Is an infinite series of random numbers possible?

[Loren Booda]

Is an infinite series of [nonrepeating] random numbers possible?

That is, can the term "random" apply to a [nonrepeating] infinite series?

It seems to me that Cantor's logic might not allow the operation of [nonrepeating] randomization on a number line approaching infinity.

 

[dodo]

I'm not sure of an answer, but allow me to put two thought-experiments on the table to discuss.

Experiment A: it will produce a rational approximation to a real in the interval (0,1]. Expressed in base 2, the number is constructed as follows: throw a fair coin, and choose the first binary digit (at position 2^-1) accordingly (maybe heads=1, tails=0). Each next coin throw will give you another digit. When you get tired, or consider that your approximation is good enough, stop throwing the coin and use the fraction so far. If you have two of these fractions and are worried that they may be equal, calculate extra digits for each number (with more coin throws) until they differ.

Experiment B: produces a random number, not uniformly distributed and possibly repeated (the latter could be solved by throwing away a repeated number and producing a new one), in the interval [0,pi). Proceed as in experiment A, but definitely terminate the experiment at the first 0 digit (tails); then return pi times your fraction.

 

[archangel95]

I would think the proof would be very similar to "Are there infinite prime numbers?"

 

[micromass]

Define random.
Define infinite series. (I'm pretty sure you meant sequence here)

 

[Loren Booda]

Define random.
Define infinite series. (I'm pretty sure you meant sequence here)

Thanks, I did mean sequence, and by "infinite" I intend for the count of random numbers in that sequence to approach infinity.

By "random" I mean eventually exhausting all relations among numbers.

__________


Is the cardinality of random numbers the same as that for reals?

Might random numbers by their nature tend toward infinity?

 

[Dashin]

I would think the proof would be very similar to "Are there infinite prime numbers?"

Not entirely ^^

You see, Prime Numbers are NOT random. They are connected by the fact they can only be divided by themselves and one.

It's incredibly unlikely if a proof to either of these were found, that it'd be similar to prove the other one.

 

[mathman]

Thanks, I did mean sequence, and by "infinite" I intend for the count of random numbers in that sequence to approach infinity.

By "random" I mean eventually exhausting all relations among numbers.

__________


Is the cardinality of random numbers the same as that for reals?

Might random numbers by their nature tend toward infinity?

If it is a sequence, by definition it will be countable. The numbers themselves can be confined to a given interval, since there are (non-countable) infinite numbers available.

 

[Loren Booda]

Please let me amend my original questions:

Is an infinite set of exclusively non repeating random numbers possible?

Is "random" analogous to "exhausting all relations among numbers"?

__________


Is the cardinality of random numbers the same as that for reals?

Might random numbers by their nature tend toward infinity more rapidly than reals?

 

[TylerH]

You have to define random. By the definition of an infinite sequence, there will always be some generating function that represents your sequence (even if it can only be expressed as a mapping). I suppose you could say that it is random if the generating function can't be expressed in terms of certain types of other functions, like elementary functions.

But until you specifically define random, by the reasoning that my choice of generating function is random, I can say that {2, 4, 6, ...} is random; because I thought of it randomly.

 

[SteveL27]

Please let me amend my original questions:

Is an infinite set of exclusively non repeating random numbers possible?

Is "random" analogous to "exhausting all relations among numbers"?

__________


Is the cardinality of random numbers the same as that for reals?

Might random numbers by their nature tend toward infinity more rapidly than reals?

What do you mean by a random number? Is 6 a random number? Is pi a random number?

Do you perhaps mean non-computable or non-definable number?

 

[Loren Booda]

You have to define random. By the definition of an infinite sequence, there will always be some generating function that represents your sequence (even if it can only be expressed as a mapping). I suppose you could say that it is random if the generating function can't be expressed in terms of certain types of other functions, like elementary functions.

But until you specifically define random, by the reasoning that my choice of generating function is random, I can say that {2, 4, 6, ...} is random; because I thought of it randomly.

What do you mean by a random number? Is 6 a random number? Is pi a random number?

Do you perhaps mean non-computable or non-definable number?

The set of random numbers exhausts integral order and corresponding numerical magnitude. Random numbers may be generated by exchanging orders with magnitudes.

 

[MathematicalPhysicist]

I know it's sound philosophical, but how would you generate a random sequence of numbers?

I mean if you want to 'generate' something, you'll have to use some apparatus (either your mind, computer, cesium atom) and you don't know what makes the apparatus pick its numbers.

I don't believe there's something like random numbers.

 

[dodo]

The set of random numbers exhausts integral order and corresponding numerical magnitude. Random numbers may be generated by exchanging orders with magnitudes.

Perhaps you question is, are there permutations of infinite length? (Just trying to guess your meaning, though.)

 

[Loren Booda]

I don't believe there's something like random numbers.

Can random numbers be defined but not generated?

Perhaps you question is, are there permutations of infinite length? (Just trying to guess your meaning, though.)

I'm not sure. How about the permutation between order and number which I mentioned previously?

 

[SteveL27]

The set of random numbers exhausts integral order and corresponding numerical magnitude. Random numbers may be generated by exchanging orders with magnitudes.

You're using some words in ways that are unfamiliar to me.

* You referred to "the set of random numbers," but I don't know what a random number is. Can you give some examples of random numbers? Is 6 a random number? Pi?

* What do you mean "exhausts integral order?"

* What do you mean by generating random numbers by "exchanging orders with magnitudes?" Can't imagine what that means. Order is the relation by which 5 < 6, for example. Magnitude is the relation that ignores the difference between -5 and 5. How do you exchange these properties? And how do use these ideas of order and magnitude to generate random numbers? Do you mean particular random numbers or the entire set of random numbers? And what set is that?

I know you have something in mind but it's difficult for me to understand what you mean.

Are you referring to numbers that are generated unpredictably? Flip a bit if a cosmic ray passes through a particular square millimeter in the next millisecond?

Or by "random" do you mean algorithmic randomness? A number is random it's incompressible via a finitely describable algorithm? So any number we can name is not random, but we know that there must be uncountably algorithmically random numbers. Those are the non-constructable or non-definable numbers. [There's a subtle difference between those two concepts?

Do any of those questions make sense in terms of what you're trying to do?

 

[Loren Booda]

You're using some words in ways that are unfamiliar to me.

* You referred to "the set of random numbers," but I don't know what a random number is. Can you give some examples of random numbers? Is 6 a random number? Pi?

* What do you mean "exhausts integral order?"

* What do you mean by generating random numbers by "exchanging orders with magnitudes?" Can't imagine what that means. Order is the relation by which 5 < 6, for example. Magnitude is the relation that ignores the difference between -5 and 5. How do you exchange these properties? And how do use these ideas of order and magnitude to generate random numbers? Do you mean particular random numbers or the entire set of random numbers? And what set is that?

I know you have something in mind but it's difficult for me to understand what you mean.

Are you referring to numbers that are generated unpredictably? Flip a bit if a cosmic ray passes through a particular square millimeter in the next millisecond?

Or by "random" do you mean algorithmic randomness? A number is random it's incompressible via a finitely describable algorithm? So any number we can name is not random, but we know that there must be uncountably algorithmically random numbers. Those are the non-constructable or non-definable numbers. [There's a subtle difference between those two concepts?

Do any of those questions make sense in terms of what you're trying to do?

I appreciate your patience, SteveL27. Upon consideration, your arguments make a lot of sense.

In this regard, allow me to modify my previous speculations to concentrate on a specific random number generator:

1. Start with an exclusive irrational number.

2. Limit its fractional part in digits by its integer value.

3. The resulting string is a random number.

 

[SteveL27]

I appreciate your patience, SteveL27. Upon consideration, your arguments make a lot of sense.

In this regard, allow me to modify my previous speculations to concentrate on a specific random number generator:

1. Start with an exclusive irrational number.

2. Limit its fractional part in digits by its integer value.

3. The resulting string is a random number.

What is an "exclusive" irrational number. What does it mean to limit its part in digits by its integer value? Do you mean just take the part to the right of the decimal point?

Can you give an example?

Say I start with my favorite irrational, e, the base of the natural log. e = 2.7128...

Are you saying that .7128... is a random number? But it isn't, in either sense of the word.

* It's not unpredictable. In fact you can can write down the well-known algorithm e = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + ... + 1/n! + ... and crank out as many digits as you like. It's completely deterministic, the opposite of random.

* It's not algorithmically random, in fact the finite string (sum as n goes from 0 to infinity of 1/n!) is a small number of symbols that precisely defines e.

Is e an "exclusive" irrational by your definition? Or do you mean something else?

Do you perhaps mean that the digits of e are normal, in the sense that any block of n digits occurs equally as often as any other block? It's not known if e is normal, but is normality the characteristic you're interested in?

 

[Loren Booda]

SteveL27,

I am beginning to believe that "random number" is an oxymoron.

Numbers are defined by value, order and relation overall, while randomness violates those quantities together.

Random number sequences manifest near infinite uncertainty and zero probability.

There exist tests for non-random numbers, but not for random numbers.

No physical measurement has confirmed a random number.

The set of random numbers and the set of non-random numbers seem mutually exclusive. Is the set of reals their union?

__________

[Strike "exclusive," please.]

Some example outputs of my pseudorandom generator (I am assuming normality for irrationals):

3.162277660...

becomes .162

2.718281828...

becomes .71

A true random number generator is an impossibility, since it requires both non-determinism (randomness) and determinism (predictiveness).

 

[chiro]

SteveL27,

I am beginning to believe that "random number" is an oxymoron.

Numbers are defined by value, order and relation overall, while randomness violates those quantities together.

Random number sequences manifest near infinite uncertainty and zero probability.

There exist tests for non-random numbers, but not for random numbers.

No physical measurement has confirmed a random number.

The set of random numbers and the set of non-random numbers seem mutually exclusive. Is the set of reals their union?

__________

[Strike "exclusive," please.]

Some example outputs of my pseudorandom generator (I am assuming normality for irrationals):

3.162277660...

becomes .162

2.718281828...

becomes .71

A true random number generator is an impossibility, since it requires both non-determinism (randomness) and determinism (predictiveness).

It might help if you think about randomness in different orders of entropy. Entropy is probably the best and most useful way to quantify randomness: maximum entropy on all orders means that there is no advantage you have for having any past information at all since a maximum value on all orders (independent, first order conditional, second order conditional, etc) would mean that the analysis of all past values with respect to each other would not give you an advantage.

The best way to describe this is to have all of these distributions be uniform because this distribution is the one that maximizes entropy. If you do this for all possible conditional probabilities, then you will get a distribution that is purely random.

From this distribution you will get hints about the kinds of processes that you could construct.

If you want something random, but not purely random then you don't have to do anywhere near as much work, but if you want a process that is random the best way possible, you need to construct the above system and from there decide what kind of process would really emulate this distribution.

 

[Loren Booda]

It might help if you think about randomness in different orders of entropy. Entropy is probably the best and most useful way to quantify randomness: maximum entropy on all orders means that there is no advantage you have for having any past information at all since a maximum value on all orders (independent, first order conditional, second order conditional, etc) would mean that the analysis of all past values with respect to each other would not give you an advantage.

The best way to describe this is to have all of these distributions be uniform because this distribution is the one that maximizes entropy. If you do this for all possible conditional probabilities, then you will get a distribution that is purely random.

From this distribution you will get hints about the kinds of processes that you could construct.

If you want something random, but not purely random then you don't have to do anywhere near as much work, but if you want a process that is random the best way possible, you need to construct the above system and from there decide what kind of process would really emulate this distribution.

S=κln|Ω|

S=entropy

κ=Boltzmann's constant

ln=natural logarithm

Ω=number of states

__________


1. Does true randomness accompany a transfinite number of states?

2. Is information about states restricted by a finite speed of light?

3. Can multiple states interfere, e.g. achieve minimum entropy?
.

 

[chiro]

1. Does true randomness accompany a transfinite number of states?

2. Is information about states restricted by a finite speed of light?

3. Can multiple states interfere, e.g. achieve minimum entropy?
.

Could you please explain what you mean by transfinite? I get the feeling its a set-theoretic term but I don't want to make an erroneous judgment.

For number 2, I have to reiterate that the above talks about randomness with respect the distributions of the states that are related to each other and are independent of the actual process itself.

If you want to answer your question you need to consider further constraints that relate to a specific process. Again the above considers a general process with a particular property.

I am not a physicist, but what you need to do is specifically outline how a law or a relationship between variables modifies the properties of the distributional information of the various joint distributions which then modifies the entropy, and from that you can get an idea of the real measure of the random nature of the process.

You should note that the physical systems are not purely random in the sense I have described above, because there is a lot of deterministic features known that we utilize and exploit everyday for various purposes. If the world were absolute truly random and completely unpredictable, then the order that we observe and make use of every day would not be present.

For 3, I wish to say that minimization of entropy gives an indication of order while maximization of entropy gives us an indication of disorder. Physicists and natural scientists usually have a goal of finding order and this directly relates to entropy.

The other thing is that you don't just want to consider the entropy characteristics of the original distributions, but also those of possible transformations of the data and hence the possible transformations of the distributions that 'make sense'. Make sense depends on both mathematical ideas and domain ideas, but for mathematical ones you want to consider at the very minimum convergence and probably topology and differentiability as well.

If you want an example of entropy minimization, think of entanglement. Instead of having two objects that would be classified as purely random, instead what we see is a reduction of entropy from that case since one has a direct effect on the other and this result shows a form of order that would otherwise not be seen in a purely random system.

In fact it is this property of minimum entropy in a variety of circumstances that has allowed us to obtain formulas like the ones you find in your science textbooks: it is this ability to quantify this order accurately enough that allows us to even understand this system we call reality or the 'universe', and I imagine that as time goes by we will be able to find transformations of our distributional representations that allow us to see even more order and subsequently a way to quantify it, just as Newton quantified gravity.

 

[Loren Booda]

Could you please explain what you mean by transfinite? I get the feeling its a set-theoretic term but I don't want to make an erroneous judgment.

For number 2, I have to reiterate that the above talks about randomness with respect the distributions of the states that are related to each other and are independent of the actual process itself.

If you want to answer your question you need to consider further constraints that relate to a specific process. Again the above considers a general process with a particular property.

I am not a physicist, but what you need to do is specifically outline how a law or a relationship between variables modifies the properties of the distributional information of the various joint distributions which then modifies the entropy, and from that you can get an idea of the real measure of the random nature of the process.

You should note that the physical systems are not purely random in the sense I have described above, because there is a lot of deterministic features known that we utilize and exploit everyday for various purposes. If the world were absolute truly random and completely unpredictable, then the order that we observe and make use of every day would not be present.

For 3, I wish to say that minimization of entropy gives an indication of order while maximization of entropy gives us an indication of disorder. Physicists and natural scientists usually have a goal of finding order and this directly relates to entropy.

The other thing is that you don't just want to consider the entropy characteristics of the original distributions, but also those of possible transformations of the data and hence the possible transformations of the distributions that 'make sense'. Make sense depends on both mathematical ideas and domain ideas, but for mathematical ones you want to consider at the very minimum convergence and probably topology and differentiability as well.

If you want an example of entropy minimization, think of entanglement. Instead of having two objects that would be classified as purely random, instead what we see is a reduction of entropy from that case since one has a direct effect on the other and this result shows a form of order that would otherwise not be seen in a purely random system.

In fact it is this property of minimum entropy in a variety of circumstances that has allowed us to obtain formulas like the ones you find in your science textbooks: it is this ability to quantify this order accurately enough that allows us to even understand this system we call reality or the 'universe', and I imagine that as time goes by we will be able to find transformations of our distributional representations that allow us to see even more order and subsequently a way to quantify it, just as Newton quantified gravity.

For number 1, by "transfinite" I attempt to attain the absolute limit of states where the system is purely random.

For number 2, I should have asked whether pure randomness can occur in a finite universe.

The relations within the equation for statistical entropy I gave are standard, and only simplistic logarithmic information derives from them.

States approaching infinity on a microscopic level are just as likely on a macroscopic level.

For number 3, quantum entropy would either involve destructive interference with change to the negative, or would involve constructive interference with change to the positive.

I believe statistical mechanics gives classical transformations of our distributional representations.

For entropy to have a true "law," I guess that entanglements must transfer their statistics instantaneously (rather than at a finite speed of c) and thermally.

 

[chiro]

For number 1, by "transfinite" I attempt to attain the absolute limit of states where the system is purely random.

For number 2, I should have asked whether pure randomness can occur in a finite universe.

This is a very good question.

I think the answer is going to be yes because the state-space does not have to necessarily define the nature of the process.

The thing you have to remember is that a process can take in a finite-state space and map it to a finite-state space like say relating the history of die rolls to the next one in terms of probability.

The state-space itself is fixed, but the process could go on forever and ever infinitely and although you have a process with a finite state space, it doesn't mean that the properties of the underlying process itself are not purely random.

Think of the problem of whether you could define a coin-toss process so that every new toss has no arbitrage chance of being biased in any way given the entire history of the process. If you think you can define a process that is unpredictable in this nature, your answer is yes. If you think you can't your answer is no.

I haven't shown a proof, but if I were to give one I would try and show that there always exists a process that given N observations, that the complete joint distribution of the N+1 observation given all the prior observations has maximal entropy (i.e. all values are the same). If this is shown, then you have proven that your answer is a resounding yes.

The relations within the equation for statistical entropy I gave are standard, and only simplistic logarithmic information derives from them.

The thing about entropy though is that you need to consider not just non-conditional entropy but conditional entropy as well.

To give you an example, imagine that you have a process corresponding to an infinite periodic sequence where one period consists of {0,1,2,3,4,5} in that order and repeats forever.

Now if you try and calculate P(X = a) for a = {0,1,2,3,4,5} you will always get 1/6 which implies maximal entropy.

But for this process we know that P(X_n+1 = a| X_n = (a-1) MOD 6) = 1 because of the periodicity. The entropy of this joint distribution is zero which implies absolute determinism.

From this statement although the non-conditional entropy is maximal, the joint ones are completely the opposite and through this we have found complete order which is exhausted at this level and thus the process is deterministic.

The order of the process if it exists, will be hidden somewhere in the conditional joint distributions, not in the non-conditional probability distribution.

States approaching infinity on a microscopic level are just as likely on a macroscopic level.

You need to be cautious about this: it may turn out depending on what you define as macroscopic that there is a different kind of entropy than what you would find on a microscopic level and that you may get some kind of dimensionality reduction when you consider a particular macroscopic space vs a microscopic space. If this occurs (especially dimensionality reduction), you are going to get some kind of different in classification of states between the two spaces.

Again, it needs to be defined what the macroscopic space is and if possible, the mapping between states in different spaces. I think you'll find that because of the way things are seen macroscopically in comparison to microscopically, there will be some kind of significant reduction in the state space for the macroscopic classifications in contrast to the microscopic ones which will look more like a projection than a bijection.

For number 3, quantum entropy would either involve destructive interference with change to the negative, or would involve constructive interference with change to the positive.

I'm not sure what you mean by this statement.

I believe statistical mechanics gives classical transformations of our distributional representations.

The thing I am more interested in is not just the non-joint distribution but the joint distribution in a general case. As I said above, order is not found in non-conditional statistical measures or distributions: it's like taking data sorting it out and putting it into separate buckets without considering how the relativity between data impacts the order found in the process.

For entropy to have a true "law," I guess that entanglements must transfer their statistics instantaneously (rather than at a finite speed of c) and thermally.

This is a very interesting point that I am going to have think about. Thanks.

 

[Loren Booda]

chiro,

Please explain the general meaning of the symbols and variables used in P(X_n+1 = a| X_n = (a-1) MOD 6) = 1.

Thanks for your dedication.

 

[chiro]

chiro,

Please explain the general meaning of the symbols and variables used in P(X_n+1 = a| X_n = (a-1) MOD 6) = 1.

Thanks for your dedication.

This is just a way of describing 1st order conditional properties for the staircase process {0,1,2,3,4,5} that keeps repeating in a periodic way.

I'll expand it out for all states and then you should see how I used the mod function: Here we go:

P(X_(n+1) = 0 | X_(n) = 5) = 1
P(X_(n+1) = 1 | X_(n) = 0) = 1
P(X_(n+1) = 2 | X_(n) = 1) = 1
P(X_(n+1) = 3 | X_(n) = 2) = 1
P(X_(n+1) = 4 | X_(n) = 3) = 1
P(X_(n+1) = 5 | X_(n) = 4) = 1

All other probabilities for all other first order conditional combinations are zero and you can show this by various probability identities and exhaustion of the probability space.

The X_(n+1) refers to the "n+1"th observation for the process and the X_(n) refers to the "n"th observation for the process. You could for this example associate n as a time parameter of a one-dimensional process.

The above process can only take on the values {0,1,2,3,4,5} which means we only have to consider going from one value in this list to another value in this list.

 

[Loren Booda]

Gedankenexperiments
__________

Please consider whether each of the following pairs is relatively entropic:

1. The big bang singularity and its imminent nonsingularity
2. Electron self-energy at a point and a spatial perspective
3. A cosmologist observing his self-inclusive universe
4. A closed universe and the black holes within
5. A quantum measurement and its measuring device
6. A vacuum of virtual particles
7. Turbulence at temperature T→∞
8. Black bodies at temperature T
__________

How many conditional entropies would there be given N non-conditional entropies?
__________

S=∫ΔQ/T

The article at http://en.wikipedia.org/wiki/Negative_temperature#Examples reads:

""Since we started with over half the atoms in the spin-down state, initially this drives the system towards a 50/50 mixture, so the entropy is increasing, corresponding to a positive temperature. However, at some point more than half of the spins are in the spin-up position. In this case, adding additional energy reduces the entropy, since it moves the system further from a 50/50 mixture. This reduction in entropy with the addition of energy corresponds to a negative temperature."

 

[chiro]

How many conditional entropies would there be given N non-conditional entropies?

I will answer this part here and discuss my thoughts on the thought experiments later.

In terms of how many conditional entropies there are, this depends on the state-space.

Usually if we were to study the system classically, then we probably would have considered things in terms of spatial locality.

What I mean by this is that in a classical context, matter or whatever kind of components that make up what we call matter, energy or whatever would have been treated in the way that things that are more local spatially would have more of an effect on the physical properties of that matter and thus affect everything from temperature to everything else and thus affect its entropy. The easiest way to think about this in terms of the local analytic viewpoint is to consider modelling a dynamical system like a fluid. Although you get all kinds of chaotic effects, typically by modelling everything as a continuum where everything is modeled in a way where all the local effects add up to produce the final cglobal behaviour.

In fact anything modeled with standard calculus will use this idea that by knowing local changes (usually in the form of a derivative), then the global changes can be found by how local changes accumulate and this is why when systems can be modeled this way, why calculus is so useful because it gives us a framework for doing exactly this.

But now you have to consider the situation when you are analyzing things that are not spatially 'close' or local (in the context I mentioned above).

When this happens, we need to consider not only spatially local effects but things that are 'non-local'. And this is the kind of thing that needs to considered in quantum mechanics and a lot of experiments are working on trying to understand this very thing.

In a classical way of analysis (physically that is), this is not only completely foreign in terms of our intuitive understanding and experience, it is a lot harder to deal with mathematically.

In terms of your questions, I can only answer them without using physical constraints: in other words, I will try and attack these from a mathematical viewpoint and not from one which is considered by physicists and this might be a little dissappointing but it will still, in my opinion give you some more understanding.

The thing for all your problems is that as a general rule, order is found when entropy is minimized.

Now the thing is that what we call 'time' is only one kind of order. Depending on the system, there are most likely going to be 'many' kinds of orders. In classical physics, time itself has a very good order to it in such a way that the conditional entropies in this context are very highly minimized in a way that the models give us something that is highly predictable which is just a result of a very low conditional entropy in the context of the system with regard to various conditional measures.

Intuitively, with calculus, the way we order things is always in terms of locality with respect to some form of a variable that is usually temporal or spatial in nature (often a system that involves a mix of the two). In terms of temporal, it has grown slowly from observation, first from a macroscopic level and then to a microscopic level but the idea is the same: relate local changes in space and time to a process and use calculus to model some form of global behaviour of the physical world.

But in the context of a general high state-space, highly complex general process, it may have many different kinds of orders and one order will typically hide a lot of information about the system in general in such a way that although the mathematical conclusions are correct, the interpretation may be very limited and in some ways detract from an otherwise higher understanding.

There is no problem with finding orders, but it needs to be considered that there might be other orders either from the raw system itself or from a transformed variant that gives an insight that can not be seen from the existing order that has been either chosen or subsequently discovered.

I will have to look up some of these things specifically later on to see what they correspond to mathematically, but the key issue in the above is to first define the states and then slowly describe the conditional distributions that are derived mostly from how these states interact.

You may find that interactions are constrained between specific parts of the system in the same kind of manner that you get local interactions spatially in terms of classical physics, but the thing is that the constraints are not-spatially local or even temporally (in the way that we see it) local and in this context you need to use a different way of analyzing the system.

Mathematically the way to describe the conditional entropies would be firstly to define the collection of all possible conditional probability distributions and then define an entropy for that distribution. You could also define things like relative entropies as well and all of this can be found in an information theory book, and I recommend Thomas and Cover's Information Theory 2nd edition book which you can buy on Amazon if you are really interested (also look on Wikipedia if you just want definitions and not something as formal).

What you'll find in highly ordered systems (no matter what the order) is that the majority of joint distributions have entropies of zero (or close enough to zero) that the order is easily determinable. If this is not the case, then it takes a lot more work (and subsequently 'appears' random).

I will look at the article in a little while: for now I hope the above has helped you.

 

[chiro]

Gedankenexperiments

S=∫ΔQ/T

The article at http://en.wikipedia.org/wiki/Negative_temperature#Examples reads:

""Since we started with over half the atoms in the spin-down state, initially this drives the system towards a 50/50 mixture, so the entropy is increasing, corresponding to a positive temperature. However, at some point more than half of the spins are in the spin-up position. In this case, adding additional energy reduces the entropy, since it moves the system further from a 50/50 mixture. This reduction in entropy with the addition of energy corresponds to a negative temperature."

This is the very thing underlying the whole motivation behind thermodynamics since we always assume entropy to either stay the same or increase for a particular kind of system, and we associate the study of energy 'in general' to having this sort of property.

It should be noted that this framework of physics grew out of the exercise of studying 'heat' and unsurprisingly many of the ways that we generate energy (that is later converted to electrical energy) is to this very day, still generated by heat. With coal we generate heat and that goes to energy, with nuclear we do the same thing and convert that to energy. The same thing for petroleum based forms of energy as well. It's all based on heating stuff up and converting it to energy.

There are exceptions for example using say hydrodynamic power stations, or wind power amongst other things but for most of the energy generation we just create enough heat so that we can create steam and drive a turbine which in my mind is absolutely ridiculuous but that's the way it is.

The result of creating more 'order' out of a system has been done in the lab (specifically look for experiments done previously at the Australian National University and there are probably others) but not to the extent where we would do it so much on a 'large scale'.

I'll have a look at the actual definitions of how they define heat in detail later on if you like, but it is true that a decrease in the entropy characteristics with how 'heat' is defined in terms of that particular entropy measure (which is going to be implicitly defined by the physical model which I need to read) will result in a decrease of the associated quantity as it's defined.

 

[chiro]

Please consider whether each of the following pairs is relatively entropic:

1. The big bang singularity and its imminent nonsingularity

Now I want a disclaimer: this is my own opinion. If it helps you then more power to you, but keep in mind that it's just an opinion that will based on a written/spoken argument that relates to what I've said previously in this thread and is not strictly mathematical in a formal sense.

For 1, this is a very good question. Here are my thoughts on this:

Many people have advocated that the 2nd law of thermodynamics should hold in the context of describing the physical universe with respect to the order that we call 'time' in that entropy should always be increasing if not at least staying the same.

To me I would say that this is only 'half-right' and in some ways misleading because an ever increasing entropy for a system means that the system gets 'more chaotic' if this happens for every form of entropy.

People talk about plates breaking, experiments with heat and other things that show a good argument for the entropy increase scenario and non-surprisingly time itself is defined by the 2nd law of thermodynamics (it's one way, but it's a very important definition in physics).

But if you consider all the different kinds of entropies that exist, I see evidence that the above is clearly not true. We have a lot of order in terms of some known approximations in physics and other scientific systems. Look around and just see the order that exists on our planet in terms of life-forms behaving with one another and in terms of any phenomena that has a high amount of stability with respect to its environment. In other words, some things in some contexts are producing situations where things become 'more-ordered' rather than 'more disordered'.

This leads me to infer that the 2nd law of thermodynamics applied across the board to represent the entire universe is faulty in its reasoning because if this was the case the universe would be in every respect, in a complete and utter state of chaos and this is not the case.

So with respect to entropies again I have to state that in a complex system there are going to be many different kinds of orders and I imagine when comparing and contrasting the different entropies of initial big bang and other states that the same argument needs to be applied. We currently do not have many different orders and when we start to get more insight and hence more different orders, we will start to explore this idea more clearly and more deeply.

2. Electron self-energy at a point and a spatial perspective

I will have to read up on this.

3. A cosmologist observing his self-inclusive universe

Again this has to do with the order that they are trying to apply. A cosmologist has a very different set of orders that they are considering with respect to a physicist that is studying macroscopic things at probably the level of the atmosphere at the high end or even with respect to a physicist that is studying how atoms behave in a very controlled environment.

To answer your question, you have to specify the kinds of orders used and because of this I can't give a decent answer to your question because it's too broad.

4. A closed universe and the black holes within

This is a very interesting question.

For this to be answered we would need to know how information is exchanged between things inside a black hole (in the event horizon) and things beyond the horizon.

If it turns out that information is exchanged (I think this is in debate currently) then that will make a huge difference with how we form constraints for the joint distributions and entropies and it also means we have to consider a system that is much much larger and more complex.

If things are completely isolated, then this simplifies things dramatically but again with Hawking idea of evaporation from black-holes I have a feeling that if the theory is correct or even if the idea is correct in terms of some form of radiation, then this means that essentially there is 'communication' (information exchange) going on and this needs to be taken into account.

Also if there is kind of entanglement that is not spatio-temporally local (i.e. action at a distance in the context of between two different space-time boundaries) then this would make it even more broader.

5. A quantum measurement and its measuring device

Ahh the measurement problem.

In terms of the measurement problem, again this is going to relate to any analysis of the joint distributions with respect to anything that is associated with the device.

It doesn't make our problem any easier because we will need to consider orders that are much much harder to extrapolate from the properties of our system than we currently do now, but again the idea of finding orders is the same except we are considering it in a different context.

You need to note that you will need to look at different orders other than the standard ones mentioned if they indeed do exist.

One thing I will mention though is that if there is some kind of arbitrage mechanism that exists to keep things stable then this could be used to formulate the properties of the various distributions and test it experimentally. I am not a physicist though.

The idea behind arbitrage in the way I am describing is that the system would have to account in whatever way it can so that there is not enough determinism in the system to produce a particular point of instability. If the system was weak enough so that it could be exploited to create instability that was detrimental to the function of the system itself, then this would cause a kind of 'system-wide turbulence' that would be utterly destructive.

It's my opinion, but it's based on the idea of creating a system that doesn't essentially 'blow up inadvertendly'.

6. A vacuum of virtual particles

I don't understand enough about Quantum Theory to even give something even remotely useful for this particular question.

7. Turbulence at temperature T→∞

The thing that is missing for this question is the definition for the order known as 'turbulence'. I don't know of a single definition that is specific enough to define this. You can't analyze something you can not describe adequately. If you give me something specific enough, I'll try to answer your question.

8. Black bodies at temperature T

I am not a physicist, so I would need a bit more clarification about what you are asking.

 

[Loren Booda]

Blackbodies at temperature T -- http://en.wikipedia.org/wiki/Black_body A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence.

A black body in thermal equilibrium (that is, at a constant temperature) emits electromagnetic radiation called black-body radiation. The radiation is emitted according to Planck's law, meaning that it has a spectrum that is determined by the temperature alone, not by the body's shape or composition.

A black body in thermal equilibrium has two notable properties:

It is an ideal emitter: it emits as much or more energy at every frequency than any other body at the same temperature.
It is a diffuse emitter: the energy is radiated isotropically, independent of direction.

An approximate realization of a black body is a hole in the wall of a large enclosure. Any light entering the hole is reflected indefinitely or absorbed inside and is unlikely to re-emerge, making the hole a nearly perfect absorber. The radiation confined in such an enclosure may or may not be in thermal equilibrium, depending upon the nature of the walls and the other contents of the enclosure.

A vacuum of virtual particles -- http://en.wikipedia.org/wiki/Vacuum_state In quantum field theory, the vacuum state (also called the vacuum) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. Zero-point field is sometimes used as a synonym for the vacuum state of an individual quantized field.

According to present-day understanding of what is called the vacuum state or the quantum vacuum, it is "by no means a simple empty space", and again: "it is a mistake to think of any physical vacuum as some absolutely empty void." According to quantum mechanics, the vacuum state is not truly empty but instead contains fleeting electromagnetic waves and particles that pop into and out of existence.

The presence of virtual particles can be rigorously based upon the non-commutation of the quantized electromagnetic fields. Non-commutation means that although the average values of the fields vanish in a quantum vacuum, their variances do not. The term "vacuum fluctuations" refers to the variance of the field strength in the minimal energy state, and is described picturesquely as evidence of "virtual particles".

It is sometimes attempted to provide an intuitive picture of virtual particles based upon the Heisenberg energy-time uncertainty principle:

Δ E Δ t ≥ h-bar ,

(with ΔE and Δt energy and time variations, and h-bar the Planck constant divided by 2π) arguing along the lines that the short lifetime of virtual particles allows the "borrowing" of large energies from the vacuum and thus permits particle generation for short times.

Anthropic entropic principle -- I hypothesize that, rather than observers (life) be where entropy density is high, they exist where entropy density is low. The act of observation itself could rely on semicoherent radiative interaction, and so tend participants toward lower entropy density.

A cosmologist observing his self-inclusive universe -- I believe this could be modeled by your staircase algorithm, chiro, where observation cycles to the event horizon and back, and as speculated by early relativists, observers could see themselves gravitationally imaged about the circumference.

Anentropy -- I think that entropy depends not only on the states of a configuration, but also on the network of interconnections (entanglement) between states." Anentropic" by nature of retrospection, this latter "pattern memory" potentially surpasses entropy's information exponentially in magnitude.

Reciprocity of entropy -- In practice, the inequality in the second "law" of thermodynamics may be the crux of the argument against it being a true law. This law may be violated for a nonisolated system. But might it also not hold in general time-like spacetime?