PHASES IN TRAFFIC

The analogy between a gas-liquid transition and the laminar-jammed transition of traffic was pointed out many times (e.g. (7,17)). The description of traffic in the well-known 2-fluid-model (23) assumes the existence of two phases; and all simulation models which use spatial queues (e.g. (24,26,25)) will display two phases because of the definition of the dynamics. The two phases in models with queues are however much easier to understand than the phases in more realistic models.

Abbildung 1: (a) Schematic representation of the gas-liquid transition in one dimension. (b) State of the gas-fluid model as a function of the density and the temperature $T$. (c) Space-time plots for different parameters. Space is horizontal; time increases downward; each line is a snapshot; vehicles move from left to right; fast cars are green, slow cars red. for all plots.

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In a gas-liquid transition, one observes the following (see also Fig. 1(a) left):

In the gas phase, at low densities, particles are spread out throughout the system. Distances between particles vary, but the probability of having two particles close to each other is very small.

In the liquid phase, at high densities, particles are close to each other. There is no crystalline structure as in solids, but the density is similar and in some cases (e.g.water) even higher in the liquid than in the gas phase. Because of the fact that the particles are so close to each other, it is difficult to compress the fluid any further.

In between, there is the so-called coexistence state, where gas and liquid coexist. In typical experiments in gravity, the liquid will be at the bottom and the gas will be above it. Without gravity, as well as for example within clouds, droplets form within the gas and remain interdispersed. In clouds, small droplets will eventually merge together into bigger droplets (coagulation), which will fall out of the cloud as rain. Without gravity, the droplets will just merge but never fall out. The final state of the system is having one big droplet of liquid, surrounded by gas.

If a system in the coexistence state is compressed, more droplets form and/or existing ones grow, but the density both inside and outside the droplets remains constant. That is, the system reacts by allocating more space to the high density phase, but not by changing the density either of the gas or the liquid phase. Let us call those two densities and . Eventually, all the space is used up by the liquid. At this point, the system will be homogeneous again and remain so if density is increased further.

The kinetics of the droplet formation (e.g. (27)) is ruled by a balance between surface tension and vapor pressure. Since surface tension pulls the droplet together, it increases the pressure inside the droplet. This interior pressure pushes water molecules out of the droplet. Vapor pressure outside the droplet is the balancing force - it pushes particles into the droplet.

Surface tension and thus interior pressure depend on the droplet radius - the smaller the droplet, the larger the surface tension and thus the interior pressure. The result is that, when coming from small densities, there is a regime, starting at , where large droplets would already be stable, but small droplets are not. That is, if the system were in equilibrium, there would be a coexistence between gas and droplets. But when coming from low density, the homogeneous gaseous phase can survive for some time. This super-critical gas is thus meta-stable. A direct consequence of meta-stability is hysteresis: When coming from low densities, it is possible to go beyond and still remain in the gaseous phase. Only after some waiting time will, by a fluctuation, some particles get close enough to each other to start the formation of a droplet.

When increasing temperature $T$ in a gas/liquid system, the 2-phase structure will eventually go away. This happens via and approaching each other and eventually meeting (see Fig. 1(b)). That is, depending on the temperature $T$, a fluid system will either display transitions from gas to coexistence and from coexistence to liquid, or there will be no transition at all.

We will now move on to describe the supporting evidence for our claims. As is typical in computational science, our evidence is based on computer simulations. It is however backed up by generic knowledge about phase transitions as they are well understood in physics.