Understanding Entropy and Fluctuation: The 2nd Law of Thermodynamics Explained

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Discussion Overview

The discussion revolves around the 2nd law of thermodynamics, specifically focusing on the concept of entropy, its implications, and the idea of recurrence in thermodynamic systems. Participants explore various interpretations of entropy, the potential for entropy to decrease over time, and the statistical mechanics that underpin these concepts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes entropy as a measure of disorder and discusses its implications in the context of gas molecules mixing and potentially unmixing over time.
  • Another participant introduces the concept of the "recurrence paradox" and questions the time it would take for a certain number of gas molecules to unmix.
  • There is a query about whether the recurrence paradox has been resolved, leading to a mention of statistical mechanics providing a mathematical definition of entropy.
  • Participants speculate on the time required for unmixing, suggesting it could be extremely long and possibly factorial in relation to the number of molecules.
  • One participant asserts that the guesses regarding the time for recurrence are correct, emphasizing that the recurrence time increases much faster than linear with particle number.

Areas of Agreement / Disagreement

Participants express differing views on the implications of entropy and the recurrence paradox. While some agree on the mathematical underpinnings provided by statistical mechanics, there remains uncertainty regarding the practical implications and the exact nature of recurrence time.

Contextual Notes

There are unresolved questions regarding the specific time scales involved in the recurrence paradox and the assumptions underlying the estimates of recurrence time based on the number of molecules.

japplepie
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The 2nd law of thermodynamics state that entropy increases with time and entropy is just a measure of how hard it is to distinguish a state from another state (information theoretical view) or how hard it is to find order within a system (thermodynamic view). There are many ways to view entropy but these are the two that I find most pleasing and they are actually equivalent.

Let's consider a box with 2 kinds of identical but distinguishable (but enough to interfere with the interactions) gas molecules which are initially separated; after a while they mix and become more disorderly due to the random motion of molecules. This seems to agree with the 2nd law of thermodynamics.

But, after a very long time, the randomness would eventually create a fluctuation where the gas would unmix and lead back to the initial state; where they are separated.

Does this mean that entropy could decrease after a long time?
 
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Thats called "recurrence paradox". Can you estimate how long it takes for, say, 10 gas molecules to unmix? 100 molecules, 10^23 molecules?
 
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has it ever been resolved?
 
japplepie said:
has it ever been resolved?

Yes, through the methods of statistical mechanics. These give us a crisp mathematical definition of entropy free of the somewhat fuzzy "how hard?" in your original post, and yield the laws of thermodynamics as statistical predictions.

Statistical mechanics might be the most unexpectedly cool thing in physics. Quantum mechanics and relativity are cool too, but even people who don't know them know they're cool; stat mech comes as a surprise.
 
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What's the resolution?
 
japplepie said:
What's the resolution?
Try to answer my questions and you will know the answer.
 
DrDu said:
Try to answer my questions and you will know the answer.

It would take so long it would almost never happen ?

Is the time proportional to (number of molecules)! ?

Both of those are just guesses.
 
japplepie said:
It would take so long it would almost never happen ?

Is the time proportional to (number of molecules)! ?

Both of those are just guesses.

Yes, the guesses are correct, although the recurrence time increases much faster than linear with particle number.
If you are an aspiring physicist, you should be able to estimate the recurrence time.
 
DrDu said:
Yes, the guesses are correct, although the recurrence time increases much faster than linear with particle number.
If you are an aspiring physicist, you should be able to estimate the recurrence time.


No, what i mean't was factorial growth.
 

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