12. Classical Statistical Mechanics Part 1

MIT 8.333 Statistical Mechanics I: Statistical Mechanics of Particles, Fall 2013
Instructor: Mehran Kardar

This is the first of three lectures on Classical Statistical Mechanics.

License: Creative Commons BY-NC-SA

[alexanderyelland]

This lecture follows Chapter 3.7 through 3.9 in the "Kinetic Theory of Gases" for anyone following along in Kardar's textbook "Statistical Physics of Particles"

[混沌-py]

《粒子的统计力学（共26讲）》

第三章 气体动力学理论

00:00:00 守恒律 （续前 / 完）
00:42:57 零阶流体动力学
01:14:53 一阶流体动力学 （未完待续）

[G-gu]

Chapter 1: Introduction to Course and Boltzmann Equation
- Timestamp: 0:00-1:06
- Summary: Introduction to MIT OpenCourseWare and the lecture's focus on the Boltzmann equation for describing motions in a dilute gas.

Chapter 2: Boltzmann Equation and Notation
- Timestamp: 1:06-2:33
- Summary: Detailed explanation of the Boltzmann equation, including time derivatives, velocity, and external forces. Introduction of shorthand notation for derivatives.

Chapter 3: Collision Operator and Boltzmann Equation Dynamics
- Timestamp: 2:33-4:24
- Summary: Examination of the collision operator's role in the Boltzmann equation and its impact on gas particle dynamics, integrating over particle momentum.

Chapter 4: Conservation in Collisions and Hydrodynamic Equations
- Timestamp: 4:24-9:15
- Summary: Discussion on conservation in collisions, leading to simplification of the Boltzmann equation. Introduction of hydrodynamic equations to describe slow variables.

Chapter 5: Derivation of Hydrodynamic Equations
- Timestamp: 9:15-21:48
- Summary: Detailed derivation of hydrodynamic equations, exploring conserved quantities like momentum and kinetic energy, and their implications in the Boltzmann equation.

Chapter 6: Average Kinetic Energy and Energy Density
- Timestamp: 21:48-34:24
- Summary: Calculation of average kinetic energy and energy density in a gas. In-depth look at the kinetic energy part of the Boltzmann equation and its role in fluid dynamics.

Chapter 7: Linearization of Hydrodynamic Equations
- Timestamp: 34:24-40:49
- Summary: Linearization of hydrodynamic equations to simplify their solution, focusing on quantities conserved during collisions like momentum and energy density.

Chapter 8: Temperature and Pressure in Gas Dynamics
- Timestamp: 40:49-53:00
- Summary: Relationship between temperature, pressure, and gas dynamics. Discussion on the impact of local fluctuations on pressure and temperature.

Chapter 9: Perturbations and Equilibrium in Gas Dynamics
- Timestamp: 53:00-59:51
- Summary: Analysis of how perturbations in gas dynamics, like shear velocity and sound waves, evolve and reach equilibrium, exploring solutions to the linearized hydrodynamic equations.

Chapter 10: Boltzmann Equation and Relaxation to Equilibrium
- Timestamp: 59:51-1:13:01
- Summary: Examination of the Boltzmann equation's ability to drive systems towards equilibrium, investigating specific scenarios and modes that do not relax to equilibrium.

Chapter 11: Improved Solution to the Boltzmann Equation
- Timestamp: 1:13:01-1:25:37
- Summary: Development of an improved solution to the Boltzmann equation, incorporating linearization and relaxation times, to better capture the dynamics of gas particles and energy flow.

[williamdorrell]

Maybe not the best place to ask this question, but anyway:

When he chooses an f function to input to the zeroth order hydrodynamic equations he specifies the local equilibrium, however it seems there are two choices for this function. One is as he writes, one is a more general version, the version he writes has the extra characteristic that {f, H} = 0. This does not seem a necessary condition, all that he says he requires of this function is that the RHS of the Boltzmann eqn is 0, it seems like this extra poisson bracket = 0 starts to make the LHS =0 too?

Anyway, great lectures!

[danielebrandi]

came for the statistical mechanics stayed for the handwriting