  We extend the recently developed non-gaussian thermodynamic formalism
\cite{tre98} of a (presumably strongly turbulent) non-Markovian medium to its
most general form that allows for the formulation of a consistent thermodynamic
theory. All thermodynamic functions, including the definition of the
temperature, are shown to be meaningful. The thermodynamic potential from which
all relevant physical information in equilibrium can be extracted, is defined
consistently. The most important findings are the following two: (1) The
temperature is defined exactly in the same way as in classical statistical
mechanics as the derivative of the energy with respect to the entropy at
constant volume. (2) Observables are defined in the same way as in Boltzmannian
statistics as the linear averages of the new equilibrium distribution function.
This lets us conclude that the new state is a real thermodynamic equilibrium in
systems capable of strong turbulence with the new distribution function
replacing the Boltzmann distribution in such systems. We discuss the ideal gas,
find the equation of state, and derive the specific heat and adiabatic exponent
for such a gas. We also derive the new Gibbsian distribution of states. Finally
we discuss the physical reasons for the development of such states and the
observable properties of the new distribution function.
