Toward a canonical-Hamiltonian and microscopic field theory of classical liquids

The old problem of describing the classical dynamics of simple liquids, modeled by standard many-particle Hamiltonians with two-body interactions, in terms of canonically conjugated collective variables, which are field variables, is taken up in this paper where both periodic and free boundary conditions are considered. Particle densities or their Fourier components are well known to be the natural collective coordinates of the problem but what about their conjugate variables? An original application of the method of generating functions of the type of Hamilton's principal function is developed to construct their conjugated momenta which turn out to be momentum potentials. The many-particle Hamiltonians are then written in terms of these two fields. The kinetic part turns out to be exactly that of a compressible, irrotational and inviscid fluid modulo the minor change of the variables, i.e., particle density and momentum potential instead of mass density and velocity potential. The potential energy is non-local in density since it expresses the mutual interactions of the particles in the initial Hamiltonian. This property is absent from any theory of classical fluids. The resulting equations of motions are (1) the continuity equation for the time derivative of the density and (2) a generalized Bernoulli equation for the time derivative of the momentum potential, generalized in the sense that it is non-local in density. To our knowledge this is a new equation in this field. Several applications and generalizations of this equation are also discussed in the text as well as the question of the thermodynamic versus the continuum limits.