Figure 1.
Single-Gradient Model in d = 2
(A) Variation of the estimated threshold position with averaging time, with xT = 2 μm and λ = 2 μm.
(B) Variation of the width as a function of averaging time.
(C) Data collapse of the width at large τ for a range of parameter values. Full line shows the prediction of Equation 7 with k2d = 0.40 and α = 2.5.
(D) w(τ) as a function of decay length, with xT = 2 μm. Results for three different averaging times are shown: ×, τ = 10 s; circle, τ = 15 s; and +, τ = 22.5 s. The full line shows the prediction from Equation 7. At large λ, the simulation results deviate from the prediction since the assumption that L ≫ λ is no longer valid.
(E) Plot of the probability distribution for measuring the threshold at position x with an averaging time τ = 45 s. The full line shows a normal distribution.
(F) Scaling of the crossover time, τ×, according to Equation 13.
In (A), (B), and (E), the standard parameter values given in the text were used. In (C) and (F), * indicates the standard parameter values. For the other datasets, one parameter value was changed as follows: open circle, D = 0.5 μm2s−1; open square, J = 6.25 μm−1s−1; ×, Δx = 0.02 μm; closed circle, μ = 1 s−1; +, μ = 0.11 s−1; open diamond, xT = 1 μm; and inverted triangle, xT = 3 μm.
Figure 2.
(A) The mean threshold position fluctuates about L/2 due to the symmetry of the system.
(B) Variation of the width w as a function of averaging time.
(C) Data collapse of the width as a function of averaging time, at long times, for a range of parameter values. The full line shows Equation 19 with k~2d = 0.63 and Α~ = 2.5. * indicates the standard parameter values. For the other datasets, parameter values were changed as follows: open circle, D = 0.5 μm2s−1; open square, J = 9 μm−1s−1; ×, Δx = 0.02 μm; closed circle, μ = 1 s−1; +, μ = 0.25 s−1; diamond, L = 7.5 μm; and inverted triangle, L = 15 μm and Δx = 0.02 μm.
(D) Plot of width as a function of decay length for averaging times: ×, τ = 30 s; open circle, τ = 45 s; and +, τ = 60 s. The full line shows the prediction from Equation 19.