PHASE TRANSITIONS IN DETERMINISTIC MODELS

Only stochastic models can display spontaneous transitions between homogeneous and coexistence states. The nature of the transition can however also become clear in deterministic models. We will discuss these similarities first for a deterministic car following model and then for deterministic fluid-dynamical models.

Car Following Models

For the model of Eq. (2), it has been shown (9) that the homogeneous solution of the model is linearly unstable for densities where , where is the first derivative of the function , and is the gap. The instability sets in for intermediate densities; for low and high densities all models are stable in the homogeneous (laminar or jammed) state. For intermediate densities, one can select the curve and the parameter such that the model either has unstable ranges, or not.

If all parameters including the density are such that the homogeneous solution is not stable, then the system rearranges itself into a pattern of stop-and-go traffic, corresponding to the coexistence state. The density of the laminar and the jammed phase in the coexistence state are independent from the average system density, that is, if in that state system density goes up, it is reflected in the jammed phase using up a larger fraction of space.

The type of the instability is similar to the better-known instability of Eq. (3). However, once the instability is triggered in Eq. (3), it will just grow exponentially, and no stable 2-phase solution is found (e.g. (38)).

Fluid-Dynamical Models

Standard Lighthill-Whitham theory, of the type

(21)

with a strictly convex flow-density-curve , results in a 1-phase model, meaning that shocks smear out over time. When has linear sections, then in those sections shock waves are marginally stable, in the sense that disturbances to those shocks are neither amplified nor dissipated away.

Fluid-dynamical theory, of the type

(22)

and

(23)

can, depending on the choice of parameters including the $V(\rho)$-curve, either be a 1-phase/1-state or a 2-phase/3-state model (39). For example, the homogeneous solution of the model with and is linearly unstable at densities where , where is the first derivative of with respect to $\rho$ (39). This is similar to the instability condition in Sec. 6.1; note that and are, albeit related, not the same.

As pointed out before, these models are deterministic, so in no situation will these models display stochastic transitions.