Computational Lab in Physics:
Monte Carlo Evaluation of Distribution
Functions.
Monte Carlo simulation of
a decay of a J/y meson (charmanticharm) into an e+e- pair.
Probability Distribution Functions:
Stochastic Processes in Physics

Stochastic variables:

Variables that fluctuate from one realization
of a system to another.
Thermal effects.
 Manufacturing uncertainties.
 Quantum processes.


Simple example: The 1-D Random Walk.
2
Random Walk and Stochastic
Processes


Physical System is characterized by parameters that
vary randomly.


RW: two discrete values: ±1 distance units
 + is right, - is left
Want to calculate a global parameter that can be
evaluated or measured.


RW: The particle will take N steps. At each step, there is a
50/50 chance for the particle to move right or left.
RW: Total initial displacement from the origin after N steps.
General problem: Predict the probability that the
global variables possess a specific value when
averaged over all trial experiments.

RW: Probability that the walk terminates at a given
displacement from the origin.
3
Making a random walk program in root:
int numberOfSteps = 40;
int numberOfRealizations = 1e4;
TH1D* mRandomWalkHisto = new TH1D
("mRandomWalkHisto","Random
Walk",2*numberOfSteps+1,-numberOfSteps0.5,numberOfSteps+0.5)
4
Code for loop
TRandom3 rnd3(0); // initialize random number generator with unique seed
for (int iRealization = 0;
iRealization<numberOfRealizations;
++iRealization) {
int position = 0;
for (int iStep = 0; iStep<numberOfSteps; ++iStep) {
double a = rnd3.Rndm(); //random number between 0-1
double step = 1;
if (a<0.5) step=-1;
// at this point, 50% of the time step will be 1
// and 50% of the time step will be -1
position += step;
} // loop over steps
//cout << "Realization " << iRealization << ", position " << position <<
endl;
mRandomWalkHisto.Fill(position);
}// loop over realizations
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Result: Random Walk Histogram
6
Examples of Histograms of
Random Distributions
7
Homework 8, Random-Walk

Code the Random Walk program in ROOT.


Modify it to use a 2-D Histogram to do a 2-D random
walk with unit length steps in which the angle that the
walker describes with respect to any fixed axis is a
uniformly distributed random variable on [0, 2p]. Use 40
steps, and also use unit width. (30 points)
Chapter 10 Exercises


1. Decay of monoenergetic pions. t=2.6 x 10-8
s. E=200 MeV. Sample of 108 pions. How many
survive after 20 m? (40 points)
2. Same as 1, but with a Gaussian distribution
of energies: mE=200 MeV, sE=50 MeV. (30
8
points)
Relativistic Kinematics

You are given:




Kinetic Energy, Pion Mas
And we know: E = g mc 2 , KE = E - mc 2
Therefore:
E
KE + mc 2 KE
g= 2=
= 2 +1
2
mc
mc
mc
From which we can get the pion
velocity:
b = 1-1/ g 2
9
Lorentz Boost transformations

Given the boost velocity b, and the
corresponding Lorentz factor g, if we
measure the time and space
separation of two events in one
frame, the corresponding
separations in the frame moving
with velocity b are given by:
æ cDt ö æ g
ç
÷ = çç
è Dx ø è bg
bg
g
öæ
ö
cDt
'
֍
֏ Dx ' ֿ
ø
10