Lattice-based walks and the Monte Carlo method for addressing mass, thermal and elasticity problems

In this paper, we review the recent developed method based around lattice-based random walks and the Monte Carlo method. This method, which is now called the Lattice Monte Carlo method, permits complex phenomenological problems in diffusion, thermal conductivity and elasticity to be addressed. It is shown how the effective mass diffusivity, thermal diffusivity/thermal conductivity and the bulk modulus in composites can be calculated and also how concentration profiles and temperature profiles can be determined in situations where the diffusivity depends on position and concentration and the thermal conductivity depends on position and temperature respectively.