Computational Physics Lectures: Introduction to Monte Carlo methods
Contents
Monte Carlo methods, plan for the lectures
Monte Carlo: Enhances algorithmic thinking!
Domains and probabilities
Monte Carlo methods, tossing a dice
Stochastic variables
Stochastic variables and the main concepts, the discrete case
Stochastic variables and the main concepts, the continuous case
The cumulative probability
Properties of PDFs
Important distributions, the uniform distribution
Gaussian distribution
Exponential distribution
Expectation values
Stochastic variables and the main concepts, mean values
Stochastic variables and the main concepts, central moments, the variance
First Illustration of the Use of Monte-Carlo Methods, integration
First Illustration of the Use of Monte-Carlo Methods, integration
First Illustration of the Use of Monte-Carlo Methods, variance in integration
Monte-Carlo integration, meaning of variance
First Illustration of the Use of Monte-Carlo Methods, integration
Important aspects of Monte-Carlo Methods
Why Monte Carlo integration?
Why Monte Carlo integration?
Why Monte Carlo integration? Example
Monte Carlo integration, simple example
Monte Carlo integration, simple example
Monte Carlo integration, simple example
Monte Carlo integration, simple example
Monte Carlo integration, algorithm
Monte Carlo integration, simple example, the program
Monte Carlo integration, simple example and the results
Testing against the trapezoidal rule for a one-dimensional integral
Second example, particles in a box
Second example, particles in a box
Second example, particles in a box
Second example, particles in a box, discussion
Second example, particles in a box, discussion
Simple demonstration using python
Brief Summary
Probability Distribution Functions
Probability Distribution Functions
The three famous Probability Distribution Functions
Probability Distribution Functions, the normal distribution
Probability Distribution Functions, the normal distribution
Probability Distribution Functions, the cumulative distribution
Probability Distribution Functions, other important distribution
Probability Distribution Functions, the binomial distribution
Probability Distribution Functions, Poisson's distribution
Probability Distribution Functions, Poisson's distribution
Meet the covariance!
Meet the covariance in matrix disguise
Meet the covariance, uncorrelated events
Numerical experiments and the covariance
Numerical experiments and the covariance
Numerical experiments and the covariance, actual situations
Numerical experiments and the covariance, our observables
Numerical experiments and the covariance, the sample variance
Numerical experiments and the covariance, central limit theorem
Definition of Correlation Functions and Standard Deviation
Definition of Correlation Functions and Standard Deviation
Definition of Correlation Functions and Standard Deviation
Definition of Correlation Functions and Standard Deviation
Definition of Correlation Functions and Standard Deviation, sample variance
Definition of Correlation Functions and Standard Deviation
Example of the central limit theorem
Random Numbers
Random Numbers, better name: pseudo random numbers
Random number generator RNG
Random number generator RNG and periodic outputs
Random number generator RNG and its period
Random number generator RNG, other examples
Random number generator RNG, other examples
Random number generator RNG, RAN0
Random number generator RNG, RAN0
Random number generator RNG, RAN0
Random number generator RNG, RAN0
Random number generator RNG, RAN0 code
Properties of Selected Random Number Generators
Properties of Selected Random Number Generators
Properties of Selected Random Number Generators
Simple demonstration of RNGs using python
Properties of Selected Random Number Generators
"Correlation function and which random number generators should I use":"https://github.com/CompPhysics/ComputationalPhysics/tree/master/doc/Programs/RNGCodes/autocorrelation.cpp"
"Correlation function and which random number generators should I use":"https://github.com/CompPhysics/ComputationalPhysics/tree/master/doc/Programs/RNGCodes/autocorrelation.cpp"
Correlation function and which random number generators should I use
Which RNG should I use?
How to use the Mersenne generator
Improved Monte Carlo Integration
Change of Variables
Change of Variables
Transformed Uniform Distribution
Exponential Distribution
Exponential Distribution
Exponential Distribution
Normal Distribution
Normal Distribution
Normal Distribution
Normal Distribution
Normal Distribution
Importance Sampling
Importance Sampling
Importance Sampling
Importance Sampling
Importance Sampling
Importance Sampling
Importance Sampling, a simple example
Importance Sampling, a simple example, a simple plot
Importance Sampling, a simple example, the code part
Importance Sampling, a simple example, and the results
Acceptance-Rejection Method
Acceptance-Rejection Method
Acceptance-Rejection Method
Monte Carlo Integration of Multidimensional Integrals
Monte Carlo Integration of Multidimensional Integrals
Monte Carlo Integration of Multidimensional Integrals
Monte Carlo Integration of Multidimensional Integrals
Monte Carlo Integration of Multidimensional Integrals
Brute Force Integration
Importance Sampling
Python codes
Python codes, importance sampling
Correlation function and which random number generators should I use
The following Python code plots the results for the correlation function from the above program.
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