In mechanics ($\mathrm dV/\mathrm dx = 0$) in equilibrium, what about thermodynamics?

Question

In mechanics we say that net force acting on a system is zero ($\mathrm dV/\mathrm dx = 0$) in equilibrium. In a similar way, which quantity should remain constant in thermodynamics?

Answer 1

Thermodynamics is more likely to use a steady state equilibrium than mechanics is. And that makes the comparison challenging.

This is even more likely if an actual fluid and not just energy or entropy is in flux, which is common as fluid mechanics and thermodynamics are often used together, and have significant areas of overlap, meaning either could legitimately claim certain topics are part of it. (In fact, I’d say a pump running and flows and pressures settling down is a classic example). Fields such as thermalhydraulics cannot rightly be called subfields of either, but are certainly more defined and smaller than both. In this way thermodynamics and fluid mechanics are similar to electrodynamics.

And steady state is different from a mechanical system in equilibrium with constant velocity because thermodynamic steady state equilibria still have a forcing function and a driven, which is not true for $ \sum~F_i=a=0~,~ v=k$ in dynamics. Therefore, two dynamic systems are matched in their dynamics. For example, temperature differences directly drive heat flow, pressure differences directly drive mass flow, voltage differences directly drive current, so they must match. Nothing drives a constant-velocity object.

SHM is the only thing in mechanics that immediately comes ti my mind as comparable. But a wide variety of steady state equilibria in thermodynamics, without oscillations, are determined.

We could achieve a steady state where:

1. $\sum q_i = 0 ~,~ \frac{dT}{dt}=0$ for heat flows and temperatures. For a solid even.

2. $\frac{dP}{d\vec{s}} = 0$ for pressure

3. $\frac{dQ}{d\vec{s}} = 0$ for flows (and not in any oscillation, at every point in a river for example)

The interconnected nature also means that very often all of the above are balanced with connections between them. Viscosity and density, which determine steady state flow equilibria, are functions of pressure and temperature for example. But heat balance and hence pressure and temperature may change if the fluid dynamics vary.