# Do identical starting conditions always lead to identical outcomes?

My friend and I are discussing whether or not physical phenomena are deterministic. Let's say, for example, that we have a 3-dimensional box with balls inside of it upon which no gravitational forces are acting. The balls each have their own size, mass, starting position and starting velocity. After a given amount of time, the balls will have changed positions and possibly also velocities due to movement and possible collisions with other balls.

The question is, does the same initial state always lead to the same state after a given amount of time? In other words, if we have two boxes of the same size with the same number of balls of the same size, starting at the same positions with the same initial velocities, will the balls inside of each box be at the same positions, as we would expect from a deterministic system, or would there be any randomness involved?

• There's no randomness in the idealized situation you've described. Why would you expect randomness? Of course, in quantum mechanics the situation is different. – DanielSank Oct 7 '17 at 23:51
• More on non-uniqueness in Newtonian mechanics: Norton's dome and its equation. – Qmechanic Oct 8 '17 at 10:31
• A mention of Laplace's Demon is mandatory. Great minds think alike ;-). – Peter - Reinstate Monica Oct 9 '17 at 11:59
• There is a video of a nice experimental setup showing a related real-world answer here. I can assure you that it is well-vetted and experimentally well controlled. The starting conditions and interventions are "identical" (within reason) in every run, and still the outcome is fairly unpredictable. The actual experiment starts at 0:40. – Peter - Reinstate Monica Oct 9 '17 at 12:09
• Certainly there's a possibility of differing radioactive decay in the two runs, imparting different forces to the balls. – David Schwartz Oct 9 '17 at 23:51

There are several layers to this, so I get to have fun uncovering them.

The first layer is simple Netwonian Mechanics. If we assume Netwonian mechanics applies, and that the universe only consists of this box and its contents, and the contents of the box are set up exactly the same every time, then the resulting positions of the balls as time goes on is deterministic. It will be exactly the same, every single time.

However, it gets more interesting. Newtonian mechanics can be chaotic. A chaotic system is sensitive to initial conditions. A slight perturbation of the setup can yield drastically different results. Perhaps you put one of the balls in the wrong place: off by 0.5mm. This can cause the collisions to occur differently, and lead to drastically different results. A classic example of this is the double pendulum. In many regions, its motion is very sensitive to initial conditions. In this sense the box is unpredictable but deterministic. There's only one way the balls can move, but it's impossible to predict because properly predicting it would require infinitely precise measurements, and we don't have any way of measuring things like that.

Which brings us to widening our universe. Up to this point, we only considered a universe containing this box and this box alone. But there are outside influences on real world boxes. For example, there are gravitational forces being applied. Literally speaking, the position of Jupiter could affect the position of these balls colliding around by subtly changing the velocities of the balls.

Of course, what I just said sounds like astrology, so I should back off a bit. In practical scenarios, Jupiter is not going to noticeably affect the results. In a truly chaotic system, all inputs matter, but in our practical box, forces like friction are eventually going to make the system highly predictable. There's no need to go to a fortune teller to find the alignment of the planets before doing this experiment in real life!

But we are good at making experiments which are closer and closer to these ideal chaotic environments. So we can ask ourselves what happens as we push this to the extreme. What happens when we make an experiment so refined that Jupiter is having an effect. Well, we also start seeing other effects: quantum effects. Quantum effects will perturb the setup, just like failing to perfectly set up all of the balls, or failing to account for the gravitational effects of Jupiter. These effects are tiny, so in any practical situation, you will not observe them. However, they are there. And what's interesting about them is that, to our best knowledge, they are truly random. We know of no way to predict the effects of quantum interactions at a particle by particle level. As far as we can tell, their effects are truly nondeterministic, and so your box is nondeterministic too.

But, taking a step back, if you look at the sum total of many trillions of quantum interactions occurring each and every second, the results are statistically predictable. If you take the laws of quantum mechanics, and apply them to incredibly large non-coherent bodies (like a billiard ball or a box), you find that the equations simplify out to Newtonian mechanics (more or less). So unless you carefully craft your box and balls with the expressed intent of detecting the nondeterministic effects of quantum mechanics, you will find that the balls behave very much deterministically (although if you build a chaotic system, they may still not be predictable).

• Thank you for a great answer. You mention, towards the end, that because of the law of large numbers, quantum effects average out to Newtonian physics for large objects such as a billiard ball or a box, and so, our system is deterministic. My question is: Is it truly deterministic? Or will there always be little tiny (although possibly unobservable) deviations from the Newtonian expectations of the positions of the balls and their velocities? – Sandi Oct 8 '17 at 8:04
• @Sandi In theory, there will be tiny deviations. In practice, other factors which you did not account for will dominate such errors. As for whether it is truly deterministic, that's actually an ontological question. Science does not answer the question of what reality is. It explores how we can model and predict reality. It is entirely possible that quantum mechanics is not actually a good model of the real universe, we just don't know it yet. – Cort Ammon - Reinstate Monica Oct 8 '17 at 16:27
• I think that a box with mostly elastically colliding balls (e.g. metal or billard balls) is a good example for a truly and utterly non-deterministic system if we consider quantum effects. The reason is that the influence that variations in the setup conditions have grows exponentially (or worse?) with the number of collisions. As is the nature of non-linear growth, one arrives at the quantum level after just a few dozen collisions. (The Newtonian impact would be pragmatically identical, the way you mentioned: Unmeasurably small differences in surface or setup would lead to big differences.) – Peter - Reinstate Monica Oct 9 '17 at 11:57
• Bravo for the "unpredictable but deterministic" distinction. That was a succinct light bulb of clarity. – Randall Stewart Oct 9 '17 at 17:42
• @tparker Yes, there are some highly contested corner cases which are capable of confusing the issue. If I needed to end the answer with Newtonian mechanics, rather than building it up as a straw man before going on to other higher fidelity models, it might be important to mention points like that where integration and differential equations start to get murky. – Cort Ammon - Reinstate Monica Jan 12 '18 at 21:36

In Newtonian Mechanics there is no randomness involved once you know all the initial data. In fact, let $M$ be the phase space of a classical system. The points of $M$ are pairs $(q,p)$ of coordinates and momenta.

The time evolution of the system is described by ordinary differential equations on $M$, and once you know the initial conditions, by the existence and uniqueness theorem for solutions of ODE's you get the result that there is a single path in phase space corresponding to the sequence of states parametrized by time, in other words, a single association

$$t\mapsto (q(t),p(t)).$$

The trouble is that if the system has a huge number of particles the problem becomes extremely hard to tackle. Because of this one studies systems like this with Statistical Mechanics and then start dealing with means and so forth. But if you knew all the initial data and could solve the equations of Classical Mechanics (existence is guaranteed, but it might be very hard as I said), there is a unique path of evolution with no randomness whatsoever.

EDIT: Let's tackle this from a different point of view. In Quantum Mechanics a system is described by a state space $\mathcal{E}$. The elements of $\mathcal{E}$ are vectors called state vectors which we denoted like $|\varphi\rangle$. It turns out that if you know the state of the system, meaning that you know what state vector describes it, you still don't have full information about the system.

One example: consider a single particle with spin $1/2$. The spin can be either up or down. If the spin of the particle is up, the state of the system is $|\uparrow\rangle$ and if it is down the state is $|\downarrow\rangle$. These states are simple to understand, but that's not all. The most general state is $|\varphi\rangle = a |\uparrow\rangle + b |\downarrow\rangle$ and in this state everything you can say is that there is a probability of $|a|^2$ that when the spin is measured it will be up and $|b|^2$ that when the spin is measured it will be down.

And that is not all. Even in the state $|\uparrow\rangle$ you can't know the $x$ and $y$ components of spin. You just know the $z$ component is $1/2$. All you can get are probabilities.

So in QM even if you know the state, you don't know it all. There is randomness that is part of the theory.

Nevertheless the evolution is deterministic. Given one initial state, there is precisely a single evolution. But that's not the point. The system will evolve to some other state like these I've examplified, and in the state there will be inacessible information about the system. Again, deterministic evolution guarantees that you can evolve an initial state in a unique manner, but even if you know the state, you can't know it all.

Classical Mechanics isn't like that. In a Classical System you can know both position and momentum exactly in each state. Every observable is a function of position and momentum, hence you know any physical quantity if you know the state. Together with the fact that the evolution in time is unique, if you know the initial state you know it all.

Again you need to know exactly: (i) the initial conditions, (ii) the solution to the equations of motion. It is guaranteed to exist, not guaranteed to be easy to know.

• Thank you for your answer. I guess I would like to know how it would work "in the real world". If we had an actual box floating around in space with actual balls in it, would there be any randomness involved? I heard quantum mechanics being mentioned. How does quantum mechanics affect the example? – Sandi Oct 8 '17 at 0:33
• @Sandi take a look on the edit. I gave a QM example, see if it makes it clearer. – user1620696 Oct 8 '17 at 1:15
• “by the existence and uniqueness theorem for solutions of ODE's” – it's important to add that this requires Lipschitz continuity. – leftaroundabout Oct 8 '17 at 14:57

I want to emphasize something that is implied in the other answers but not said explicitly:

Your problem is not as clearly defined as you may think.

Why?

• If your question is about real life, then because perfectly precise measuring is impossible, the problem makes no sense: you can never know for real if the balls are in the same positions as in the previous experiment, nor if they behaved in the same way. Unless your question is statistical, but then as mentioned above you would need to define your problem more precisely (tolerances for measurements, for instance; but then, as mentioned by Cort Ammon, we know of systems which are chaotic, which means that you cannot prescribe certain precision for the initial state and expect the outcome to be within the same precision).

• If the question is about physics, then you are putting it in some theoretical framework, and each has its own answer (in Newtonian Mechanics the answer would be yes, in Quantum Mechanics the answer would be no). To correlate the theory with real life you would need experiment, and then you go back to the first bullet.

For your given example, the results in the "ideal" case, would be deterministic. However, where it fails, is the fact that we can't give the balls in one box the "same position and velocity" of the corresponding balls in the other box - except to a certain degree of accuracy! With a time long enough, even the small differences in starting position and velocities, will eventually lead the corresponding balls to be in different states.

Quantum mechanics returns measurement of the state that you tried to gain and result in outcomes that you are not fully able to measure because of the small varience in measurement and what it is actually ie Copenhagen etc, if you threw those measurements into a non linear model it would be grossly different from reality a system such as a pendulum the varience would not be as significant but still would be measurably different and its behaviour is absolutely determined as function of a differential equation. The world is a crazy place and our model we have now should be determinative if we throw a state into it, but in principle this is physically not possible unless your Laplace's demon and are able to have a perception of a complete overall state of everything at one moment in time and truly throw it into its own model and thus able to determine the past present and future.