A very brief argument shows that time reversal is impossible – in principle and not just for statistical reasons. The same argument proves that initial conditions and physical laws do not represent a starting point for exact calculations of the evolution of physical systems, not even under idealized conditions.
We look at two interacting objects A and B that are moving relative to each other, between two time points T0 and T2.
The question is: Is the time reversal of this real physical process still a real physical process?
The answer is no. The reversed process is not possible in reality, and this applies in any case, regardless of which types of objects and interactions are involved and in which environment they are – even if all physical laws that apply to this process are time-symmetrical.
Justification:
Let T1 be a point in time that lies between T0 and T2. A is at position PA(T1), B is at position PB(T1).
Since the propagation speed of the interaction is finite, the following holds:
The effect from B, which A is subjected to at time T1, does not originate from B at position PB(T1) but from a position where B was before, just as long before as it took the effect to get from there to PA(T1). We call this position PB(forward).
Now we look at the backwards running process. Again we stop it at time T1. If we consider the process to be real, then again it applies that the effect that B exerts on A at time T1 does not emanate from B at position PB(T1) but from a position where B was before. But now B's direction of motion is reversed, and this means that B must be moved along its trajectory in the opposite direction, to a position we call PB(backward).
In any case it applies:
PB(forward) ≠ PB(backward)
A necessary condition for a time-symmetrical process is that the acceleration exerted on the objects involved is identical in both time directions at any point in time.
However, since the positions, from which the state of motion of A and B is changed by the other object, are different for both time directions, this condition is obviously not fulfilled here.
Thus the backwards running process is no real physical process.
Notes
1. The definition of the above scenario is so general that it includes every possible physical process: in any process there must be objects that interact with each other so that they change each other's states of motion. It is irrelevant which interactions these are and how they are formulated: in any case, the objects are sources of the effects (carriers of the "charges"), and the effect that an object exerts on other objects depends on the position of the object.
2. Even if it were not possible to verify the condition "identical acceleration in both time directions at any point in time" in the usual physical representation of the scenario, nonetheless in any case there is a representation in which it can be validated.
3. The widespread belief that from the time symmetry of the equations that describe a physical process follows the time symmetry of this process itself is refuted by our argument.
4. An interesting question is to what extent the argument affects the assessment of some scenarios that play an important role in the discussion of time reversal, such as the scenario in which all molecules of a gas are initially in a tiny sub-volume of a closed container and are distributed throughout the whole container after a while. I suspect that in this case, too, the deviations (angle changes of the trajectories) that follow from our argument – even if they are extremely small – escalate after a relatively short period of time to such an extent that they abolish the time-reversibility of the process: the molecules will not gather again in the tiny sub-volume – not even if the conditions at the end of the process serve exactly as starting conditions of the time-reversed process. (That's just a guess, though; I didn't calculate how long one would actually have to let the process run to prevent this outcome.)
5. According to thermodynamic arguments, time reversal is improbable. Our argument proves that it is impossible.
Further inference:
Accurate knowledge of initial conditions and laws is generally considered a safe starting point for calculating the evolution of physical systems. Only limitations of measuring and calculating then prevent exact results.
However, our argument shows that there is a fundamental (absolute) limitation: even a Laplacian demon with infinite resources of space, time and information would not be able to carry out an exact calculation on the basis of initial conditions and laws, because he would not even be able to start with it: In order to determine the effects to which any of the objects of the system is initially subjected, all other objects would have to be put back to the positions from which these effects emanate, and obviously this cannot be done exactly, because the same applies to all objects.