  Starting from the microscopic description of a normal fluid in terms of any
kind of local interacting many-particle theory we present a well defined step
by step procedure to derive the hydrodynamic equations for the macroscopic
phenomena. We specify the densities of the conserved quantities as the relevant
hydrodynamic variables and apply the methods of non-equilibrium statistical
mechanics with projection operator techniques. As a result we obtain
time-evolution equations for the hydrodynamic variables with three kinds of
terms on the right-hand sides: reversible, dissipative and fluctuating terms.
In their original form these equations are completely exact and contain
nonlocal terms in space and time which describe nonlocal memory effects.
Applying a few approximations the nonlocal properties and the memory effects
are removed. As a result we find the well known hydrodynamic equations of a
normal fluid with Gaussian fluctuating forces. In the following we investigate
if and how the time-inversion invariance is broken and how the second law of
thermodynamics comes about. Furthermore, we show that the hydrodynamic
equations with fluctuating forces are equivalent to stochastic Langevin
equations and the related Fokker-Planck equation. Finally, we investigate the
fluctuation theorem and find a modification by an additional term.
