Continuum limit: is it really justifiable?

[drkatzin]

Usually in statistical physics, when your system has a large number N of particles, you take the continuum limit -- you let $N\rightarrow\infty$, and convert sums to integrals (with an appropriate normalization factor).

My understanding is that as a finite number tends to infinity, the infinity is still countable. What confuses me is the jump to uncountable infinity that allows us to use the continuum. Is there a rigorous math way to explain why this is ok, or is it just a physicist's way of saying countable infinity $\approx$ uncountable infinity? (To me, this seems absolutely unjustifiable, even in an approximation. The concepts are fundamentally different.)

Thanks!

 

[Vanadium 50]

We do it because it works - i.e. it agrees with observation. It may not be mathematically rigorous, but it is well-defined.

 

[Dale]

. Is there a rigorous math way to explain why this is ok

As the number of particles goes to infinity the error from making the continuum approximation goes to 0.

 

[Mute]

Usually in statistical physics, when your system has a large number N of particles, you take the continuum limit -- you let $N\rightarrow\infty$, and convert sums to integrals (with an appropriate normalization factor).

My understanding is that as a finite number tends to infinity, the infinity is still countable. What confuses me is the jump to uncountable infinity that allows us to use the continuum. Is there a rigorous math way to explain why this is ok, or is it just a physicist's way of saying countable infinity $\approx$ uncountable infinity? (To me, this seems absolutely unjustifiable, even in an approximation. The concepts are fundamentally different.)

Thanks!

You will hopefully find the answer in this thread sufficient: https://www.physicsforums.com/showthread.php?t=407934 (the relevant post is a remark that a continuous function is completely determined by its values on the rationals, which are countable).

Some further comments, though:

The limit you are describing, $N \rightarrow \infty$, is called the thermodynamic limit, and although it is similar to the continuum limit, they are not quite the same. The thermodynamic limit addresses the case of what happens when you take the number of "particles" of a system to infinity, but those particles may still have a discrete set of locations.

In this context, the approximation of sums with integrals can perhaps be viewed as a consequence of the Euler-Maclaurin formula, which enables one to approximate sums as integrals, at very least deriving an asymptotic expression, which is generally what one does in statistical mechanics. The change of variables from sum indices to a more physical variable will introduce the density of states.

As for the continuum limit, this is the limit in which your system actually becomes a continuum e.g., taking the lattice spacing to zero. Often this is done in conjunction with the thermodynamic limit while holding the density of the system fixed. Because the number of particles is generally still taken to infinity, the integral theorem I mentioned above still applies.

 

[drkatzin]

You will hopefully find the answer in this thread sufficient: https://www.physicsforums.com/showthread.php?t=407934 (the relevant post is a remark that a continuous function is completely determined by its values on the rationals, which are countable).

Some further comments, though:

The limit you are describing, $N \rightarrow \infty$, is called the thermodynamic limit, and although it is similar to the continuum limit, they are not quite the same. The thermodynamic limit addresses the case of what happens when you take the number of "particles" of a system to infinity, but those particles may still have a discrete set of locations.

In this context, the approximation of sums with integrals can perhaps be viewed as a consequence of the Euler-Maclaurin formula, which enables one to approximate sums as integrals, at very least deriving an asymptotic expression, which is generally what one does in statistical mechanics. The change of variables from sum indices to a more physical variable will introduce the density of states.

As for the continuum limit, this is the limit in which your system actually becomes a continuum e.g., taking the lattice spacing to zero. Often this is done in conjunction with the thermodynamic limit while holding the density of the system fixed. Because the number of particles is generally still taken to infinity, the integral theorem I mentioned above still applies.

Thank you Mute! This makes a lot of sense.