If you pour a cup of tomato sauce into a 50 lb bag of chili powder and thoroughly mix, the spiciness of the combination will be very nearly equal to the spiciness of chili powder alone. This exemplifies one of three regimes which, together, I believe, account for all action–>effect pairs. I’ll call these regimes Limit Approach, Limit Meet, and Trend Reversal. For those reading at university or office, rudimentary graphs accompany the descriptions and these will hopefully contribute to the semblance of serious work being done, strengthening the immunity of this post against the cursory glances of professor or supervisor.

When a particular effect results from an action, one expects that the effect’s magnitude will grow with the strength of the action causing it. For example, adding chili powder to tomato sauce makes it spicy; adding more makes it spicier. This idea is shared by all three regimes in the preliminary region, where the strength of the action is growing from zero but is still relatively small. What differentiates one regime from another has only to do with the behavior of an effect as the strength of its associated action draws near and/or exceeds a critical level.

Limit Approach

As the name blatantly states, this category applies to action–>effect pairs where the magnitude of the effect only approaches a theoretical limit, even as the strength of the action increases with reckless abandon. This is the regime in which the chili powder example belongs. No matter how much chili powder is added, the presence of some tomato sauce necessarily means that the spiciness of the combination will never match the spiciness of chili powder alone. Similarly, the quest to reach absolute zero can only yield very cold temperatures approaching the limit, even as more energy is dedicated to the endeavor.

Limit Meet

Here, the effect reaches a limit and, once there, remains stationary, regardless of the continued strengthening of the action. One example that comes readily to mind is boiling water. The water temperature in a tea kettle increases with an increasingly hot burner until the water begins to boil, at which point it has reached its temp limit, even as the burner gets hotter. Considering action–>effect pairs belonging to this regime can lead to some conclusions imparting the scale of the implications: assuming the metal isn’t allowed to evaporate into nothingness, the water in a pot set on a surface as hot as a sun spot will be the same temperature as if the pot were on an ordinary stove top.

Trend Reversal

This is probably the most interesting regime and, refreshingly, neither of the examples I’ve picked for it have anything to do with temperature or spiciness.

There are degrees of memory loss. In mild cases, you remember enough to know you forgot, and before long you’re singing Modest Mouse. As you forget more things, you may become more aware of just how much you’re forgetting. Up to and including this point, this regime is just like the others: the magnitude of the effect (awareness that things have been forgotten) is increasing with the strength of the associated action (degree to which things have been forgotten). But it takes a surprising turn when a critical level is reached and you’ve forgotten so profoundly that you don’t realize you’ve forgotten anything at all. It’s tempting to interpret this example as pertaining solely to the extreme cases of a healthy individual versus an Alzheimer’s patient, but its applicability reaches further. For instance, in a mild case you might remember that you have a dentist appointment and also realize that you’ve forgotten the scheduled time, whereas if the case were more severe you wouldn’t remember the appointment at all and would, therefore, be oblivious to your having forgotten about it.

The plot below is shown as a discrete-time sample only because it makes more intuitive sense for the y-axis variable to be represented this way than as a continuous-time function. Granted, it’s debatable as to whether the right-most line on the awareness plot should really be on the negative side of the y-axis and not instead a dot on the x-axis. The intent was to show an immediate and complete reversal in the trend of increasing awareness, but I admit that the absence of awareness is not negative awareness. Read on for an example that reverses more explicitly.

Another action–>effect pair belonging to Trend Reversal is a balloon tethered to a small weight and continually being filled with helium as it rises. The velocity of the balloon increases with increasing volume of helium until the balloon bursts. Here, the trend quite literally reverses, with both the effect (velocity) and the balloon remnants now moving in the opposite direction. A unique aspect of this example is that the strength of the action (quantity of helium in balloon) cannot increase indefinitely because it depends on a variable (the balloon) that is shared with its effect, so that when the balloon bursts, the strength of the action immediately returns to zero.