  In previous work we provided the explicit form of the nonlinear PDEs,
subjected to the appropriate boundary conditions, which have to be satisfied by
transport coefficients for systems out of Onsager's region. Since the proposed
PDEs are obtained without neglecting any term present in the balance equations
(i.e., the mass, momentum, and energy balance equations), we propose them as a
good candidate for describing also transport in thermodynamic systems in
turbulent regimes. As a special case, we derive the nonlinear PDEs for
transport coefficients when the thermodynamic system is subjected to two
thermodynamic forces. In this case, the obtained PDE is, in thermodynamical
field theory (TFT), analogous to Liouville's equation in Riemannian (or
pseudo-Riemannian) geometry. The validity of our model is tested by analyzing a
concrete example where Onsager's relations manifestly disagree with experience:
transport in Tokamak-plasmas. More specifically, we compute the electron mass
and energy losses in turbulent FTU (Frascati Tokamak Upgrade)-plasmas. We show
the agreement between the theoretical predictions and experimental
observations. This approach allows for predicting the values of the Bohm and
the gyro-Bohm coefficients. To the best of our knowledge, it is the first time
that such coefficients have been evaluated analytically. The aim of this series
of works is to apply our approach to the Divertor Tokamak Test facility (DTT),
to be built in Italy, and to ITER.
