# If a system is deterministic, will it still be deterministic if time is reversed?

If you were to drop a ball, it would be easy to calculate when it will hit the ground and how much energy will be absorbed by the ground (let's assume there's no air resistance and the ball does not bounce).

If you were to then reverse time, would energy from the ground gather rapidly toward the location of the ball, creating a powerful but microscopic impulse capable of launching the ball to its original height?

If the answer is yes, then the time reversed system should in fact be deterministic.

If the answer is no, then there could not possibly be any way to predict when the ball will bounce or how high it could go. The system would not be deterministic.

Which result is closer to reality?

• The dynamics of a perfectly isolated system is reversible (both at the classical and quantum level). In concrete situations, this is only sometimes a good assumption/approximation. The coupling with an environment is usually the cause for irreversibility, such as in thermodynamics and quantum measurements. In your example, you are not considering a "dynamical" theory for the ground; this results in an effective dissipative theory that is irreversible (e.g. the ground acts as a thermodynamic bath that irreversibly dissipates energy). – yuggib Oct 8 '16 at 9:58

## 1 Answer

If the equations in whatever deterministic theory you are using are reversible in time, then you can use the current state to predict both the future and the past your system just as easily, because it's all in the mathematics. However, deterministic equations can suffer from sensitivity to initial conditions. Generally, that happens in both directions of time too, so you can only predict the weather about as far into the future as you can into the past. You might be able to predict how much heat a bouncing ball will lose, and thereby predict the height it was dropped from if you know how many bounces it took, but not if you wait until the ball comes to a stop because stopping a ball is not going to happen using equations that are time reversible.