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In the study of the intricate dynamics of many-body systems, it is often convenient, or actually unavoidable, to resort to simpler approximate descriptions. For quantum-mechanical many-body systems of bosons it is possible to use effective one-particle equations to track the microscopic evolution of many-particle states in appropriate regimes. One of these regimes is the so called mean-field limit where the gas density rho=N/V is assumed to be large. A key open problem is to show that the many-body dynamics can be controlled in terms of an effective equation when the thermodynamic limit, i.e., large V and constant rho, is taken before the mean-field limit of large rho.

While at present time a satisfactory solution to this problem seems out of sight, in a recent joint work with J. Froehlich, P. Pickl, and A. Pizzo we propose a modest contribution in this direction by considering an interacting Bose gas of bosonic atoms at temperature zero in the regime of large volume, large density, but small ratio V/rho. We derive an effective non-linear equation for the time evolution of coherent order one excitations above the ground state of the gas and provide an explicit error bound in terms of density and volume. The effective equation allows an immediate discussion of the dispersion of low-energy excitations. For repulsive potentials we recover Bogolyubov's well-known formula for the speed of sound in the gas, and for attractive potentials we show a dynamical instability.