From Osher Doctorow Ph.D.
The logistic equation is in calculus terminology: 1) dx/dt = c1x(c2 - x) where c1, c2 > 0. When x and t are taken to be in the interval [0, 1] (as for example when x is taken to be a percentage of the population), then c2 can be replaced by 1: 2) dx/dt = c1x(1 - x) This is in fact the usual Rare Event scenario on [0, 1] scales. To extend and generalize the logistic equation, which is very useful for biological growth, to growth in physics and psychology and astrophysics and so on, the simplest general way would be to replace (2) by: 3) dx/dt = c1xf(x, t)(h(x, t) - xg(x, t)) where f(x, t) and g(x, t and h(x, t) are functions that depend on x and t. The case where f(x, t) = g(x, t) = h(x, t) = 1 is the ordinary logistic equation. Since x is some arbitrary variable that grows and/or contracts, why not replace x in a particular scenario by dx/dt? Then we get: 4) d^2x/dt^2 = c1(dx/dt)f(x, t)[h(x, t) - dx/dt g(x, t)] Since the equation has both enough dx/dt's and even has a second derivative d^2x/dt^2, the next simplest choice for f(x, t), h(x, t), and g(x, t) would be either a constant or a linear function of x or a linear function of t. In the bracket, RET tells us that the form of a difference suggests that one term is causal and the other is caused (with probability), so since having t or x in both terms would factor them out into f(x, t), the simplest case is to assign x to one of the two terms in the brackets and t to the other. At this stage, let us compare this picture in (4) with the difference k3 from xp and Et in quantum theory, remembering that xp = k1, Et = k2, xp - Et = k1 - k2 and we define k3 = k1 - k2: 5) mx(dx/dt) - (1/2)m(dx/dt)^2 t = k3 which factors to: 6) m(dx/dt)[x - (1/2)dx/dt] = k3 Equations (4) and (6) are identical except for k3 and d^2x/dt^2 if we set the constant m = f(x, y) and h(x, t) = x and g(x, t) = (1/2)t. This is one of the two ways that we wanted to select h and g (we could have put t and x in opposite terms) except for the constant 1/2, and it is certainly satisfactory to have f as a constant m. We also have the added "bonus" that either space (x) exerts a probable causal influence on time t or vice versa. What do we do about k3 versus d^2x/dt^2, or using differential operator notation, Dtt(x)? It certainly is not unimaginable that the acceleration Dtt (x) of x is constant. It is roughly similar to a constant applied force. We could propose to generalize the resulting equation beyond constant accelerations and just consider that constant k3 holds in certain scenarios, but like special relativity we might as well begin with something constant. So we have another way of deriving a very general "hyper-logistic" equation of growth across several domains. Osher Doctorow Ph.D.