The Mystery of Disorder: Why Entropy Follows Strict Rules ⚡ экспресс
The authors established two new 'precursor' majorization relations (a way to compare distributions by how evenly spread they are), from which the properties of supermodularity and subadditivity follow for any sum of concave functions on the majorization lattice. This allowed them to rigorously prove that Tsallis entropies (for all α) and Rényi entropies (for α > 1) possess these properties in a lattice-theoretic sense, with subadditivity being always strict. As a bonus, analogous results for Shannon entropy were confirmed. The work reveals a common algebraic nature behind the behavior of uncertainty measures—just as symmetry principles underlie conservation laws in physics.
Entropy is a measure of disorder. Two shuffled decks of cards, dumped into one pile, won’t become more chaotic than the sum of their individual disorders. This is the rule of subadditivity: the whole is no more chaotic than its parts. And if you add a joker, it’ll stir up more confusion in an already shuffled deck than in an ordered one—that’s supermodularity, where a system reacts more sharply to something new the more “smeared out” it already is.
Mathematicians have proven that all functions glued together from smoothly convex “humps” (concave curves) behave this way. Ordinary entropy, used in computers, and its exotic variants—all are sums of such humps. So they automatically obey both inequalities. The most striking part—the inequalities turned out to be strict. When you mix any non-identical systems, disorder always increases beyond the sum, as if merging two different decks gives birth to an unpredictable new chaos. This discovery, built on the work of Ludwig Boltzmann and John von Neumann, matters for physics, data compression, and even the study of black holes and the mysteries of the Big Bang.
🎯 Entropy was born in the 19th century out of attempts to improve steam engines, and later snuck into computer science as a measure of a message’s surprise.
🎬 In Isaac Asimov’s story “The Last Question,” entropy threatens the very existence of the Universe—and only a supercomputer manages to reverse it.