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In quantum mechanical systems, fundamental laws are time-symmetric. But this does not hold good for entropy. If entropy was really time-symmetric it would violate the 2nd law of thermodynamics.

At first I thought the question was really similar to Is time a measurement of Entropy? but then found it majorly different. So, when a particle accounts for the case of time-symmetric property where all the fundamental laws hold true (time considering in backward motion), then its entropy should go backward as it was the then-case. But the system energy at the present reality goes in the opposite direction (increases with time) always.

How can entropy really account for the case ?

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    $\begingroup$ Open systems are not composed of fixed amount of particles. Closed systems of macroscopic size are too complex to describe particle-by-particle. $\endgroup$
    – Paul Kolk
    Commented Sep 15 at 6:33
  • $\begingroup$ @Poutnik yeah I didnt include the breaking of CPT symmetry which occur under weak interactions here $\endgroup$
    – user146560
    Commented Sep 15 at 6:50
  • $\begingroup$ @PaulKolk I surely mean here Closed system , since no matter exchange concept was included in my qn $\endgroup$
    – user146560
    Commented Sep 15 at 6:52
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    $\begingroup$ Well its my wrong it would have been CP,T symmetries . Since T(reversal symmetry) is violated when CP violations occur @Poutnik $\endgroup$
    – user146560
    Commented Sep 15 at 6:54
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    $\begingroup$ CPT is the only combination of C, P, and T that is observed to be an exact symmetry of nature at the fundamental level. $\endgroup$
    – Poutnik
    Commented Sep 15 at 6:55

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This is a comment rather than an answer. In the Dog-Flea problem (Ehrenfest Urn diffusion) one dog has fleas (minimum entropy) and after sitting next to another dog these jump between them and come to equilibrium and entropy is at a maximum. However, if we wait the fleas will all be found on the first dog again, so now entropy has decreased at least for a short time. This time is short as jumping back and forth still goes on and so a second and third and so on recurrence can occur. Thus entropy can increase and decrease as time flows normally, so it seems reasonable to propose that the same could happen if time is reversed.

(This recurrence proved a great problem for Boltzmann in his kinetic description of thermodynamics because Poincare showed that a dynamic system could return arbitrarily close to its starting position, i.e. recur and this is never observed, for example a dye dissolved in solution is never seen to reform the drop it began with. As it happens calculations show that the recurrence time soon becomes longer than the age of the universe with even moderate numbers of particles (fleas), so Boltzmann's answer to Poincare was 'you wait'. Recurrence can now be seen in wavepacket experiments where only a few oscillators are excited and should be seen in experiments on single molecules.)

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  • $\begingroup$ If the rate of increase of entropy of a closed system is say n*dS then if some particle (A)—> y(dS-d(dS)) then to conserve the net entropy change some particle say in B state should (n-y)(dS+d(dS)) . Means some particles tend to move in forward direction forwards which would oppose poincáre recurrence theorem.In closed systems, when one part of the system reduces its entropy change, another part must increase it to maintain the overall entropy balance. The local compensation will always oppose pointcare recurrence to ever reach even when time reaches infinity. $\endgroup$
    – user146560
    Commented Sep 15 at 15:18
  • $\begingroup$ Whatever the arguments are if you do the monte-carlo simulation you can see repeated recurrences. You can also get the same result from probability theory and see it experimentally as I mentioned. $\endgroup$
    – porphyrin
    Commented Sep 15 at 15:33
  • $\begingroup$ But I think my question was really for short time span thought-experiment . But I agree with you when time tends to infinity. Your explanation and effort infact needs to be praised . $\endgroup$
    – user146560
    Commented Sep 15 at 17:02

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