Statistical Mechanics: Algorithms and Computations

Go to Course: https://www.coursera.org/learn/statistical-mechanics

Introduction

### Course Review: Statistical Mechanics: Algorithms and Computations on Coursera #### Overview If you’re curious about modern physics and have an interest in algorithms, "Statistical Mechanics: Algorithms and Computations" is a fantastic course to consider. This course provides an engaging experience that intertwines classical and quantum physics with practical programming through Python. Whether you're a novice looking to dip your toes into the world of statistical mechanics or a seasoned physicist seeking to refine your computational skills, this course offers a comprehensive, hands-on approach to learning. #### Course Structure The course is structured over ten weeks, each focusing on different fundamental concepts of statistical mechanics. The format includes weekly lectures and tutorial videos, supplemented by downloadable Python programs to reinforce your understanding of the material. Peer-graded assignments are a key component of the curriculum; spanning from weeks 1 to 9, participants engage in practical tasks that deepen their comprehension of the lectures. These assignments account for 50% of your final grade, while the remaining half comes from a final exam, solidifying your understanding of the concepts taught throughout the course. #### Syllabus Highlights **1. Monte Carlo Algorithms & Introduction to Basics** In the first week, students familiarize themselves with Monte Carlo techniques through a fun, interactive tutorial using a pebble game. This hands-on approach kicks off the learning journey with essential concepts such as detailed balance and the Metropolis algorithm, providing a solid foundation for subsequent weeks. **2. Hard Disk Model** Week 2 dives into the hard-disk model and its historical significance in molecular dynamics. Participants learn the distinction between direct and Markov-chain sampling, bridging concepts from classical mechanics to statistical mechanics, essential for any aspiring physicist. **3. Entropic Interactions** By week 3, the course delves into entropic interactions using the clothe-pin model. It emphasizes statistical mechanics considerations, encouraging students to apply theoretical knowledge to practical problems, thus linking concepts with real-world applications. **4. Sampling & Integration** In week 4, topics spotlight Maxwell and Boltzmann distributions, critical for understanding the fundamental principles governing statistical behavior and interactions at the particle level. **5-7. Quantum Statistical Mechanics** The subsequent weeks explore quantum mechanics, beginning with density matrices and path integrals, progressing through advanced topics like the properties of bosons and the phenomenon of Bose-Einstein condensation. These sections are insightful, demonstrating how quantum statistical mechanics interconnects with algorithmic sampling techniques. **8. The Ising Model** Approaching classical physics anew in week 8, the Ising model illustrates magnetic spin interactions, which is pivotal for comprehending phase transitions. The course does well to highlight different sampling algorithms related to these models, emphasizing their practical applications. **9. Dynamic Monte Carlo & Optimization Techniques** In the penultimate week, students explore dynamic Monte Carlo methods and simulated annealing, showcasing the optimization strategies grounded in physical principles with wide-ranging applications. **10. Conclusion & Review** The final week is an engaging summary, merging historical perspectives with modern statistical theories, such as Lévy stable distributions and their relevance to the central limit theorem. #### Why You Should Take This Course 1. **Hands-On Learning**: The combination of theory and programming ensures a practical understanding of the concepts, enhancing retention and real-world application. 2. **Comprehensive Syllabus**: The well-structured content covers a breadth of topics within statistical mechanics, from basic principles to complex quantum mechanics, making it suitable for various skill levels. 3. **Community Engagement**: The peer-graded assignments foster community involvement and collaborative learning, allowing you to gain insights from fellow participants. 4. **Flexibility**: With accessible materials and quizzes, you can study at your own pace while still benefiting from structured guidance. 5. **Experts Lead the Course**: The instructors’ expertise in both physics and computational methods ensures high-quality content delivered engagingly. #### Recommendation I wholeheartedly recommend the "Statistical Mechanics: Algorithms and Computations" course on Coursera to enthusiasts of physics and aspiring scientists. Whether you’re looking to underpin your academic knowledge with practical programming skills or wishing to explore the captivating world of statistical mechanics through an algorithmic lens, you'll find this course both enlightening and enjoyable. By the end, not only will you grasp essential principles of statistical mechanics, but you’ll also possess an arsenal of computational tools to tackle a variety of scientific challenges. Enroll today, and begin your journey!

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.

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.

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.

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!

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!

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.

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.

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.

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 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!)