I can't believe how dense certain people are. After many years that Sean Carroll had to localize the elementary, childish mistakes in his reasoning, he is still walking around and talking about the "arrow of time mystery" and the "Past Hypothesis":
Reverse Times Square 2013. In the real world, objects moving forward and backward in time can't co-exist, a well-defined logical arrow of time indicating what evolves from what has to exist everywhere. That's also why none of the clowns above managed to unbreak an egg. Exercise for you: Can the video above be authentic or was it inevitably computer-reversed afterwords? You should be able to decide by looking at details of the video. For the advanced viewers: Which parts were edited?
Moreover, since its very birth, statistical physics was a set of methods to deal with the laws of physics that were time-reversal- (or CPT-) symmetric and it always "got off the ground" beautifully. There is absolutely nothing new about these matters in 2013. In fact, nothing really qualitative has changed about statistical physics for a century or so.
At least, you see some progress in Carroll's talk about these matters because in the quote above, he at least seems to admit that to reconstruct the past, you need to use somewhat different methods than to predict the future. For years, we would hear him saying the – completely wrong – statement that the retrodictions follow the very same procedure as predictions.
However, his ideas about the way how retrodictions work are still completely wrong.
He thinks that to retrodict the past, "we need to conditionalize over trajectories that also started in a low-entropy past state". But that's not true. In fact, the correct method to retrodict the past – one based on the Bayesian inference – automatically implies that the entropy was lower in the past than it is now.
I have already explained this thing about 20 times on this blog but the reason why I keep on repeating the simple argument is that I want to be absolutely certain that I am accurate when I say that every physicist who still fails to understand these issues is either a fraudster or a complete moron avoiding fundamental knowledge whatever it costs.
To predict the future, you take the initial state – either a point in the classical phase space or the pure state vector in quantum mechanics; or (if you deal with incomplete knowledge) the probability distribution on the phase space or the density matrix in quantum mechanics – and evolve them in the future direction. In general, this tells you the probabilities of different final states whose interpretation is straightforward (frequentism is good enough).
However, to retrodict the past, you need to follow a reverse procedure. The possible initial states are different hypotheses \(H_i\) as understood by Bayesian inference. These hypotheses – initial states – predict (i.e. imply) different observations (observed in the present). The actual observations are evidence that may be used to adjust the probabilities of different hypotheses (initial states).
It's important to realize that just like we can't predict the exact state in the future – only probabilities may be reconstructed – we can't retrodict the exact microstate in the past. We may only retrodict probabilities of different states but we have to do it right. When we do it right, we automatically see that the entropy of the state in the past had to be lower, just like the second law of thermodynamics always said.
I borrow the proof that the entropy of the initial state was lower from the December 2012 article, Information and Heat. Let's consider the evolution from the ensemble \(A\) to ensemble \(B\). By ensembles, I mean some states roughly given by their macroscopic properties which may be represented by one of many "practically indistinguishable" microstates. Only when we consider statistically large collections of atoms or other degrees of freedom, we may derive some non-probabilistic statements such as "one entropy is higher than the other".
The probability of the evolution from \(A\) to \(B\) is\[
P(A\to B) = \sum_{a\in A}\sum_{b\in B} P(a)\cdot P(a\to b)
\] where \(a,b\) go over microstates included in the ensembles \(A,B\), respectively. This notation is valid both for classical physics and quantum mechanics (where the microstates refer to basis vectors in an orthonormal basis of the ensemble subspace); in quantum mechanics, we could write the expression in a more basis-independent way using traces, too. If we compute the probabilities for ensembles, we must sum the probabilities over all possible final microstates \(b\) because we're calculating the probability of "one possible \(b\)" or "another possible \(b\)" and the word "OR" corresponds to the addition of probabilities.
On the other hand, we have included \(P(a)\) which denotes the probability that the initial microstate is a particular \(a\), because the probability of evolution \(P_{a\to b}\) is only relevant when the initial state actually is the particular \(a\). Now, we will simplify our life and assume that all microstates \(a\) are equally likely so \(P(a)=1/N(A)\), the inverse number of microstates in the ensemble \(A\). This "egalitarian" assumption could be modified and different microstates in the ensemble could be given diifferent prior probabilities but nothing would qualitatively change.
It's also possible to calculate the probability of the time-reversed process \(B^*\to A^*\). The stars mean that the signs of the velocities get inverted as well; in quantum field theory, the star would represent the complete CPT conjugate of the original ensemble because CPT is the only reliably universal symmetry involving the time reversal (but the star may also mean T whenever it is a symmetry). Clearly, the probability is\[
P(B^*\to A^*) = \sum_{a^*\in A^*} \sum_{b^*\in B^*} P(b^*) P(a\to b)
\] where I used that \(P(a\to b)=P(b^*\to a^*)\) from the time-reversal symmetry (or, more safely, CPT symmetry) of the evolution followed by the microstates. Note that the two probabilities are almost given by the same formula but in the latter one, we see the factor \(P(b^*)=1/N(B^*)=1/N(B)\) in our simplified "equal odds" setup. At any rate, the ratio is\[
\frac{P(A\to B)}{P(B^*\to A^*)} = \frac{P_a}{P_b} = \frac{1/N(A)}{1/N(B)} = \exp\left(\frac{S_B-S_A}{k}\right)
\] where I used \(N(A)=\exp(S_A/k)\) – the inverted formula from Boltzmann's tomb – and similarly for \(B\). But now realize that \(k\) is extremely tiny in the macroscopic SI-like units while \(S_B-S_A\), the entropy change, is finite. So the ratio \((S_B-S_A)/k\) is equal either to plus infinity or minus infinity. It follows that one of the probabilities, \(P(A\to B)\) or \(P(B^*\to A^*)\), is infinitely (even exponential of infinity) times greater (i.e. more likely) than the other one which means that only one of the processes may occur with a nonzero probability (just like probabilities in general, both are less than or equal to one which means that the larger one equals at most one and the smaller one is infinitely smaller than that, i.e. zero).
If you look at the sign, the only possible (nonzero probability) transition between ensembles is one for which the final entropy is greater than the initial one. If \(S_B-S_A\gt 0\), the numerator \(P(A\to B)\) is infinitely greater than the denominator \(P(B^*\to A^*)\) which means that only the numerator is allowed to be nonzero. In other words, \(A\to B\) may occur while \(B^*\to A^*\) is prohibited.
Again, I have proven that the evolution in which the entropy increases is hugely more likely than the reverse process – the ratio of their probabilities is \(\exp(\Delta S/k)\) where \(\Delta S\) is the entropy difference between the initial and final state. To prove such a thing, I didn't have to assume anything about the Earth, the Solar System, the Universe, or cosmology because these – totally universal – calculations in statistical physics and their conclusions (that were already known to thermodynamics) have nothing to do with these large objects. If we were talking about the entropy change in a closed lab, we have only discussed the degrees of freedom (atoms...) inside the lab. Whatever the rest of the Universe was doing had no impact whatsoever on the calculation (and the resulting odds).
The calculation above is not hard but you may say that this particular general derivation above – presented in a way optimized by your humble correspondent, to say the least – is a bit unfamiliar to students who don't go through clear enough courses. Perhaps it is a bit ingenious. It may be OK if people happen to be ignorant about it.
However, what I find amazing is that people like Carroll are clearly ignorant about certain facts indicating that they must completely misunderstand the scientific method. Why? Sean Carroll is clearly convinced that the laws of physics as he knows them imply that the entropy of the young Universe had to be higher than the current entropy. So this is a prediction of those laws of physics and those methods to calculate that he is proposing. Because he also agrees that the entropy of the young Universe was actually lower, it must mean that his laws of physics or methods or both have been falsified. They're dead because they have made a wrong prediction (in this case, infinitely many exponentially bad predictions about pretty much everything!). Once they're dead, they can't be made "undead" again. This is a trivial observation that everyone who uses the scientific method in any context simply has to realize. Because Sean Carroll doesn't realize that, he can't possibly understand what the scientific method is.
If there is a contradiction between the empirical data – low entropy of the Universe in the past – and your theory, your theory has to be abandoned. You have to throw at least pieces of it and, perhaps, replace them with other pieces. Just adding "extra stuff" to your theory can't possibly help. If the theory was able to produce a wrong prediction (at least with a high enough confidence, like 99.9999%), then it has been falsified and making the theory more extensive (by the supposedly helpful additions such as "the cosmological realization measure" or the "Past Hypothesis") just can't solve the problem.
I am also amazed by the brutal distortion of the history below – Sean Carroll must know very well that it is a distortion:
It's important to realize that this second law or H-theorem apply to any moment of the history and any state. It doesn't matter whether the year "now" is 2013 or less or more. The proof of the H-theorem doesn't depend on "now" or the current year at all. It always implies that the entropy increases. It will increase between 2013 and 2014 and it was increasing between 2012 and 2013, too. If \(t_1\lt t_2\), then \(S(t_1)\lt S(t_2)\). One may have different emotional attachments to these two periods of time (perhaps we were not yet born at time \(t_1\) while we fell in love at \(t_2\)) but physics – and the proof of the H-theorem – just isn't about emotions. It is about facts and the laws of physics and about mathematically valid propositions and implications – and those are (perhaps except for the facts that may be interpreted as time-dependent things) independent of the time, independent of the current year.
I am also stunned by the fact that among the hundreds of readers who were probably going through the blog entry at Sean Carroll's blog, there was not a single person who would understand these basic issues about statistical physics and who would kindly explain to Sean Carroll how utterly stupid he was being and how superfluous all his concepts and proposed hypotheses are.
Incidentally, some news related to time: Strontium lattice clocks may improve the accuracy of the state-of-the-art atomic clocks by a stunning factor of 100! See BBC, Nature.
Cosmology and the Past HypothesisHis Santa Cruz, California talk about this non-problem took no less than 3.5 hours. Imagine actual people sitting and listening to this utter junk for such a long time. I can't even imagine that. He has even coined some new phrases that describe the very same misconception of yours.
If there is one central idea, it’s the concept of a “cosmological realization measure” for statistical mechanics. Ordinarily, when we have some statistical system, we know some macroscopic facts about it but only have a probability distribution over the microscopic details. If our goal is to predict the future, it suffices to choose a distribution that is uniform in the Liouville measure given to us by classical mechanics (or its quantum analogue). If we want to reconstruct the past, in contrast, we need to conditionalize over trajectories that also started in a low-entropy past state — that the “Past Hypothesis” that is required to get stat mech off the ground in a world governed by time-symmetric fundamental laws.As talented enough students learn when they are college juniors or earlier, statistical mechanics explains thermodynamic phenomena by statistically analyzing large collections of atoms (or other large collections of degrees of freedom) and doesn't depend on cosmology (or anything that is larger than the matter whose behavior we want to understand) in any way whatsoever. It's the atoms, short-distance physics, that determines the behavior of larger objects (including the whole Universe), not the other way around!
Reverse Times Square 2013. In the real world, objects moving forward and backward in time can't co-exist, a well-defined logical arrow of time indicating what evolves from what has to exist everywhere. That's also why none of the clowns above managed to unbreak an egg. Exercise for you: Can the video above be authentic or was it inevitably computer-reversed afterwords? You should be able to decide by looking at details of the video. For the advanced viewers: Which parts were edited?
Moreover, since its very birth, statistical physics was a set of methods to deal with the laws of physics that were time-reversal- (or CPT-) symmetric and it always "got off the ground" beautifully. There is absolutely nothing new about these matters in 2013. In fact, nothing really qualitative has changed about statistical physics for a century or so.
At least, you see some progress in Carroll's talk about these matters because in the quote above, he at least seems to admit that to reconstruct the past, you need to use somewhat different methods than to predict the future. For years, we would hear him saying the – completely wrong – statement that the retrodictions follow the very same procedure as predictions.
However, his ideas about the way how retrodictions work are still completely wrong.
He thinks that to retrodict the past, "we need to conditionalize over trajectories that also started in a low-entropy past state". But that's not true. In fact, the correct method to retrodict the past – one based on the Bayesian inference – automatically implies that the entropy was lower in the past than it is now.
I have already explained this thing about 20 times on this blog but the reason why I keep on repeating the simple argument is that I want to be absolutely certain that I am accurate when I say that every physicist who still fails to understand these issues is either a fraudster or a complete moron avoiding fundamental knowledge whatever it costs.
To predict the future, you take the initial state – either a point in the classical phase space or the pure state vector in quantum mechanics; or (if you deal with incomplete knowledge) the probability distribution on the phase space or the density matrix in quantum mechanics – and evolve them in the future direction. In general, this tells you the probabilities of different final states whose interpretation is straightforward (frequentism is good enough).
However, to retrodict the past, you need to follow a reverse procedure. The possible initial states are different hypotheses \(H_i\) as understood by Bayesian inference. These hypotheses – initial states – predict (i.e. imply) different observations (observed in the present). The actual observations are evidence that may be used to adjust the probabilities of different hypotheses (initial states).
It's important to realize that just like we can't predict the exact state in the future – only probabilities may be reconstructed – we can't retrodict the exact microstate in the past. We may only retrodict probabilities of different states but we have to do it right. When we do it right, we automatically see that the entropy of the state in the past had to be lower, just like the second law of thermodynamics always said.
I borrow the proof that the entropy of the initial state was lower from the December 2012 article, Information and Heat. Let's consider the evolution from the ensemble \(A\) to ensemble \(B\). By ensembles, I mean some states roughly given by their macroscopic properties which may be represented by one of many "practically indistinguishable" microstates. Only when we consider statistically large collections of atoms or other degrees of freedom, we may derive some non-probabilistic statements such as "one entropy is higher than the other".
The probability of the evolution from \(A\) to \(B\) is\[
P(A\to B) = \sum_{a\in A}\sum_{b\in B} P(a)\cdot P(a\to b)
\] where \(a,b\) go over microstates included in the ensembles \(A,B\), respectively. This notation is valid both for classical physics and quantum mechanics (where the microstates refer to basis vectors in an orthonormal basis of the ensemble subspace); in quantum mechanics, we could write the expression in a more basis-independent way using traces, too. If we compute the probabilities for ensembles, we must sum the probabilities over all possible final microstates \(b\) because we're calculating the probability of "one possible \(b\)" or "another possible \(b\)" and the word "OR" corresponds to the addition of probabilities.
On the other hand, we have included \(P(a)\) which denotes the probability that the initial microstate is a particular \(a\), because the probability of evolution \(P_{a\to b}\) is only relevant when the initial state actually is the particular \(a\). Now, we will simplify our life and assume that all microstates \(a\) are equally likely so \(P(a)=1/N(A)\), the inverse number of microstates in the ensemble \(A\). This "egalitarian" assumption could be modified and different microstates in the ensemble could be given diifferent prior probabilities but nothing would qualitatively change.
It's also possible to calculate the probability of the time-reversed process \(B^*\to A^*\). The stars mean that the signs of the velocities get inverted as well; in quantum field theory, the star would represent the complete CPT conjugate of the original ensemble because CPT is the only reliably universal symmetry involving the time reversal (but the star may also mean T whenever it is a symmetry). Clearly, the probability is\[
P(B^*\to A^*) = \sum_{a^*\in A^*} \sum_{b^*\in B^*} P(b^*) P(a\to b)
\] where I used that \(P(a\to b)=P(b^*\to a^*)\) from the time-reversal symmetry (or, more safely, CPT symmetry) of the evolution followed by the microstates. Note that the two probabilities are almost given by the same formula but in the latter one, we see the factor \(P(b^*)=1/N(B^*)=1/N(B)\) in our simplified "equal odds" setup. At any rate, the ratio is\[
\frac{P(A\to B)}{P(B^*\to A^*)} = \frac{P_a}{P_b} = \frac{1/N(A)}{1/N(B)} = \exp\left(\frac{S_B-S_A}{k}\right)
\] where I used \(N(A)=\exp(S_A/k)\) – the inverted formula from Boltzmann's tomb – and similarly for \(B\). But now realize that \(k\) is extremely tiny in the macroscopic SI-like units while \(S_B-S_A\), the entropy change, is finite. So the ratio \((S_B-S_A)/k\) is equal either to plus infinity or minus infinity. It follows that one of the probabilities, \(P(A\to B)\) or \(P(B^*\to A^*)\), is infinitely (even exponential of infinity) times greater (i.e. more likely) than the other one which means that only one of the processes may occur with a nonzero probability (just like probabilities in general, both are less than or equal to one which means that the larger one equals at most one and the smaller one is infinitely smaller than that, i.e. zero).
If you look at the sign, the only possible (nonzero probability) transition between ensembles is one for which the final entropy is greater than the initial one. If \(S_B-S_A\gt 0\), the numerator \(P(A\to B)\) is infinitely greater than the denominator \(P(B^*\to A^*)\) which means that only the numerator is allowed to be nonzero. In other words, \(A\to B\) may occur while \(B^*\to A^*\) is prohibited.
Again, I have proven that the evolution in which the entropy increases is hugely more likely than the reverse process – the ratio of their probabilities is \(\exp(\Delta S/k)\) where \(\Delta S\) is the entropy difference between the initial and final state. To prove such a thing, I didn't have to assume anything about the Earth, the Solar System, the Universe, or cosmology because these – totally universal – calculations in statistical physics and their conclusions (that were already known to thermodynamics) have nothing to do with these large objects. If we were talking about the entropy change in a closed lab, we have only discussed the degrees of freedom (atoms...) inside the lab. Whatever the rest of the Universe was doing had no impact whatsoever on the calculation (and the resulting odds).
The calculation above is not hard but you may say that this particular general derivation above – presented in a way optimized by your humble correspondent, to say the least – is a bit unfamiliar to students who don't go through clear enough courses. Perhaps it is a bit ingenious. It may be OK if people happen to be ignorant about it.
However, what I find amazing is that people like Carroll are clearly ignorant about certain facts indicating that they must completely misunderstand the scientific method. Why? Sean Carroll is clearly convinced that the laws of physics as he knows them imply that the entropy of the young Universe had to be higher than the current entropy. So this is a prediction of those laws of physics and those methods to calculate that he is proposing. Because he also agrees that the entropy of the young Universe was actually lower, it must mean that his laws of physics or methods or both have been falsified. They're dead because they have made a wrong prediction (in this case, infinitely many exponentially bad predictions about pretty much everything!). Once they're dead, they can't be made "undead" again. This is a trivial observation that everyone who uses the scientific method in any context simply has to realize. Because Sean Carroll doesn't realize that, he can't possibly understand what the scientific method is.
If there is a contradiction between the empirical data – low entropy of the Universe in the past – and your theory, your theory has to be abandoned. You have to throw at least pieces of it and, perhaps, replace them with other pieces. Just adding "extra stuff" to your theory can't possibly help. If the theory was able to produce a wrong prediction (at least with a high enough confidence, like 99.9999%), then it has been falsified and making the theory more extensive (by the supposedly helpful additions such as "the cosmological realization measure" or the "Past Hypothesis") just can't solve the problem.
I am also amazed by the brutal distortion of the history below – Sean Carroll must know very well that it is a distortion:
That would stand in marked contrast to the straightforward Boltzmannian expectation that any particular low-entropy state is both preceded by and followed by higher-entropy configurations.Ludwig Boltzmann has never believed that the entropy was higher in the past than it is today. In fact, he was the history's most important man who has demonstrated just the opposite – that the increase of the entropy is a universal property of any system with many degrees of freedom. He has proven this second law of thermodynamics using the tools of statistical physics – where the second law is represented by the H-theorem, a more waterproof and transparent demonstration of why the second law has to be right.
It's important to realize that this second law or H-theorem apply to any moment of the history and any state. It doesn't matter whether the year "now" is 2013 or less or more. The proof of the H-theorem doesn't depend on "now" or the current year at all. It always implies that the entropy increases. It will increase between 2013 and 2014 and it was increasing between 2012 and 2013, too. If \(t_1\lt t_2\), then \(S(t_1)\lt S(t_2)\). One may have different emotional attachments to these two periods of time (perhaps we were not yet born at time \(t_1\) while we fell in love at \(t_2\)) but physics – and the proof of the H-theorem – just isn't about emotions. It is about facts and the laws of physics and about mathematically valid propositions and implications – and those are (perhaps except for the facts that may be interpreted as time-dependent things) independent of the time, independent of the current year.
I am also stunned by the fact that among the hundreds of readers who were probably going through the blog entry at Sean Carroll's blog, there was not a single person who would understand these basic issues about statistical physics and who would kindly explain to Sean Carroll how utterly stupid he was being and how superfluous all his concepts and proposed hypotheses are.
Incidentally, some news related to time: Strontium lattice clocks may improve the accuracy of the state-of-the-art atomic clocks by a stunning factor of 100! See BBC, Nature.
The "Past Hypothesis" nonsense is alive and kicking
![The "Past Hypothesis" nonsense is alive and kicking]()
Reviewed by DAL
on
July 11, 2013
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