I always thought, on the basis of Boltzmann's equation, that a state of high entropy is stabler (more probable) than a state of low entropy, Am I wrong?
Tullio
True, but there are subtleties!
I assume you mean this Boltzmann equation ( there are several ):
S = k ln ( W )
where S is entropy, k is a constant of proportionality ie. Boltzmann's constant, and W is the number of states accessible to the system ( disorder, roughly - or more precisely the number of distinct microstates consistent with a given macrostate ), and 'ln' refers to natural logarithm - the introduction of which essentially tames the writing out of the many orders of magnitude that exist in real systems.
The equation is a definition of entropy based upon state counting.
So if one has a system with some macroscopic parameters ( temperature, pressure, volume..... whatever ), then any actual specific set of values of these parameters is going to be consistent with a certain number of distinct and equally probable particle states which you are not individually observing. This means that if you re-arrange these underlying particle states amongst themselves then you will not observe any difference in the macroscopic parameters. Systems of course are continually transitioning between these microstates as the particles interact, but overall you won't notice this with a large-scale/coarse measurement like say pressure ( which is the summation of alot of little hits against a surface ). It is a fact that systems of large numbers of particles tend to behave this way. ( The logic is much the same if one considers an ensemble - meaning a large number of similiarly prepared systems, a fraction of which are in the states of interest ). I've assumed that each of the distinct microstates is equally likely - but you can adjust the calculations with weightings if not. So if you have more microstates ( higher W ), then by the above definition S will be larger too, and of course more microstates contribute more to the numerator of any probability calculation ( the denominator being some total state count - or a total ensemble count ).
Note that if a system is in equilibrium, it's macroscopic parameters do not vary - by definition - and we can make the a priori assumption that each microstate is equally likely. If not in equilibrium, then that is not so.....
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Thanks Mike, that helps quite a bit. So is the universe considered open or closed or isolated..? I understand it's flat, and possibly even just one of many in parallel, but ... how did the energy from the big bang get in? Is energy leaking out? When you mentioned everything occupying the lowest possible quantum state, would that then be like the pre-big bang singularity?
Yep..... they're goodies alright! :-)
Some feel that the universe is the 'ultimate free lunch', which I think means they don't really know.
Some say that interior of black holes 'bud off' universes, implying ours is a bud from some other universe.
Any answer presupposes some knowledge of what went 'before' or 'outside' singularities! Initial conditons ( like a bank account balance ) tend to determine the future, but singularities wipe out knowledge ( and banks too ). There is some fascinating conjecture about the fate of entropy with black holes, given that they are 'one way', meaning that : it would seem that any system with some amount of order/disorder is not going to partake in any physical process ( beyond gravitational effect from it's mass/energy, electromagnetic effect from it's charge, or some dynamic effect from it's angular momentum ) once it crosses the event horizon and thus is going to be lost to any accounting of states.
Quote:
What does 'increase the measure of their occupied quantum states' mean? Are more states occupied closer to the ground state, or does the measure of unoccupied states increase in quantity (in the steam engine example)?
See below for the state counting stuff...
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
I always thought, on the basis of Boltzmann's equation, that a state of high entropy is stabler (more probable) than a state of low entropy, Am I wrong?
Tullio
True, but there are subtleties!
I assume you mean this Boltzmann equation ( there are several ):
S = k ln ( W )
.....
Cheers, Mike.
Yes, that's the one which appears on his tomb in Wien. Ludwig Boltzmann committed suicide in Duino in 1906. Duino is a small seaside village where Dante is believed to have passed and Rainer Maria Rilke wrote his "Elegies". Now it hosts a World College sponsored by the United Nations.
Tullio
I always thought, on the basis of Boltzmann's equation, that a state of high entropy is stabler (more probable) than a state of low entropy, Am I wrong?
Tullio
True, but there are subtleties!
I assume you mean this Boltzmann equation ( there are several ):
S = k ln ( W )
.....
Cheers, Mike.
Yes, that's the one which appears on his tomb in Wien. Ludwig Boltzmann committed suicide in Duino in 1906. Duino is a small seaside village where Dante is believed to have passed and Rainer Maria Rilke wrote his "Elegies". Now it hosts a World College sponsored by the United Nations.
Tullio
I always thought, on the basis of Boltzmann's equation, that a state of high entropy is stabler (more probable) than a state of low entropy, Am I wrong?
Tullio
True, but there are subtleties!
I assume you mean this Boltzmann equation ( there are several ):
S = k ln ( W )
.....
Cheers, Mike.
Yes, that's the one which appears on his tomb in Wien. Ludwig Boltzmann committed suicide in Duino in 1906. Duino is a small seaside village where Dante is believed to have passed and Rainer Maria Rilke wrote his "Elegies". Now it hosts a World College sponsored by the United Nations.
Tullio
Tullio, Lei non ha letto l'Inferno di Dante ?
Nel mezzo del cammin di nostra vita
Mi ritrovai per una selva oscura
Question: what relationship exists between "selva oscura" and "La noche sombre de l'alma" of Saint John of the Cross? See also Georges Bernanos's "Je suis moi meme nuit", in "Diary of a country priest". But this is metaphysics, no longer physics.
I always thought, on the basis of Boltzmann's equation, that a state of high entropy is stabler (more probable) than a state of low entropy, Am I wrong?
Tullio
True, but there are subtleties!
I assume you mean this Boltzmann equation ( there are several ):
S = k ln ( W )
.....
Cheers, Mike.
Yes, that's the one which appears on his tomb in Wien. Ludwig Boltzmann committed suicide in Duino in 1906. Duino is a small seaside village where Dante is believed to have passed and Rainer Maria Rilke wrote his "Elegies". Now it hosts a World College sponsored by the United Nations.
Tullio
Tullio, Lei non ha letto l'Inferno di Dante ?
Nel mezzo del cammin di nostra vita
Mi ritrovai per una selva oscura
Question: what relationship exists between "selva oscura" and "La noche sombre de l'alma" of Saint John of the Cross? See also Georges Bernanos's "Je suis moi meme nuit", in "Diary of a country priest". But this is metaphysics, no longer physics.
Tullio,Io no so, non posso descrivere questo relatio, neri pensieri fat rompere mio cuore.
La nuit sombre de l'ame, c'est dechirant, je ne puis en parler.
It is said that universe should have a point of beginning :the Big Bang. The general relativity is unable to tell us how the universe began, but the theories of singularity mention that in the first times of the universe , it should exist a moment when it was so tiny that only the quantum mechanics can give some explanation.
On the basis of classical theory of gravity relying on "real" Time-Space, we can only suppose two behaviors for the Universe : or it existed eternally or it began by a singularity in a determined moment of the past. But in quantum theory of gravity, there is a third option : the ambiguous time. In the "real" time, Universe has a beginning and an end in singularities. Singularities being the borders of Time-Space , the laws are anihilated on those points. But in ambiguous time, singularity and border are absent. Then we deduce that perhaps what we call the ambiguous time is much more fundamental, and what we call the real time is just a convention or a fiction we have created to describe what we suppose "credible" about the universe.
Given a scientific theory is a mathematical paradigm we made to explain our observations , and it exists only in our mind, so this question that which one of the two, real time or ambiguous time is "real" doesn't matter, the question is which one can give a simpler description.
For example by using the integral of directions along with "unlimited", we can calculate the probability of the expansion of the Universe with the same quotient in all directions and at the moment when the mass of universe reached its present amount.
Based on the condition of "unlimited", we can deduce that the Universe should have began with the least possible "non-uniformity", valid in the principle of uncertainty...
I always thought, on the basis of Boltzmann's equation, that a state of high entropy is stabler (more probable) than a state of low entropy, Am I wrong?
Tullio
True, but there are subtleties!
I assume you mean this Boltzmann equation ( there are several ):
S = k ln ( W )
.....
Cheers, Mike.
Yes, that's the one which appears on his tomb in Wien. Ludwig Boltzmann committed suicide in Duino in 1906. Duino is a small seaside village where Dante is believed to have passed and Rainer Maria Rilke wrote his "Elegies". Now it hosts a World College sponsored by the United Nations.
Tullio
Tullio, Lei non ha letto l'Inferno di Dante ?
Nel mezzo del cammin di nostra vita
Mi ritrovai per una selva oscura
Question: what relationship exists between "selva oscura" and "La noche sombre de l'alma" of Saint John of the Cross? See also Georges Bernanos's "Je suis moi meme nuit", in "Diary of a country priest". But this is metaphysics, no longer physics.
Tullio,Io no so, non posso descrivere questo relatio, neri pensieri fat rompere mio cuore.
La nuit sombre de l'ame, c'est dechirant, je ne puis en parler.
Yes, silence is better. But cheer up, you are not alone.
Tullio
There is some fascinating conjecture about the fate of entropy with black holes...
Thanks Mike. But I'd still like very much to believe that humans posses the ability to evaluate equilibria, and then make decisions regarding action upon particular states, and that such decisions are based on notions of 'right and wrong' rather than 'hot and cold.' Is this belief an illusion? Or can maths be used to show that right = hot and wrong = cold ?
Clearly there are more dimensions than just 3 spatial ones. All due respect to Hawking and others, if the structure of these dimensions (however many) accommodates singularities, then isn't all information necessarily preserved on both sides of an event horizon, and in the distance between the horizon and the singularity, why shouldn't it take something longer to traverse that distance than it takes, say, for a photon to go from the core of a star to the surface?
On the basis of classical theory of gravity relying on "real" Time-Space, we can only suppose two behaviors for the Universe : or it existed eternally or it began by a singularity in a determined moment of the past.
But in fact we know that classical theories are all of them only approximately true, and we know that at scales smaller than the so-called Planck length the classical theory of gravity does not hold. So the best interpretation is to say that neither of the imagined possibilites is relevant
Quote:
But in quantum theory of gravity, there is a third option : the ambiguous time. In the "real" time, Universe has a beginning and an end in singularities. Singularities being the borders of Time-Space , the laws are anihilated on those points. But in ambiguous time, singularity and border are absent.
Some quantum theories are like that, others are not. Until we have a quantum gravity that works it is just speculation to say any more than 'time may be ambiguous'.
Quote:
Given a scientific theory is a mathematical paradigm we made to explain our observations , and it exists only in our mind, so this question that which one of the two, real time or ambiguous time is "real" doesn't matter, the question is which one can give a simpler description.
That is the *second* criteria we use to choose a theory.
The first, and most important, is that a theory generates predictions which are testable and which provide a genuine 'risk' that the theory may be disproved. Without that it is not a theory in the proper scientific sense.
If we choose between untestable theories on the basis of preference we are doing philosophy (or maybe theology), we are doing not science in my opinion. That is not a problem so long as we understand that we are doing science philosophy inspired by science rather than science itself.
If scientists were as exact in their language as they like to pretend, they would not talk of 'string theory' but of 'the string approach', and similarly all the other quantum gravity 'theories'.
RE: I always thought, on
)
True, but there are subtleties!
I assume you mean this Boltzmann equation ( there are several ):
S = k ln ( W )
where S is entropy, k is a constant of proportionality ie. Boltzmann's constant, and W is the number of states accessible to the system ( disorder, roughly - or more precisely the number of distinct microstates consistent with a given macrostate ), and 'ln' refers to natural logarithm - the introduction of which essentially tames the writing out of the many orders of magnitude that exist in real systems.
The equation is a definition of entropy based upon state counting.
So if one has a system with some macroscopic parameters ( temperature, pressure, volume..... whatever ), then any actual specific set of values of these parameters is going to be consistent with a certain number of distinct and equally probable particle states which you are not individually observing. This means that if you re-arrange these underlying particle states amongst themselves then you will not observe any difference in the macroscopic parameters. Systems of course are continually transitioning between these microstates as the particles interact, but overall you won't notice this with a large-scale/coarse measurement like say pressure ( which is the summation of alot of little hits against a surface ). It is a fact that systems of large numbers of particles tend to behave this way. ( The logic is much the same if one considers an ensemble - meaning a large number of similiarly prepared systems, a fraction of which are in the states of interest ). I've assumed that each of the distinct microstates is equally likely - but you can adjust the calculations with weightings if not. So if you have more microstates ( higher W ), then by the above definition S will be larger too, and of course more microstates contribute more to the numerator of any probability calculation ( the denominator being some total state count - or a total ensemble count ).
Note that if a system is in equilibrium, it's macroscopic parameters do not vary - by definition - and we can make the a priori assumption that each microstate is equally likely. If not in equilibrium, then that is not so.....
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: Thanks Mike, that helps
)
Yep..... they're goodies alright! :-)
Some feel that the universe is the 'ultimate free lunch', which I think means they don't really know.
Some say that interior of black holes 'bud off' universes, implying ours is a bud from some other universe.
Any answer presupposes some knowledge of what went 'before' or 'outside' singularities! Initial conditons ( like a bank account balance ) tend to determine the future, but singularities wipe out knowledge ( and banks too ). There is some fascinating conjecture about the fate of entropy with black holes, given that they are 'one way', meaning that : it would seem that any system with some amount of order/disorder is not going to partake in any physical process ( beyond gravitational effect from it's mass/energy, electromagnetic effect from it's charge, or some dynamic effect from it's angular momentum ) once it crosses the event horizon and thus is going to be lost to any accounting of states.
See below for the state counting stuff...
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: RE: I always thought,
)
Yes, that's the one which appears on his tomb in Wien. Ludwig Boltzmann committed suicide in Duino in 1906. Duino is a small seaside village where Dante is believed to have passed and Rainer Maria Rilke wrote his "Elegies". Now it hosts a World College sponsored by the United Nations.
Tullio
RE: RE: RE: I always
)
Tullio, Lei non ha letto l'Inferno di Dante ?
Ariane
RE: RE: RE: RE: I
)
Nel mezzo del cammin di nostra vita
Mi ritrovai per una selva oscura
Question: what relationship exists between "selva oscura" and "La noche sombre de l'alma" of Saint John of the Cross? See also Georges Bernanos's "Je suis moi meme nuit", in "Diary of a country priest". But this is metaphysics, no longer physics.
RE: RE: RE: RE: Quote
)
Tullio,Io no so, non posso descrivere questo relatio, neri pensieri fat rompere mio cuore.
La nuit sombre de l'ame, c'est dechirant, je ne puis en parler.
Ariane
It is said that universe
)
It is said that universe should have a point of beginning :the Big Bang. The general relativity is unable to tell us how the universe began, but the theories of singularity mention that in the first times of the universe , it should exist a moment when it was so tiny that only the quantum mechanics can give some explanation.
On the basis of classical theory of gravity relying on "real" Time-Space, we can only suppose two behaviors for the Universe : or it existed eternally or it began by a singularity in a determined moment of the past. But in quantum theory of gravity, there is a third option : the ambiguous time. In the "real" time, Universe has a beginning and an end in singularities. Singularities being the borders of Time-Space , the laws are anihilated on those points. But in ambiguous time, singularity and border are absent. Then we deduce that perhaps what we call the ambiguous time is much more fundamental, and what we call the real time is just a convention or a fiction we have created to describe what we suppose "credible" about the universe.
Given a scientific theory is a mathematical paradigm we made to explain our observations , and it exists only in our mind, so this question that which one of the two, real time or ambiguous time is "real" doesn't matter, the question is which one can give a simpler description.
For example by using the integral of directions along with "unlimited", we can calculate the probability of the expansion of the Universe with the same quotient in all directions and at the moment when the mass of universe reached its present amount.
Based on the condition of "unlimited", we can deduce that the Universe should have began with the least possible "non-uniformity", valid in the principle of uncertainty...
Ariane
RE: RE: RE: RE: Quote
)
Yes, silence is better. But cheer up, you are not alone.
Tullio
RE: There is some
)
Thanks Mike. But I'd still like very much to believe that humans posses the ability to evaluate equilibria, and then make decisions regarding action upon particular states, and that such decisions are based on notions of 'right and wrong' rather than 'hot and cold.' Is this belief an illusion? Or can maths be used to show that right = hot and wrong = cold ?
Clearly there are more dimensions than just 3 spatial ones. All due respect to Hawking and others, if the structure of these dimensions (however many) accommodates singularities, then isn't all information necessarily preserved on both sides of an event horizon, and in the distance between the horizon and the singularity, why shouldn't it take something longer to traverse that distance than it takes, say, for a photon to go from the core of a star to the surface?
RE: On the basis of
)
But in fact we know that classical theories are all of them only approximately true, and we know that at scales smaller than the so-called Planck length the classical theory of gravity does not hold. So the best interpretation is to say that neither of the imagined possibilites is relevant
Some quantum theories are like that, others are not. Until we have a quantum gravity that works it is just speculation to say any more than 'time may be ambiguous'.
That is the *second* criteria we use to choose a theory.
The first, and most important, is that a theory generates predictions which are testable and which provide a genuine 'risk' that the theory may be disproved. Without that it is not a theory in the proper scientific sense.
If we choose between untestable theories on the basis of preference we are doing philosophy (or maybe theology), we are doing not science in my opinion. That is not a problem so long as we understand that we are doing science philosophy inspired by science rather than science itself.
If scientists were as exact in their language as they like to pretend, they would not talk of 'string theory' but of 'the string approach', and similarly all the other quantum gravity 'theories'.
River~~
~~gravywavy