Dry Screening Stratification: A Particle Climbing and Settling Problem
This Demonstration is an idealization of dry screening stratification. The plot shows an end-on view of a black-bordered shaker box that extends into the screen along its border. The box houses variously sized ideal cubes of varying mass. Each of these cubes moves with a speed that is itself assumed to be a function of a given cube's mass and size. These speeds are depicted by the various colors in the starting grid configuration. The Demonstration endeavors to show the climbing and settling nature of these cubes; specifically, it seeks to show the stratification profile that emerges when particles such as these ideal cubes are steadily shaken and then climb and settle under the influence of gravity while interacting with one another or with barriers. Ideal conditions and linearity are assumed to prevail in every respect.
Cubes of various size and mass arranged randomly in a container manifest, owing to their various positions above the ground, a range of potential energies. At rest, the arrangement is largely energetically stable; when disturbed, instabilities arise as foundations shift. Any force applied to the container, such as that which arises from the vibrational stress imposed upon the shaker box, translates into and throughout the container and catalyzes these instabilities; in response, dislodged cubes are forced to attain new lowest possible energy states (stability). In an environment of repeated collision and deflection as in the shaker box, individual cube mobilities localize this stability to well-defined regions consistent with the strata of a stratification profile. This is the fundamental physical premise that guides the Demonstration.
All the cubes in the shaker box are assumed to be involved in ongoing collisions, either with neighboring cubes or barriers. At initialization the starting material is divided into cells (cells represent cubes), and each cube is randomly assigned one of six possible speeds. Importantly, this speed is assumed to be a function of the cube's mobility, which in this problem is qualified by cube mass and size. The automaton neighborhood is a von Neumann neighborhood, or five cells in the form of a plus sign (+), and this neighborhood's center cell (CC) is defined to be the point of impact for cube-to-cube, head-on, vibration-induced collisions. Cube movement is constrained to be orthogonal along the directions defined by the neighborhood. Relative cube speeds are the basis of all collisions: within any local neighborhood, cubes with greater speeds (mobility) are defined to travel in the respective Cartesian negative-axis direction, while cubes with lesser speeds are defined to travel in the respective Cartesian positive-axis direction. Thus a possible head-on collision may occur when, relative to the CC, a northern neighbor's speed is greater than a southern neighbor's speed (this scenario is north moving south in the direction of the negative axis, and south moving north in the direction of the positive axis), or when an eastern neighbor's speed is greater than a western neighbor's speed (this scenario is east moving west in the direction of the negative axis, and west moving east in the direction of the positive axis).
You can see from the rule and the orthogonality constraint that no other neighborhood speed configurations can result in a head-on collision. When either of these head-on collision configurations are sensed by the search, update, feedback cellular automaton (SCA), a rule is enforced: head-on collisionorthogonal deflection. The rule simulates its right-hand side (RHS) deflection by way of an incremental clockwise rotation of the neighborhood (referred to as a "von Neumann swap" atop the shaker box plot). In words, the rule says, "whenever cubes come together, they are to separate." The RHS of the rule enters the problem computationally by way of the speeds that were assigned to all the cubes as a function of their mobility (mobility is delimited by mass and size). The left-hand side of the rule enters the computation by way of the head-on collision definitions that are also based on speed. As concerns conservation principles, the deflecting neighborhood rotation is done so as to conserve cube particle number. Furthermore, though deflection entirely alters local speed arrangements, no net change is introduced in momentum relative to the center of mass of the local collision neighborhood; and since the collisions are ideally assumed to be perfectly elastic, local energy is conserved as well (local speeds are simply rearranged, but in no way altered, which simulates the elastic nature of the collisions that the Demonstration assumes). Cube spin-angular momentum, a very real aspect of the problem, is assumed to converge to zero mean by the central limit theorem and randomization. Cube interactions with the shaker box's sides follow a similar logic as do the cube-to-cube interactions just described, but here, instead of orthogonal deflection following a head-on collision, collisions with a side force a complete recoil swap of the CC and the neighbor situated across the CC and opposite the side with which contact was made. The same conservation considerations with respect to cube particle number, linear and angular momentum, and energy are enforced.
The final profile shows an energetically stable configuration. The cube speeds have been "decomposed" into a spectrum that is a global non-collision arrangement. In keeping with the relative speed definitions above, the slower top strata cubes are moving north, and the faster bottom strata cubes are moving south; the cubes are tending to separate all along the axis—this is the signature of the SCA's rule; the separation tendency per se is characteristic of stratification. From the vantage point of the Demonstration's main abstraction, namely that cube speed is a function of cube mass and size, the slower, sluggish, less mobile cubes ascended to the top of the profile, while the more mobile ones descended toward the bottom. Assuming that cube collisions free up the cubes so that (predominantly) gravity may then catalyze settling (collide/mixdeflect/settle), a practical, physical explanation of the dynamics that foster the final profile from the second vantage point might be that the more mobile cubes tend to attain to their lowest energy state in relatively better response to the force of gravity than the larger, more massive ones, which are inhibited by the friction set up by their larger size and by inertia. As more relatively mobile cubes lower their energy (at the expensive of shaker box vibrational energy) and ultimately attain to their (ever-changing) new ground state, commensurately more bases are formed by these cubes all throughout the profile upon which relatively less mobile cubes may climb (again, at the expense of shaker box vibrational energy) or float (when size precludes climbing and settling) in an energetically stable manner. These bases are not static; they shift levels in lockstep with the new ground states that are set by cube arrangement destabilizations (see premise), and appear as a continuum as the cube arrangement moves from randomness toward some measure of stratification (especially early in the settling process), thus they are hard to follow early on in the SCA output (it helps to use the "color scheme" control here). Nevertheless, these bases are "string-like discrete" in the final profile in that they appear to manifest as the interfaces between the strata.