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Do you know of an elementary proof for the second law of thermodynamics, for example, from the Newton laws or perhaps some particular model in which it is equivalent/reduces to it?

My naive concept of entropy was it was some concave function of all velocities of a system, for example, $min(\{v\})$. Is it correct at all?

Peter Mortensen
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konto
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    To my knowledge there is no relationship whatsoever between Newton's 2nd law of motion and the 2nd law of thermodynamics. But I'm all ears. – Bob D Jan 28 '23 at 22:20
  • @BobD I used it for the instance because its simple enough. 2nd law of thermodynamics uses the entropy that i dont understand, so I tried to turn the situation and understant it using this law. – konto Jan 28 '23 at 22:25
  • You can think of entropy as something similar to randomness or messiness. Just consider your room. Without your energy input cleaning it up periodically it will get more and more messy over time. – slebetman Jan 29 '23 at 23:18
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    Observe that Newton's second law is unchanged by the transformation $t \mapsto -t$ (assuming $F$ does not explicitly depend on $t$). The same cannot be said of the 2nd law of thermodynamics. As such, it cannot be possible to derive the latter from the former. – Charles Hudgins Jan 30 '23 at 00:33
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    Information theory is a better approach. Basically, the number of states labelled with high entropy is insanely insanely larger than the ones labeled low entropy. Then you just need operations to cause a somewhat random walk in the space and you'll end up in the high entropy states with near certain probability. There is nothing stopping you from getting into the low entropy states, but it is like dropping a cork into the ocean, picking a random spot in it, then spotting the cork right there in exactly 10 years. The cork is somewhere, but the low entropy states are almost none of the ocean – Yakk Jan 30 '23 at 00:49
  • @CharlesHudgins It is possible, if we include additional assumptions on initial conditions. Symmetry of newtonian equations of motion is not really a problem. In fact adiabatically isolated many-particle newtonian systems prepared in certain way (random choice of initial conditions, no fine tuning of initial conditions) do behave in accordance with non-decrease of entropy. – Ján Lalinský Jan 30 '23 at 10:47

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You cannot derive the second law of thermodynamics from Newton's laws. Boltzmann's H-theorem was intended to do that, but it's not an actual theorem: the proof is flawed. However, although it's not a theorem, it works in reality. Go figure.

John Doty
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Reversible Newtonian mechanics is not enough; it can describe both physical behaviour (systems in non-equilibrium state spontaneously relaxing to equilibrium state) and reversed behaviour we don't observe (systems spontaneously leaving equilibrium state and going into the non-equilibrium state).

To select the "correct" behaviour of mechanical systems consistent with 2nd law, we need additional assumption, such as assumption on the initial conditions that lead to the "correct" behaviour. Most "random" initial conditions will lead to physical behaviour. But initial conditions may be fine-tuned to lead to unphysical behaviour, where the system evolves towards states of lower entropy.

Ján Lalinský
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