Derivation of the Euler Equation from Hamiltonian Systems with Negligible Random Noise

The Euler equation of conservation law has been one of the fundamental equations in fluid dynamics since its discovery two centuries ago. Although there are disputes regarding its maximum range of applicability, it has been firmly established in suitable region. In principle, it is a logical consequence of Newtonian mechanics and a rigorous derivation of it from Newton’s equation should be possible. Certainly some scaling has to be chosen and Euler equation is exact only in the scaling limit. This problem indeed is much harder than it appears, as Euler equation involves thermodynamical quantities such as pressure and temperature while classical Hamiltonian systems are characterized by the pair potential. So there is a link via classical statistical mechanics which does not enter explicitly in the classical Hamiltonain systems. In other words, to prove Euler equation from classical Hamiltonian systems one in a certain sense justifies Boltzmann’s principle from classical mechanics. So far no one knows how to achieve this except in some artificial systems. The known results in this direction include: a heuristic derivation of Euler equation by assuming some strong ergodic property and all equilibrium states being Gibbs; one dimensional hard rod systems; Lanford’s theorem in the small time and low density limit. (For a review see [3].) In this note, I shall report some recent progress in this direction done in collaboration with S. Olla and S. R. S. Varadhan [2].