Introduction to the Lattice Boltzmann Method

Lecture Overview

• Date: Winter term 2022-2023
• Method: Introduction to the Lattice Boltzmann (LB) Method
• Important Note: Lecture is being recorded; refrain from speaking if sensitive material is to be discussed.
• Mask Policy: Please wear masks if feeling unwell.

Key Objectives

• Understand the Lattice Boltzmann Method with a focus on the Boltzmann equation.
• Relate the Boltzmann equation to the Navier-Stokes equations.
• Derive the Ether theorem which describes the direction of time progression.

The Boltzmann Equation

• The Boltzmann equation describes the statistical behavior of a thermodynamic system away from equilibrium.
• The equation involves the Collision integral and conserved quantities.
• Understanding the Boltzmann equation is crucial before delving into the Lattice Boltzmann equation.
• Key Concepts:
  • Probability density function, Collision operator, Conservation of mass, energy, and momentum.

Three Views of the World

1. Atomistic View:

   • Focuses on distinct particles (atoms/molecules).
   • Describes particle motion using Newton's laws and binary forces.
   • Important for understanding the dynamics at the microscopic level.

2. Continuous View:

   • Described by the Navier-Stokes equations.
   • Assumes a fluid's properties can be described as a continuous field.
   • Involves terms for velocity, pressure, and diffusion.
   • Conservation of mass, momentum, and energy.

3. Kinetic/Mesoscopic View:

   • Based on the Boltzmann transport equation (BTE).
   • Considers indistinguishable particles.
   • Focuses on statistical mechanics rather than individual particle dynamics.
   • Probability density function describes the distribution of particles in phase space.

Kinetic Theory

• The Boltzmann transport equation includes:
  • Time derivative of probability density function
  • Advection terms based on microscopic velocities
  • Collision operator accounting for particle interactions
• Collision Assumptions:
  • Only binary collisions are considered.
  • Assumes molecular chaos where colliding particles are uncorrelated prior to collision.

Collision Dynamics

• Collision conservation laws:
  • Mass, momentum, energy, and the number of particles are conserved during collisions.
• The Collision integral expresses how particles gain or lose their states during interactions.

Ether Theorem

• Describes how systems evolve toward equilibrium.
• Key Findings:
  • When gain equals loss in collisions, the system reaches equilibrium.
  • H (or Ether) function is defined as an integral of the probability density function multiplied by its logarithm.
  • The derivative of H with respect to time shows the approach to equilibrium (entropy).

Final Remarks

• Homework assignments will include coding tasks based on the theories discussed.
• Students are encouraged to form groups for collaborative learning.
• Next lecture will delve further into the implications of the Ether theorem and the Boltzmann equation.