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Werner Krauth
In this course you will learn a whole lot of modern physics (classical and quantum) from basic computer programs that you will download, generalize, or write from scratch, discuss, and then hand in. Join in if you are curious (but not necessarily knowledgeable) about algorithms, and about the deep insights into science that you can obtain by the algorithmic approach.
Register for this course and see more details by visiting:
OpenCourser.com/course/sysnal/statistical
What's inside
Syllabus
Monte Carlo algorithms (Direct sampling, Markov-chain sampling)
Dear students,
welcome to the first week of Statistical Mechanics: Algorithms and Computations!
Here are a few details about the structure of the course: For each week, a lecture and a tutorial videos will be presented, together with a downloadable copy of all the relevant python programs mentioned in the videos. Some in-video questions and practice quizzes will help you to review the material, with no effect on the final grade. A mandatory peer-graded assignment is also present, for weeks from 1 to 9, and it will expand on the lectures' topics, letting you reach a deeper understanding. The nine peer-graded assignments will make up for 50% of the grade, while the other half will come from a final exam, after the last lecture.
In this first week, we will learn about algorithms by playing with a pebble on the Monte Carlo beach and at the Monaco heliport. In the tutorial we will use the 3x3 pebble game to understand the essential concepts of Monte Carlo techniques (detailed balance, irreducibility, and a-periodicity), and meet the celebrated Metropolis algorithm. Finally, the homework session will let you understand some useful aspects of Markov-chain Monte Carlo, related to convergence and error estimations.
Read more
Syllabus
Monte Carlo algorithms (Direct sampling, Markov-chain sampling)
Dear students,
welcome to the first week of Statistical Mechanics: Algorithms and Computations!
Here are a few details about the structure of the course: For each week, a lecture and a tutorial videos will be presented, together with a downloadable copy of all the relevant python programs mentioned in the videos. Some in-video questions and practice quizzes will help you to review the material, with no effect on the final grade. A mandatory peer-graded assignment is also present, for weeks from 1 to 9, and it will expand on the lectures' topics, letting you reach a deeper understanding. The nine peer-graded assignments will make up for 50% of the grade, while the other half will come from a final exam, after the last lecture.
In this first week, we will learn about algorithms by playing with a pebble on the Monte Carlo beach and at the Monaco heliport. In the tutorial we will use the 3x3 pebble game to understand the essential concepts of Monte Carlo techniques (detailed balance, irreducibility, and a-periodicity), and meet the celebrated Metropolis algorithm. Finally, the homework session will let you understand some useful aspects of Markov-chain Monte Carlo, related to convergence and error estimations.
Here are a few details about the structure of the course: For each week, a lecture and a tutorial videos will be presented, together with a downloadable copy of all the relevant python programs mentioned in the videos. Some in-video questions and practice quizzes will help you to review the material, with no effect on the final grade. A mandatory peer-graded assignment is also present, for weeks from 1 to 9, and it will expand on the lectures' topics, letting you reach a deeper understanding. The nine peer-graded assignments will make up for 50% of the grade, while the other half will come from a final exam, after the last lecture.
In this first week, we will learn about algorithms by playing with a pebble on the Monte Carlo beach and at the Monaco heliport. In the tutorial we will use the 3x3 pebble game to understand the essential concepts of Monte Carlo techniques (detailed balance, irreducibility, and a-periodicity), and meet the celebrated Metropolis algorithm. Finally, the homework session will let you understand some useful aspects of Markov-chain Monte Carlo, related to convergence and error estimations.
Read more
Hard disks: From Classical Mechanics to Statistical Mechanics
In Week 2, you will get in touch with the hard-disk model, which was first simulated by Molecular Dynamics in the 1950's. We will describe the difference between direct sampling and Markov-chain sampling, and also study the connection of Monte Carlo and Molecular Dynamics algorithms, that is, the interface between Newtonian mechanics and statistical mechanics. The tutorial includes classical concepts from statistical physics (partition function, virial expansion, ...), and the homework session will show that the equiprobability principle might be more subtle than expected.
Entropic interactions and phase transitions
After the hard disks of Week 2, in Week 3 we switch to clothe-pins aligned on a washing line. This is a great model to learn about the entropic interactions, coming only from statistical-mechanics considerations. In the tutorial you will see an example of a typical situation: Having an exact solution often corresponds to finding a perfect algorithm to sample configurations. Finally, in the homework session we will go back to hard disks, and get a simple evidence of the transition between a liquid and a solid, for a two-dimensional system.
Sampling and integration
In Week 4 we will deepen our understanding of sampling, and its connection with integration, and this will allow us to introduce another pillar of statistical mechanics (after the equiprobability principle): the Maxwell and Boltzmann distributions of velocities and energies. In the homework session, we will push the limits of sampling until we can compute the integral of a sphere... in 200 dimensions!
Density matrices and Path integrals (Quantum Statistical mechanics 1/3)
Week 5 is the first episode of a three-weeks journey through quantum statistical mechanics. We will start by learning about density matrices and path integrals, fascinating tools to study quantum systems. In many cases, the Trotter approximation will be useful to consider non-trivial systems, and also to follow the time evolution of a system. All these topics, including the matrix-squaring technique, will be reviewed in detail in the homework session, where you will also study the anharmonic potential.
Note that previous knowledge of quantum mechanics is not really necessary to go through the next three weeks. Follow us in our journey through algorithms and physics, and don't forget to ask on the forum if you have any doubt!
Note that previous knowledge of quantum mechanics is not really necessary to go through the next three weeks. Follow us in our journey through algorithms and physics, and don't forget to ask on the forum if you have any doubt!
Lévy Quantum Paths (Quantum Statistical mechanics 2/3)
In Week 6, the second quantum week, we will introduce the properties of bosons, indistinguishable particles with peculiar statistics. At the same time, we will also go further by learning a powerful sampling algorithm, the Lévy construction, and in the homework session you will thoroughly compare it with standard sampling techniques.
Bose-Einstein condensation (Quantum Statistical mechanics 3/3)
At the end of our quantum journey, in Week 7, we discuss the Bose-Einstein condensation phenomenon, theoretically predicted in the 1920's and observed in the 1990's in experiments with ultracold atoms. In the path-integral framework, an elegant description of this phenomenon is in term of permutation cycles, which will also lead to a great sampling algorithm, to be discussed in the homework session.
Ising model - Enumerations and Monte Carlo algorithms
In Week 8 we come back to classical physics, and in particular to the Ising model, which captures the essential physics of a set of magnetic spins. This is also a fundamental model for the development of sampling algorithms, and we will see different approaches at work: A local algorithm, the very efficient cluster algorithms, the heat-bath algorithm and its connection with coupling. All of these will be revisited in the homework session, where you will get a precise control over the transition between ordered and disordered states.
Dynamic Monte Carlo, simulated annealing
Continuing with simple models for spins, in Week 9 we start by learning about a dynamic Monte Carlo algorithm which runs faster than the clock. This is easily devised for a single-spin system, and can also be generalized to the full Ising model from Week 8. In the tutorial we move towards the simulated-annealing technique, a physics-inspired optimization method with a very broad applicability. You will also revisit this in the homework session, and apply it to the sphere-packing and traveling-salesman problems.
The Alpha and the Omega of Monte Carlo, Review, Party
The lecture of Week 10 includes the alpha and the omega of our course. First we repeat the experiment of Buffon's needle, already performed in the 18th century, and then we touch the sophisticated theory of Lévy stable distributions, and their connection with the central limit theorem. In the tutorial there will be time for a review of the entire course material, and then a little party is due, to celebrate the end of the course!
(There is no homework session for Week 10, but don't forget that the final exam is still there!)
(There is no homework session for Week 10, but don't forget that the final exam is still there!)
Good to know
Good to know
Know what's good ,
what to watch for , and
possible dealbreakers
Recommended for students and professionals in physics who want to gain a deep understanding of modern physics through practical algorithms
Emphasizes algorithmic approaches to provide insights into scientific concepts, particularly for those interested in algorithms
Led by Werner Krauth, a renowned physicist recognized for significant contributions to the field
Covers advanced topics such as quantum statistical mechanics, the Ising model, and dynamic Monte Carlo algorithms
Includes peer-graded assignments and a final exam, providing opportunities for assessment and feedback
May not be suitable for complete beginners in physics, as it assumes some foundational knowledge
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Reviews summary
Highly praised statistical mechanics course
Learners largely positive about this top-notch course on Statistical Mechanics, focusing on Monte Carlo methods. Engaging assignments, well explained videos, and well-designed exercises help students master the content. The course provides a good introduction for those who need fundamental algorithms in statistical mechanics but not a background in physics. It is challenging but super informative, with subtle and efficient algorithms and many fields that borrow ideas from statistical mechanics.
Assignments and exercises help master the content
"Engaging assignments and very insightful algorithms to learn!"
"Very well designed course with well thought out syllabus and nice demonstrations."
Course is challenging but very informative
"It's not easy. But you can learn a lot and have much fun if you complete the course!"
"Really challenging course but super informative. Highly recommend"
Excellent course materials and well-explained videos
"High quality, very well explained"
Course emphasizes computational tools and algorithms
"Emphasis on computation and sampling algorithms"
"This course teaches you the fundamentals of statistical mechanics focusing mostly on the algorithms."
Quantum mechanics section needs improvement
"The section on quantum mechanics would benefit from a good introduction of quantum mechanics for the uninitiated."
Too much homework and peer-correction assignments
"although the homework were illuminating, there was a bit too much homework and peer-correction."
"It would be better if they could consolidated the assignments into 5 homework every other week and we could do only 2 peer-corrections instead of 3."
Activities
Be better prepared
before
your course. Deepen your understanding
during
and
after
it. Supplement your coursework and achieve mastery of the topics covered
in Statistical Mechanics: Algorithms and Computations with these
activities:
Organize and consolidate course content
Show steps
Organizing and consolidating your course materials will help you retain information more effectively.
Show steps
-
Review your notes, assignments, quizzes, and exams.
-
Identify key concepts and ideas.
-
Create a system for organizing your materials.
Solve Monte Carlo exercises
Show steps
Give yourself extra practice to strengthen your understanding of Monte Carlo algorithms.
Show steps
-
Work through example exercises provided in the course materials.
-
Attempt practice problems at the end of each chapter or section.
-
Seek out additional practice problems or exercises online.
Explore Monte Carlo tutorials
Show steps
Watching a video tutorial will serve as a secondary reinforcement of the lecture material on Monte Carlo algorithms.
Show steps
-
Look for tutorials from reputable sources such as Coursera, edX, or YouTube channels like Khan Academy or The Great Courses.
Four other activities
Expand to see all activities and additional details
Show all seven activities
Connect with professionals
Show steps
Seeking guidance from experienced professionals can provide you with valuable insights and support.
Show steps
-
Attend industry events or conferences.
-
Reach out to professionals on LinkedIn or other social media platforms.
-
Ask your professors for recommendations.
Create a comprehensive resource list
Show steps
Compiling a list of relevant resources will provide you with a valuable reference for future learning.
Show steps
-
Search for articles, books, websites, and other resources on Monte Carlo algorithms.
-
Evaluate the quality and relevance of each resource.
-
Organize the resources into a coherent structure.
-
Document your findings in a clear and concise manner.
Develop a Monte Carlo simulation
Show steps
Putting your knowledge into practice by developing your own simulation will deepen your understanding of the concepts.
Show steps
-
Choose a problem or question that interests you and can be solved using a Monte Carlo simulation.
-
Design and implement your simulation.
-
Test and refine your simulation.
-
Document your simulation.
Develop a presentation on a specific topic
Show steps
Creating a presentation will help you synthesize your knowledge and improve your communication skills.
Show steps
-
Choose a specific topic that you are interested in.
-
Research your topic thoroughly.
-
Develop an outline for your presentation.
-
Create visual aids to support your presentation.
-
Practice your presentation.
Organize and consolidate course content
Show steps
Organizing and consolidating your course materials will help you retain information more effectively.
Show steps
Show steps
Show steps
- Review your notes, assignments, quizzes, and exams.
- Identify key concepts and ideas.
- Create a system for organizing your materials.
Solve Monte Carlo exercises
Show steps
Give yourself extra practice to strengthen your understanding of Monte Carlo algorithms.
Show steps
Show steps
Show steps
- Work through example exercises provided in the course materials.
- Attempt practice problems at the end of each chapter or section.
- Seek out additional practice problems or exercises online.
Explore Monte Carlo tutorials
Show steps
Watching a video tutorial will serve as a secondary reinforcement of the lecture material on Monte Carlo algorithms.
Show steps
Show steps
Show steps
- Look for tutorials from reputable sources such as Coursera, edX, or YouTube channels like Khan Academy or The Great Courses.
Four other activities
Expand to see all activities and additional details
Show all seven activities
Connect with professionals
Show steps
Seeking guidance from experienced professionals can provide you with valuable insights and support.
Show steps
Show steps
Show steps
- Attend industry events or conferences.
- Reach out to professionals on LinkedIn or other social media platforms.
- Ask your professors for recommendations.
Create a comprehensive resource list
Show steps
Compiling a list of relevant resources will provide you with a valuable reference for future learning.
Show steps
Show steps
Show steps
- Search for articles, books, websites, and other resources on Monte Carlo algorithms.
- Evaluate the quality and relevance of each resource.
- Organize the resources into a coherent structure.
- Document your findings in a clear and concise manner.
Develop a Monte Carlo simulation
Show steps
Putting your knowledge into practice by developing your own simulation will deepen your understanding of the concepts.
Show steps
Show steps
Show steps
- Choose a problem or question that interests you and can be solved using a Monte Carlo simulation.
- Design and implement your simulation.
- Test and refine your simulation.
- Document your simulation.
Develop a presentation on a specific topic
Show steps
Creating a presentation will help you synthesize your knowledge and improve your communication skills.
Show steps
Show steps
Show steps
- Choose a specific topic that you are interested in.
- Research your topic thoroughly.
- Develop an outline for your presentation.
- Create visual aids to support your presentation.
- Practice your presentation.
Career center
Learners who complete Statistical Mechanics: Algorithms and Computations will develop knowledge and skills
that may be useful to these careers:
Machine Learning Engineer
Machine Learning Engineer
Machine Learning Engineers design, develop, and maintain machine learning models. They use a variety of statistical techniques to train and evaluate models. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate complex systems and how to use statistical techniques to analyze data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of machine learning models.
Data Analyst
Data Analyst
Data Analysts use statistical techniques to analyze data and extract insights. They work in a variety of industries, including finance, healthcare, and technology. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate complex systems and how to use statistical techniques to analyze data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of complex systems.
Physicist
Physicist
Physicists study the fundamental laws of nature. They use a variety of mathematical and statistical techniques to analyze data and develop theories. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate complex systems and how to use statistical techniques to analyze data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of matter and energy.
Statistician
Statistician
Statisticians collect, analyze, and interpret data. They work in a variety of fields, including healthcare, education, and government. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate complex systems and how to use statistical techniques to analyze data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of complex systems.
Data Scientist
Data Scientist
Data Scientists use statistical techniques to analyze data and extract insights. They work in a variety of industries, including finance, healthcare, and technology. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate complex systems and how to use statistical techniques to analyze data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of complex systems.
Quantitative Analyst
Quantitative Analyst
Quantitative Analysts use mathematical and statistical techniques to analyze financial data. They develop models to predict the performance of stocks, bonds, and other financial instruments. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate financial markets and how to use statistical techniques to analyze financial data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of financial markets.
Actuary
Actuary
Actuaries use mathematical and statistical techniques to assess risk and uncertainty. They work in a variety of industries, including insurance, finance, and healthcare. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate complex systems and how to use statistical techniques to analyze data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of risk and uncertainty.
Epidemiologist
Epidemiologist
Epidemiologists use statistical techniques to study the spread of disease. They work in a variety of settings, including academia, industry, and government. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate complex systems and how to use statistical techniques to analyze data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of complex epidemiological systems.
Biostatistician
Biostatistician
Biostatisticians use statistical techniques to analyze data in the field of biology. They work in a variety of settings, including academia, industry, and government. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate complex systems and how to use statistical techniques to analyze data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of complex biological systems.
Financial Engineer
Financial Engineer
Financial Engineers develop and implement financial models. They use a variety of mathematical and statistical techniques to analyze financial data. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate financial markets and how to use statistical techniques to analyze financial data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of financial markets.
Econometrician
Econometrician
Econometricians use statistical techniques to analyze economic data. They work in a variety of settings, including academia, industry, and government. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate complex systems and how to use statistical techniques to analyze data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of complex economic systems.
Operations Research Analyst
Operations Research Analyst
Operations Research Analysts use mathematical and statistical techniques to solve problems in a variety of industries, including manufacturing, logistics, and healthcare. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate complex systems and how to use statistical techniques to analyze data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of complex systems.
Software Engineer
Software Engineer
Software Engineers design, develop, and maintain software applications. They use a variety of programming languages and tools to create software that meets the needs of users. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate complex systems and how to use statistical techniques to analyze data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of complex systems.
Financial Analyst
Financial Analyst
Financial Analysts work to assess the performance of stocks, bonds, and other financial instruments. They use a variety of statistical techniques to analyze data. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate financial markets and how to use statistical techniques to analyze financial data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of financial markets.
Risk Analyst
Risk Analyst
Risk Analysts use mathematical and statistical techniques to assess risk and uncertainty. They work in a variety of industries, including finance, insurance, and healthcare. The course Statistical Mechanics: Algorithms and Computations will help you develop the skills necessary to be successful in this role. You will learn how to use Monte Carlo algorithms to simulate complex systems and how to use statistical techniques to analyze data. This course will also provide you with a strong foundation in the theory of statistical mechanics, which is essential for understanding the behavior of risk and uncertainty.
For more career information including salaries, visit:
OpenCourser.com/course/sysnal/statistical
Reading list
We've selected eight books
that we think will supplement your
learning. Use these to
develop background knowledge, enrich your coursework, and gain a
deeper understanding of the topics covered in
Statistical Mechanics: Algorithms and Computations.
Is the course textbook and provides a comprehensive overview of the topics covered in the course. It valuable resource for both students and instructors.
Provides a comprehensive overview of Monte Carlo methods in statistical physics. It valuable resource for both students and instructors.
Save
Provides a comprehensive overview of quantum statistical mechanics. It valuable resource for both students and instructors.
Provides a comprehensive overview of simulated annealing. It valuable resource for students and instructors who are interested in this topic.
Provides a comprehensive overview of statistical mechanics. It valuable resource for students and instructors who are interested in this topic.
Provides a comprehensive overview of quantum mechanics. It valuable resource for students and instructors who are interested in this topic.
Save
Provides a comprehensive overview of classical mechanics. It valuable resource for students and instructors who are interested in this topic.
Provides a comprehensive overview of electromagnetism. It valuable resource for students and instructors who are interested in this topic.
For more information about how these books relate to this course, visit:
OpenCourser.com/course/sysnal/statistical
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Good to know
Recommended for students and professionals in physics who want to gain a deep understanding of modern physics through practical algorithms
Emphasizes algorithmic approaches to provide insights into scientific concepts, particularly for those interested in algorithms
Led by Werner Krauth, a renowned physicist recognized for significant contributions to the field
Covers advanced topics such as quantum statistical mechanics, the Ising model, and dynamic Monte Carlo algorithms
Includes peer-graded assignments and a final exam, providing opportunities for assessment and feedback
May not be suitable for complete beginners in physics, as it assumes some foundational knowledge
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