# Has anyone theorized a connection between entropy and quantum uncertainty?

I apologize if this kind of idle theorizing is frowned upon here, but I was wondering if it is possible that the Second Law of Thermodynamics is a consequence of quantum uncertainty.

I've heard entropy of a system defined as the number of micro-states that it can have to correspond to the macro-states it has. So that definition makes it sound like entropy is simply losing information. As we know, entropy increases as time goes on. Now this seems contradictory to me; we know more as time goes on, not less.

Is it possible that, because you can gain more and more information about a system as time goes on as you can interact with it more that, some information needs to be "hidden" from you. And that this process of losing information is entropy?

P.S. I know that what I "know" about any system does not approach the limits set out by the uncertainty principle. But as System A interacts with System B, over time System A's state is more influenced by System B and in that sense System A has gained knowledge of System B.

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Your premise "we know more as time goes on" cannot be generalized to all thermodynamic systems. Consider this experiment: You have two boxes with white and red balls which are next to each other but with a plate separating them. In the beginning all white balls are on the left, all red on the right. So you know exactly where all white and red balls are. Now pull out the plate and shake the system. The balls will mix, entropy is increased and the we know less about the positions now. Nothing is hidden, the entropy just increased over time and this is not a quantum effect. –  Alexander Nov 29 '11 at 22:01
You do know more if you heard and saw the shaking. That represents additional measurements of the system. We just don't know more about the answer to particular question you asked. In fact, the more the box was shaken, the more waves hit your body. So shaking the box increases the amount of information about the box and the thing that was shaking it that is present in your body. –  Joe Nov 29 '11 at 23:24
Just to clarify what I mean, quantum certainty is about measuring quantities, not exactly knowing things like where is the red ball or blue ball. Your example conflates quantum uncertainty with the human concept of knowing facts. Keep in mind you don't know what particles are in the red and blue balls, or which particles left or came into them. –  Joe Nov 30 '11 at 2:18
To determine the position of objects is a measurement. And one of the standard examples for the Heisenberg principle as well. I am sorry that I do not see a real question here and would recommend you to think about your problem/theory again and might try to ask a more well defined question here on this site. –  Alexander Nov 30 '11 at 14:13
My question is if any physicists have devised a theory that derives entropy as a result of quantum uncertainty. In your example you site one piece of information that you don't have because of the passage of time. I am saying you gain and lose information. (For example, observing the shaking let's you know the density of the box and balls better.) What you do is remind us that entropy has occurred. A better question would be, if you shake a box and lose no information about the position of the balls, have you gained information? –  Joe Nov 30 '11 at 15:08
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This paper considers and relates uncertainty relations, and entropic relations in an information-theoretical sense (amongst other things). Maybe it is possible to extend that to entropic relations in a physical sense.

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The third law of thermodynamics has a purely quantum-mechanical origin. However, the second law applies equally to classical and quantum-mechanical systems. For instance, it applies to an ideal gas. This tells us that the second law doesn't depend logically on quantum mechanics.

So that definition makes it sound like entropy is simply losing information. As we know, entropy increases as time goes on. Now this seems contradictory to me; we know more as time goes on, not less.

There is a difference between the information content of a system and the information that can actually be extracted from a system by measurements. This is true both classically and quantum-mechanically. In comments, Alexander gave the example of the red and white balls mixing. The system is classical, and in theory, you could observe the final state and use Newton's laws to evolve the system backward in time and find out the initial state. In practice, this fails, because the system is chaotic, so its evolution either forward or backward in time is extremely sensitive to the initial conditions. To extrapolate by a time $t$, you need measurements that have a precision that grows exponentially with $t$, and for large $t$ this rapidly becomes impossible.

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