What's the meaning for the derivatives for temperature and pressure? If we view the temperature and pressure as the function of time and space,

$$T = T(x,y,z,t) \quad ; \quad P = P(x,y,z,t)$$

then what's the meaning for the following derivatives?

$$\nabla T \quad, \quad \dfrac{\partial T}{\partial t} \quad ; \quad \nabla P  \quad ,\quad \dfrac{\partial P}{\partial t} $$

Can we apply these derivatives to describe the nonequilibrium phenonema?
 A: Formally
$$
\nabla = (\partial_x, \partial_y, \partial_z)
$$
and the vectors
$$
\nabla T, \qquad\qquad \nabla P
$$
are the (spatial) gradients of the function $T$ and $P$. They are a sort of generalization of the derivative in one dimension, and they describe the rate of change of $T$ and $P$ in the 3 dimensional space. One property of the gradient is that it points towards the direction of maximum slope.
You can try to visualize this in the following way: suppress one dimension and consider just the plane $(x,y)$; in the $z$ axis plot the temperature. Since we expect the temperature to be a continuous function, then you can imagine the plot of the function, say, $T(x,y)$ as being a big sheet covering the $(x,y)$ plane, with mountains and valleys in the $z$ direction; then the gradient $\nabla T = (\partial_x T, \partial_y T)$ at every point of this sheet points towards the most inclined direction.
$\partial_t$ gives instead the variation in time of these quantities; from the principles of thermodynamics you can expect that after long time the conditions of the system will be dictated by homogeneity and increase of entropy. Hence you have an evolution of the system, so their time derivative is nonzero (but it tends to zero as time goes to infinity).
The way non-equilibrium thermodynamics is constructed is the following: you consider small regions of space, where you can consider the thermodynamic function, e.g. $T$, to be well defined in certain small area where equilibrium thermodynamics applies. Then you consider the effect of, say, the temperature difference between close areas of space (so that $\nabla T \neq 0$) in a similar way in which you would discuss the behaviour of bodies at a given temperature. This will induce a variation in time of these quantities (ie $\partial_t T \neq 0$), that modifies $\nabla T$ and so on.
For more details, I suggest you digging into any book on this topic.
Two caveats:

*

*Since matter is made of atoms, the "small regions" of space are to be intended small with respect to the macroscopic dimensions, but still containing a large number of molecules/ atoms


*The actual equations that describe the variation of temperature etc, involve also higher derivaties of these fields, like $\nabla^2 T = \nabla \cdot \nabla T$ and so on; you have to know a bit of calculus in order to follow the math
