Filtration, flow in narrow channels and traffic flow are examples of processes subject to blocking when the channel conveying the particles becomes too crowded. If the blockage is temporary, which means that after a finite time the channel is flushed and reopened, one expects to observe a maximum throughput for a finite intensity of entering particles. We investigate this phenomenon by introducing a queueing theory inspired, circular Markov model. Particles enter a channel with intensity λ and exit at a rate μ. If N particles are present at the same time in the channel, the system becomes blocked and no more particles can enter. After an exponentially distributed time with rate the blockage is cleared and the system resets to an empty channel. We obtain an exact expression for the steady state throughput (including the exiting blocked particles) for all values of N. For N  =  2 we show that the throughput assumes a maximum value for finite λ if . The time-dependent throughput either monotonically approaches the steady state value, or reaches a maximum value at finite time. We demonstrate that, in the steady state, this model can be mapped to a previously introduced non-Markovian model with fixed transit and blockage times. We also examine an irreversible, non-Markovian blockage process with constant transit time exposed to an entering flux of fixed intensity for a finite time and we show that the first and second moments of the number of exiting particles are maximized for a finite intensity.

Particle conveying channels may be bundled together. The limited carrying capacity of the constituent channels may cause the bundle to be subject to blockages. If coupled, the blockage of one channel causes an increase in the flux entering the others, leading to a cascade of failures. Once all the channels are blocked, no additional particles may enter the system. If the blockages are of finite duration, the system reaches a steady state with an exiting flux that is reduced compared to the incoming one. We propose a stochastic model consisting of N channels, each with a blocking threshold of N particles. Particles enter the system's open channels according to a Poisson process, with an equally distributed input flux of intensity Λ. In an open channel the leading particle exits at a rate μ and a blocked channel unblocks at a rate [Formula: see text], where [Formula: see text]. We present and explain the methodology of an analytical description of the behavior of bundled channels. This leads to exact expressions for the steady-state output flux, for [Formula: see text], which promises to extend to arbitrary N and N. The results are applied to compare the efficiency of conveying a particulate stream of intensity Λ using a single, high capacity (HC) channel with multiple channels of a proportionately reduced low capacity (LC). The HC channel is more efficient at low input intensities, while the multiple LC channels have a higher throughput at high intensities. We also compare [Formula: see text] coupled channels, each of capacity N  =  2 with the corresponding number of independent channels of the same capacity. For [Formula: see text], if [Formula: see text], the coupled channels are always more efficient. Otherwise the independent channels are more efficient for sufficiently large Λ.

We model a particulate flow of constant velocity through confined geometries, ranging from a single channel to a bundle of Nc identical coupled channels, under conditions of reversible blockage. Quantities of interest include the exiting particle flux (or throughput) and the probability that the bundle is open. For a constant entering flux, the bundle evolves through a transient regime to a steady state. We present analytic solutions for the stationary properties of a single channel with capacity N ≤ 3 and for a bundle of channels each of capacity N = 1. For larger values of N and Nc, the system's steady state behavior is explored by numerical simulation. Depending on the deblocking time, the exiting flux either increases monotonically with intensity or displays a maximum at a finite intensity. For large N we observe an abrupt change from a state with few blockages to one in which the bundle is permanently blocked and the exiting flux is due entirely to the release of blocked particles. We also compare the relative efficiency of coupled and uncoupled bundles. For N = 1 the coupled system is always more efficient, but for N > 1 the behavior is more complex.

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We introduce the Iris Billiard, consisting of a point particle enclosed by a unit circle enclosing a central scattering ellipse of fixed elongation (defined as the ratio of the semi-major to the semi-minor axes). When the ellipse degenerates to a circle, the system is integrable, otherwise it displays mixed dynamics. Poincaré sections are displayed for different elongations. Recurrence plots are then applied to the long-term chaotic dynamics of trajectories launched from the unstable period-2 orbit along the semi-major axis i.e., one that initially alternately collides with the ellipse and the circle. We obtain numerical evidence of a set of critical elongations at which the system transitions to global chaos. The transition is characterized by an endogenous escape event, E , which is the first time a trajectory launched from the unstable period-2 orbit misses the ellipse. The angle of escape, θ esc and distance of closest approach, d min of the escape event are studied, and are shown to be exquisitely sensitive to the elongation. The survival probability that E has not occurred after n collisions is shown to follow an exponential distribution.