A finite-size dynamic-scaling approach for the diffusion front of particles

The scaling behavior of the diffusion front, originated by the random motion of particles under a concentration gradient, is studied by means of the Monte Carlo method and solving the diffusion equation. Simulations are performed on the square lattice using confined geometries of size MxL, where the gradient is established along the M direction while periodic boundary conditions are set along the L direction. A dynamic scaling Ansatz is proposed such as the width of the front [w(M,t)] scales as w(M,t) approximately M(alpha)f(t/M(alpha/beta)), where alpha and beta are the roughness and growing exponents, respectively. This proposal is based on the fact that the development of w is constrained by the gradient, that decays as M(-1), in contrast to the standard Family-Vicsek scaling Ansatz where correlations are constrained by the lateral dimension of the sample. It is found that the roughness exponent exhibits a systematic dependence on the sample size that can be rationalized in terms of a finite-size correction. Extrapolation to the thermodynamic limit gives alpha=4/7, in excellent agreement with theoretical predictions linking the diffusion system to the percolation problem. The evaluation of the growing exponent gives beta=0.30+/-0.02, leading us to the conjecture beta=2/7 for the exact value.