Let n(x,t) denote the density of random walkers at position x ∈ Z at time t ∈ Z. At each time step t a walker makes a move x → x + Δ, where Δ = + 1 or Δ = −1 with probability 1/2.

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(a) First derive the discrete time and space mean-field equation that describes the time evolution of n(x, t). Assume the continuous time and space approximation, i.e. let t,x = R, and derive the partial differential equation according to which n(x,t) evolves. (b) Now assume that walkers enter and leave the x-axis according to the rate function g(x,t). By use of the Fourier transform dw dk n(x,t) = n(k,w)eikx+iwt 2 express n(x, t) in terms of the Fourier transform of g(x,t). Consider the time signal N(t) describing the evolution of the density of walkers at position xo > 0, i.e. N(t) = n(xo, f). Assume that walkers can only enter or leave the system at x = 0 and that they do so in an uncorrelated manner. This is represented by assuming where x(t) is a white noise signal. (c) Show that 0 2 log(PR(T)) 5 -6 -7 g(x,t) = 8(x)x(t), 00 dk (w) eikxo (w) = 12 iw + k L 0.5 1 1.5 2 2.5 2.5 log(7) to 3 3.5 ++ 4.5 5 (d) Show that for w < <1/(2x0) the signal N(t) has a 1/f power spectrum in question (a).