Does a broken symmetry lead to the second law of Thermodynamics? (Noether's Theorem)

Question

So, Noether's Theorem states that for every conservation law, there is a corresponding symmetry. If that symmetry is broken, the conservation law no longer stands. Can this logic be applied to entropy? The second law of thermodynamics states that entropy increases over time, so it's not conserved. Does that mean there is some symmetry that has been broken? If so, what is that symmetry?

Answer 1

Not exactly. Noether's theorem is one of the most widely misunderstood results in mathematical physics. What is says is:

1st) The other way: For every (continuous) symmetry there is a conservation law.

Caveats:

c1) Symmetry is defined up to a boundary term in the Lagrangian

c2) The symmetry is off-shell (independent of the dynamical variables being solutions or not).

2nd) Then there is a conservation law that is on-shell (once the equations of motion are applied)

3rd) The theory must be derivable from an action principle by means of a Lagrangian

On top of that, the logic you're trying to apply is the inverse one:

If a quantity is not conserved, there must be a broken symmetry. That's certainly not necessarily true. We know from classical logic that, $$ p\Rightarrow q $$ is not equivalent to, $$ q\Rightarrow p $$ what is true is that, $$ p\Rightarrow q\Leftrightarrow\lnot q\Rightarrow\lnot p $$ In other words, if I'm Belgian, I'm certainly European; but if I'm European, I'm not necessarily Belgian.