# Derivation of lattice Boltzmann equation via analytical characteristic integral

## TL;DR

This paper introduces the analytical characteristic integral (ACI) lattice Boltzmann theory, rigorously deriving LB equations from the BGK Boltzmann equation, clarifying the role of collision number, and analyzing factors affecting LB method accuracy.

## Contribution

It provides a rigorous derivation of LB equations from kinetic theory using characteristics, and redefines the characteristic parameter as collision number rather than relaxation time.

## Key findings

- ACI LB theory supports existing LB models including LBGK.
- Collision number is the key parameter, not relaxation time.
- Accuracy depends on the evolution of equilibrium distribution along characteristics.

## Abstract

A lattice Boltzmann (LB) theory, analytical characteristic integral (ACI) LB theory, is proposed in this paper. ACI LB theory takes Bhatnagar-Gross-Krook (BGK) Boltzmann equation as the exact kinetic equation behind Navier-Stokes continuum and momentum equations and constructs LB equation by rigorously integrating BGK-Boltzmann equation along characteristics. It's a general theory, supporting most existed LB equations including the standard lattice BGK (LBGK) equation inherited from lattice-gas automata, whose theoretical foundation had been questioned. ACI LB theory also indicates that the characteristic parameter of LB equation is collision number, depicting the particle-interacting intensity in the time span of LB equation, instead of traditionally assumed relaxation time, and the over relaxation time problem is merely a manifestation of temporal evolution of equilibrium distribution along characteristics under high collision number, irrelevant to particle kinetics. In ACI LB theory, the temporal evolution of equilibrium distribution along characteristics is the determinant of LB method accuracy and we numerically prove it.