Why are strongly-correlated systems treated with equilibrium conditions?

Why do you expect that an interacting system will not be able to reach equilibrium?

What is true is that you have to deal with interacting systems differently than you deal with non-interacting systems. Introductory Stat Mech courses deal only with non-interacting systems, and develop Boltzmann, Fermi-dirac or Bose-Einstein statistics in this framework. When dealing with interacting systems, things get tougher. The many-body Hamiltonian is no longer just a product of single-particle states. The interaction terms affect the partition function, and add a level of complexity to the problem.

For many weakly or moderately interacting systems, there are perturbative, field theoretic and other approaches to extracting different statistical quantities. In some of these systems (like the Fermi liquid, for instance) interactions can be magicked away by the wave of a wand, and the system can be pictured as containing essentially non-interacting creatures (called quasiparticles).

 

Hello Gerenuk,

When systems are strongly correlated such as an electron gas (eg The Jellium Model). The state distribution of the electrons will be dependent on temperature not though Fermi-Dirac statistics, but some other distribution function due to correlation effects.

Modey3

 

Temperature is a well defined quantity for virtually any macroscopic system. Take a chunk of manganese oxide and set it on your countertop; it will eventually come to thermal equlibrium with the environment.

From a microscopic stat mech point of view, the canonical partition function is calculated via
[tex]Z = \sum_\psi \langle \psi | e^{-\beta H} | \psi \rangle[/tex]
where the sum is carried out over a complete set of many-body wavefunctions, regardless of whether the system is interacting or not. For a non-interacting system, the Hamiltonian is generally writable as a sum of terms which are all single-particle terms. In that case, this greatly simplifies the calculation of the partition function as it will be just the single particle partition function to the power N.

 

Oh why bother with two compartments? Take a container with water. If you apply stat mech without thinking, the water should spread out of the container and flow along the floor. Why doesn't it do so? Because the container has walls.

See the point?

Moral: Do not do stat mech without thinking!

 

What you are saying is that your partitioned container has two isolated sections. I had imagined it with a partition that went up only to the liquid level. But if the container is completely partitioned, then each partition becomes an isolated system and knows nothing of the other one. So there is no real problem there once you recognize this.

In any case, I don't understand the relevance of the boundaries on role of interactions. I think perhaps you chose a poor example to illustrate your question.

Now, I'm not saying that a temperature can be defined for any interacting system. I believe you are correct that this is not true in general, but not that this is untrue in the case of every interacting system. What is true is that temperature can be defined for many types of interacting sytems. It is only those systems that are specifically dealt with using tools of equilibrium thermodynamics. Interacting systems that do not equilibrate must be studied using non-equilibrium techniques.

 

The reason your two containers of water don't reach equilibrium is because of the wall - the containers are not allowed to exchange particles even though they have different chemical potentials. For this to work as an analogy to temperature, you would have to have the two systems thermally isolated from each other so they could not exchange energy.

Stongly correlated systems have been studied quite a bit experimentally. If they didn't reach thermal equilibrium with their environment, I would think this would have been noticed, as it would have huge implications.

There are interactions which break stat mech; specifically, a long range interaction like gravity is problematic for stat mech because the potential energy from it is not extensive. But for systems of charges, as long as the system has no net charge, then the electric interaction is not enough to break extensivity.

There are certainly systems which exhibit long range correlations that are treated just fine with stat mech. Take a ferromagnet, for instance. Certainly some care has to be taken when interpreting what the ground state and low-lying excitations are, but that does not mean stat mech is useless in this case. A magnet certainly reaches thermal equilibrium with its environment.

A diatomic molecule has very strong interactions between the two atoms that form the molecule. Yet in spite of the strong interaction, stat mech applies very well to a diatomic gas.

So I don't understand why you think thermal equilibrium wouldn't apply to strongly correlated systems? Strong correlation doesn't mean strong interactions... by "strong interaction" I mean that the potential energy of the interactions is overwhelming. Strong correlation means (rougly speaking) that the energy scales of different types of energy come in at about the same energy scale, so that different types of order or disorder can end up competing. This is why phase diagrams for strongly correlated systems can be quite rich, because minor changes in pressure or concentration affect which phase "wins" (see the phase diagram for Pu, for instance).

 

Gerenuk,

We can always use stat mech to study a system. If the system is in equilibrium, we use equilibrium stat mech. Else, we must use techniques of nonequilibrium stat mech.