# Maximum precision of deterministic measurements

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.)

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HUP holds true even at the macroscopic level; we just happen to need less precision to "find" the entity. – Ignacio Vazquez-Abrams Jun 24 '12 at 5:19

## 1 Answer

The question of the theoretical limit to classical measurement is interesting, and it is contained in the general thermodynamic formalism. To get a position measurement of n-bits when the environment is at temperature T, you need to dump at least $nk\log2$ entropy in the environment.

The argument is by Liouville's theorem. In order to measure the position of a particle to n-bit accuracy using other particles, you need to reduce the phase space volume of the measured particle by a factor of $2^n$, which means you need to increase the phase space volume of everthing else by the same factor (to make the volume of the phase space occupied by possible microstate at least stable--- it can only grow).

This is the restriction. At zero temperature, there is no limit to an arbitrarily accurate classical measurement--- you can take some little tiny particles that are moving really fast and bounce them off the moving object (modeling light bouncing off the object) and detect the bounced objects and calculate the position from their incoming direction. If there is no thermal jitter, you can do this classically completely accurately without any limit. The thermodynamic bound only says that you need to generate a certain irreducible amount of heat in order to get the bouncing particles' positions accurately recorded in your device.

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Strictly speaking, the $nk\log 2$ of entropy only has to be created when resetting the device to its original state after making the measurement. If you're only planning on making a finite number of measurements you can (in theory) get away with not making any heat at all. – Nathaniel Jun 24 '12 at 9:18
@Nathaniel: That's sort of true, you can increase the phase space of the device instead of the environment. But then you don't know what bits you measured, because you transferred the uncertainty from the particle to the device. If the device is self-aware, I am not sure of whether it can learn it's own state by introspection in some way which doesn't require dumping entropy outside. – Ron Maimon Jun 24 '12 at 17:59