# Can we write down a dynamical law of physics which is totally non-deterministic?

In classical mechanics, $F=ma$ tells us how to evolve a system at time $t=t_0$ to $t=t_0+dt$.

In quantum mechanics, the Schrodinger equation gives us a similar recipe.

These equations are, in a certain sense, completely deterministic. Is it possible that nature only appears to be deterministic because the only language we know how to express physics is math (particularly equations), which (not to offend statisticians) seems to be particularly apt at describing deterministic systems?

In other words, are there possible time-evolution laws that are both non-deterministic and falsifiable?

If not, is determinism not falsifiable?

• deterministic chaos is probably an interesting case: deterministic (even simple) equations produce unpredictable time evolutions. – scrx2 Nov 17 '15 at 21:49

"are there possible time-evolution laws that are both non-deterministic and falsifiable?"

Yes, they are called stochastic (differential) equations. The classic example is the Langevin equation, which is Newton's law with a random force.

• Interesting. I still feel that this is deterministic in some sense. By running Monte Carlo simulations, etc. one could in principle map out the probability distribution of this particle as a function of time. In this sense, it almost as deterministic as QM – hwlin Feb 23 '13 at 3:59
• Yes, the probability distribution evolves deterministically. Whatever your non-deterministic mechanics are you can describe it by a deterministically evolving probability distribution on some appropriate space of states. – Michael Brown Feb 23 '13 at 4:19
• @hwlin - To elaborate Michael's point a little bit, if something is not deterministic once, we can always take a massive ensemble of similar systems and make some kind of a statistical inference which is true on average but not necessarily in every case. This is how the whole of statistical mechanics came about, and leading from that, stochastic processes and non-equilibrium stat mech as well. So essentially, take a large enough sample of your non-deterministic thing, and we have the tools to tell you what will happen. :) – Kitchi Feb 23 '13 at 9:57
• To elaborate a bit more. The Langevin equation for a single system is $\tilde{F}=m\tilde{a}$, which is non deterministic because both $\tilde{F}$ and $\tilde{a}$ are random. If we take averages on both sides we recover the deterministic $F=ma$ with $F=\langle\tilde{F}\rangle$ and $a=\langle\tilde{a}\rangle$. Thus the ensemble behaves deterministically because we are averaging out the random fluctuations, but each individual system continues being non-deterministic. – juanrga Feb 24 '13 at 13:55

$F=ma$ only applies to a special class of classical systems. It does not apply to non-deterministic classical systems for which more general equations of motion are needed:

Poincaré resonances and the extension of classical dynamics

Poincaré resonances and the limits of trajectory dynamics

The same about the Schrödinger equation except that any ordinary textbook on QM already explains you in what situations you cannot use the Schrödinger equation to describe the evolution of the system under study

The quantum version of the above extension of classical dynamics is covered in The Liouville Space Extension of Quantum Mechanics