# Setting of the canonical ensemble for actual interacting systems

The canonical ensemble formalism is usually derived by considering a given small system (the system under study) weakly coupled to a huge system (the thermal bath), so that the microcanonical ensemble computations can be applied to the pair of system, regardless of the details of the interaction between the system and the bath.

However, the applicability of the canonical ensemble to a system does rely (?) on the existence of an actual interaction between the system and a bath, so that studying, e.g. the Ising model with say, periodic boundary condition with hamiltonian

$$H = -J \sum_{}\sigma_i \sigma_j$$

in the canonical ensemble actually means, from a physical point of view studying a larger system with fixed hamiltonian

$$H = -J \sum_{}\sigma_i \sigma_j + H_\text{interaction} + H_\text{bath}$$

where $$H_\text{interaction}$$ has an explicit expression, acts on the physical sites of the system, etc. It is not immediately obvious what are the most general forms of $$H_\text{interaction}$$ and $$H_\text{bath}$$ that are acceptable (i.e. such that the canonical ensemble predictions are indeed recovered), nor that the actual form of the interaction with the bath has no physical effect: for instance, $$H_\text{interaction}$$ can be small and still induce correlations (?) between sites of the Ising lattice.

So my question is what does it mean concretely to couple a system to a heat bath, so that the canonical ensemble formalism makes sense for a given hamiltonian system ? What is the most general form of the coupling to a thermal bath that preserves the physics that we get from canonical ensemble computations ?

A rather general consideration usually given in statistical mechanics books is that the energy of the system of interest is proportional to its volume, whereas the interaction energy (with the bath) is proportional to the size of the contact surface. This means that for a sufficiently big system (i.e., in thermodynamic limit) the contact interaction can be neglected, in the same way we neglect the residual interactions within the system itself - although these are necessary for establishing the thermodynamic equilibrium, they are negligible ehen describing the properties of this equilibrium.

I think there are two related questions here.

First, we might simply assume that the microcanonical ensemble and its associated probability distribution offer an appropriate description of the joint system + bath. We can then ask the question: Under what circumstances will the system be well-described by a canonical ensemble? The general answer to this question is that the joint system must satisfy two conditions:

1. The bath should be much larger than the system; that is, it should have many more degrees of freedom than the system. This implies, among other things, that the energy of the bath will be much greater than that of the system (at least for a typical sample from the microcanonical distribution of the joint system).

2. The interaction Hamiltonian $$H_{interaction}$$ should be small in some sense relative to the system Hamiltonian $$H_{system}$$.

These conditions may sound a bit imprecise. However, at this level of generality, where the characteristics of the system and bath are left mostly unspecified, I don't think that that's necessarily a bad thing. However, if we have a specific joint system in mind, then we can always try to flesh out these conditions in more detail.

Now, if (1) and (2) hold, and we assume that the joint system is describe by a microcanonical ensemble, then standard arguments found in intro stat mech textbooks can be used to show that the system may be described by a canonical ensemble.

However, this isn't the whole story. We can ask another question: When are we justified in using the microcanonical ensemble to describe the joint system in the first place? For classical Hamiltonian systems, the standard answer is that the joint system must be ergodic and chaotic, or approximately ergodic and chaotic in some sense. Roughly, ergodicity means that over the course of its evolution, the joint system visits every state available to it consistent with its total energy being constant (more precisely, it comes arbitrarily close to all such states), and the term chaotic refers to a sensitive dependence of the joint system's evolution on its initial state.

It's difficult to prove that a given system is ergodic and chaotic. To my knowledge, certain classes of dynamical billiards are some of the few types of systems which have been rigorously proven to be ergodic and chaotic. However, what is clear is that conditions (1) and (2) alone aren't enough to establish chaos or ergodicity (and therefore to justify the microcanonical ensemble). For example, we can satisfy (1) and (2) with a joint system with $$H_{interaction}=0$$. For such a joint system, the system and bath will be noninteracting, and so the individual energies of the system and the bath will each be constant. This joint system is not ergodic, since it will only visit states where the system and bath energies separately remain unchanged.

In summary, if we assume the validity of the microcanonical description for the joint system, then conditions (1) and (2) ensure that the system will be described by a canonical ensemble. However, to justify our microcanonical assumption, we need to further establish that the joint system is ergodic and chaotic, which is a nontrivial task to say the least.

• These conditions seem rather restrictive. If we imagine our Ising spins living on the atoms of a crystal lattice, with interaction strength J depending on the distance between adjacent atoms (to have coupling), the heat bath of the canonical Ising model is naturally interpreted, I suppose, as the mechanical degrees of freedom of the lattice (and the mechanical energy thereof as "heat"), which are not that many wrt the number of spins, and not necessarily ergodic, if the crystal is harmonic for instance. Feb 14 at 17:35
• If there are a large # of spins, then the argument in @RogerVadim's answer will at least establish that any subset of spins which is itself large (yet small relative to the total # of spins) can be described by a canonical ensemble (assuming again that the whole system is described w/ a MC ensemble). As for the question of ergodicity of such a system, I definitely don't know enough to say. It's at least conceivable that while the mechanical d.o.f. could be harmonic and therefore nonergodic in isolation, the spins + mechanical d.o.f. might still be approximately ergodic when coupled together. Feb 15 at 0:28