Randomness in Evolution: Valid and Invalid Usage

[Reality Check]

The quantum mechanical wave function is a random variable as it describes the probability amplitudes at different points in space. Since the Schrödinger equation takes the wave function its argument, it is a probabilistic equation.

The wave function in the Schrödinger equation is interpreted as a probability amplitude because when we take a measurement it enters as a magnitude squared. However the wave function changes in time according to the deterministic Schrödinger equation.

 

[mijopaalmc]

Wrong. The wavefunction is a probability distribution (or really, the square root of one), not a random variable. All these posts and you can't even get the basic mathematical terminology right?

The time evolution of the wavefunction is totally deterministic, as it obeys a differential equation.

Any measurable function of a random variable (which is described by a probability distribution is itself an random variable. The wave function and the its square are therefore both random variables.

 

[jimbob]

Surprisingly, I'd agree with that, but several people have been proposing "Laplacian Determinism", i.e.

An intellect which at any given moment knew all the forces that animate Nature and the mutual positions of the beings that comprise it, if this intellect were vast enough to submit its data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom: for such an intellect nothing could be uncertain; and the future just like the past would be present before our eyes.

Interesting. I'm pretty sure I can prove on the basis of physical law that such an "intellect" is literally impossible.

Of course Laplace wrote that in 1814, and even then I think this hypothetical intellect would have to have been larger than and outside the universe.

I have always understood it as similar to the statement, "God does not play dice", which I have understood was supporting the idea of a completely determanistic (and predetermined) universe, where any (pseudo)random events or behaviour only look random because of incomplete information.

 

[Reality Check]

Any measurable function of a random variable (which is described by a probability distribution is itself an random variable. The wave function and the its square are therefore both random variables.

Here is the Wikipedia article on wave functions. If you read it you will see that a wave function is not defined using random variables. For example

• The wave function for a particle in 1 spacial dimension is a function of x
• The wave function for a particle in three spacial dimensions is a function of x, y and z.

The outputs of a wave function are complex numbers that are interpreted as probability amplitudes (the Copenhagen interpretation).

 

[mijopaalmc]

Here is the Wikipedia article on wave functions. If you read it you will see that a wave function is not defined using random variables. For example

• The wave function for a particle in 1 spacial dimension is a function of x
• The wave function for a particle in three spacial dimensions is a function of x, y and z.

The outputs of a wave function are complex numbers that are interpreted as probability amplitudes (the Copenhagen interpretation).

I read the article before I posted.

It's really quite simple: if something is described by a probability distribution it is a random variable. This doesn't mean it can necessarily take on any value with any probability, because that is not how a random variable is defined.

 

[Reality Check]

I read the article before I posted.

It's really quite simple: if something is described by a probability distribution it is a random variable. This doesn't mean it can necessarily take on any value with any probability, because that is not how a random variable is defined.

It's really quite simple: if something is not described by a probability distribution it is not a random variable.

Wave functions are functions of the phase space under consideration, e.g. the wave function for a particle in three spacial dimensions is a function of x , y and z. X, y and z are the spacial dimensions. They are not probability distributions. Therefore the wave function is not a described by probability distributions and is not a random variable.

A basic definition of a random variable is a function that maps events from a random process into numbers. The Schrödinger equation is not a random process - it is a partial second differential equation and deterministic.

An analogy would be the Gaussian function. It is definitely deterministic. But you can use it to represent a probability distribution. Does that make the Gaussian function a random variable?

 

[sol invictus]

Any measurable function of a random variable (which is described by a probability distribution is itself an random variable. The wave function and the its square are therefore both random variables.

The wavefunction is not a function of a random variable - you're very very confused.

After kicking and screaming about this nonsense for so many months (or is it years?) you still don't understand even the first and most basic probability-theoretic definitions?

 

[sol invictus]

Of course Laplace wrote that in 1814, and even then I think this hypothetical intellect would have to have been larger than and outside the universe.

I have always understood it as similar to the statement, "God does not play dice", which I have understood was supporting the idea of a completely determanistic (and predetermined) universe, where any (pseudo)random events or behaviour only look random because of incomplete information.

If the intellect doesn't obey the usual laws of physics that statement is perfectly consistent in a quantum world as well. Deterministic non-local hidden variable theories can explain all of QM (but they are non-local).

If the intellect does obey the laws of physics - e.g. if it's a big computer with all that data - I think I can prove it impossible, even in a classical world, because of chaos.

 

[Belz...]

You are completely missing the point. Quantum randomness has an orderly classical (or semiclassical) limit.

Does it ?

 

[articulett]

The wavefunction is not a function of a random variable - you're very very confused.

After kicking and screaming about this nonsense for so many months (or is it years?) you still don't understand even the first and most basic probability-theoretic definitions?

Bingo.

And it's been over a year since he started a thread supposedly seeking to understand how evolution can be described as non-random.

Sad, but true. He and Jimbob were making the same arguments then, btw.

 

[jimbob]

If the intellect doesn't obey the usual laws of physics that statement is perfectly consistent in a quantum world as well. Deterministic non-local hidden variable theories can explain all of QM (but they are non-local).

If the intellect does obey the laws of physics - e.g. if it's a big computer with all that data - I think I can prove it impossible, even in a classical world, because of chaos.

Agreed and, if it were within the universe, it would have to have a perfect model of itself.

Do you see my point that some people are arguing that there is no randomness in the system of natural selection, and that if we made an accurate enough model, we could predict all selective events (with 100% certainty). In other words I believe their contention is that randomness in natural selection is simply an illusion due to inadequate information.

Hey, the only difference I wanted to make at the start was that organisms with beneficial traits tend to reproduce, whilst those with adverse traits tend to not reproduce. This was apparently confusing compared to saying that organisms with beneficial traits reproduce whilst those with adverse traits don't.

 

[mijopaalmc]

The wavefunction is not a function of a random variable - you're very very confused.

After kicking and screaming about this nonsense for so many months (or is it years?) you still don't understand even the first and most basic probability-theoretic definitions?

Actually I understand the basic probability-theoretic definitions much better than you do. The wave function (or more properly the square of its absolute value) take spatial-temporal coordinates and maps the to a set of probabilities, making it a random variable by definition.

 

[mijopaalmc]

Does it ?

Yes

 

[sol invictus]

Actually I understand the basic probability-theoretic definitions much better than you do.

If so, you're certainly not demonstrating it.

The wave function (or more properly the square of its absolute value) take spatial-temporal coordinates and maps the to a set of probabilities, making it a random variable by definition.

Go back and re-read your textbook.

At least in the Copenhagen interpretation of QM, the wavefunction squared is a probability distribution on a random variable (the position of a particle if we are discussing the wave function in the position basis). It is not a random variable itself, nor does it depend on one. It is a function of position and time - which are coordinates, not variables.

 

[cyborg]

Do you see my point that some people are arguing that there is no randomness in the system of natural selection, and that if we made an accurate enough model, we could predict all selective events (with 100% certainty).

What is it about "if everything is the same everything is the same," exactly that is particularly hard to understand?

 

[jimbob]

Because the universe isn't like this.

As far as I understand it, the best interpretation about quantum uncertainty isn't just that we can't know both a particle's position and momentum beyond with an accuracy more than Hbar/2, but that the universe itself doesn't "know". It isn't decided, i.e. it is random.

In other words, should we have five identical universes when everything is the same at one point in time, they would differ later on.

 

[cyborg]

In other words, should we have five identical universes when everything is the same at one point in time, they would differ later on.

I'll make it really simple for you.

Universe Ux

Events {A, B}

U1: AA
U2: AB
U3: BA
U4: BB

Now, give me a U5 that IS NOT the same as U1-4 after two events.

Also please tell me under what circumstances would the sequence of A/B events would be inadequate to describe the sequence of A/B events in any arbitrary Universe.

What is it about "if everything is the same everything is the same," exactly that is particularly hard to understand?

 

[jimbob]

I am not sure that I understand you, unless you are saying that if identical cstarting conditions included some non-physical reason that made fuure random events happen the same in the future.

With the snooker ball example, is the motion of the ball after twenty collisions predetermined at the time of the first collision?

 

[cyborg]

I am not sure that I understand you

The same event is the same event.

 

[jimbob]

So is the motion of the snookerball after its twentieth collison predetermined at the time of the first collision?