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There isn't too much to explain:

We know that all fundamental forces are reversible then where does the irreversibility come from?


Edit: The following is edit based on comments:

Consider a block of wood and you just make it slide on a desk, it will move a little bit and then stop. It stops because of molecular forces as surfaces are rough, of course, they aren't uniform surfaces at all. And then bonds break so we say that friction is at the molecular level. So if we need quantum mechanics for explaining these things but apart from that, We know that friction that makes the block stop is at the molecular level. This means, that energy can transfer from one system to another at the molecular level. These are random motions and that is even though all forces are conservative at the fundamental level, it may turn out that energy can dissipate as heat, which we can't recover.

I'm asking what's the origin of this macroscopic irreversibility, Why we can't recover the energy which gets lost if at the fundamental level these forces are reversible?


Stating from Wikipedia,

Time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, $T:t\rightarrow -t$

Stating R. Shankar, "You will no way of knowing if the projector is running forwards or backward."

Further Stating Wikipedia,

Since the second law of thermodynamics states that entropy increases as time flow toward the future, in general, the macroscopic universe does not show symmetry under time reversal.

Now consider that I'm studying the microscopic universe, So I would expect time-reversal symmetry. (We can't tell if the picture running forward or backward). Now lets I start adding more elementary particles in my system, When does it so happen that I can tell that this picture is actually running forward.

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    $\begingroup$ Have you read Wikipedia on the arrow of time? en.wikipedia.org/wiki/Arrow_of_time $\endgroup$
    – Allure
    Jun 29, 2021 at 4:39
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    $\begingroup$ More on Loschmidt's paradox. $\endgroup$
    – Qmechanic
    Jun 29, 2021 at 6:01
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    $\begingroup$ Following the discussion in the comments to my answer, I think the question needs to be clarified. What does irreversibility mean here? $\endgroup$ Jun 29, 2021 at 11:47
  • $\begingroup$ Probably not enough for an answer but maybe useful food for thought: When rolling a large number of dice over and over, why is it more likely to get from a configuration where all the numbers are equal to one where they aren't than the other way around? The underlying rule is time-reversible (for any two configurations of the dice A and B, it's as likely to get from A to B as from B to A), so where did the apparent irreversibility come from? $\endgroup$
    – aekmr
    Jun 29, 2021 at 18:14
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    $\begingroup$ Comment on the extended question: 1) friction is not a conservative force; 2) once we talk about energy transferred into heat, we essentially invoke the second law of thermodynamics and the entropy increase. $\endgroup$ Jun 29, 2021 at 19:59

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There's a distinction between microscopic reversibility and macroscopic reversibility. Or if you will, a difference between something being irreversible in theory versus irreversible in practice. (Or absolutely irreversible versus probabilistically irreversible.)

A hopefully relatable analogy:

Imagine that you have a large number of coins in front of you. They all start heads-up (obverse visible). Now imagine that at each "step" you choose a coin randomly and flip it. That is, if it's heads-up, you make it heads-down, and if it's heads-down you make it heads-up. Each step is reversible. If you flip a coin heads-down in one step, you can flip it heads-up in the next. But actually running the experiment will accord with your (likely) intuition -- if you choose coins at random, the coins become a random (approximately equal) distribution of heads-up and heads-down. Even though each individual step is reversible, on the macroscopic scale the combination of steps is not: if you start with an all heads-up state, you never go back to that same state.

Theoretically, you could. It's possible that you just so happen to randomly get a streak where you pick only those coins which are heads-down, and flip them heads-up. Or vice versa: only select heads-up coins and flip them heads-down. But since you're picking randomly, that's a very, very low probability case. And it gets even less probable the more coins you have to flip.


Physics systems are similar. Most macroscopic systems are composed of a large number of individual particles/elements. While the individual interactions of the particles are reversible (like the individual coin flips), on a global, macroscopic scale the system isn't. Indeed, all of those interactions could theoretically run in just the right way to revert the system to exactly the previous state, but the probabilities of that are small. You could be talking about $10^{20}$ or $10^{30}$ particles, each of which need to be properly reversed. While the chance of any individual interaction being reversed could be quite high, the chance that all of the interactions are reversed in just the right way to put the macroscopic system back into a previous state is mind-bogglingly low.

There's different formulations of entropy, but in many that's what entropy is -- it's the measure of the "probability" of the state (Boltzmann's entropy formula). When someone says that things proceed from low entropy states to high entropy states, they're basically saying that things proceed from a low probability states to higher probability states. But the second law of thermodynamics is a statistical one, not an absolute one. "Entropy always increases" is of a slightly different character than "energy cannot be created or destroyed". It's not a hard-and-fast rule which can never be broken, it's just $10^{20}$ to 1 odds that it won't be.

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    $\begingroup$ Welcome and +1 for the example with coins which highlights that the fundamental reason boils down to statistics and has little to do with the details of the underlying physical laws. In fact, one does not even need to know what entropy is to understand the reasoning: The "highly ordered" macrostate A = ">99.9% coins heads-up" consists of much fewer microstates than the "disordered" macrostate B = "49.95% to 50.05% coins heads-up". So even with perfectly reversible dynamics making steps to a randomly chosen new microstate, we're much more likely to step from A to B than from B to A. $\endgroup$
    – aekmr
    Jun 29, 2021 at 17:51
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    $\begingroup$ +1. I was thinking in terms of forces observed in collections of particles/atoms/bodies, i.e. dissipative forces. These prevent the kinetic energy gained during a state-change to be completely retained so as to effect the reverse state-change. So the ball falling h metres will never quite bounce back up h metres nor the dislocation advancing one interatomic distance after yielding will always lose enough of its kinetic energy to prevent recovery of the plastic yield strain when the load is reversed. Like R.M. says, it's the difference between individual and collection interactions. $\endgroup$
    – Trunk
    Jun 29, 2021 at 18:36
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    $\begingroup$ The coin analogy breaks imo. Say, there's a universe of 1 trillion coins, with the law of physics being : "Flip a random coin every 1 second". If you run this law on any initial combination of coins in both forward time and reverse time, both time evolutions would tend to produce higher and higher entropy outcomes. However, in Newtonian mechanics, given some initial combination of positions and velocities, the laws of physics are guaranteed to produce lower and lower entropies in one of the directions of time. $\endgroup$
    – Ryder Rude
    Jun 30, 2021 at 2:10
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    $\begingroup$ Suppose we take our universe's particles' current positions, but flip each particles' velocity vector in the opposite direction. This is our initial state of positions and velocities. Now, statistics would have us conclude that this system would produce higher and higher entropies as it evolves forward in time, as higher entropy states are simply more likely. However, running the actual laws of physics on this system would yield lower and lower entropies as we move forward in time. $\endgroup$
    – Ryder Rude
    Jun 30, 2021 at 2:20
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    $\begingroup$ I have already gone through this example which is also pointed out by Blundell, Thermal Physics, But this answer again as most of given, tells the same story. Taking statistical mechanics (equal prior as granted and then making a statement that system will settle to most likely state) but I'm not going toward that. As pointed out by Ryder, Suppose these coins are changing state by an equation that is time-reversal invariant, If I make the system evolve in ever time of course I go back to initial state. $\endgroup$ Jun 30, 2021 at 5:32
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Irreversibility comes from the thermodynamics: the probability that we return to the same state in any reasonable amount of time is extremely small. In more technical terms: the entropy is increasing. The proof that the irreversible macroscopic behavior can emerge from the reversible microscopic behavior is known as the Boltzmann H-theorem. (At the time of Boltzmann the result was considered rather controversial, and sometimes cited among the reasons that drove Boltzmann to the suicide.) Note that although statistical physics textbooks usually use gases as examples, the results of statistical mechanics and thermodynamics are by far more general, applicable to most macroscopic systems.

There is also interesting intermediate case of collapse and revival, in the systems that are big, but not quite big to by considered in thermodynamic limit.

Update: Loschmidt paradox
@josephh mention in their answer Loschmidt paradox: if all the velocities in the system are reversed (it implies also reversing all the angular momenta and spins), the system should return to its initial state. The paradox was originally intended as a critique of the H_theorem (more precisely, as a critique of Boltzmann equation). In reality, there is no paradox: if we could indeed reverse all the velocities and the angular momenta, the system would indeed evolve into its initial state. In practice, we do not have means of controlling all the particles in any macroscopic system. Note that some experimental techniques, such as spin echo, explicitly use this very idea.

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    $\begingroup$ This does not answer the question I think Roger. He appears to be asking why at the quantum level we have reversibility but not a the macroscopic level. $\endgroup$
    – joseph h
    Jun 29, 2021 at 5:52
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    $\begingroup$ @josephh I think this is precisely the reason: in classical physics we also have reversibility at microscopic level, but not when we deal with huge numbers of molecules (which si what macroscopic systems are). $\endgroup$ Jun 29, 2021 at 7:25
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    $\begingroup$ I agree with Joseph, Your answer doesn't seems to get my problem. $\endgroup$ Jun 29, 2021 at 10:57
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    $\begingroup$ @YoungKindaichi how do you define irreversibility? $\endgroup$ Jun 29, 2021 at 11:00
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    $\begingroup$ I understood the question in the same way as Roger did. Hence, the assertion "There isn't too much to explain:" seems to be false :p $\endgroup$ Jun 29, 2021 at 14:10
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This question is asking why, if at the quantum realm of particles, processes can happen in reverse (particle interactions obey time reversal transformations), why does macroscopic matter (which is also made up of these particles) behave irreversibly?

This apparent contradiction, namely that the thermodynamic arrow of time (entropy) points in one direction, though particle interactions do not follow this rule, is the subject matter of what is known as Loschmidt's paradox. Whether there is a resolution to this paradox$^1$ is debatable, and from the above link

"Loschmidt's paradox, also known as the reversibility paradox, irreversibility paradox or Umkehreinwand, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behaviour of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict, hence the paradox."

"Any process that happens regularly in the forward direction of time but rarely or never in the opposite direction, such as entropy increasing in an isolated system, defines what physicists call an arrow of time in nature. This term only refers to an observation of an asymmetry in time; it is not meant to suggest an explanation for such asymmetries. Loschmidt's paradox is equivalent to the question of how it is possible that there could be a thermodynamic arrow of time given time-symmetric fundamental laws, since time-symmetry implies that for any process compatible with these fundamental laws, a reversed version that looked exactly like a film of the first process played backwards would be equally compatible with the same fundamental laws, and would even be equally probable if one were to pick the system's initial state randomly from the phase space of all possible states for that system."

Current research in dynamical systems offers one possible mechanism for obtaining irreversibility from reversible systems.

"The central argument is based on the claim that the correct way to study the dynamics of macroscopic systems is to study the transfer operator corresponding to the microscopic equations of motion. It is then argued that the transfer operator is not unitary (i.e. is not reversible) but has eigenvalues whose magnitude is strictly less than one; these eigenvalues corresponding to decaying physical states."

Though this method has various problems and works well for only a handful of models that have exact solutions.

$^1$ Another popular resolution to this paradox is to consider that $CPT$ invariance is an exact symmetry, but $CP$ and $T$ are not. Therefore, it's possible that this asymmetry invoked the second law of thermodynamics (since the universe is primarily dominated by matter instead of anti-matter).

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  • $\begingroup$ Worth noting that Loschmidt paradox was originally intended as a critique of H-theorem. In fact, Loschmidt and Boltzmann's were friends. $\endgroup$ Jun 30, 2021 at 9:11
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then where does the irreversibility come from?

You mean macroscopic irreversibility, our inability to set up or observe a macroscopic process that retraces past states of a known spontaneously occurring process in reverse order. For example, we can't prepare or find examples of a process that is like teacup cooling down, but in reverse order.

This macroscopic irreversibility has a common explanation: during such a reverted process, entropy of the whole supersystem (system+environment) would have to decrease, which is extremely improbable for the supersystem to do. It could do it if we could somehow revert all velocities (and magnetic field components) at some time, but we can't do it in practice and it does not happen spontaneously (that would break the fundamental equations).

We know that all fundamental forces are reversible

Our best theories have fundamental equations that are reversible, meaning when velocities are reversed, the system retraces its past states. We can perform the reversal in real experiment in simple cases like spins precessing in magnetic field, and in that case, the system can be said to be reversible. But in general, we cannot do it.

For example, we can't revert velocities of all gas molecules or, in case of radiating charged particle, revert magnetic component of radiation to make it all go back and be absorbed by the particle.

In case we cannot do the reversal and it does not happen naturally either (equations do not predict such reversal), it is microscopically irreversible in practice, and then it is natural that the system appears macroscopically irreversible too.

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  • $\begingroup$ +1 excellent definition of irreversibility! Essentially, it is tied to the information loss? $\endgroup$ Jun 29, 2021 at 16:16
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    $\begingroup$ I do not think information loss is necessarily the reason. Even if could find the microstate and coudl calculate all trajectories in both directions of time, irreversible process would still be irreversible, because we can't reverse the velocities in practice. $\endgroup$ Jun 29, 2021 at 19:34
  • $\begingroup$ For instance, Consider Joule expansion, Also consider somehow I manage to revert the velocities of all gas molecules, does the gas go back where it came from? That's is concentrate to where it starts from? $\endgroup$ Jun 30, 2021 at 5:40
  • $\begingroup$ @YoungKindaichi yes, in classical-mechanical model, if all velocities are reversed at some time, the system will retrace past states in reverse order. If the model takes into account EM interaction, direction of magnetic field everywhere has to be reversed just like the velocities to accomplish this. This is practically impossible to do. $\endgroup$ Jun 30, 2021 at 12:30
  • $\begingroup$ @JánLalinský If the theory allows the reverse process, then why it doesn't see in the nature? $\endgroup$ Jun 30, 2021 at 12:40
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Assume there are a bunch of particles in the universe. Assign a randomised set of positions and velocities to each particle, set $S={p_i, v_i}$. Note that when you assign velocity values to the particles, you are implicitly choosing a positive direction for the time axis. For example, a velocity of $3ms^{-1}$ means that the particle moves $3m$ in $1s$ of your chosen positive time direction. Equivalently, the same particle moves $-3m$ in $1s$ of the chosen negative time direction. So by assigning the velocity values, you've randomly chosen one direction of time to be positive, and its opposite to be negative.

The law of entropy says that, corresponding to each of these randomised sets of positions and velocities, there is a preferred direction of time. If you allow the laws of physics to run on each of these sets, you will get a preferred direction of time in which the entropy will increase.

The exception is the set corresponding to the minimum possible entropy. For this set, the entropy would increase in both directions of time. Hence, there will be no preferred direction.

The laws of physics still do not prefer any direction of time. For each set $S={p_i, v_i}$, which has, say, the negative time direction as its preferred direction (the direction of entropy increase), there is another set $S'={p_i, -v_i}$, which as the positive direction of time as its preferred direction.

This follows from the fact that the laws of physics are time-symmetric. Set $S'={p_i, -v_i}$ has been obtained by reversing the velocities of the particles in set $S={p_i, v_i}$. So, watching $S'$ evolve in positive time is equivalent to watching $S$ evolve in negative time. Since we assumed that $S$ prefers the negative direction as its "entropy increase direction", it follows that the entropy of $S'$ increases in the positive direction of time.

Since the number of sets which prefer the positive direction of time is equal to the number of sets which prefer the negative direction, the laws of physics give no overall preference to either of the directions. It's the sets, i.e the particular combination of position and velocity states, which have a preferred time direction.

The reason we do not observe broken plates re-assembling themselves in the direction of time that is perceived by human consciousness, is that the particular set of positions and velocities of our universe has the preferred time direction which is the same as the direction perceived by human consciousness. For this reason, we do not observe entropy decreasing processes in this direction of time.

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The reversibility of the dynamics at a fundamental level does not imply an equal probability of the initial conditions. In all the cases where we observe irreversible behavior, we have systems with a large (huge) number of degrees of freedom. Loschmidt's paradox misses the key point that in a mechanical system with an infinite number of degrees of freedom it quite simple to have an irreversible behavior starting with perfectly reversible equations of motion.

A simple example is a simple harmonic oscillator coupled with an infinite elastic string. The initial motion of the oscillator will induce traveling waves that will subtract energy from the oscillator irreversibly damping its motion. It is true that by inverting all velocities after some time the system should trace back its evolution towards its starting state. However, this would be a very atypical starting condition. Almost all the neighbor configurations would not go back to a neighbor of the starting state. Notice that, even for moderately large systems, "almost all" is indistinguishable from "all" in practice.

Notice that there is nothing special with classical mechanics. The same considerations apply to quantum system evolution.


Postscript

It is attempting to relate the irreversibility of macroscopic dynamical systems to entropy. However, entropy is not an explanation in itself. Actually, things should go the other way around: from the irreversible dynamical behavior, one should find a convenient way of encoding it into entropy.

Such encoding opens another problem: which entropy? It is well known that entropy is a name corresponding to many non-equivalent concepts. The irreversible dynamics of macroscopic systems is not restricted to thermodynamic or statistical systems. Therefore a more comprehensive concept than Clausius or Gibbs-Shannon entropy is required. I think that topological entropy defined for generic dynamical systems is the proper concept if one would like to relate the effective irreversible dynamics of macroscopic systems to entropy. It is handy that recently Addabbo and Blackmore were able to establish a dynamically grounded hierarchy of entropies where the topological entropy appears as the most general and Clausius' entropy as the most particular case.

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  • $\begingroup$ "However, this would be a very atypical starting condition. Almost all the neighbor configurations would not go back to a neighbor of the starting state. Notice that, even for moderately large systems, "almost all" is indistinguishable from "all" in practice." - this is essentially the statement that the number of possible states accessible to the system is large, due to our inability to control the initial conditions (random initial state). I think this is restating the Boltzmann's argument without using words H-theorem and entropy. $\endgroup$ Jun 30, 2021 at 9:40
  • $\begingroup$ @RogerVadim No, it is not the same as stating that the number of states is large. Even for non-interacting systems, that number is huge, maybe larger. What really matters is something more and more connected with dynamics. One can speak in terms of mixing processes, chaotic dynamics, or alike. But it is more than H-theorem. It may be entropy, but not the thermodynamic entropy, as I noticed in my postscript. $\endgroup$
    – GiorgioP
    Jun 30, 2021 at 10:19
  • $\begingroup$ thermodynamic entropy is just Shannon's entropy, under the assumption if equal probability of microstates. Although historically it was the other way round: Shannon called his quantity entropy on advice of von Neumann (if I am not mistaken), because it already existed in physics $\endgroup$ Jun 30, 2021 at 11:08
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    $\begingroup$ @RogerVadim Not really, in general. One needs the thermodynamic limit to recover the right convexity and extensiveness properties. Moreover, Shannon entropy is defined for every probability distribution, even not depending on energy. Thermodynamic entropy requires the connection with energy. $\endgroup$
    – GiorgioP
    Jun 30, 2021 at 11:24
  • $\begingroup$ Anyhow +1 - I think your answer is interesting, even if we do not agree about the meaning of some terms. $\endgroup$ Jun 30, 2021 at 11:29
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Physical events are determined by two separate factors: the dynamical laws (usually some form of partial differential equation, which is time symmetric), and the boundary conditions, specifying what happens on the boundary of the region in which the laws apply. (Even a solution over all space is often dependent on the behaviour 'at' infinity, which has to be assumed or asserted.) It is very common when discussing the laws of physics to forget about the importance of the boundary conditions, but they are an essential ingredient to find any solution, and they are frequently the source of unexamined assumptions that result in paradox and misunderstanding.

If the laws are symmetric to time reversal, then the arrow of time must come from the boundary conditions. In particular, we generally consider starting configurations (the past boundary of a region of spacetime) with low entropy. Given the past boundary has low entropy (compared to the typical entropy of the states it is free to occupy), then it becomes statistically almost certain that the entropy will increase with time. However, if we decided to assert that the future boundary was low entropy, and tried to figure out what prior behaviour led to this state, then again, virtually all possible pasts would have higher entropy. Time flows away from any point on the boundary at which low entropy is asserted.

Thus, the observation of a universal arrow of time in nature is a consequence of the early universe somehow having an extremely low entropy - something which is on the face of it vanishingly unlikely! The Big Bang was like a fireball of hot gases that rapidly expanded and cooled, condensing as huge masses of unburnt fuel, ready to power the universe's subsequent history.

Without this low-entropy beginning, allowing us to put a low-entropy patch of boundary at the past edge of our region of interest, and so get interesting things going on, physics would be boring. Almost all states would start and end with high-entropy, and nothing much would change in between. You would have a box of gas that just sat there, not moving, not changing. That's the 'Heat Death' of the universe.

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We have irreversibility because the initial conditions of the universe were a low entropy state. If the universe reaches an equilibrium state (heat death), then the arrow of time will disappear and there will be no way to tell past from future.

Example of effectively irreversible behavior in a reversible system

It is very easy to get effectively irreversible evolution of system from reversible dynamics. I made a JSFiddle that demonstrates it:

https://jsfiddle.net/WaterMolecule/q0mLy8av/11/

Here is a box containing 16 particles. The particles obey Newton's laws with no friction (reversible dynamics). The particles are initially arranged in a simple pattern and given the same initial velocity along the x-axis. The first particle is also given a very small velocity along the y-axis. Without this small y-velocity, the particles would stay perfectly ordered. However, this small velocity makes the first particle bounce against its neighbor at a slight angle to the horizontal. Eventually this particle collides with the others and the disorder grows in time. After many seconds, the system becomes fully disordered and there is no trace of the original pattern. Despite being fully reversible, you will not see the particles return to the initial state in your lifetime: the evolution is effectively irreversible until equilibrium is reached.

While the disorder is increasing in the system, the amount of disorder serves as the "arrow of time". This is the state our universe is in. We started from a very ordered arrangement at the big bang and are moving toward equilibrium. Currently, we can measure time by the increasing disorder, but measuring time will no longer be possible when the heat death of the universe is reached (if that is the ultimate fate of the universe).

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Synchronicity is Improbable

This is a different way of describing Roger Vadim's answer, but hopefully it will help. Imagine a billiard table. With a single ball, it is easy to imagine typical scenarios of motion, and their reversals. With two balls, it is also fairly easy to imagine that time-reversal is effectively impossible to detect (especially if the table is infinitely large). However, once there are 3 balls on the table, something special happens: there are now a large number of typical, common scenarios whose time-reversal is manifestly unlikely. The simplest one to imagine is when two balls are sitting at a location touching, and a third ball strikes them at the same time, coming to a stop and transferring all its momentum to the targets, which fly off at angles away from each other. There is nothing special about this scenario, until you try to run it in reverse. The reason you will suspect that the inverse scenario is backwards-time is because it is extremely easy to push one ball into two, but it is extremely difficult to time two balls to collide with a single ball, transferring all their momentum to it. This requires exquisitely precise timing and positioning. While any regular at your local pool hall can set up the forward scenario with ease, the reverse may be nearly impossible even if you get two professional billiards players to cooperate on it.

Improbability Grows Exponentially

As the number of objects in the system increases, the number of these improbable sequences goes up exponentially, because each new object can multiply the number of existing improbable sequences. Thus, when we are talking about "microscopic" vs "macroscopic" scales, really, we are talking about fundamental particle number. And once you get to 3, you are basically already in the "macroscopic" realm.

Now, you may not have noticed it, but the 3-billiard ball example above is a simplified model of the sliding wood-block example. The initially moving ball is the wood block, and the initially static balls are the desk. Their final motion is "heat". If we increase the number of target balls to a standard 15 ball pool rack, the number of possible outcomes quickly becomes intractable. And yet, they are all consistent with a general idea of: "moving object dissipates kinetic energy into heat via friction". But what should we make of the time-reversal of these traces? Technically speaking, they are possible! A universe in which one of them occurs does not automatically invalidate QM. But if all possible traces are equally likely, then it is easy to see why we don't observe blocks of wood spontaneously sliding around desks from a concentration of local thermal energy. Obviously, the number of ways for the collision to go forwards is much larger than the number of ways for it to run backwards...or is it? After all, we can just reverse all the momentum arrows, so the forward and backward traces should be equal in number, right?

Macrostates

Instead of the block sliding on wood, imagine that it is in space, and drifts into a gas cloud. If the cloud is big enough (or the momentum is small enough), the block will eventually dissipate all its KE into the cloud as heat (it will raise the temperature of the gas cloud). Now, the time-reversal would be for a gas cloud to have a block of wood, then spontaneously eject it in some direction, while simultaneously cooling.

To pinpoint exactly why the forward scenario is boring, and the backwards is magical, we just need to look at two states: one where the block is outside the cloud with momentum, and one where the block is inside the cloud with no momentum. These are the macrostates of interest. If we are told that these states are causally connected and asked which comes first, our intuition will tell us that the block moves from outside in. But does the math tell us that?

Yes. Yes, it does. As the block moves towards the gas cloud, the cloud just sits there being warm: its constituent particles are bouncing around randomly, not changing the temperature of the gas by a meaningful amount (we presume that in both states the gas is in thermal equilibrium, or close enough). Once the block hits the gas, there are many, many ways for the gas to absorb its momentum and convert it to heat (convert the unified momentum of the block into the randomized momentum of the gas molecules). However, there are only a few ways for the thermal motion of the gas molecules to conspire to push a static block out of the gas.

Again, we need to start small. If we just consider a single gas molecule that hits the block, the probability that it will push the block towards the external end state instead of away from it is 50%. But if we look at two molecules which hit the block, there are several possibilities:

  1. Both molecules push the block towards the external state
  2. Both molecules push the block away from the external state
  3. The molecules tend to cancel each other out

Because of the high-dimensional nature of the momentum space, it is not trivial to give the precise probabilities for each outcome, but hopefully it is clear that even with just two gas molecules, there are more ways for the external state to not happen than there are for it to happen. And each additional molecule that we add to the computation reduces the probability of case 1 and increases the probability of case 3.

So even though every microstate which leads from the external moving block to the internal static block is individually reversible, there simply aren't enough of the reversed states for this outcome to be likely. The vast majority of microstates correspond to the gas molecules hitting the block randomly with a near net-zero force. These swamp the comparatively tiny number of states in which the gas molecules are synchronized in their momentum to push the block out of the cloud.

Low-Entropy States are Few

Ultimately, the improbability of time-reversed processes doesn't really depend on much physics at all. It can all be deduced mathematically using statistics. If we take a finite sequence of integers, and permute it randomly, what are the odds that all the numbers in the first half will be smaller than all the numbers in the second half? It's not large, but also not absurdly tiny. It only requires a fairly coarse-grained sorting function to achieve this (gravity operating on rocks in a gently-shaken bucket will suffice). But what is the probability that the sequence will become strictly ascending? Well, there is only one way for that state to occur, no matter how big the list is. Which means that the bigger the list, the more improbable this state becomes. This would correspond to a minimum-entropy state in a physical system. The probability of the system entering this state from a random walk becomes inconceivably small as the size of the list gets large (especially if the transition function is not a global permutation, but rather a bunch of local permutations, so that it takes a while for a number to move from one end of the list to the other).

Now, what is the probability of a state in which the largest number occurs after the smallest number? It's easy to see that virtually every microstate occurs in this distribution, making this macrostate very high-entropy.

The macrostate in which the thermal motions of gas molecules produce a consistent net force on a block is extremely low-entropy, because there are only a tiny number of microstates which can produce this macrostate. The vast majority of macrostates will produce a net zero force, for the same reason that most random shuffles of a card deck will not deal you a straight flush in a game of poker.

All we need to know to obtain this statistical result is that thermal motion is effectively random. Since random walks toward a low-entropy state are highly improbable, the shape of the state transition space itself effectively defines the arrow of time. Random processes tend to push systems towards the most heavily populated macrostates, not the least-populated. And those macrostates are the ones which look like "boring thermal equilibrium", not "objects shooting spontaneously out of a static cloud".

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No fundamental force is reversible. If you push a still electron, it will not only gain a kinetic energy, but also will radiate soft modes with the probability 1. This is an eloquent example of irreversibility of a fundamental (electromagnetic) interaction. If the energy loss is small, then it may look as reversible, but actually it is not.

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Consider a block of wood and you just make it slide on a desk, it will move a little bit and then stop.

It is because we are making the block of wood slide. As a macroscopic object, there are a truly large number of atoms that we are influencing to move the same direction. As it moves along the desk, atoms are jostled randomly, converting that uniform motion to random motion or heat. The reverse process would be all of those atoms of the wood that are moving (heat) to move in the same direction to cause the block of wood to move. It is possible, but so unlikely that you could almost count the likelihood as zero.

Think of a new deck of cards that are in order. We recognize that order as we recognize the block of wood moving across the surface of the desk. Now imaging putting the cards into a perfect shuffling machine. The ordered deck is just as likely as any other deck, but because there are 52 cards, there is a 1 in 80658175170943878571660636856403766975289505440883277824000000000000 chance that any random shuffling will produce an ordered deck.

A 3.5cm block of wood has about 100,000,000,000,000 molecules (using 3.5nm for size of lignin) in contact with the desk. For the desk to spontaneously accelerate the block of wood, it would have to be in some kind of synchronicity to have a majority of the vibrating molecules jostle the wood in the same direction over and over again. The problem is that the surfaces of both the block of wood and the desk are being jostled randomly by other atoms in themselves and randomly jostling those atoms as well.

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