Abstract:
It is well-known that the Liouville equation of statistical mechanics is restricted to systems where the total number of particles (N) is fixed. In this paper, we show how the Liouville equation can be extended to systems where the number of particles can vary, such as in open systems or in systems where particles can be annihilated or created. A general conservation equation for an arbitrary dynamical variable is derived from the extended Liouville equation following Irving and Kirkwood's2 technique. From the general conservation equation, the particle number conservation equation is obtained that includes general terms for the annihilation or creation of particles. It is also shown that the grand canonical ensemble distribution function is a particular stationary solution of the extended Liouville equation, as required. In general, the extended Liouville equation can be used to study nonequilibrium systems where the total number of particles can vary.

Abstract:
Small perturbation of the Liouville equation under smooth initial data is considered. Asymptotic solution which is available for a long time interval is constructed by the two scale method.

Abstract:
An Ansatz for the Poincar\'e metric on compact Riemann surfaces is proposed. This implies that the Liouville equation reduces to an equation resembling a non chiral analogous of the higher genus relationships (KP equation) arising in the framework of Schottky's problem solution. This approach connects uniformization (Fuchsian groups) and moduli space theories with KP hierarchy. Besides its mathematical interest, the Ansatz has some applications in the framework of quantum Riemann surfaces arising in 2D gravity.

Abstract:
We present solutions to the classical Liouville equation for ergodic and completely integrable systems - systems that are known to attain equilibrium. Ergodic systems are known to thermal equilibrate with a Maxwell-Boltzmann distribution and we show a simple derivation of this distribution that also leads to a derivation of the distribution at any time t. For illustrative purposes, we apply the method to the problem of a one-dimensional gravitational gas even though its ergodicity is debatable. For completely integrable systems, the Liouville equation in the original phase space is rather involved because of the group structure of the integral invariants, which hints of a gauge symmetry. We use Dirac's constrained formalism to show the change in the Liouville equation, which necessitates the introduction of gauge-fixing conditions. We then show that the solution of the Liouville equation is independent of the choice of gauge, which it must be because physical quantities are derived from the distribution. Instead, we derive the solution to the classical Liouville equation in the phase space where the dynamics involve ignorable coordinates, a technique that is akin to the use of the unitarity gauge in spontaneously broken gauge theories to expose the physical degrees of freedom. It turns out the distribution is time-independent and precisely given by the generalized Gibbs ensemble (GGE), which was solved by Jaynes using the method of constrained optimization. As an example, we apply the method to the problem of two particles in 3D interacting via a central potential.

Abstract:
The notion of Laplace invariants is transferred to the lattices and discrete equations which are difference analogs of hyperbolic PDE's with two independent variables. The sequence of Laplace invariants satisfy the discrete analog of twodimensional Toda lattice. The terminating of this sequence by zeroes is proved to be the necessary condition for existence of the integrals of the equation under consideration. The formulae are presented for the higher symmetries of the equations possessing integrals. The general theory is illustrated by examples of difference analogs of Liouville equation.

Abstract:
The most general vortex solution of the Liouville equation (which arises in non-relativistic Chern-Simons theory) is associated with rational functions, $f(z)=P(z)/Q(z)$ where $P(z)$ and $Q(z)$ are both polynomials, $\deg P<\deg Q\equiv N$. This allows us to prove that the solution depends on $4N$ parameters without the use of an index theorem, as well as the flux quantization~: $\Phi=-4N\pi(sign \kappa)$.

Abstract:
Canonical coordinates for both the Schroedinger and the nonlinear Schroedinger equations are introduced, making more transparent their Hamiltonian structures. It is shown that the Schroedinger equation, considered as a classical field theory, shares with the nonlinear Schroedinger, and more generally with Liouville completely integrable field theories, the existence of a "recursion operator" which allows for the construction of infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket. The approach may provide a good starting point to get a clear interpretation of Quantum Mechanics in the general setting, provided by Stone-von Neumann theorem, of Symplectic Mechanics. It may give new tools to solve in the general case the inverse problem of Quantum Mechanics.

Abstract:
Suggestions concerning the generalization of the geometric quantization to the case of nonlinear field theories are given. Results for the Liouville field theory are presented.

Abstract:
The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the Liouville equation are presented and then used to solve specific boundary value problems. The results are compared with exact solutions satisfying the same boundary conditions. All three discretizations are on four point lattices. One preserves linearizability of the equation, another the infinite-dimensional symmetry group as higher symmetries, the third one preserves the maximal finite-dimensional subgroup of the symmetry group as point symmetries. A 9-point invariant scheme that gives a better approximation of the equation, but significantly worse numerical results for solutions is presented and discussed.