Use of quantum ideas in classical statistical physics

[spaghetti3451]

When we study a classical system of distinguishable particles, we use parameters $\epsilon_{j}$ for the energy states and $n_{j}$ for the number of particles in $\epsilon_{j}$. But clearly, the energy states are not discrete in classical systems. Surely, this is nonsensical. Why are we doing this then?

 

[chrisbaird]

The energy states in a quantum system are not discrete either if the states are unbound, or if states are bound to an essentially infinitely wide potential, such as how we get bands of electron states bound to a crystal.

But more to your question, strictly speaking this is misleading. You should instead speak about the density of occupied particle states in ϵ_j (energy density, not spatial density). To talk about the actual total number of particles, you would have to integrate in energy over some interval the density of occupied states.

 

[Bill_K]

Why are we doing this then?

Because it's a heckuva lot easier to deal with discrete states, and take the continuum limit at the end.

 

[chrisbaird]

Yes, I have noticed that physicists (even the famous ones) often do what is easiest but not the cleanest to a mathematician, switching between discrete states and continuums and back and forth (all legal if you do the limits right).

 

[vanhees71]

The reason for using quantum arguments in classical statistics simply is that you cannot so easily get the right answer from classical considerations alone, and that's not a mathematical but a physical problem, and it's also not so much related to the discreteness of the spectra of some observables.

First of all, in classical physics, we don't have a natural unit for the phase-space volume. In quantum theory we know it's given by $h^{6N}=(2 \pi \hbar)^{6N}$. Thus, one can divide the phase-space volume in hypercubes with this volume and do statistics by "counting".

Second, you must apply the notion of indistinguishability of particles to avoid the Gibb's paradox.

Third, the entropy is bounded from below if there's a gap between the ground state (vacuum/quasi-particle vacuum) and the lowest excited state, from which follows Nernst's theorem of heat (the third Law of thermodynamics).

I guess there are a lot more examples, where you need a minimal version of quantum mechanics when doing classical physics. Thus, it's much better to learn statistical physics after having heard a bit about quantum theory before and consider the (quasi-)classical limit of quantum statistics to derive classical statistics.

This is also an important fundamental step: It explains, how a classical world appears for macroscopic objects although the underlying principles of our world is quantum on the fundamental level.