Conclusion

Abstract

This book has two thrusts relating to vortex filaments in statistical equilibrium: (1) mathematical and computational approaches and (2) applications. The first thrust emphasizes the complementary nature of analysis and computation. It is wrong to assume that, because finite difference or element methods for solving the Navier–Stokes’ equations or other PDEs are able to simulate vortex systems with high fidelity, that analytical approaches or Monte Carlo are obsolete. In high dimensional systems, numerical PDEs are not a reliable or tractable way of modeling systems. Furthermore, Monte Carlo, while valuable, cannot possibly replace analysis because parameter spaces of realistic systems are too large. Also, only analysis can provide rigorous proof of the existence of phenomena, particularly phase transitions which require a continuum, observed in simulation. Meanwhile, analysis alone runs into tractability problems that frequently require approximation, hence the need for complementary computational approaches. Indeed, in the preceding chapters analytical results have often been motivated by computational results.