双向超车双车道模型中的自发对称性破缺
Spontaneous symmetry breaking in a two-lane model for bidirectional overtaking traffic
摘要：
首先，我们考虑一个单向通量ω_bar的车辆，其中每一个是其'自然'的速度v从分布P（v）的特点。交通流被建模为一个收集直的“世界线”在时间-空间平面，超车事件表示的一个固定的排队时间τ施加在超车车辆。该几何模型展示了队列形成，并允许计算自然速度v的车辆的有效平均速度w= v（v）。其次，我们将该模型扩展到两个相对的车道A和B。我们认为，在一个车道的排队时间τ是由在对向车道的交通密度。在合理的附加假设的基础上，我们建立了一组方程，耦合的两个车道，可以数值求解。似乎在控制参数ω_bar的临界值ω_bar_c以上，车道之间的对称性自发地被破坏：存在慢车道，其中在最慢的车辆后面形成长排，以及快车道，其中由于相反方向上的排之间的宽间距而容易超车。研究了模型的一个变形，其中空间车辆密度ρ bar而不是流量ω bar是控制参数。还考虑了两个通道中的不相等通量ω_bar_A和ω_bar_B。该模型所表现出的对称性破缺现象，尽管在现实交通中很难以纯粹的形式观察到，但仍然表明了这种交通的趋势。
英文摘要：
First we consider a unidirectional flux ω_bar of vehicles each of which is characterized by its `natural' velocity v drawn from a distribution P(v). The traffic flow is modeled as a collection of straight `world lines' in the time-space plane, with overtaking events represented by a fixed queuing time tau imposed on the overtaking vehicle. This geometrical model exhibits platoon formation and allows, among many other things, for the calculation of the effective average velocity w=ϕ(v) of a vehicle of natural velocity v. Secondly, we extend the model to two opposite lanes, A and B. We argue that the queuing time τin one lane is determined by the traffic density in the opposite lane. On the basis of reasonable additional assumptions we establish a set of equations that couple the two lanes and can be solved numerically. It appears that above a critical value ω_bar_c of the control parameter ω_bar the symmetry between the lanes is spontaneously broken: there is a slow lane where long platoons form behind the slowest vehicles, and a fast lane where overtaking is easy due to the wide spacing between the platoons in the opposite direction. A variant of the model is studied in which the spatial vehicle density ρ_bar rather than the flux ω_bar is the control parameter. Unequal fluxes ω_bar_A and ω_bar_B in the two lanes are also considered. The symmetry breaking phenomenon exhibited by this model, even though no doubt hard to observe in pure form in real-life traffic, nevertheless indicates a tendency of such traffic.
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