Finite Time Thermodynamics and Lorentz Invariance. Study of an Example

We use an example to study how finite-time thermodynamics (with finiteness constraints along with a criterion for determining optimal solutions) works when we change our space-time reference frame and move by relative motion. The example we have chosen concerns a hyperbolic conservation equation for a single chemical component, in one dimension of space. The relativistic study requires us to consider four-vectors, which can also be seen as the general necessity, of quantum origin, of placing ourselves in an open system. This is the case for entropy in the pair (entropy, entropy flow), or for the system considered in the pair (resource, resource flow). Lorentz invariance then concerns entropy production in particular, and not scalar entropy alone. The usual relativistic relations of length contraction and time dilation extend to resources and their flows: we can combine the two classes of relations and demonstrate complementary type relations between resources and their flows on the one hand, and the times and spaces concerned on the other, i.e. between the two classes of finiteness: if times are finite, so are resources, which can be neither zero nor infinite.