The arrow of time, especially that introduced by the second law of thermodynamics, has been at the centre of a paradox, a paradox that comes about as follows.....
The fundamental equations of physics are deterministic and by and large reversible in time*; that is, if one replaces the time variable t with -t then the essential form of the fundamental equations remains the same. It is therefore impossible to tell from the dynamics constrained by these equations whether time is going backward or forward; the very concepts of backwards and forwards time doesn't exist at the level of the fundamental laws. And yet our macroscopic experience of reality is that there is very definitely a direction to time: For if we play the film of macroscopic reality backwards it presents a very different aspect to the forwards film, not least because of the second law of thermodynamics, a law which tells us that the disorder of an isolated system always increases as time progresses forwards. But - and here's the rub - the reversibility of the fundamental equations of physics, at least at the microscopic level, means that these equations have as many "forward" solutions as they have "backward" solutions, all of which obey the same equations; this follows because the replacement of t with -t implies a one-to-one relation between backward and forward solutions. Therefore for every solution where entropy is increasing there must also be a solution where it is correspondingly decreasing. Therefore if we randomly select boundary conditions then the probability of selecting a solution where entropy is decreasing will be equal to the probability of selecting a boundary condition where entropy is increasing
Why then the clear macroscopic distinction between the backwards and forwards film and the ubiquity of the second law? This question is known as "Loschmidts's paradox". Apparently only recently has this paradox has been addressed professionally via the Fluctuation Theorem; see here. One issue I would have with this theorem is that it takes "cause & effect" as axiomatic concepts. But I'm not so sure that cause & effect are robust enough notions to be used in an axiomatic way (See my post here for more).
At one level the second law is fairly self evident: Given a macroscopic system state, a state which is usually understood in terms of the systems macroscopic statistical properties (e.g temperature, pressure, volume, mass. distribution, density etc), then the entropy, S, of the system is defined by:
S = k Log W
...where k is Boltzmann's constant and W is the number of microscopic combinations consistent with the macrostate as defined by the value of its statistical variables. If we take this statistical weight definition of disorder and we imagine a dynamic where some sort of random walk between microstates is in operation then migration to a higher a disorder (that is migration to macrostates with a higher statistical weight) is a fairly self evident outcome. But then random walk isn't deterministic let alone reversible.
I thought I would have quick think about Loschmidt's conundrum myself and to this end I've come up with what follows.
***
We start by imagining a set of concentric 3D spheres; that is a set of nested spheres so that the whole picture looks like a large onion. (Actually it would be nearer the truth to imagine an n-dimensional set of nested spheres, but the 3D picture of the onion should suffice). Let each sphere of the onion represent a macroscopic state where the magnitude of the surface area of a sphere represents the magnitude of the disorder of the macrostate - that is, the area of each sphere of the onion quantifies the number of microscopic combinations states consistent with the macrostate. Hence, the larger the nested sphere the greater the disorder of the macrostate it represents.
We can represent the dynamic development of a system in time by a path in the space enclosed by the onion. For all we know at this stage such a path could zig-zag around in the space of the onion any-old-how. But if we are dealing with deterministic equations where the exact microstate of the system determines both the direction and speed of development at each point on the path it is then not possible for any two development paths to cross..... this follows because at the crossing point there would be a choice of directions to move in and this would mean that the microstate then doesn't determine the speed and direction of motion through the onion. This contradicts the stipulation of strict determinism. Hence, none of the possible paths through the onion will cross.
However, for all we know at this stage even these non-crossing pathways could still violate the second law and there remains the possibility of a development path which at some point could move toward the centre of the onion thus implying a decrease in entropy in time. But if we require the stipulation that disorder can only increase in time then it means that all motions in the onion could only ever be outwards.
So, given the following assumptions:
Assumption 1: The second law of thermodynamics
Assumption 2: That the fundamental laws are deterministic,
then we have concluded this:
Conclusion: All motions in the onion are outward motions and these motions form a set of non-crossing streamlines.
But this conclusion leads to a contradiction. For if we follow these streamlines back toward the centre of the onion then because they are non-crossing stream lines this means that each sphere has an equal number of development lines passing through it. But if the point where each streamline crosses one of concentric circles of the onion designates a single microscopic combination then it follows that the disorder value remains constant both in backwards and forwards time. This is clearly an anti-physical conclusion. So what is wrong?
***
Because by requirement disorder must increase as we move outwards through the onion then the number of development lines crossing the spheres of the onion must also increase as we move outwards. The only way that can happen is if the paths form a branching pattern as they move outward; that is they bifurcate into one or more branches as does a tree when we follow its limbs upwards and outwards. The consequences of this "tree" picture are this:
ONE) The forward time development paths can not be deterministic; for if we take any branch point then it means that the microstate at that point doesn't fully determine where the system will advance to in the next move on its outward path. A branch point, by definition, gives a choice of ways.
TWO) And yet the time reversed motion may still remain deterministic: Following a simple branching pattern backwards will always take us back to the same place; (or if there is an element of in-determinism in going backwards, following the branching pattern backwards will take us to fewer possible places than if when we move forward in time).
THREE) In this branching pattern scenario cause & effect turn out to be meaningful ideas: Looking forward in time into a branching pattern we find that there are many possible routes by which the systems could have traveled backwards in time to the current state. We therefore are unable to attribute the current state to a definite cause emanating from the future. But because looking backwards we do find that the current state of affairs would have developed from a restricted number of starting states the idea that today's state had a fairly specific cause rooted in the past makes more sense.
FOUR) The branching pattern means that in looking forward in time we find a far greater number of possibilities open to the future than the possibilities in the past from which the current macrostate could have evolved from. The progressive branching as we move away from the centre of the onion ensures there are far more possible paths into the future than there are back into the past. Therefore if we are choosing possible paths at random then there is a far greater probability of the system increasing in disorder than in it increasing its order.
The foregoing tells us that the dynamic which controls our world cannot be deterministic. This suggests that the apparent random jumps in the quantum mechanical state vector are real events.
Footnote:
* See the link below for a qualified view on the reversibility of the laws of physics:
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