5 ms·

> Therefore, a system transitioning from state to state is much more likely to be in one that looks chaotic, and very, very unlikely to ever go back to a state

by roberto 11y ago

> Therefore, a system transitioning from state to state is much more likely to be in one that looks chaotic, and very, very unlikely to ever go back to a state that is non-chaotic. This one thing I don't understand about the 2nd law of thermodynamics: given enough time won't the system go back to a more organized state simply by chance? And in that case, wouldn't the total entropy be reduced?

TTPrograms 11y ago
Yes, that can happen, but with overwhelmingly small probability. Say if you consider the probability that all the gas atoms in a box spontaneously are in one half of the box, the chance is approximately (1/2)^N, where N is on the order of Avogadro's number.
nabla9 11y ago
You understand it just fine. What may be nonintuitive is the incredibly low change of that happening even in microscopic scale. Difference between macroscopic 'organized states' vs 'unorganized states' just mind mindbogglingly huge.
TeMPOraL 11y ago
As others said, your understanding is fine - you just need to gain the appreciation to how much more unordered states are there for a system to be in than the ordered ones. My physics professor used to underscore it by drawing diagrams that show a line that suddenly goes from almost 0 to "the tip of this line is somewhere in the next galaxy". Or to give you a somewhat similar problem - consider a 32x32px 8bit image (a standard Windows icon since version 3.0). The image consists of 1024 pixels, each capable of representing a different color from the set of 256 colors. How many possible pictures like this are there? Or, as you'd ask in thermodynamics, in how many states such a system can be? It's 256 states per pixel raised to 1024th power (just like 6-digit binary number has 2^6 possible values). It's 256^1024. You know how much is it? 10907481356194159294629842447337828624482641619962326924318327861897213318491192 95216264234525201987223957291796157025273109870820177184063610979765077554799078 90629884219298953860982522804820515969685161359163819677188654260932456012129055 39018863010179002525357999172000100796000265358368009052978058809523505016301954 75653911005312364560014847426035293551245843928918752768696279344088055617515694 34994540667782514081490061610592025643850457801332649356583604724240738244281224 51315177575191648992263657437224322773680750276278830452065017927617009456991684 97257879683851737049996900961120515655050115561271491492515342105748966629547032 78632150573082843022166497032439613863525162640951616800542762343599630892169144 61811874063953106654048857394348328774281674074953709935118687563599703901170218 23616749458620969857006263612082706715408157066575137281027022310927564910276759 16052087830463241104936456875492096732298245918476342738379027244843801852697776 49410727156115804346908274593399919614142427414105991174260605564837637563145276 11362658628383368621157993638020878537675545336789915694234433955666315070087213 53547025567031200413072549583450835743965382893607708097855057891296790735278005 49356215610907958451729541159729274798775277385600082041185589300047777487277618 53813510493840581861598652211605960308356405941821189714037868726219481498727603 65361629885617482241303348543878532402475141941718301228107820972930353737280457 43720952287036227763639452908698062584223551485075710396193874496298668081887696 62815778153079393179093143648340761738581819563002994422790754955061288818308430 07964869323217915876591803556521615711540299212027615560787310793747746684152836 29877086994501520312318625942030856938389446570613462367042340268211029589549511 97087076546186622796294536451620756509351018906023773821539532776208676978589731 96633030889330466516943618507835064156833694453005143749131129883436726523859540 49042734559287239495252271846174043678547546104743770197680255766058810380772707 07717942221977090385438585844095492116099852538903974655703943973086090930596963 36076752996493841459818570596375456149735582781362383328890630900428801732142480 86639626713335280092327583508730596141187237814221014601986157473868550968960891 89180441339558524822867541113212638793675567650340362970031930023397828465318547 23824423202801518968966041882297600081543761065225427016359565087543385114712321 4227266605403581781469090806576468950587661997186505665475715792896 This much. (Source: [0]). Now ask yourself, how many of those pictures represent a letter "A". Quite a lot probably, but nowhere near that much. In all those 256^1024 pictures, you have every possible representable letter A, as well as any other possible Unicode character. In those images are all your most cherished private photos (or at least their thumbnails), and also the photos of all things that you'd wish happened but didn't. A thumbnail of every possible photo of the universe is there as well. There's also something else. Something much more frequent than all other image I've just mentioned taken together. It's the noise. The things we don't recognize, the things we consider uninteresting. I ask you, use your intuition - if you were to create a random Windows icon every second, how soon would you expect to get one depicting a letter "A"? And now realize we were talking about silly icons that are probably barely visible on your screen. There are 6.02 x 10²³ atoms in 12 grams of carbon. I.e. in a tip of your pencil. 602000000000000000000000 atoms. Those are your pixels. And if you want to compute the number of states this pile of atoms can be in, that is the number that goes in your exponent. The base is some combination of possible positions, orientations and velocities, and then probably something else I'm forgetting right now. And almost all of those states are noise. That's how the Second Rule works. [0] - http://www.wolframalpha.com/input/?i=256%5E1024 http://www.wolframalpha.com/input/?i=256%5E1024
- prewett 11y ago
  The Digital Library of Babel (https://libraryofbabel.info/ https://libraryofbabel.info/) was posted here a while ago, which is a tangible way to search through this space. The library comes from a book by Jorge Luis Borges, which has all possible books, meaningful and not. The people in the book are searching for books with meaning, but the chance of finding one is really small. The digital library lets you do the search electronically. Kind of fun to play around with.
  - Nadya 11y ago
    It also has a link to the Universal Slideshow. Which let's one browse every image (of a given size) that represents what TeMPOral was talking about. Ever since I discovered that site I've been slightly obsessed with it. In the "This is really interesting" sort of way. It makes one think how everything is noise. We just happen to find meaning in some of it.
  - dsmolovich 11y ago
    It's funny though as I had a very similar idea some years ago sticking much with virtual machines. I thought that if VM's disk image is just a file represented by sequence of zeroes and ones and what if someone would generate a final number of those images starting from 0...0 to 1...1 and then probe them all on a given emulated hardware what would be a chance to discover some viable/useful software? How would we detect it? How would we know or reverse engineer it to understand what it could be used for and how to use it, interact with it etc.? So I ended up thinking that it would be a project similar to SETI but on a smaller scale (let's say if the image size would be limited to just 1Gb).
- vardump 11y ago
  > There are 6.02 x 10²³ atoms in 12 grams of carbon. I.e. in a tip of your pencil. Pretty big pencil.
  - javajosh 11y ago
    we're dealing with big numbers so an order of magnitude or two doesn't matter much.
pdkl95 11y ago
As entropy is usually a probabilistic concept, a "more organized" future is very very unlikely, but not impossible. Numberphile has a video[1] that discusses the Poincaré recurrence[2] time for the universe. So while it may be possible, you probably have top wait about 10^10^10^10^2.8 "Plank times, millenia, or whatever"[3] for a state to occur. [1] http://www.numberphile.com/videos/longest_time.html http://www.numberphile.com/videos/longest_time.html [2] https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theor... [3] The units used in the associated paper. With a number that large, it doesn't really matter what unit of time you use.
effie 11y ago
If the model is Hamiltonian and the phase space finite, then any state will get eventually reapproached. For systems with infinite phase space (infinite volume suffices) or non-Hamiltonian systems, such conclusion may not be true.

by roberto 11y ago

TTPrograms 11y ago

Yes, that can happen, but with overwhelmingly small probability. Say if you consider the probability that all the gas atoms in a box spontaneously are in one half of the box, the chance is approximately (1/2)^N, where N is on the order of Avogadro's number.

nabla9 11y ago

You understand it just fine. What may be nonintuitive is the incredibly low change of that happening even in microscopic scale. Difference between macroscopic 'organized states' vs 'unorganized states' just mind mindbogglingly huge.

TeMPOraL 11y ago

As others said, your understanding is fine - you just need to gain the appreciation to how much more unordered states are there for a system to be in than the ordered ones. My physics professor used to underscore it by drawing diagrams that show a line that suddenly goes from almost 0 to "the tip of this line is somewhere in the next galaxy". Or to give you a somewhat similar problem - consider a 32x32px 8bit image (a standard Windows icon since version 3.0). The image consists of 1024 pixels, each capable of representing a different color from the set of 256 colors. How many possible pictures like this are there? Or, as you'd ask in thermodynamics, in how many states such a system can be? It's 256 states per pixel raised to 1024th power (just like 6-digit binary number has 2^6 possible values). It's 256^1024. You know how much is it? 10907481356194159294629842447337828624482641619962326924318327861897213318491192 95216264234525201987223957291796157025273109870820177184063610979765077554799078 90629884219298953860982522804820515969685161359163819677188654260932456012129055 39018863010179002525357999172000100796000265358368009052978058809523505016301954 75653911005312364560014847426035293551245843928918752768696279344088055617515694 34994540667782514081490061610592025643850457801332649356583604724240738244281224 51315177575191648992263657437224322773680750276278830452065017927617009456991684 97257879683851737049996900961120515655050115561271491492515342105748966629547032 78632150573082843022166497032439613863525162640951616800542762343599630892169144 61811874063953106654048857394348328774281674074953709935118687563599703901170218 23616749458620969857006263612082706715408157066575137281027022310927564910276759 16052087830463241104936456875492096732298245918476342738379027244843801852697776 49410727156115804346908274593399919614142427414105991174260605564837637563145276 11362658628383368621157993638020878537675545336789915694234433955666315070087213 53547025567031200413072549583450835743965382893607708097855057891296790735278005 49356215610907958451729541159729274798775277385600082041185589300047777487277618 53813510493840581861598652211605960308356405941821189714037868726219481498727603 65361629885617482241303348543878532402475141941718301228107820972930353737280457 43720952287036227763639452908698062584223551485075710396193874496298668081887696 62815778153079393179093143648340761738581819563002994422790754955061288818308430 07964869323217915876591803556521615711540299212027615560787310793747746684152836 29877086994501520312318625942030856938389446570613462367042340268211029589549511 97087076546186622796294536451620756509351018906023773821539532776208676978589731 96633030889330466516943618507835064156833694453005143749131129883436726523859540 49042734559287239495252271846174043678547546104743770197680255766058810380772707 07717942221977090385438585844095492116099852538903974655703943973086090930596963 36076752996493841459818570596375456149735582781362383328890630900428801732142480 86639626713335280092327583508730596141187237814221014601986157473868550968960891 89180441339558524822867541113212638793675567650340362970031930023397828465318547 23824423202801518968966041882297600081543761065225427016359565087543385114712321 4227266605403581781469090806576468950587661997186505665475715792896 This much. (Source: [0]). Now ask yourself, how many of those pictures represent a letter "A". Quite a lot probably, but nowhere near that much. In all those 256^1024 pictures, you have every possible representable letter A, as well as any other possible Unicode character. In those images are all your most cherished private photos (or at least their thumbnails), and also the photos of all things that you'd wish happened but didn't. A thumbnail of every possible photo of the universe is there as well. There's also something else. Something much more frequent than all other image I've just mentioned taken together. It's the noise. The things we don't recognize, the things we consider uninteresting. I ask you, use your intuition - if you were to create a random Windows icon every second, how soon would you expect to get one depicting a letter "A"? And now realize we were talking about silly icons that are probably barely visible on your screen. There are 6.02 x 10²³ atoms in 12 grams of carbon. I.e. in a tip of your pencil. 602000000000000000000000 atoms. Those are your pixels. And if you want to compute the number of states this pile of atoms can be in, that is the number that goes in your exponent. The base is some combination of possible positions, orientations and velocities, and then probably something else I'm forgetting right now. And almost all of those states are noise. That's how the Second Rule works. [0] - http://www.wolframalpha.com/input/?i=256%5E1024 http://www.wolframalpha.com/input/?i=256%5E1024

prewett 11y ago
The Digital Library of Babel (https://libraryofbabel.info/ https://libraryofbabel.info/) was posted here a while ago, which is a tangible way to search through this space. The library comes from a book by Jorge Luis Borges, which has all possible books, meaningful and not. The people in the book are searching for books with meaning, but the chance of finding one is really small. The digital library lets you do the search electronically. Kind of fun to play around with.
- Nadya 11y ago
  It also has a link to the Universal Slideshow. Which let's one browse every image (of a given size) that represents what TeMPOral was talking about. Ever since I discovered that site I've been slightly obsessed with it. In the "This is really interesting" sort of way. It makes one think how everything is noise. We just happen to find meaning in some of it.
- dsmolovich 11y ago
  It's funny though as I had a very similar idea some years ago sticking much with virtual machines. I thought that if VM's disk image is just a file represented by sequence of zeroes and ones and what if someone would generate a final number of those images starting from 0...0 to 1...1 and then probe them all on a given emulated hardware what would be a chance to discover some viable/useful software? How would we detect it? How would we know or reverse engineer it to understand what it could be used for and how to use it, interact with it etc.? So I ended up thinking that it would be a project similar to SETI but on a smaller scale (let's say if the image size would be limited to just 1Gb).
vardump 11y ago
> There are 6.02 x 10²³ atoms in 12 grams of carbon. I.e. in a tip of your pencil. Pretty big pencil.
- javajosh 11y ago
  we're dealing with big numbers so an order of magnitude or two doesn't matter much.

pdkl95 11y ago

As entropy is usually a probabilistic concept, a "more organized" future is very very unlikely, but not impossible. Numberphile has a video[1] that discusses the Poincaré recurrence[2] time for the universe. So while it may be possible, you probably have top wait about 10^10^10^10^2.8 "Plank times, millenia, or whatever"[3] for a state to occur. [1] http://www.numberphile.com/videos/longest_time.html http://www.numberphile.com/videos/longest_time.html [2] https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theor... [3] The units used in the associated paper. With a number that large, it doesn't really matter what unit of time you use.

effie 11y ago

If the model is Hamiltonian and the phase space finite, then any state will get eventually reapproached. For systems with infinite phase space (infinite volume suffices) or non-Hamiltonian systems, such conclusion may not be true.