Trasporto ottimo, sistemi viventi

Cardin, Franco

Matematica, cultura e società, Série 1, Tome 2 (2017) no. 3, pp. 327-341.

Riassunto (VO)

In questa rassegna si traccia un incontro con la teoria del trasporto ottimo, fornendo alcune notizie sulla sua nascita, sulla sua rinnovata riscoperta e conseguenti inattese nuove applicazioni. Si parlerà di trasporto ottimo per il sistema arterioso, della legge di Kleiber, delle radici degli alberi, e infine, della diffusione della muffa Physarum Polycephalum e delle sue caratteristiche geodetiche. Il filo conduttore delle sezioni, apparentemente scollegate, è in realtà sempre il problema di Monge-Kantorovich, a partire dalla sua lettura originale statica, passando per una lettura dinamica di tipo stazionario, fino ad una lettura dinamica di tipo non stazionario (nel caso delle muffe), tendente asintoticamente nel tempo a Monge-Kantorovich. L'esposizione che segue si accompagna ad un utilizzo di tecniche di tipo basilare. Scogli in teoria della misura e analisi funzionale sono solo accennati e ci si cimenta nel mantenere fruibile il racconto per più possibili lettori interessati o incuriositi.

Abstract

This review concerns the optimal transport theory, providing some news about its birth, its renewed rediscovery and the resulting unexpected new applications. We are talking about optimal transport for the arterial system, Kleiber's law, tree roots, and finally the diffusion of the Physarum Polycephalum mold and its geodetic features. The central thread of our tale is the problem of Monge-Kantorovich, starting from its original static formulation, passing through a stationary dynamic reading, up to a dynamic non stationary transient phase (in the case of molds), tending asymptotically in time to the stationary Monge-Kantorovich setting. The exposition is accompanied by the use of very basic techniques. Measure theory and functional analysis are merely mentioned and we try to keep the story as readable as possible to interested people.

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