# Lattice Boltzmann

Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased
Second Law of Thermodynamics

# But!

Boltzmann looked at the microscopic world...

## Boltzmanns assumes

Most important microscopic interactions are

ideal / reversible / fully elastic

## But...

That contradicts the 2nd law of Thermodynamics

# Why?

Boltzmanns theory is accomplished!

## Macro-irreversibility

Boltzmann's problem
macroscopic irreversibility
based on
microscopic reversibility

# Quiet a deal breaker!

We believe the macroscopic world to follow the
Rules of Thermodynamics

This problem is called Loschmidt's paradox, but Boltzmann "solved" it.

1. There are many microstates leading to one macrostate

2. A lot of microstates have high entropy

3. Very few of microstates have low entropy

4. A lot of them lead to higher entropy

5. Very few of them lead to lower entropy

Every process occurring in nature proceeds in the sense in which the sum of the entropies [...] is nearly always increased
Boltzmann's Second Law of Thermodynamics

# What's right?

Boltzmann grounds his assumptions on molecular chaos

## Contraposition

Loschmidt et al.:
chaos assumption is wrong
there is order on microscale

• Simplicity
• Ergodicity (mostly used in Simulated Annealing)
• Natural Parallelization

# Ergodicity

Inspect one particle for a long time
$$\Leftrightarrow$$
Inspect multiple particles for a short time

# Particle Movement...

With microscale laws

# No! Macroscopic Model

### Smoothed Particle Hydrodynamics (SPH)

 Particles

### Smoothed Particle Hydrodynamics (SPH)

 Particles Kernel / Neighborhood

### Smoothed Particle Hydrodynamics (SPH)

 Particles Kernel / Neighborhood Macroscopic Model

### Smoothed Particle Hydrodynamics (SPH)

 Particles Kernel / Neighborhood Macroscopic Model Fluid / Gas

### Smoothed Particle Hydrodynamics (SPH)

 Particles Kernel / Neighborhood Macroscopic Model Fluid / Gas

# Lattice Gas

Works differently...

( Lattice Gas = Lattice Boltzmann prototype )

# No Free Movement

Lattice Gas discretizes position and velocity drastically

### The Lattices

 2D Rectangular Lattice (D2Q4) 2D Hex Lattice (D2Q6) 3D Lattice (D2Q26) 3D Neighbors

# Discret Velocities

There are only a few possible velocities

# HPP Method (1973)

The first Lattice Gas prototype

# Pauli Principle

HPP uses a kind of exclusion principle like for electrons in atoms

# 1 Particle per Direction

 $$\Leftrightarrow$$

 $\mathcal{P}($$) = \ldots • Conserves Mass • Conserves Momentum # Collisions ## Collision Operator  \mathcal{C}($$)$ = , $\mathcal{C}($$) = • Conserves mass • Conserves momentum ## Collision remark It is unclear which particle is which after a collision. But we assume Ergodicity! ## The Whole HPP$$S_{t+1} = E S_t = \mathcal{P}\mathcal{C} S_t$$E = time evolution Operator St = Lattice State at time t ## "Mean" Process # Waves in HPP # HPP Problems • Noise • Conservation of Momentum per row / column • No solution to Navier-Stokes # HPP Improvements Several methods try to solve the diseases of HPP # FHP Method (1986) Very similar to HPP but with a hex grid # FHP Hexgrid # FHP Propagation  \mathcal{P}($$)$ = $\ldots$
• Conserves Mass
• Conserves Momentum

 $\;\;\mathcal{C}($$)= • Conserves mass • Conserves momentum ## FHP Collision  \;\;\mathcal{C}($$)$=
• Conserves mass
• Conserves momentum