1. ST325
JOURNAL OF THE AMERICAN STATISTICAL
ASSOCIATION
NUMBER 247 SEPTEMBER 1949 VOLUME 44
THE MONTE CARLO METHOD
NICHOLAS METROPOLIS AND S. ULAM
LOS ALAMOS LABORATORY
The Monte Carlo Method..

2. SESSION 1
Presenter…
Mohomed Abraj.

3. Introduction
Classical statistical physics is a well understood
subject which poses, however, many difficult problems
when a concrete solution for interacting systems is
sought. In almost all non-trivial applications, analytical
methods can only provide approximate answers.
Numerical computer simulations are, therefore, an
important complementary method on our way to a deeper
understanding of complex physical systems such as
(spin) glasses and disordered magnets or of
biologically motivated problems such as protein
folding

4. Cont…
Quantum statistical problems in condensed matter or
the broad field of elementary particle physics and
quantum gravity are other major applications which,
after suitable mappings, also rely on classical simulation
techniques. we shall confine ourselves to a survey of
computer simulations based on Markov chain Monte
Carlo methods which realize the importance sampling
idea.

5. Method and Approach
 We shall present here the motivation and a
general description of a method dealing with a
class of problems in mathematical physics.
 The method is, Essentially, a statistical approach to
the study of differential equations ,or more
generally, of integro-differential equations that occur
in various branches of the natural sciences.

6.  ALREADY in the nineteenth century a sharp
distinction began to appear between two different
mathematical methods of treating physical phenomena.
 Problems involving only a few particles were studied in
classical mechanics, through the study of systems of
ordinary differential equations.
 For the description of systems with very many particles,
an entirely different technique was used, namely, the
method of statistical mechanics. In this latter approach,
one does not concent rate on the individual particles but
studies the properties of sets of particles

7. PHYSICAL SCALES FOR DILUTE GASES
Collision
Collision
Molecular
Diameter
System Size
Gradient Scale
Quantum scale Kinetic scale Hydrodynamic scale
T/T
DSMC is the dominant
numerical algorithm at the
kinetic scale
DSMC applications are expanding to multi-scale problems

8. Divide the system into cells and generate particles in each cell
according to desired density, fluid velocity, and temperature.
From density, determine number of particles in cell volume, N,
either rounding to nearest integer or from Poisson distribution.
Assign each particle a position in the cell, either uniformly or
from the linear distribution using the density gradient.
From fluid velocity and temperature, assign each particle a
velocity from Maxwell-Boltzmann distribution P(v; {u,T}) or
from the Chapman-Enskog distribution .}),,,{;( TTP uuv

9.  The Hamiltonian Function that is commonly used
representing the energy of the model when using Monte
Carlo Methods
 we would see the great importance of Monte Carlo
methods applied in Physics. Furthermore,
 Monte Carlo methods also play significant role in
quantum dynamics, physical chemistry, and related
applied fields.
 In quantum dynamics, Quantum Monte Carlo methods
solve the multi-body problems for quantum system. In
experimental particle physics.
 Monte Carlo Methods are use for designing detectors,
understanding their behavior and comparing experimental
data to theory.

10. Presenter

11. Particle Markov chain Monte Carlo
methods
Markov chain Monte Carlo and sequential Monte Carlo
methods have emerged as the two main tools to sample from
high dimensional probability distributions. Although
asymptotic convergence of Markov chain Monte Carlo
algorithms is ensured under weak assumptions, the
performance of these algorithms is unreliable when the
proposal distributions that are used to explore the space are
poorly chosen and/or if highly correlated variables are
updated independently.

12. CHARACTERISTICS OF MARKOV CHAIN
Irreducible Chain
Aperiodic Chain
Stationary Distribution
 Markov Chain can gradually forget its initial state
 eventually converge to a unique stationary distribution
 invariant distribution
Ergodic average


n
mt
tXf
mn
f
1
)(
1

13. TARGET DISTRIBUTION
 Target Distribution Function
 (x)=ce-h(x)
h(x)
in physics, the potential function
other system, the score function
c
normalization constant
 make sure the integral of (x) is 1
Presumably, all pdfs can be written in this form

14.  Each technique has it advantages and disadvantages ,Broadly, for
complex systems that may be subject to change later, the Monte-
Carlo method is preferred because of its flexibility. For simpler
systems, or studies to get a ‘feel’ for a problem, analytical methods
may suffice
 The decision as to whether the modeller should use analytical (e.g.
deterministic equations) or simulation (i.e. Monte-Carlo) methods
may be influenced by the following factors:
 Complexity
 Scope
 Accuracy.
 Future development
 Application

15. 15
SOME ADVANTAGES OF MC
Often the only type of model possible for complex systems
 Analytical models frequently infeasible
Process of building simulation can clarify understanding of
real system
 Sometimes more useful than actual application of final
simulation
Allows for sensitivity analysis and optimization of real system
without need to operate real system
Can maintain better control over experimental conditions than
real system
Time compression/expansion: Can evaluate system on slower
or faster time scale than real system

16. 16
SOME DISADVANTAGES OF MC
 May be very expensive and time consuming to build
simulation
 Easy to misuse simulation by “stretching” it beyond the
limits of credibility
 Problem especially apparent when using commercial
simulation packages due to ease of use and lack of familiarity
with underlying assumptions and restrictions
 Slick graphics, animation, tables, etc. may tempt user to
assign unwarranted credibility to output
 Monte Carlo simulation usually requires several (perhaps
many) runs at given input values
 Contrast: analytical solution provides exact values

17. Accuracy of Result..
 The Monte-Carlo simulation method is a type of
sampling procedure, thus any output is not exact but a
statistical estimate whose accuracy depends on the
number of missions or failures generated. For example if
mission parameters are of prime important (e.g.
probability of mission survival failure free) then the
number of missions to be simulated is the important
parameter. The number of system failures generated is
not necessarily important, e.g. if in 1000 mission
simulated only 5 system failures are generated, mission
reliability is none the less reasonably well established.
However, if MTBF(mean time between failures)
estimates are the prime consideration then a sufficient
number of system failures must be simulated to yield the
desired accuracy.

18. THANK YOU