Noether's Invariance for Heat?

Question

So we know from the first law of thermodynamics:

$$ dU = dQ + dW $$

where $U$ is internal energy, $Q$ is heat and $W$ is work done. From Noether's theorem we also know that conservation of internal energy relates to time translational invariance

$$ dU = 0 \implies \text{Time Translational Invariance}$$

I haven't seen this done yet but what invariance does one gain when

$$dQ = 0 \implies ?$$

I suspect it should be some invariance of spatial thermodynamic probabilities? In the reversible case? But how do I get that from Noether's theorem? — More Anonymous, Jan 08 '23 at 03:38
I'm not well versed in Noether's theorem, but from the little I know of it the conserved quantities seem to be state functions, whereas heat is a path function. I don't know that Noether's theorem would apply to something like Q. Is it even possible to come up with an action for a system that is coupled to surroundings? — scmartin, Jan 08 '23 at 04:02
If I think of the action as a minimization of an integral in the reversible case my answer does seem plausible. I would be willing to accept an answer which explains why the premise of the question is misled — More Anonymous, Jan 08 '23 at 04:16
Does this answer your question? Can heat be thought as a type of Noether invariant? — GiorgioP-DoomsdayClockIsAt-90, Jan 08 '23 at 05:25
FWIW, the original Noether's theorem (and its Noether current formula) assume an action formulation. Related: https://physics.stackexchange.com/q/51534/2451 — Qmechanic, Jan 08 '23 at 06:10

score 1 · Answer 1 · answered Jan 08 '23 at 03:59

I shall be providing a partial answer. We will restrict ourselves to the reversible case:

$$ Q = TdS$$

But, we also know from Gibbs entropy formula:

$$S = -k_B\sum_i p_i \ln p_i$$

Thus, $dS = 0$ implies,

$$ 0 = \sum_i (\ln p_i + 1 ) dp_i$$ Using $\sum_i p_i = 1$ $$ 0= \sum_i \ln p_i dp_i $$

Since $p_i \leq 1$ implies,

Either all the energy is in a particular energy state or there there is there is an invariance of thermodynamic probabilities.

Note: This derivation does not does apply in the irreversible case.

Noether's Invariance for Heat?

1 Answers1