Mechanical equilibrium: thermodynamics and classical mechanics

Question

A similar question was asked here but mine is a bit different.

In thermodynamics, a mechanical equilibrium is defined as a uniform pressure (for a fluid).

In classical mechanics, equilibrium is defined by: sum of external forces equals to zero.

The link between the two is that no external forces work at an equilibrium. Why do we use pressure for the definition of thermodynamics, while it does not cover solid mechanics (because stress tensor is not a scalar tensor)? Would it be possible to give a global definition, compatible in both frameworks? I'm thinking of "an equilibrium is a state where all the macroscopic velocities are 0 with respect to an inertial frame". Unfortunately, this does not constitute a definition similar to common definitions of equilibrium in thermodynamics, which involve conjugate intensive and extensive quantities.

Answer 1

The definitions are equal: Sum of external forces zero, sum of external torques is zero. This comes from classical mechanics.

For a perfect ideal fluid, the external force density is the pressure gradient: $\mathbf f = -\nabla p$, and therefore, uniform pressure in a fluid means no external force on it, and then it is in mechanical equilibrium. So, its more convinient for thermodynamics, to define mechanical equilibrium in terms of pressure. But this definition is completely equivalent to the definition from classical mechanics. The definition of mechanical equilibrium is also valid for continuum mechanics.

There is a catch therefore. We can do thermodynamics of "everything". So, in general, the state of a system includes: generalized displacement, generalized force, temperature. May include others. For an hydrostatic system this becomes volume, pressure and temperature. Therefore, the way you define mechanical equilibrium for general thermodynamical systems, might result in a different definition from classical mechanics. And more, since this is too general, there may not be a definition of mechanical equilibrium which remains valid for every possible general thermodynamical system.

Also, what you said about the velocities is not correct. For example: A gas on thermodynamical equilibrium has a velocity distribution of particles/molecules/whatever but it is on mechanical equilibrium too: Sum of external torques and forces in all molecules are zero. So, I cannot take a reference frame such that the whole system has zero velocity, but system is in mechanical equilibrium.