Fluctuational theory of media with manifest space-time inhomogeneity

A nonlinear determined thermodynamics of media with essential violation of $(\mathbf{r},t)$ homogeneity of intensive variables and their derivatives is constructed. Either equations of the ideal liquids (IL) or ideal liquid with thermal conductivity (ILTC) are taken as balance equations in it. As basis variables, $\mu (\mathbf{r},t)$ and $T(\mathbf{r},t)$ are chosen. The hypothesis of local equilibrium is given in the form of the Gibbs–Duhem relation; conjugate coordinates are $\rho (\mathbf{r},t)$ and $\sigma (\mathbf{r},t)$, and the local potential is $P(\mu ,T)$. The potential of velocities $\nu _i(\mathbf{r},t)$ enters through the substantial derivative. The variational principle is formulated; in the case of the ILTC there naturally occurs a local decrease in the pruduction of entropy $z_2(t)=z_2^0\exp \left (-\frac {2t} {\bar t}\right )$ where $\bar t$ is the relaxation time.