Comment on ‘ Garden of Eden states in traffic model revisited ’

Recently, Huang and Lin suggested a combination of two successfull mean-field theories, the 2-cluster approximation and paradisical mean-field, for the Nagel-Schreckenberg cellular automaton model of traffic flow. We argue that this new approximation is inconsistent since it violates the Kolmogorov conditions. In a recent work [1] Huang and Lin have studied a combination of two mean-field theories, the 2-cluster approximation [2, 3, 4] and paradisical mean-field (PMF) [5, 4], for the Nagel-Schreckenberg cellular automaton model of traffic flow [6] (for a review, see [7]). The suggested combined theory yields results for the flow-density relation which are worse compared to Monte Carlo simulations than those of each individual approximation alone. The authors concluded that the success of paradisical mean-field theory is accidental and cannot be improved systematically, in contrast to the cluster approximation. In this comment we will show that the combined theory as suggested by Huang and Lin is inconsistent since it violates the elementary Kolmogorov conditions. This also invalidates the conclusions about the success of paradisical mean-field theory. Here we assume translational invariance so that the probabilities P τ,τ ′ do not depend on the position. An important consequence are the so-called Kolmogorov consistency conditions [8] which for the 2-cluster approximation read τ2 P τ1,τ2 = P τ1 = τ2 P τ2,τ1 (2) where P τ is the probability to find a cell in state τ. Especially for τ = x, denoting an empty cell in the notation of [1], we have P x = 1 − ρ where ρ is the total density of cars. Since also P 1 + P 2 = ρ, instead of eq. (2) in [1] one has more precisely