of 19th century physics was to describe and understand the equations governing atomic magnets. The probability that state S occurs is given by the Boltzmann probability density function P(S) = e-E(S)/kT/ Z, where Z is the normalizing constant (partition function) sum e-E(A)/kTover all states A, k is Boltzmann's constant, T is the absolute temperature (in degrees Kelvin), and E(S) is the energy of the system in state S.

Ising model proposed to describe magnetism in crystalline materials. Also models other naturally occurring phenomena including: freezing and evaporation of liquids, protein folding, and behavior of glassy substances.

Ising model.The Boltzmann probability function is an elegant model of magnetism. However, it is not practical to apply it for calculating the magnetic properties of a real iron magnet because any macroscopic chunk of iron contains an enormous number atoms and they interact in complicated ways. TheIsing modelis a simplified model for magnets that captures many of their important properties, including phase transitions at a critical temperature. (Above this temperature, no macroscopic magnetism, below it, systems exhibits magnetism. For example, iron loses its magnetization around 770 degrees Celsius. Remarkable thing is that transition is sudden.)referenceFirst introduced by Lenz and Ising in the 1920s. In the Ising model, the iron magnet is divided into an N-by-N grid of cells. (Vertex = atom in crystal, edge = bond between adjacent atoms.) Each cell contains an abstract entity known asspin. The spin siof cell i is in one of two states: pointing up (+1) or pointing down (-1). The interactions between cells is limited tonearest neighbors. The total magnetism of the system M = sum of si. The total energy of the system E = sum of - J sisj, where the sum is taken over all nearest neighbors i and j. The constant J measures the strength of the spin-spin interactions (in units of energy, say ergs). [The model can be extended to allow interaction with an external magnetic field, in which case we add the term -B sum of skover all sites k.] If J > 0, the energy is minimized when the spins are aligned (both +1 or both -1) - this modelsferromagnetism. if J < 0, the energy is minimized when the spins are oppositely aligned - this modelsantiferromagnetism.

Given this model, a classic problem in statistical mechanics is to compute the expected magenetism. Astateis the specification of the spin for each of the N^2 lattice cells. The expected magnetism of the system E[M] = sum of M(S) P(S) over all states S, where M(S) is the magnetism of state S, and P(S) is the probability of state S occurring according to the Boltzmann probability function. Unfortunately, this equation is not amenable to a direct computational because the number of states S is 2N*Nfor an N-by-N lattice. Straightforward Monte Carlo integration won't work because random points will not contribute much to sum. Need selective sampling, ideally sample points proportional to e-E/kT. (In 1925, Ising solved the problem in one dimension - no phase transition. In a 1944 tour de force, Onsager solved the 2D Ising problem exactly. His solution showed that it has a phase transition. Not likely to be solved in 3D - see intractability section.)

Metropolis algorithm.Widespread usage of Monte Carlo methods began with Metropolis algorithm for calculation of rigid-sphere system. Published in 1953 after dinner conversation between Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller. Widely used to study equilibrium properties of a system of atoms. Sample using Markov chain using Metropolis' rule: transition from A to B with probability 1 if E 0. When applied to the Ising model, this Markov chain is ergodic (similar to Google PageRank requirement) so the theory underlying the Metropolis algorithm applies. Converges to stationary distribution.

ProgramCell.java,State.java, andMetropolis.javaimplements the Metropolis algorithm for a 2D lattice.Ising.javais a procedural programming version. "Doing physics by tossing dice." Simulate complicated physical system by a sequence of simple random steps.

Measuring physical quantities. Measure magnetism, energy, specific heat when system has thermalized (the system has reached a thermal equilibrium with its surrounding environment at a common temperature T). Compute the average energyand the average magenetizationover time. Also interesting to compute the variance of the energy orspecific heat = - 2, and the variance of the magnetization orsusceptibility = - 2. Determining when system has thermalized is a challenging problem - in practice, many scientists use ad hoc methods.

Phase transition.Phase transition occurs when temperature Tcis 2 / ln(1 + sqrt(2)) = 2.26918). Tcis known as the Curie temperature. Plot magnetization M (average of all spins) vs. temperature (kT = 1 to 4). Discontinuity of slope is signature ofsecond order phase transition. Slope approaches infinity. Plot energy (average of all spin-spin interactions) vs. temperature (kT = 1 to 4). Smooth curve through phase transition. Compare againstexact solution. Critical temperature for which algorithm dramatically slows down. Below are the 5000th sample trajectory for J/kT = 0.4 (hot / disorder) and 0.47 (cold / order). The system becomes magnetic as temperature decreases; moreover, as temperature decreases the probability that neighboring sites have the same spin increasing (more clumping).

Experiments.

Start will above critical temperature. State converges to nearly uniform regardless of initial state (all up, all down, random) and fluctuates rapidly. Zero magnetization.

Start well below critical temperature. Start all spins with equal value (all up or all down). A few small clusters of opposite spin form.

Start well below critical temperature. Start with random spins. Large clusters of each spin form; eventually simulation makes up its mind. Equally likely to have large clusters in up or down spin.

Start close to critical temperature. Large clusters form, but fluctuate very slowly.

Exact solution for Ising model known for 1D and 2D; NP-hard for 3d and nonplanar graphs.

Models phase changes in binary alloys and spin glasses. Also models neural networks, flocking birds, and beating heart cells. Over 10,000+ papers published using the Ising model.

Java Simulation --- Calculate PI using Random Numbers

import java.util.*;

public class CalculatePI2

{

public static boolean isInside (double xPos, double yPos)

{

boolean result;

double distance = Math.sqrt((xPos * xPos) + (yPos * yPos));

if (distance < 1)

result = false;

return(distance < 1);

}

public static double computePI (int numThrows)

{

Random randomGen = new Random (System.currentTimeMillis());

int hits = 0;

double PI = 0;

for (int i = 0; i