How Does Penrose Calculate Entropy in Cycles of Time?

In summary, entropy as state counting is a concept in thermodynamics that measures the disorder or randomness in a system based on the number of possible states. It is calculated by counting microstates and using the formula S = k ln W. Entropy and energy are related, as any change in energy will also cause a change in entropy. The second law of thermodynamics states that entropy will always tend to increase in a closed system. However, local entropy decrease can occur when energy is transferred within the system, resulting in a decrease in the number of possible microstates in a specific part of the system.
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bob012345
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Sir Roger Penrose in his book Cycles of Time on page 19 states the result of a calculation of probability of mixing red and blue balls as an illustration of entropy as state counting and the Second Law. He assumes an equal number of each. There is a cube of 10^8 balls on an edge subdivided into smaller cubes of 10^5 on an edge. He states each smaller cube looks uniformly purple if the ratio of red/blue balls is between 0.999 and 1.001. Then he states there are around 10^23,570,000,000,000,000,000,000,000 different arrangements of all the balls that give the appearance of uniform purple and some 10^46,500,000,000,000 different arrangements of the "original configuration in which the blue is entirely on top and the red entirely on the bottom".

If anyone has read the book, I am seeking help understanding how he gets to those numbers. Not necessarily a complete solution but a hint on where to start. Thanks and Happy New Year!
 
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bob012345 said:
Sir Roger Penrose in his book Cycles of Time on page 19 states the result of a calculation of probability of mixing red and blue balls as an illustration of entropy as state counting and the Second Law. He assumes an equal number of each. There is a cube of 10^8 balls on an edge subdivided into smaller cubes of 10^5 on an edge. He states each smaller cube looks uniformly purple if the ratio of red/blue balls is between 0.999 and 1.001. Then he states there are around 10^23,570,000,000,000,000,000,000,000 different arrangements of all the balls that give the appearance of uniform purple and some 10^46,500,000,000,000 different arrangements of the "original configuration in which the blue is entirely on top and the red entirely on the bottom".

If anyone has read the book, I am seeking help understanding how he gets to those numbers. Not necessarily a complete solution but a hint on where to start. Thanks and Happy New Year!
My plan is to start by adapting the random walk problem to the sub blocks and then use Sterlings approximation for the large factorials.
 
  • #3
bob012345 said:
where to start
Start with 32 white balls and 32 black balls in a 4x4x4 box ...
bob012345 said:
Sterlings approximation
The guy's name is Stirling, James Stirling :smile:
 
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BvU said:
Start with 32 white balls and 32 black balls in a 4x4x4 box ...
The guy's name is Stirling, James Stirling :smile:
Thanks. Don't know why I used an 'e'!

The number of states for 32/32 in a 64 ball cube is 64!/(32!32!) or 1.822X 10^18. I don't know how to compare that to all other possibilities. If I don't know any sub cube is mixed evenly, which is what I'm trying to show is most probably as N gets large, there could be also the other combinations such as 31/33, 30/34, 29,35 ...1/63. Then wouldn't I have to sum all the combinations. Or is there a straightforward formula for that?
 
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I reproduced Penrose's number which is just the Stirling approximation of 1x10^24! That seems a slight oversimplification from how he set up the problem but he did say the number of states was 'around' that number. The difference between 24 and 23.57 in the exponent comes from the 1/e in Stirlings formula. Penrose ignored the multiplicative factor of ~ 10^12 since that would just bury a 1 halfway among the zeros.
 
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I don't understand the same part either, did you find out how the value 10^46,500,000,000,000 was calculated?
 
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:welcome:

Not good to start at the tail end of a four year old thread. Better to start your own fresh one...

Am I right guessing you can find the 10^23.... number but not the 10^46... number.?

##\ ##
 
  • #8
BvU said:
:welcome:

Not good to start at the tail end of a four year old thread. Better to start your own fresh one...

Am I right guessing you can find the 10^23.... number but not the 10^46... number.?

##\ ##
Thank you for your replying.
Yes.
10^23… must be approximate value of (10^24)!

But I have no idea for 10^46…

I will follow your advice and start a new thread.
 

1. What is entropy as state counting?

Entropy as state counting is a concept in thermodynamics that refers to the measure of the disorder or randomness in a system. It is based on the idea that the higher the number of possible states a system can have, the more disordered it is.

2. How is entropy calculated using state counting?

Entropy is calculated by counting the number of possible microstates (or arrangements of particles) that a system can have, and then using a mathematical formula to convert it into a macroscopic value. This formula is S = k ln W, where S is the entropy, k is the Boltzmann constant, and W is the number of microstates.

3. What is the relationship between entropy and energy?

Entropy and energy are related in that any change in energy in a system will also result in a change in entropy. In a closed system, energy will naturally tend to disperse and increase entropy, as the number of possible microstates increases with energy.

4. How does entropy relate to the second law of thermodynamics?

The second law of thermodynamics states that the total entropy of a closed system will always tend to increase over time. This is because as energy is transferred and transformed within a system, it will naturally lead to an increase in the number of possible microstates and hence, an increase in entropy.

5. Can entropy ever decrease?

While the second law of thermodynamics states that the total entropy of a closed system will always increase, it is possible for the entropy of a specific part of the system to decrease. This is known as local entropy decrease and occurs when energy is transferred from one part of the system to another, resulting in a decrease in the number of possible microstates in that part of the system.

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