A Fluctuating Lattice Boltzmann Method for the Diffusion Equation
  We derive a fluctuating lattice Boltzmann method for the diffusion equation. The derivation removes several shortcomings of previous derivations for fluctuating lattice Boltzmann methods for hydrodynamic systems. The comparative simplicity of this diffusive system highlights the basic features of this first exact derivation of a fluctuating lattice Boltzmann method. 
Alexander J. Wagner and Kyle Strand
physics.comp-ph cond-mat.soft