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I know in an isolated system entropy always tends to increase, but what about the speed of that increase (e.g. the acceleration of entropy-the derivative of its speed)? Is there any law or relation giving the rate of that increase? Can it be decreased? (I am not asking here about a way to decrease entropy itself but its rate of increase-is it always increasing, too, is it constant, or can it be decreased?) Also, is there any difference in the answer of the question if it is asked for isolated and for non-isolated (both closed and open) systems?

Can anybody show me any links to work done on the subject? Or if there isn't any such work explain me what are the reasons why nobody has tackled the issue up until now (e.g. impossibility of experimental verification, lack of theoretical framework to put the question in, difficulty to make a sound mathematical model and so on)?

Thank you in advance.

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  • $\begingroup$ I believe your first statement should say "isolated system". A closed system can exchange heat with its surroundings, so it can be cooled and experience a decrease in entropy (although this might just be different definitions). $\endgroup$ – John Mar 3 '17 at 21:11
  • $\begingroup$ In addition, the entropy will not increase indefinitely. It will reach a maximum eventually, so I guess a quick answer to your question is that yes, the rate of increase will eventually decrease to zero. $\endgroup$ – John Mar 3 '17 at 21:12
  • $\begingroup$ Thank you for the remark, I will correct it in a minute. Just wait a little bit. The mistake is on my side. $\endgroup$ – Yordan Yordanov Mar 3 '17 at 21:15
  • $\begingroup$ Is it better now? $\endgroup$ – Yordan Yordanov Mar 3 '17 at 21:20
  • $\begingroup$ Entropy is only always increasing for isolated systems. There is no such restrictions on entropy if your system is either open or closed. Also, I have just said that the rate of increase will decrease to zero eventually. $\endgroup$ – John Mar 3 '17 at 21:26
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From a thermodynamical point of view, you may consider the following: 1) Entropy is only defined for a system in the state of equilibrium. 2) An isolated system in equilibrium cannot depart from it spontaneously. Therefore, the entropy of an isolated system is constant.

From a quantum mechanical point of view: the Von Neumann entropy is invariant under unitary transformations, and since time evolution for isolated systems is unitary, the entropy remains constant.

However, I understand your question. Imagine a bottle with some gas trapped inside that you might prepare as an isolated system in equilibrium with an entropy $S_1$. Then, if the gas is released to let fill the entire room, after some time, it will reach again the equilibrium with an entropy $S_2>S_1$. As I said before, you can only have entropy at the initial and the final states, because there is no equilibrium intermediate state, therefore there's no rate of change in the entropy. In addition, that system had changed its entropy because it wasn't isolated, there had to be some external agent to open the bottle. Right?

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  • $\begingroup$ Actually I am trying to ask (I seem to have problem editing my question), if there is a system where entropy is increasing (due to the flow of energy and/or matter) can there be a model where the rate of this entropy increase is decreasing while the flow is continuing with the same rate-can there be some mechanism able to in a way "counter" the increase of entropy and allow the system to accumulate more Gibbs free energy in itself? Or is the Gibbs free energy always destined to decrease in time? $\endgroup$ – Yordan Yordanov Mar 3 '17 at 21:45
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In the absence of mass transfer, the local rate of generation of entropy in a system is proportional to the square of the velocity gradient (actually, the double dot product of the rate of deformation tensor with itself) and to the square of the temperature gradient. See Transport Phenomena by Bird, Stewart, and Lightfoot, John Wiley, NY, 2nd Edition, 2002, p372, problem 11D.1 for the exact relationship for the local rate of entropy generation. Since this relationship involves squares of the gradients, the terms involved make positive definite contributions to the entropy generation rate.

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