If R(t) is constant, or is following some repeating pattern, then we can say that the Reactor is in a Persistent State. The following questions arise in regard to Persistent States:-
For example, in a plane, the Manhattan distance between the point P1 with coordinates (x1, y1) and the point P2 at (x2, y2) is
The Manhattan Distance between two points is not dependent on the route taken. This is illustrated in the figure below; the three routes, a, b and c between P1 and P2 have the same length.
If we monitor the Manhattan Distances between Reactor Compositions at different times and find a period during which all of the distances are zero, we can therefore say that the Reactor is in a Persistent State.
It would be unduly restrictive to insist that subsequent states must be identical, and so we can identify a tolerance Manhattan Distance, Dtol, and say that the Reactor is in a Persistent State during a period if the Manhattan Distance between any two Reactor Compositions during the period is no greater than Dtol.
Reducing Dtol can be regarded as increasing the sensitivity to differences between compositions. A series of Reactor Compositions that are regarded as a single Persistent State when a particular value of Dtol is used may be separated out into different states if the value of Dtol is reduced.
It is possible that R(t) varies cyclically, such that it traverses a large region in the space of possible compositions, but returns periodically to a composition that is close to (ie within Dtol of) the original composition. This situation can also be regarded as a Persistent State.
In order to provide a mechanism enabling recognition and discrimination of Persistent States, a 'Manhattan Plot' facility has been added to SimSoup.
An example plot is included in the figure above. Each point on the triangular plot indicates the normalised Manhattan Distance between Reactor Compositions at times 'Base Time' (y axis) and 'Time' (x axis). A light tone signifies a low distance between the two Instantaneous States. Darker tones signify larger distances.
Two light toned points along a horizontal section have similar Reactor Compositions. 'Sensitivity' determines the darkness of the display for a particular distance. Increasing Sensitivity increases the darkness for a particular value of Manhattan Distance. At Sensitivity = 10, the plot has maximum darkness for any distance above one tenth of the maximum possible distance Dmax, where Dmax is calculated as twice the number of Molecules in the Reactor at 'Base Time'.
The plot can be read as follows. The point Base Time = 100,000, Time = 90,000 (towards the top right of the plot) is light grey. This signifies that the Reactor Compositions at these two times are close. Following a horizontal line leftwards from this point, we see that the plot remains light grey until about Time = 30,000. This signifies that at all times between 30,000 and 90,000, the Reactor Composition was close to that at Time = 100,000. Similarly, if we follow the plot to the right, we can see that at times between 90,000 and 100,000, the Reactor Composition was also close to that at time 100,000.
In short, after time 30,000 the Reactor is in a Persistent State in which the Reactor Composition varies only slightly.
Tracing leftwards beyond Time = 30,000, the plot becomes increasingly dark. This signifies that at these times the Reactor Composition was substantially different to that at Time = 100,000.
Note that while a horizontal line of light tone indicates a (roughly) constant Reactor Composition, a horizontal line of black does not indicate constant state; it simply signifies that the Reactor Compositions re substantially different to that at the Base Time. They may or may not be similar to each other.
In the example plot above, the Reactor Composition is in fact in a considerable state of flux during the period up to Time = 30,000. This can be seen by focusing on the horizontals for Base Time = 30,000 and earlier (bottom left of the figure). These horizontals are black except for all but the latest Times (those that are close to the diagonal). This indicates that whatever Base Time we choose during this period, its Reactor Composition is substantially different to that for earlier times.