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According to the Heisenberg principle $\Delta E\cdot\Delta t>h/2$ and since the number of microstates in a system (regardless of if we have bosons or fermions or both) depends on the energy then does this mean that for small intervals of time, the entropy of such system can be decreased?

Also wouldnt it mean that for small intervals the arrow of time could be reversed?Thanks!

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    $\begingroup$ This question really ought to be posted on Physics.SE $\endgroup$
    – Nicolau Saker Neto
    Commented Aug 23 at 3:53
  • $\begingroup$ I mean quantum thermodynamics is very closely related to chemistry. $\endgroup$
    – Root Groves
    Commented Aug 23 at 4:45
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    $\begingroup$ Wouldn't it be the other way around: reducing time uncertainty increases energy uncertainty and therefore entropy? $\endgroup$
    – Buck Thorn ♦
    Commented Aug 23 at 7:10
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    $\begingroup$ You're going to have to tie this down a lot more carefully. The Wikipedia article, amongst other caveats, notes that the meaning of both delta E and t varies with the formulation, and the area of applicability of different formulations varies. So until you tell us precisely what you mean by them in the above I would argue that the question is unanswerable. $\endgroup$
    – Ian Bush
    Commented Aug 23 at 10:01

2 Answers 2

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It is simply a consequence of probability. Over a short time, and for few particles, the laws of thermodynamics may appear to be violated.

There is a finite, but incredibly small, chance that all the molecules of air in the place you now inhabit would move to one side, leaving half the space in vacuo and half at double pressure. That would make a reservoir of potential energy, ex nihilo. However of all the ways molecules and their vectors could be arranged, that probability is infinitesimal.

J. C. Maxwell pointed out one could theoretically harness that energy, but subsequent calculations show more energy would be spent in redirecting molecules than could be gained from them.

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  • $\begingroup$ What I am saying is fundamentally different though.The total energy of the system , since the energy eigenstates are discrete is the total energy is equal to the series of density of states*each energy at a state.According to HUP for a small amount of time energy doesnt need to be conserved and if the energy somehow gets lower than it used to be then since the density of states would become higher in lower energies than in higher energies ,that means more particles would be in the same energy state which means that the # of microstates would be decreased -> spontaneous decrease in entropy. $\endgroup$
    – Root Groves
    Commented Aug 23 at 1:48
  • $\begingroup$ It is different than classical since entropy is related to temperature , a instantenous decrease in entropy due to HUP could cause the temperature to drop for small intervals of time ,which classically isnt possible. $\endgroup$
    – Root Groves
    Commented Aug 23 at 1:50
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The 'time energy uncertainty' is not really an uncertainty principle as there is no time operator in QM. It is more like a fourier relationship as time x frequency =const. It is true that uncertainty broadening of spectral lines exist (due to collisions shortening excited state lifetime) just as a femtosecond laser pulse has a broader spectrum than a nanosecond one does, but there is nothing 'deeper' here. See Atkins & Friedman 'Molecular Quantum Mechanics' for a fuller discussion.

Out of interest the change in entropy with time is discussed in the answer to this question Is Entropy time-symmetric? although this has nothing to do with quantum processes.

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