Talk:Loschmidt's paradox

Archives

No archives yet.

This page is archived by ClueBot III.

Misc

The decrease in entropy that's local to the glass accompanies an equal or (probably) larger increase in entropy associated with the energy source for melting. The article doesn't demonstrate reversability, and it seems to confirm rather than resolve the paradox. -munge 5 August 2004

Thought experiments regarding thermodynamics unfortunately suffer from a lack of verifiability. It's as much of a mistake to present a broken glass and say entropy has increased as it is to present a reformed glass and say entropy has decreased. All we can be certain of is the state of the glass. Perfect reversability also requires some exotic devices:

• a closed system
• a perfect collector of heat
• a perfect initial state for the system, which requires:
  • omniscience regarding the eternal future behavior of the system

Any one of these is hard enough to construct by itself, and our experiment needs all of them simultaneously and instantaneously!^* Two more devices are also needed if others are to verify that the system works; namely, a means of looking inside a closed system without altering its state, and immortality.

So we can't possibly demonstrate anything about entropy.

The universe, however, is a closed system at some scale. Its continuing expansion ensures that. Galaxies that have vanished past the boundary set by Hubble's constant will never affect us, nor us them. In other words, we're hopelessly trapped. We can at least find some consolation in that everything else is trapped as well.

Unfortunately, energy will constantly leak out of our bubble (due to it being a fixed size) until nothing is left inside. There might be a lot of very cold iron for a while, but it must surely collapse under gravity. Our system isn't quite closed enough.

Our system isn't quite large enough, either, though. What identity does space have when all its energy has leaked away? It may as well simply not exist anymore -- and so it doesn't. Nor do the laws of physics apply anymore, as nothing remains that they could meaningfully act upon. Our aleph-null universe is like a flashlight gone dead, waiting for somebody in aleph-one to replace the batteries, saying, "Let there be light." --Eequor 10:14, 6 Aug 2004 (UTC)

^* note that, if one is clever, one can stop as soon as the closed system has been constructed, and simply lie about the other three -- nobody can look inside anyway. This approach also avoids cosmological dilemmas.

It seems to me that perfect knowledge at any instant at all would violate Heisenberg's principle. If we could violate quantum and have omniscience, then we might be able to detect a violation in the second law. Suppose we did. Then thermo would not apply in at least some thought-universes where quantum doesn't apply. But both seem to apply in the universe of experimental physics.

The thought experiments that claim to resolve Loschmidt's paradox are the ones that appear to suffer from lack of verifiability. Macro-scale experiments that are consistent with thermo are abundant. Macro-scale violations of thermo, if possible, are the experiments that await verification. Perhaps the paradox simply stands, and the article should say so.

Speculating now: What is the harm after all in adding an arrow of time to the principles of physics? Assuming that accepting a paradox is unscientific, is resistance to adding an arrow of time a scientific attitude? If the question seems anthrocentric, is there really anything whatever about physics that is not? Just asking. -munge 7 August 2004

There are actually experiments which apparently violate the second law. See fluctuation theorem. --Eequor 18:49, 7 Aug 2004 (UTC)

The next-to-last paragraph of that article says otherwise, that FT implies 2nd law. If true (I am not convinced that article is right either), you are onto something bigger. What assumptions underlie FT? FT is a theorem of some subset of math, Maxwell, Newton, and QM? If so, the 2nd law is true if those are true, and therefore Loshmidt's is not a paradox.

Just like statistical mechanics, quantum mechanics likewise allows all sorts of weird things to happen on rare occasion. By that yardstick, thermo is thus not any less true than gravity, which is then "less a law than a guideline". IMO, physical principles called laws are actually recipes that reliably produce repeatable results and rarely, nonrepeatable anomalous ones. -munge 9 August 2004

Yeah. By the fluctuation theorem, entropy in the part of the universe we can observe tends to behave according to the second law. The second law breaks down entirely at small enough scales. The statistical nature of the fluctuation theorem leads to some interesting questions. What is the cumulative effect of all these probabilities? If large-scale changes are not forbidden, but may occur very rarely, what effects might they have on the universe? What is the behavior of a dimensionless point? Could this drive inflation?

Also note that the theorem states the average change in entropy over arbitrarily long times is zero. --Eequor 07:05, 9 Aug 2004 (UTC)

I think I'll mark that article for {{expansion}}. I'm realizing it's somewhat vague. --Eequor 07:05, 9 Aug 2004 (UTC)

I do not agree with the article. The skeptic can use whatever real or conceptual family of technology that the glass shop uses. One would need conceptual perfect heat exchangers and perfect insulators, for both fusing and measuring. As to the small scale, if there are exotic exceptions to the 2nd law, I find them counterbalanced by the exotic exceptions to time symmetry at that scale, such as the decay of the neutral kaon. As to the large scale. I find it remarkable if QM plus gravity plus electromagnetics cannot indeed predict friction, which in practice can be overcome only by the imagination. If they cannot do so, what other large-scale phenomena might they be inadequate to predict, phenomena that might well have a bearing on cosmology?

If macro-scale systems return to prior states over time, like the mystical doctrine of cycles, then once again, Eequor, you and I will reason over this, once again we will quibble. In some permutations of the future, it will be you instead of I who assert that all experimental evidence indicates the 2nd law is strongly true. In others, we both refute it and together craft a nanoprobe that harvests the zero point energy. The cycles where the relationship works out, those where it ends up in lawsuits, those in which I cheat you and those in which I lose everything...

If (like many-worlds) this is the position one must take after all to logically avoid a paradox, then I'm probably going with the paradox. -munge

It wouldn't matter if the theoretical glass shop even could reassemble the glass perfectly, atom for atom. The overall entropy of the entire system would still increase. All that is happening is that the entropy in the glass is being "exported" so to speak by the burning of fossil fuels (or whatever is being done to generate the power necissary to heat/reassemble/etc the glass.) --||bass 23:06, Apr 11, 2005 (UTC)

Out of disorder

I'm removing the following entire section from the text:

A simpler resolution of the paradox can be found if the second law of thermodynamics is not seen as absolute; rather, less a law than a guideline. If the law is revised to state that entropy tends to increase over time, a different universe emerges, in which entropy may increase in both directions of time.

The usual example is a glass or other easily breakable object. Consider a point in time where the glass is intact, perhaps in falling to the ground. This is an ordered state for the glass; it may easily remain in this state indefinitely, but once broken it cannot be put back together. Upon reaching the ground, the glass shatters: this is its disordered state.

Consider the numerous ways in which the glass may break. It may simply chip, it may break into a few pieces, it may shatter into innumerable tiny fragments. Clearly, the glass has many disordered states, compared to its one ordered state. Over time, it is inevitable that the glass will fall into one of these states, as its ordered state is only one out of many possibilities. It can be seen that ordered states tend to become disordered.

Generally, this is where speculation on the nature of thermodynamics ends, as it has been demonstrated that entropy will increase. There is another possible conclusion, however.

Brought to a proper temperature, silica or glass will melt and can be shaped. Logically, this is the origin of our unfortunate glass. Consider the shattered glass: its fragments may be collected and melted, causing its molecules to become increasingly disordered from heat. The molten glass may then be formed into a new glass, identical (or nearly so) to the original glass. As the glass cools, its molecules become more ordered.

The falling glass is once again in an ordered state. Viewing time backwards, the glass warms until it melts and loses its shape entirely. As with its shattering, there are many ways in which the glass may melt. Once again it is inevitable that the glass will transition to one of these states: there are more of them.

Loschmidt's paradox vanishes. Regardless of the direction of time, ordered states tend to become disordered. It may be said that even as order becomes disorder, order arises out of disorder.

It's criticised above, because the given example involves the introduction of extra energy to re-form the glass. However, the main reason for my removing it is that it appears to be an original theory, or an original attempt to resolve the given paradox, not pre-existing knowledge. I might be wrong, it may be a quite well known example, in which case a citation would be useful. For the time being, though, I am removing it. Crosbiesmith 21:11, 1 August 2005 (UTC)

I recall a similar argument, without the example, in A Brief History of Time. The example is ill-chosen; glass is a chaotic solid. Septentrionalis 16:15, 14 March 2006 (UTC)

fluctuation theorem

I disagree with the statement that the fluctuation theorem resolves the paradox. The fluctuation theorem assumes a system that's initially not in equilibrium. In a system that is mostly in equilibrium over a long stretch of time, fluctuations occur, and the fluctuation theorem simply describes the probability that such a fluctuation will be a certain size, and span a certain amount of time. If we observe the system to be undergoing an unusually big fluctuation at a particular time, then the fluctuation theorem tells us that the fluctuation will be unlikely to persist for a long time into the future, but it also tells us that the fluctuation is unlikely to have persisted since a long time in the past. The resolution of Loschmidt's paradox is simply that (for reasons unknown to us) our universe has an endpoint to its time coordinate (the Big Bang) that is a state of low entropy. In a universe that had a maximum-entropy Big Bang, there would be no arrow of time, and the second law of thermodynamics would be meaningless.--Bcrowell 02:03, 24 January 2006 (UTC)

Loschmidt's own idea was similar, he thought the explanation of the rise of entropy lay somehow in the initial conditions of the Universe, rather than in the H-theorem98.109.241.146 (talk) 02:24, 17 June 2013 (UTC)

Failed GA nomination

As it stands, one reference is not nearly enough for an article of this complexity. The article is barely comprehensible to a layman such as I (In the first sentence alone, "time-symmetric dynamics and a time-symmetric formalism" is almost meaningless). It might be better to describe the paradox as a conflict between two laws, describe the laws in relation to the paradox, and then state what the paradox itself is. Nifboy 07:31, 15 March 2006 (UTC)

Yeah, I don't quite understand what "time-symmetric formalism" is supposed to mean. The equations describing the laws of physics are certainly time-symmetric, but I would usually understand "formalism" to mean something like "algebraic equations" or "tensor equations", i.e. a particular convention for expressing the laws of physics mathematically. Time-symmetry is a property of the laws of physics themselves, not of the choice of mathematical notation you use to write them down - it would be perfectly possible to write down the equations for time-asymmetric laws using the same formalism as is used to write down the actual laws, for example. For another example of this use of "formalism", see this paper on expressing the Einstein-Maxwell equation "in the Newman-Penrose formalism" - I'm pretty sure they're talking about expressing the same laws of physics that are expressed by the Einstein-Maxwell equation, but with a different type of mathematical notation (the 'spinor formalism of Newman and Penrose'). Google the words "equations particular formalism" (as individual words, not a phrase) and you'll find a number of other examples of "formalism" used in this way. So can anyone justify this phrase? Hypnosifl 17:50, 23 October 2006 (UTC)

Yeah, re-reading the opening, it is probably better without "time-symmetric formalism". So I've come round to be in support of your edit.

The "formalism" is the mathematical machinery you use to represent the system, and the laws of its physics. The thing is, Boltzmann's H-theorem was getting a time-asymmetric result out of time-symmetric laws. If the time asymmetry wasn't coming from the laws, where else could it be coming from? It was coming in from the way he was representing the system mathematically -- in particular, from his assumption that the motion of each of the particles at each time step could be treated as uncorrelated from all the other motions. But this is actually destroying information as you move away from the boundary conditions. So the entropy change is not time-symmetric; the direction of the time asymmetry in turn reflecting the fact that the boundary conditions were placed on the past, not the future. Jheald 15:57, 25 October 2006 (UTC)

I do not think very highly of the idea of including a grab bag of possible "resolutions". An encyclopedia article should be more focussed than that. As it stands, it smacks too much of independent research. There are two standard sources, at least, on the resolution of the paradox: Sir James Jeans The Dynamical Theory of Gases, Jan von Plato's Creating Modern Probability, and of course the Ehrenfests, along with Martin J. Klein, Paul Ehrenfest, vol. 1: The making of a theoretical physicist.98.109.241.146 (talk) 02:27, 17 June 2013 (UTC)

Quantum mechanics

I'm cutting the following paragraph just added by Enormousdude:

However, quantum mechanical nature of interaction of microscopic particles results in less correlated states after interaction than before - thus providing mathematical explanation to asymmetry or microscopic processes (and thus, of some mactroscopic processes too) versus time reversal.

The usual unitary evolution of the system under quantum mechanics is just as time-symmetric and information preserving as the usual classical evolution of a system given by Liouville's equation.

Saying that interaction of microscopic particles under say the Schroedinger equation leads to less correlated states after interaction is simply unfounded. Jheald 20:56, 2 November 2006 (UTC)

Actually, quantum mechanics does lead to less correlated states after interaction. Although the unitary evolution of the wave function described by Schrodinger's Equation is perfectly symmetric under time reversal, the process of wave packet reduction, AKA wave function collapse, is an irreversible process. 69.248.123.28 (talk) 17:54, 18 July 2008 (UTC)

Yes, well, wave-function collapse is its own bag of worms, isn't it? linas (talk) 18:29, 2 October 2008 (UTC)

I wouldn't call it collapse of the wave-function. The main idea is that the interaction with an environment leads to decoherence and thus the dynamics of the subsystem under consideration is described by the irreversible Liouville von Neumann equation, while the full system is still described by the reversible Schroedinger equation. There are dozens of papers on the subject, and the article on the Second Law already contains similar arguments. Hweimer (talk) 11:45, 3 August 2009 (UTC)

To me it seems otiose to drag in QM when Loschmidt's paradox really has a totally independent basis. If you want to write a research paper, perhaps. But in this encyclopedia article? A one-sentence mention that some Loschmidt's paradox has inspired some theoreticcl papers in Qm, with a few references, should suffice. One should not be posting a grab bag of possible solutions unless one has been commissioned to write an entire review article.. 98.109.241.146 (talk) 02:31, 17 June 2013 (UTC)

The article contradicts itself

Furthermore, due to CPT symmetry reversal of time direction is equivalent to renaming particles as antiparticles and vice versa.

If hydrogen and antihydrogen do the same thing, they are not having opposite and therefore cancelling effects on entropy. This represents a contradiction of this formulation of Loschmidt's paradox. If on the other hand they did have a cancelling effect on each other with regards to entropy, Loschmidt's paradox would reduce to the question of baryogenesis, and thus the arrow of time exists due to the asymmetry between matter and antimatter in the universe. --75.49.222.55 01:19, 8 October 2007 (UTC)

The thing is, hydrogen and antihydrogen behave identically, although simply opposite in charge, and annihilate with eachother. Replacing all matter in the universe with all antimatter would have almost no difference, except for certain rare cases like neutral kaon decay (these processes give a very, very weak arrow of time, since they are so incredibly rare and do not account for the large-scale time asymmetry of ALL processes). Reversal of time IS the same as renaming particles as antiparticles (and also switching their handedness--this is key) on small scales. Note that as already stated, all small-scale laws seem time-symmetrical. This provides no answer to the paradox of large-scale resolution.Eebster the Great (talk) 00:39, 7 April 2008 (UTC)

Yeah, CP violation is normally invoked to explain why there's more baryonic matter in the universe; its *not* applicable to garden-variety mixing. linas (talk) 18:31, 2 October 2008 (UTC)

Global vs Local

The Big Bang setting a boundary condition and resulting in an arrow of time for thermodynamics makes no sense since increase in entropy is observed in purely local system where the boundary starting time t_0 is not necessarily the Big Bang. In physics Also see http://motls.blogspot.com/2007/06/is-cosmology-behind-second-law.html "But the second law of thermodynamics is a local law that applies to any macroscopic process even if it takes a very short time in any small but macroscopic region of space. The law shows its muscles instantly. The slow cosmological evolution has virtually no detectable influence on such local processes." "The second law of thermodynamics trivially implies that the initial state of the Universe in the past had to have a smaller entropy than its current value. But it is incorrect to revert this implication." "Analogously, cosmological evolution can't influence the validity of any robust, local laws of physics such as the second law of thermodynamics today. And saying that the cosmological evolution in the past is the primary culprit either violates locality or it contradicts other dynamical laws of physics." ThVa (talk) 18:53, 18 August 2008 (UTC)

As pointed out in the "fluctuation theorem", given any out-of-equilibrium state you can indeed use local arguments from statistical mechanics to show the entropy will tend to increase, however the exact same arguments could be used in reverse to "retrodict" that at earlier times the system was in a higher-entropy state as well, but this time-reversed application of statistical mechanics would usually be wrong, since out-of-equilibrium states were often at even lower entropy in the past. The fact that entropy was lower in the past, in contradiction with a naive application of time-reversed statistical mechanics, is what is thought to require a cosmological explanation involving a low-entropy Big Bang. Hypnosifl (talk) 05:46, 2 December 2009 (UTC)

Since the Universe as a whole is naked-eye obviously not in an equilibrium state according to the definitions of Thermodynamics, it makes no sense to define its entropy or try to apply the Second Law of Thermodynamics, in the thermodynamical sense of the concepts.98.109.241.146 (talk) 02:37, 17 June 2013 (UTC)

The informational definition of entropy, Boltzmann's, used in Stat Mech, applies to probability distributions on states, and the evolution of a probability distribution. It does not apply to an individual state such as the initial condition of the Big Bang. So I recommend omitting all such discussions entirely: they have obviously left the framework within which Kelvin, Boltzmann, and Loschmidt were operating.98.109.241.146 (talk) 02:37, 17 June 2013 (UTC)

Stosszahlansatz

The Stosszahlansatz is not the same hypothesis as Maxwell's "molecular chaos". They serve the same purpose, more or less, and hence introduce the irreversibility as the previous editor says. See also Pauli, Statistical Physics, MIT press paperback, p. 8 who says the same thing.

But it is too pejorative to say that such a hypothesis is "flawed". The hypothesis is wunderbarisch. The use of it in a derivation that professes to deduce the Second Law of Thermodyanmics from Newtonian Mechanics makes the derivation flawed. No progress in Stat MEch can be made without a statistical hypothesis somewhere. So I edited the text to be more neutral. 98.109.241.146 (talk) 01:34, 17 June 2013 (UTC)

My proposed draft, not exclusive.

Loschmidt's paradox concerns the use of Statistical Mechanics to prove the Second Law of Thermodynamics. The Second Law of Thermodynamics states that entropy always increases in real processes: all real processes are irreversible (the reversible processes, which pass from one equilibrium point to another to another \dots are merely theoretical, they cannot occur in Nature): that is, when all the effects on the environment are included. A small system interacting with the environment can shed some of its own entropy to the environment, so the system's entropy can decrease, but only provded the amount of increase in entropy which is incurred by the environment more than cancels it out, so that the total entropy increases, as required. Statistical Mechanics has the goal of deducing the laws of Thermodynamics, especailly including the Second Law, from ... Thermodynamics was already a well established physical theory, and so was Analytical Dynamics, but they used such different concepts that it was completely mysterious how they could both be true in the same world. The kinetic theory of heat had to contend against the rival fluid theory of heat. The goal of Statistical Mechanics, invented by Maxwell and followed up by Boltzmann, was to demonstrate the relations between the two, and thus justify the theory of atoms and molecules and justify the kinetic theory of heat, that heat was simply the macroscopic manifestation of the kinetic energy of the motion of the microscopic molecules. It was therefore important to start out by assuming Newton's Laws, study the trajectories in phase space of a closed, conservative dynamical system, and end up with the laws of Thermodynamics, without assuming any new laws of Physics.

Boltzmann invented another definition of entropy, via his $H$-function (which is proportional to the negative of the entropy). Boltzmann defined the entropy of a probability distribution $f$ on the micro-states (which Sir James Jeans calls `dynamical states, in contrast to the traditional `thermodynamic state', which we would now call a `macro-state') of a dynamical system. It is what we would now call an information-theoretical definition with no obvious relation to thermodynamics. But he proved it was related to the previous thermodynamic conept of entropy, and he proved his infamous $H$-theorem, that $dH\over dt$ is always zero or negative. He did not prove this by a direct calculation of $dH\over dt$, which is too hard, but made some statistical ansatzes. The statistical ansatz used is also famous: it is called the Stosszahlansatz and makes an assumption about the probabilities of different collisions of perfectly spherical molecules. Maxwell also had had to make a statistical ansatz, but his was different, and is referred to, more or less, as `molecular chaos'. After Maxwell's first formulation of what he meant by `molecular chaos' was criticised, he adopted a different formulation. The differences between these different hypotheses, all of which are statistical or probabilistic in nature, are irrelevant to this topic.

Boltzmann took Loschmidt's criticism seriously and changed his explanations. That is, Loschmidt's criticism was accepted as a valid refutation of the literal meaning of what Boltzmann wrote. But Boltzmann hadn't meant it, and explained it more clearly and at greater length. Prof. von Plato has a useful discussion of the details. In his opinion, Boltzmann's assertions should not be taken literally. They need to be interpreted in light of the context and flow of argument wich surrounds them, including other papers not yet written, and are too simple to correctly express Boltzmann's meaning, which was way ahead of its time.

Loschmidt's paradox proves that the literal meaning of many of Boltzmann's statements is untenable, and this is generally accepted. The resolution of Loschmidt's paradox has been to clarify the logic of Boltzmann's assertions, and this was done carefully and at length by Ehrenfest (a doctoral student of Boltzmann's), at Felix Klein's request, in 1911, in a review article for Klein's encyclopedia, The Conceptual Basis Foundations for Statistical Mechanics, which impressed Lorentz so much that he chose Ehrenfest, to be his successor in Holland, when he could not get Einstein. But this was already done briefly but clearly by Sir James Jeans in his standard textbook, The Dynamical ...

[Um, this draft is for physics.stackexchange and so is not encyclopedic in style. Some should be cut and some should be revised. Mañana.] == The supporting quotes from Jeans's book p. 38. James H. Jeans, The Dynamical Theory of Gases, fourth edition, 1925. Apparent Irreversibility of Motion When a gas is not in a steady state, it follows from \seq 23 (p. 24) that $dH\over dt$ must be negative. Some writers have interpreted this to mean that $H$ will continually decrease, until it reaches a minimum value, and will then retain that value for ever. A motion of this kind would, however, be dynamically irreversible, and therefore inconsistent with the dynamical equations of motion from which it professes to have been deduced. As will appear later (cf. \seqs 70---73), the truth is that we have at this point reached the limit within which the assumption of molecular chaos leads to accurate results. The motion is, in point of fact, strictly reversible, and the apparent irreversibility is merely an illusion introduced by the imperfections of the statistical method.

pp. 65f. The assumption of molecular chaos (corrected, if necessary, in accordance with \seq 69) will therefore give correct results, provided it is interpreted with reference to the basis of probability supplied by our generalised space [phase space], and provided it is understood that it gives probable, and not certain, results. If we wish to obtain strictly accurate results, the quantities calculated from it must not be regarded as applying to a single system, but must be supposed to be averaged over all the systems in the generalised space which satisfy certain conditions. \dots

We now come to what is, at first sight a paradox. Let us suppose that \dots the value of $f$ will be the same for each of these two systems and both systems are equally to be included in the average of $dH\over dt$. But the motion of the first system is exactly the reverse of that of the second system. It would therefore appear as though the values of $dH\over dt$ must be equal and opposite for the two systems, so that the average of $dH\over dt$ for these two must be zero. Since the whole of the systems corresponding to a given $f$ fall into pairs of this type, it might be inferred that the average value of $dH\over dt$ must be zero.

[This paradox is then resolved in detail in the rest of the page.]

It is interesting that Loschmidt, an important scientist in his day, is referenced in Sir James Jeans's text several times, but for his scientific contributions: not for this paradox. Perhaps this is because British scientists had already made this observation before Loschmidt published it. (ref., von Plato).

98.109.241.146 (talk) 03:09, 17 June 2013 (UTC)

Vortex tubes

Experimental proof of the Loschmidt gravito-thermal effect can be easily seen in a Ranque-Hilsch vortex tube wherein a force far greater than gravity separates a gas into measurably hotter and colder streams as it redistributes kinetic energy, just as happens in a planet's troposphere due to the force of gravity. — Preceding unsigned comment added by 110.147.139.222 (talk) 00:02, 19 March 2014 (UTC)

WP:RS? Paradoctor (talk) 02:08, 19 March 2014 (UTC)

Conclusory conclusion?

From the introduction:

"Reversible laws of motion cannot explain why we experience our world to be in such a comparatively low state of entropy at the moment (compared to the equilibrium entropy of universal heat death); and to have been at even lower entropy in the past."

I'm totally sympathetic to this line of thinking, and indeed I'd rather read this line than a pastiche, but is this really the conclusion that science has drawn, today? I would think the paradox remains unresolved to this day, including (of course) whether it really is a paradox, and above all, that it would be difficult to justify this sentence by a citeable consensus of accepted publications, as opposed to the pure reason of the reasonable man. 84.227.248.81 (talk) 20:04, 13 April 2014 (UTC)

I agree - There is a paradox here, and I wish I knew the name of it. If we consider the universe to be a closed system and our present condition to be part of a low-entropy fluctuation, then it is overwhelmingly probable that our present condition is at or very near the bottom of that fluctuation and that entropy increases in both directions of time. The idea that we are on an entropy climb from an even lower entropy state (e.g. the big bang) is overwhelmingly unlikely. PAR (talk) 16:12, 18 March 2016 (UTC)

External links modified

Hello fellow Wikipedians,

I have just modified one external link on Loschmidt's paradox. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

• Added archive https://archive.is/20121216121427/http://www.scientificblogging.com/hammock_physicist/fibonacci_chaos_and_times_arrow to http://www.scientificblogging.com/hammock_physicist/fibonacci_chaos_and_times_arrow

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

This message was posted before February 2018. After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{source check}} (last update: 5 June 2024).

• If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
• If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot (Report bug) 22:38, 14 December 2017 (UTC)