Differential Equations: Less is More

Differential equations embody the parsimony and elegance of our Universe. It is natural to expect that as phenomena rise in complexity, differential equations—ordinary (ODEs) or partial (PDEs)—should rise in their order. This expectation is not how the Universe generates its endlessly fascinating phenomena.  First or second order differential equations embrace the majority of natural phenomena. Rates of change and rates of rates of change suffice to tame complicated natural phenomena. Radioactive decay, money in a bank growing at a fixed interest rate, population growth, diffusion of heat through a rod, the waves generated by a vibrating string—whether longitudinal or transverse—diffusion of a disease through a population, the dynamics of motion as described by Newtons laws in the non-relativistic domain, the dynamic probability distribution of Brownian Motion: all these seemingly different phenomena are embraced by differential equations, and thereby elegantly unified. The Chapman-Kolmogorov equation and the Fokker-Planck equation describe the dynamic probability distributions of general stochastic processes, subject only to mild restrictions. Both require nothing more complicated than second-order differential equations. That differential equations should provide such an embarrassment of riches is unexpected and gratifying. Why third and higher orders of ODEs or PDEs are scarce in Nature is not clear to this author. It is another manifestation of the Principle of Parsimony in our Universe: faced with alternative explanations of a phenomenon, the simplest should be preferred. Einstein characteristically said it best: All things should be made as simple as possible but no simpler. And along the same lines, Einstein remarked: “Raffiniert ist der Herr Gott, aber bothaft ist er nicht,” which translates to “God is sophisticated, but he is not malicious.” This is a gift that convinces the author that we are living in the best of all possible worlds.