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

• japplepie
In summary, the conversation discusses the 2nd law of thermodynamics and its relationship to entropy, which can be viewed as a measure of disorder or information within a system. The recurrence paradox is mentioned, where after a long time, the randomness in a system could potentially lead back to its initial state. This paradox is resolved through the methods of statistical mechanics, which provide a mathematical definition of entropy and predict the laws of thermodynamics. The conversation also touches on the surprising coolness of statistical mechanics and the difficulty in estimating the recurrence time for large numbers of particles.
japplepie
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?

Thats called "recurrence paradox". Can you estimate how long it takes for, say, 10 gas molecules to unmix? 100 molecules, 10^23 molecules?

1 person
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.

1 person
What's the resolution?

japplepie said:
What's the resolution?

DrDu 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.

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.

## 1. What is entropy and how does it relate to the 2nd law of thermodynamics?

Entropy is a measure of the disorder or randomness of a system. The 2nd law of thermodynamics states that the total entropy of an isolated system will always increase over time. This means that as energy is transferred or transformed within a system, some of it will inevitably be lost as heat, resulting in an increase in overall disorder.

## 2. How does the concept of entropy apply to everyday life?

Entropy can be seen in everyday processes such as the melting of ice cubes or the rusting of metal. In both cases, the system is moving towards a state of greater disorder. Entropy also plays a role in the aging process, as living organisms gradually become more disordered over time.

## 3. Can entropy ever decrease in a system?

According to the 2nd law of thermodynamics, the total entropy of an isolated system will always increase. However, in some cases, local decreases in entropy may occur, as long as the overall trend is towards an increase in entropy.

## 4. How are entropy and energy related?

Entropy and energy are closely related, as energy is required to maintain order and decrease entropy in a system. In other words, a decrease in entropy requires an input of energy. This is why energy is constantly needed to maintain order in living organisms, and why systems tend towards higher entropy when energy is not continually supplied.

## 5. How does the concept of fluctuation tie into the 2nd law of thermodynamics?

The concept of fluctuation refers to the natural variations that occur in a system. These fluctuations can sometimes result in temporary decreases in entropy, but over time the 2nd law of thermodynamics dictates that the overall trend will still be towards an increase in entropy. Therefore, even though fluctuations may occur, the 2nd law of thermodynamics still holds true.

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