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    Welcome To Hotel Boltzmann
    By Johannes Koelman | December 30th 2009 11:22 PM | 15 comments | Print | E-mail | Track Comments
    About Johannes

    I am a Dutchman, currently living in India. Following a PhD in theoretical physics (spin-polarized quantum systems*) I entered a Global Fortune

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    Room for one please!”

    Josef was impatient. He was tired. It had been a long hot day and he needed a bath.

    Of course sir. How many nights?”

    Twenty-one nights, I will be leaving the same day in three weeks time.”

    Your name please?”

    Loschmidt. Josef Loschmidt.”

    Josef listened to the rattling of the keyboard. This young man was quick and seemed experienced.

    Hotel Boltzmann - Vienna

    The clerk handed over a key card. “You have room 13 sir, it is indicated in binary on your smartkey card. The number of bits on the card indicates the floor number, tonight you will be on the fourth floor.”

    Josef looked at his keycard. It had a small LCD display showing the four characters '1101'.

    The clerk continued: “Each day you will be moving to a higher numbered room at a higher floor. The smartkey card is programmed to switch over to your new room number every day at 10:00 AM. At any given time, the keycard will give access to the room indicated on the display and that room only. Each day the bellboy will be at your room 9:55 AM exactly and take care of your luggage.”

    Josef looked again at his keycard. Digital gadgets and impersonal automated procedures, he hated every bit of it! “Give me a room I can stay at the full twenty nights, I don't want to move.”

    Sorry sir, this is impossible. All our guests have to move according to protocol.”

    Are you saying that the next three weeks each day I have to move up to a higher floor?”

    Yes sir.”

    This is ridiculous! Put me in a room at a lower floor.”

    Sir, I have no rooms available at lower levels.”

    Then put me in a room such that I will move down during my stay.”

    Sir, such rooms do not exist.”

    Josef's irritation grew. “What is your name young man?”

    Ludwig, Sir.”

    Listen Ludwig. If there are guests in this hotel who have to move up to higher room numbers, than surely there are also guests that move down to lower numbers. Stop kidding me and give me a keycard that allows me to move to lower floors.”

    I apologize sir, but this is impossible. According to the H-theorem each and every day the majority of the guests move up in room number. None of our long-stay guests can avoid moving up.”

    Josef stared angrily at the young man behind the desk. Was he trying to exercise a practical joke? It didn't look like that. The clerk was getting visibly nervous. Despite his growing anger, Josef became curious.

    What theorem are you talking about young man?”

    The theorem on the Hotel-room-number sizes, sir. A cumbersome name for a simple theorem, I usually call it the H-theorem. It is actually easy to demonstrate that when you keep moving the guests in a sufficiently large hotel according to a fixed protocol, then these guests keep going to higher and higher room numbers. In the long run, room numbers can only grow in size, and the number of bits required to identify the location of each guest will keep increasing.”

    That is impossible.”

    That was my initial tought as well sir. But it does seem to be correct. If you look at the daily movements averaged over all guest, than the room numbers increase in length by a quarter of a bit. Every four days, the average guest gets a bit added to his room number and moves up a floor.”

    Ha, that means that the lower levels get vacated. So you must be able to give me a lower level room!”

    I beg you pardon Sir, but that is incorrect. The lower floors stay fully occupied.”

    Who are you kidding boy?”

    Sir, I do apologize for the inconvenience, but I can not avoid you moving up to higher floors.”

    Josef was puzzled. Surely this clerk was trying to fool him. What he wanted him to believe was simply preposterous. Yet, the boy certainly was not enjoying himself. The young man avoided his stare and looked down to his keyboard. A droplet trickled down his left cheek.

    Then a thought occurred to Josef: “Listen young man, I am certain that what you are telling me is logically impossible. But for sake of argument, suppose you are right. Suppose that you got some room movement scheme working here that causes guests to move to higher numbered rooms. Then I propose that you simply reverse these movements. The guests will then move down!”

    The clerk was silent for a moment. Josef looked at him with a triumphant smile on his face.

    Sir...”

    Well...?”

    Sir, I am very sorry. We indeed can reverse all the movements, but even if we do so, still the majority of the guests will move to higher numbered rooms.”

    Josef's mouth opened. He wanted to say something, but instead he had to gasp for breath. A silence followed. Josef kept his eyes fixed at the young man opposite him.

    Sir, please. I can show you the movement scheme and you can judge for yourself. Here it is...”

    The boy handed over a small piece of paper. His hand trembled. “Sir, you can take any room number as starting point, and do the calculations yourself. You might occasionally observe a few reductions in the number of bits in the room numbers, but invariably an expansion will set in and the room numbers will continue to grow.”

    Your Smartkey Card

    The rooms are numbered in binary notation. Room numbers consisting of n bits are on the n-th floor. So, room 1 is on the first floor, rooms 10 (2) and 11 (3) are on the second floor, and rooms 100 (4) thru 111 (7) are on the third floor, etc.

    Each day at 10:00 the smartkey card will change the displayed room number. The change in the room number is determined solely by the last two bits displayed. For room number ending with a '1', another '1' gets added at the end. Room numbers ending in '00', will undergo a deletion of the last bit. The remaining room numbers are those ending in '10'. These undergo a swapping of the last two bits

      ...1 → ...1(½) – you find your new room one floor up

      ...10 → ...01  (¼) – new room on the same floor (except when on the 2nd floor)

      ...00 → ...0  (¼) – you find your new room one floor down

    Half the number of guests (those in rooms ending with a '1') will move to a room number that is one bit longer, and only a quarter of the guests (those in rooms ending with '00') will move to a room numer that is one bit shorter. The remaining guest (those in rooms ending with '10') will move to a room number with the same number of bits.

    With the higher numbered rooms on the higher floors, each day the majority of your guests will enjoy a better view from their room!

    If preferred, the room movements can be reversed. This is achieved in above scheme by changing every '1' in a '0' and vice versa:

      ...0 → ...0(½) – you find your new room one floor up

      ...01 → ...10  ) – new room on the same floor (except when on the 1st floor)

      ..11 → ...1 (¼) – you find your new room one floor down

    This does not change the statistics of the size increases of the room numbers.


    Enjoy your stay!

    The Hotel Management

    Josef kept staring at the piece of paper. He read it, and read it again. Finally Josef looked up, and broke the silence. “Young man... how many rooms does this hotel have?”

    Ludwig looked up. A smile spread across his face. “Sir, it might surprise you, but my friend Georg - who is studying math - assures me this hotel has vastly fewer rooms than there are points in even the tiniest line segment.”

    Ludwig hesitated, and then continued: “Personally, I don't think the rooms in this hotel outnumber the stars in our universe. That suffices for a contraction followed by an unbounded expansion. Come to think of it, it could very well be that the universe follows a similar protocol."

    "Don't you agree, sir?”

     

    ------------------------------------------------

    More Hammock Physicist articles: The largest distance between two points. What you didn't know about E=mc2. Time's arrow. Quantum telepathy. Booting up the universe. Fibonacci chaos. Powers of six-billion. Quantum virus. The grand arena of physical reality. Game theory and the art of acting rational. Holographic hot horizons. Holographic horizons get hotter.

    Comments

    Pick a number, any number. If it is odd, double it and add one. If it is even, divide it by two.
    Therefore, even numbers are divided until they become odd, and then double plus one daily.

    "If it is odd, double it and add one. If it is even, divide it by two."

    I think it is slightly more complicated:
    - If odd: double and add one
    - if a multiple of four: divide by two
    - if neither of the above: subtract one

    Creates a reversible dynamics that seems to contain the essence of entropy increase. Starting from any finite number, a short contraction is followed by an infinite exponential expansion. Renders the second law of thermodynamics trivial almost.

    This is part two. One might want to read http://scidiv.bellevuecollege.edu/Math/InfiniteHotel.html first.

    Johannes Koelman
    Had a follow-up blog in mind, but might as well add it here. Graphics explaining the dynamics:
    Hotel Boltzmann Dynamics

    Key question to ask: is the H-theorem correct?
    If the universe wanders around in a Boltzmann hotel, will there be naked brains floating in the other rooms?
    Or does Hotel Boltzmann eliminate the Boltzmann brain problem?

    http://www.nytimes.com/2008/01/15/science/15brain.html

    I start to understand why the poor guy killed himself...

    This model is too artificial for me. IMO spacetime is behaving like the water surface: the transversal waves of light are dispersing by background noise into longitudinal ones at distance in similar way, like landscape under haze. So, the answer is, no contracting phase preceded the Big Bang, which is itself just an observational illusion. Remote observer would see our portion of Universe in the same way, like we can see it in Hubble depth field.

    A cute take on the Hilbert Hotel (but it'd probably be better had the relative references and links been provided by the author instead of the readers). Let me myself copy the idea of someone else, how many pages should the log of the hotel have since the names in there can't be moved around?

    Fred Phillips
    Georgescu-Roegen claimed Boltzmann's Ergodic Theorem is tautological; that it simply says, If you live long enough, you'll see everything.

    To which my friend Moshe Kress added Kress' Corollary: "Especially if you travel."

    The hotel problem conveniently offers daily state-changes. But you can search Boltzmann's theorem and never find a clue about how fast states change or how long it takes for a complete tour of states.

    Which takes us back to the blues song about the guy who didn't stop drinking, because "Doc says it'll kill me, but he won't say when."
    I won't even pretend to know where does this example comes from, but this little bit is bothering me:

    "If preferred, the room movements can be reversed. This is achieved in above scheme by changing every '1' in a '0' and vice versa:"

    It's probably a flaw in the example itself or a misunderstood, but isn't reversing the movement supposed to make things that would go "up" go "down" and vice versa instead of just swapping the way to represent bits?

    Johannes Koelman
    Swapping '0' and '1' in the rule for the dynamics has the effect of reversing the movements. Just give it a try. Cheers,
    Johannes
    rholley

    Reading this makes me feel uneasy.  It reminds me of the "grey town", representative of Hell in The Great Divorce by C.S.Lewis.  The inhabitants there find the others intolerable, and when a newcomer arrives, they put up another house, simply by wanting one, in order to get away from unwanted human contact.  So the city keeps on getting bigger and bigger....

    Robert H. Olley / Quondam Physics Department / University of Reading / England
    The "statistics" in the letter bit are complete BS, they depend on the space being completely occupied, but the transformations given clearly lead to spaces that are not completely occupied.

    Johannes Koelman
    "the transformations given clearly lead to spaces that are not completely occupied"
    It seems you have misinterpreted the transition rules. Can you give an example where you think non-occupied will result?
    No, my mistake, I was tired and didn't pay enough attention to the context.

    If the hotel is not infinite then over time the guests will move into rooms ending with 1.

    I'm inclined to say that as well as being infinite the hotel needs to be fully occupied, but I'm not sure about that, infinite bends my mind. But now I suspect infinite mind bending is the entire point of the exercise.

    Regardless, I still feel the core issue is with those proportions, given that we have an infinite (and for the sake of simplicity fully occupied) hotel my original accusation no longer stands. However what we now have is:
    ...1 -> ...11 (1/2 infinity)
    ...10 -> ...01 (1/4 infinity)
    ...00 -> ...0 (1/4 infinity)
    Which means you are implicitly making the invalid assertion that 1/2 infinity is more than 1/4 infinity.