What is the definition of randomness in mathematics and physics?

[SW VandeCarr]

I have sort of an offshoot question for you folks:

Would the first 100 decimal places of pi (minus the decimal point, or 314159...) be considered a series of random numbers? What about the first 100 decimal places of pi/2?

This has been discussed ad nauseum in this thread and I might re-ignite it by responding, but I'll give you the view to which I subscribe.

The decimal expansions of irrational numbers cannot be random as they are completely determined by an algorithm (see Kolmogorov). However, many use intervals of such sequences as "random numbers" because they have no apparent pattern. That is, they can pass statistical tests for randomness.

 

[Oldfart]

Thanks, SW! But wouldn't that eliminate any series of numbers as being random? This because one could (in theory) endlessly multiply or divide the digits of pi by different numbers (a determinant process), eventually yielding every possible combination of 100 digits. Or a zillion digits.

What am I missing here? Duhh...

 

[SW VandeCarr]

Thanks, SW! But wouldn't that eliminate any series of numbers as being random? This because one could (in theory) endlessly multiply or divide the digits of pi by different numbers (a determinant process), eventually yielding every possible combination of 100 digits. Or a zillion digits.

What am I missing here? Duhh...

Well for one thing, multiplication and division are algorithmic procedures, so you are taking a string produced by an algorithm and transforming it by additional algorithmic procedures. By the Kolmogorov definition, a random sequence must be generated by a random process (not pseudorandom). Such a process may from time to time generate strings that will fail a statistical test for randomness.

As was stated in this thread, you can't tell if a string is random in the Kolmogorov sense unless you know how it is generated.

 

[bbbeard]

I don't necessarily agree with this since the evolution of the wave function seems intrinsically random. How is is not intrinsic? It is a theorem that there are no hidden variables that can improve our knowledge.

I know I'm jumping in late to respond to a post you put up a month ago, but...

It seems to me that there is nothing random about the time evolution of a quantum state. If you know the initial state |psi₀>, you just have to compute all the components <k|psi₀> in the energy eigenbasis {|k>}. Then each component evolves as exp(-iω_kt) where ω_k=E_k/hbar.

In other words, the evolution is not random. The evolution is deterministic. But if you measure the energy at a later time, the result of the measurement will be probabilistic (unless the initial state is an energy eigenstate, in which case the later measurement should give the same energy).

 

[PatrickPowers]

The Oxford English Dictionary defines 'random' as: "Having no definite aim or purpose; not sent or guided in a particular direction; made, done, occurring, etc., without method or conscious choice". However, if we intend randomness as events with equal frequency probability this can't be. Think for example of the frequency of symbol sequences of a crypted text file. So I'm wondering if there exists a rigorous definition of randomness in mathematics and/or physics which can be interpreted as the above definition?

No one owns the word so it is used with several different meanings intended.

The definition I like the most is simply "unpredictable." It is perfectly OK for something to be predictable to one person with more knowledge and unpredictable to another. It is subjective.

Often it is used to mean "unpredictable with every possibility is equally likely."

Physicists tend to use it as "as far as we know no one can predict this." Sometimes they seem to be saying that "no one will ever be able to predict this" which seems to me like overreaching. I think a better definition would be "don't try to get a Phd by trying to figure this process out because we are pretty sure you won't succeed."

"Pseudorandom" may be used for something that is unpredictable the first time but repeats so it is predictable subsequently. "Stochastic" means something that is both unpredictable and doesn't repeat.

So is something that is predictable 99.99% of the time predictable or is it random? Well, if you predict "no hurricanes" every day you may be right 99.99% per cent of the time and wrong only 0.01%, but your prediction is nonetheless worthless.

 

[lavinia]

I know I'm jumping in late to respond to a post you put up a month ago, but...

It seems to me that there is nothing random about the time evolution of a quantum state. If you know the initial state |psi₀>, you just have to compute all the components <k|psi₀> in the energy eigenbasis {|k>}. Then each component evolves as exp(-iω_kt) where ω_k=E_k/hbar.

In other words, the evolution is not random. The evolution is deterministic. But if you measure the energy at a later time, the result of the measurement will be probabilistic (unless the initial state is an energy eigenstate, in which case the later measurement should give the same energy).

you are right but I just wrote too hastily and used the wrong words. Quantum mechanical amplitudes evolve according to a Markov like process.This Markov process describes the world as intrinsically random although i suppose - I don't know - there might be another description of the world where things are merely intrinsically unpredicatble. The Shroedinger equation for a free particle is a complex heat equation so it is no surprise that it describes a random process. A great description of this can be found in Feynmann's third volume of Lectures on Physics.