Maximum precision of deterministic measurements

[Runner 1]

Okay, I tend to have some weird thoughts, so bear with my odd question here.

Suppose you have a collection of particles that obey Newtonian mechanics. For simplification, all particles are identical and can be assumed to be hard spheres that collide elastically. Each particle has a position and a velocity. Now suppose we want to measure the position and velocity of a particular particle. By measurement, I mean a series of bits indicating its position, and a series of bits indicating its velocity.

The only way we can obtain information about that particle is using other hard, spherical particles. With enough types of collisions between the various particles (the "experiment" in essence), we eventually get a digital readout of both values of interest.

Now, intuitively it seems like there must be a theoretical limit to the amount of information we can get about the particle. Has someone done a calculation that determines what this is? In other words, what is the maximum amount of bits we can get that accurately describes the particle using other particles?

If you need more clarification about what I'm asking, just let me know (because I am not that great at conveying what I mean to others).

(Note this has nothing to do with HUP or quantum mechanics -- it's just an idealized classical situation.)

 

[Simon Bridge]

Question is underspecified ... how are you using the spheres to collect the information?

However - the type of calculation you want has been done in the QM case ... the classical one will differ in the value of h. You have a similar problem to the Heisenberg uncertainty - you cannot know position and momentum, simutaniously, to arbitrary precision, even in this 100% classical system, because the act of measurement (in this case, hurling a sphere at the sphere you want to measure) disturbs the system.

I suppose, in principle, you should be able to rig something where repeated measurements could give better precision ... which is why you need to specify the system you have in mind in more detail.

 

[HallsofIvy]

Okay, I tend to have some weird thoughts, so bear with my odd question here.

Suppose you have a collection of particles that obey Newtonian mechanics. For simplification, all particles are identical and can be assumed to be hard spheres that collide elastically. Each particle has a position and a velocity. Now suppose we want to measure the position and velocity of a particular particle. By measurement, I mean a series of bits indicating its position, and a series of bits indicating its velocity.

The only way we can obtain information about that particle is using other hard, spherical particles. With enough types of collisions between the various particles (the "experiment" in essence), we eventually get a digital readout of both values of interest.

Now, intuitively it seems like there must be a theoretical limit to the amount of information we can get about the particle. Has someone done a calculation that determines what this is? In other words, what is the maximum amount of bits we can get that accurately describes the particle using other particles?

If you need more clarification about what I'm asking, just let me know (because I am not that great at conveying what I mean to others).

(Note this has nothing to do with HUP or quantum mechanics -- it's just an idealized classical situation.)

On the contrary, it has everything to do with HUP. It is the basic argument behind the uncertainty principle.