I admit when the idea for this thread titled, "does randomness exist in nature?"
came up i thought it would be easy just to trawl the internet for a few examples of seemingly random events in nature such as the radioactive decay of certain elements.
Then the question naturally arises, how do we know an event is truly random, or conversely, how do we know an event is truly predetermined?
I did some hasty googling and realised I might be a bit over my head on the second question, so I'm hoping some of our more knowledgeable members will chime in, but what I vaguely gathered is that you would have to have an infinite sample size to determine if something was truly random or predetermined, since even seeming patterns still have a chance of being the outcome of a random event.
Thoughts?
Randomness test - Wikipedia
According to Wikipedia (above), the
Diehard Battery of Tests is used to determine randomness.
According to Wikipedia (above), the
Einstein-Podolsky-Rosen (EPR) paradox caused
"statistically independent" to be added to the requirement of the Heisenburg Uncertainty Principle (which derives from the Robertson Shaw Inequality). Einstein, et al, proposed that a particle of spin 1 might randomly split into two particles whose spins add up to 1. So, though the top is random, and the bottom is random, the top plus bottom always equals one. So, you can use only tops of particles or bottoms of particles, but if you mix them, some will be statistically dependent and ruin the results of the Heisenburg Uncertainty Principle.
Quantum mechanics deals with random complex numbers (complex numbers have real and imaginary parts). The
wave function (also called the state function because it determines the state of a particle)
of an electron is imaginary. Probability is the product of the wave function and its complex conjugate (this is how complex numbers are squared). Thus, the product of two imaginary wave functions creates a probability (of finding the particle in a particular place in space) real. To reiterate,
from imaginary wave functions, we get a real probability.
To do quantum mechanics math, we convert the dimensions that we know (x,y,z,time) to dimensions in probability space (called
Hilbert Space). Oddly, the probabilities also have dimensions. Also, oddly, the dimensions of space in the real world don't correspond to the dimensions of Hilbert Space.
A simple electron orbiting a single proton produces a
Hamiltonian operator (an energy operator which is like a matrix but filled with math operations, and some are complex numbers) that is
infinite columns by infinite rows. This is because each probability density corresponds to a unique position of the electron, and, as it orbits, it can be located in an infinite number of places.
Since calculating an infinite operator is too difficult,
Schroedinger's Equation substitutes the Hamiltonian operator for its eigenvalue (that's one of many values that solves the Hamiltonian operator). This makes the math much simpler.
Physicists have a shortcut notation for probabilities called
Dirak notation. It uses two probability densities. The front one is a < bracket called a bra, and the back one is a > bracket called a ket....together, they form a bracket. Variables in the bra and ket can be interchanged by taking the complex conjugate transpose. But, like matrix math, the order of the variables must be preserved.
Quantum mechanics is used to understand the small world (molecules and smaller). But,
when applied to the larger world, the random numbers combine in such a way as to make things non-random. Thus, in the big world of Newtonian physics, random motions is sometimes converted to non-random motion.
Some data has both
random and nonrandom elements.
grainy pixelated picture - Google Search
The link above shows
grainy pixilated pictures. That is, pictures that have huge square chunks to represent the average values of color in the region.
Pixilated pictures have low frequency noise. So, they can be cleaned up, somewhat by a filter. But, to really be effective, dither has to be used. Dither is random noise that is added to a signal, then the picture is fixed by averaging the dithered signal. Surprisingly, a dithered signal brings out the original picture, in surprising detail. Even details that are not captured in the pixils come out.
Dithering doesn't affect the signals. According to Fourier's theorem, any periodic wave can be split into harmonic components. Thus, a square wave can be represented by an infinite series of sine waves. All sine waves have PDFs (probability density functions) that go to infinity on the endpoints, and are rather low in the middle. As a result, dithering doesn't affect the sine waves, so it doesn't affect the entire signal. For digitally sampled signals, Fourier requires
two samples per sine wave (that is, at twice the frequency of the components of the signal).
Dithering (which is noise added to the signal) drops the noise level of signals because the PDF (Probability Density Function) of the dither
convolves (a mathematical process)
with the PDF of the noise. The process of convolution causes the noise to be spread throughout the frequency spectrum and that lowers the noise of the signal. Noise outside of the band of interest is then filtered out (for examples, noises too low pitched to hear or too high pitched to hear), which removes the noise entirely that used to be in the signal band.
Dither can be used to reverse the effects of quantization and effects of jitter (caused by timing delays, for example, those in a computer as numbers are calculated). The PDF of quantization has a characteristic staircase pattern.
Large scale dither has to be subtracted out at the end.
FINDING RANDOM NOISE IN NATURE:
Clouds are random, yet they still form patterns, so there is some non-randomness in them. Rushing water is somewhat random, though there are some effects that are not. For example Q in fluid dynamics is the flow rate = velocity times the cross sectional area must remain constant. Sometimes, at the bottom of water falls and spillways, the water, leaps higher (called
hydraulic jump), caused by disruptions in laminar flow.
Thermodynamics is filled with random events. The mean free path determines how often one air molecule bumps into another (and that is defined as temperature).
Yet, there are laws and boundaries that define thermodynamics. For example, there is the
Ideal Gas Law, which relates temperature, pressure, and volume.
Saturated steam tables show the amount of water vapor, steam, and air at given temperatures and pressures.
As I was getting degrees and advanced degrees in physics, and various types of engineering, I realized that there is an
infinite amount to know about all of this.