|
所在平台: Coursera |
课程主页: https://www.coursera.org/learn/statistical-mechanics
课程评论：没有评论
课程名称：统计力学：算法与计算 课程概述：在这门课程中，您将通过下载、概括或从头编写基础计算机程序，学习大量现代物理（经典与量子）。课程将探讨算法及其在科学深度洞察中的应用。欢迎对算法和数据驱动的科学研究感兴趣的学生参与，尽管您不必具备背景知识。 课程大纲： 1. **第一周：蒙特卡罗算法（直接采样，马尔可夫链采样）** 学习基本的蒙特卡罗技术，通过3x3棋子游戏了解详细平衡、不可约性与非周期性，介绍著名的梅特罗波利斯算法，并进行有关马尔可夫链蒙特卡罗的作业。 2. **第二周：硬盘：从经典力学到统计力学** 探讨硬盘模型，了解直接采样与马尔可夫链采样的区别，研究蒙特卡罗与分子动力学算法之间的关系，学习统计物理的基本概念。 3. **第三周：熵相互作用与相变** 使用洗衣线上的衣夹模型学习熵相互作用，即统计力学的考虑，探讨如何用算法采样配置，并演示液体与固体之间的相变。 4. **第四周：采样与积分** 深入理解采样与积分之间的联系，引入麦克斯韦和玻尔兹曼速度与能量分布，进行高维积分的采样计算。 5. **第五周：密度矩阵与路径积分（量子统计力学1/3）** 开始量子统计力学的三周旅程，学习密度矩阵与路径积分的基础，探讨时间演化及其在复杂系统中的应用。 6. **第六周：利维量子路径（量子统计力学2/3）** 介绍玻色子的特性及其奇特统计，学习强大的采样算法——利维构造，并与标准采样技术进行比较。 7. **第七周：玻色-爱因斯坦凝聚（量子统计力学3/3）** 讨论玻色-爱因斯坦凝聚现象，探讨其在路径积分框架下的优雅描述及相关采样算法。 8. **第八周：伊辛模型 - 枚举与蒙特卡罗算法** 关注经典物理中的伊辛模型，学习不同的采样算法，包括局部算法和聚类算法，探索有序与无序状态的转变。 9. **第九周：动态蒙特卡罗，模拟退火** 学习动态蒙特卡罗算法和模拟退火技术，探讨其在物理优化问题中的广泛应用。 10. **第十周：蒙特卡罗的起源与总结，庆祝派对** 重温课程内容，探讨布冯针实验及利维稳定分布的复杂理论，进行课程内容的复习，并庆祝课程的圆满完成。 请注意，第十周没有作业，但还有最终考试。
Name:Monte Carlo algorithms (Direct sampling, Markov-chain sampling)
Description: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.
Name:Hard disks: From Classical Mechanics to Statistical Mechanics
Description: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.
Name:Entropic interactions and phase transitions
Description: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.
Name:Sampling and integration
Description: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!
Name:Density matrices and Path integrals (Quantum Statistical mechanics 1/3)
Description: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!
Name:Lévy Quantum Paths (Quantum Statistical mechanics 2/3)
Description: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.
Name:Bose-Einstein condensation (Quantum Statistical mechanics 3/3)
Description: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.
Name: Ising model - Enumerations and Monte Carlo algorithms
Description: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.
Name:Dynamic Monte Carlo, simulated annealing
Description: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.
Name: The Alpha and the Omega of Monte Carlo, Review, Party
Description: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!)
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.
