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For ex. if two particals close to each other require n bits of info to describe them, why does it take n bits to describe them when they are far apart? Shouldn't the information content be the same for the macrosystem?
Can you give a specific example?if two particals close to each other require n bits of info to describe them, why does it take n bits to describe them when they are far apart?
Your question is not as concrete as you believe it to be. You are being coached to fill in the blanks.What do you not understand? I laid the question out pretty concretely
You may think you did, but you didn't. And I strongly suspect that is because you do not have a good understanding of the topic you're asking about.I laid the question out pretty concretely
What do you think the Shannon entropy is of a system of two particles close together? What do you think the Shannon entropy is of a system of two particles far apart? Please show your work.since Shannon entropy is considered information, where does this new information come from?
I am still not clear on why you are saying that the Shannon entropy of a system with two nearby particles is different from the Shannon entropy when they are more widely separated. Can you show the calculation of the entropy under both conditions?My question is, since Shannon entropy is considered information, where does this new information come from?
Because a system of two particles is not a gas.If a gas expands it's entropy increases. So how would a simplification of two gas particles not follow the same entropy increase?
Entropy is not a property of particles. Entropy is a property of a gas as a whole, when its detailed particle content is ignored. More generally, entropy is something what you can say about a big system when you don't know all the details of the small constituents comprising the big system. If physicists were superbeings who knew all the details about everything, then they would not need the concept of entropy. Metaphorically, if particles are made by God, then entropy is made by men.If a gas expands it's entropy increases. So how would a simplification of two gas particles not follow the same entropy increase?
Consider a one dimensional two particle bounded system (not a gas) that is at equilibrium that has 8 slots, each particle is a "1".So how would a simplification of two gas particles not follow the same entropy increase?
Suppose that someone gives you a closed book and tells you that this book contains only one letter, say letter "A", written at one of the pages in the book. All the other pages are empty. How much information that book contains?My question is, since Shannon entropy is considered information, where does this new information come from?
Again, why is there a cutoff in not being able to know about n particles. Also, you should be able to know the exact same amount about each particle in a 10^23 mass of particles as you can know about each particle in a 2 particle mass. HUP doesnt care how many particles you are looking at. And you cant know everything about a particle. Again bc of hup.Entropy is not a property of particles. Entropy is a property of a gas as a whole, when its detailed particle content is ignored. More generally, entropy is something what you can say about a big system when you don't know all the details of the small constituents comprising the big system. If physicists were superbeings who knew all the details about everything, then they would not need the concept of entropy. Metaphorically, if particles are made by God, then entropy is made by men.
That being said, now the answer to your question is easy. Two particles do not have an entropy in the way a gas has because, even if you can know every detail about 2 particles, you cannot know every detail about ##10^{23}## particles.
For more details, I highly recommend
https://www.amazon.com/dp/9813100125/?tag=pfamazon01-20
There is no hard cutoff (see below), but the extremes are certainly easily distinguished. A box of 1 liter of hydrogen gas at room temperature has about ##10^{22}## atoms. Your intuitions about how gases work are based on collections of that many atoms or more. Claiming that things should work exactly the same for a system of 2 atoms shows a huge failure to understand the issue.So if you have 2 hydrogen atoms bouncing around in a otherwize empty box, at a certain point when adding more H atoms the system becomes a gas?
Not a paradox, but a failure to realize that the term "gas", like most terms, does not have crisp, precise boundaries.What is the cutoff point mr Adonis? Sounds like a massive paradox to me.
So is there a slight disconnect between the concept of a systems information content and its entropy? What i dont get is that if a system gains information it should take more to describe it. But it seems like apart from each particles intrinsic values, each particle could be described by an xyz position that doesnt need more info to describe.Suppose that someone gives you a closed book and tells you that this book contains only one letter, say letter "A", written at one of the pages in the book. All the other pages are empty. How much information that book contains?
At first sight, not much. However, it really contains more information than it looks at first. The letter "A" is written at some definite place of some definite page, and the information about the exact place of the letter - is an information too. If the book contains ##N## pages, then Shannon information about the page on which the letter is written is ##{\rm ln} N##. So the bigger the book the more information it contains, even when it contains only one letter.
Expansion of a gas with a fixed number of particles is similar to an "expansion" of a book (that is, increasing the number of pages) with a fixed number of letters.
This is a classical model, not a quantum model. In a quantum model, the particle is described by a state vector in a Hilbert space; the x, y, and z components of position are parameters that pick out which particular state vector it is. But the Hilbert space itself is not the 3-dimensional space of the x, y, z position vector.each particle could be described by an xyz position
This is a somewhat different sense of the word "information" from the one you're using when you ask about the relationship between information and entropy. When Susskind talks about information not being destroyed, he is referring to quantum unitary evolution; basically he is claiming that unitary evolution can never be violated. But that just means that, as far as the quantum state of an entire system is concerned, its evolution is deterministic: if you know the state at one instant, you know it for all time. And if you know the system's exact state for all time, its entropy is always zero, by definition.Leonard suskind believed that the idea of information being destroyed was an abomination.
If one doubles pages in @Demystifier 's book then there are twice as many possible configurations for the book to be in. The letter is still only on a single page, but it now requires more information to say which page because there are twice as many of them to consider.each particle could be described by an xyz position that doesnt need more info to describe.
It's not so much the macro level thermodynamic state (a system of two particles isn't usefully viewed using the thermodynamic approximation) as the fact that the set of possible two-particle position states is the same whether the particles are close together or far apart. So the "number of pages in the book" stays the same, all that changes is which of the "pages" each "letter" (particle) is on.the two microstates (particles close and particles farther apart) might both yield the same macro level thermodynamic state for the system