Mathematics · Statistical Mechanics

A particle is moving on a one-dimensional discrete lattice with lattice spacing unity. It can jump to its nearest neighbor site every 1/5 seconds, with p being the probability to move right and q = (1 - p) being the probability to move left. Given that the particle starts at the origin (x = 0) at time t = 0 and taking p = 3/4, calculate the variance ⟨(x - ⟨x⟩)2⟩ at time t = 5 × 104 seconds.