Saturday, September 20, 2008

Traffic and Shock Waves

As explained in elementary terms here traffic can be modeled using a density function ρ(x, t) and a flux function j(ρ(x, t)), i.e. we assume that the number of cars passing through a point at a given time is a function of the density of cars at that point. Making certain continuity assumptions, we can obtain a conservation law

ρ_t + j’(ρ)ρ_x = 0

where subscripts indicate partial differentiation and j’ indicates differentiation with respect to ρ. If we make j(ρ)=4ρ(2-ρ) and start with a discontinuous initial density distribution like

1 if x <= 1
1/2 if 1< x <=3
2/3 if x > 3

Then we can show how the discontinuity persists over time and changes.

Such persisting discontinuities are called shock waves and appear as lines across which the density changes discontinuously. For example, in this figure we have lines of constant density intersecting at x = 3. The philosophical question is "what are we to make of this representation of a given traffic system?" That is, what does the system have to be like for the representation of a shock wave to be correct? My suggestion is that we need only a thin strip around x = 3 where the density changes very quickly, i.e. so quickly that a driver crossing it would have to decelerate to zero speed. Then, on the other side of the strip, the driver experiences a dramatic drop off in density, and so can accelerate again. Still, there is something a bit strange in talking about shock waves in traffic cases where the number of objects involved is so small, as opposed to fluid cases where many more fluid particles interact across a shock wave. Here, then, I would suggest that we have a case where the mathematics works, but we are less than sure what it is representing in the world.

See this New Scientist article (and amusing video) for the claim that shock waves can be observed in actual traffic experiments (summarizing this 2008 article).


gabriele contessa said...

Chris, this is a very nice case study. Could you expand on your comments at the end of the next-to-last paragraph though? I'm not sure I completely understand what your worry about "shock waves" is.

Chris Pincock said...

Gabriele, thanks for your comment. What I am interested in here is how the mathematically defined notion of a shock wave is interpreted differently across different physical models. Suppose that we define a shock wave as a discontinuity in density. These appear in both traffic models and models of compressible fluids. All I am trying to suggest with this remark is that the physical interpretation of what these models are representing can differ across models. Somewhat vaguely, I would say that the gap between the model and the genuine features of the system gives us a lot of interpretative flexibility, so that a shock wave for traffic can stand for something quite different than a shock wave for fluids. But I still trying to work out a more definite way of saying this!