Interaction between systems but energy of interaction is $0$ There is something I don't totally get when we talk about interactions.
In statistical physics, to prove the canonical probabilities, we assume that the thermostat and the system don't interact such that we can write :
$$E_{tot}=E_S+E_{T}$$
Where $E_{T}$ is the energy of the thermostat.
And we allow the energy of $S$ to vary (we can find the associated probability).
But for me if I don't have interactions, it is not possible to have a varying energy. To illustrate what I mean I will take an example :
Imagine a collision between two balls, I agree that $E=E_1+E_2$ where $E_1$ is the energy of my first ball and $E_2$ the energy of my second. But it is because there is an impact that we can have an energy transfer from one to the other. And this impact is by definition an interaction.
Can I say that as we can model the interaction with a force like this : $F_{1 \rightarrow 2}(t)=F\delta(t-t_0)$ where $t_0$ is the interaction time, the energy of interaction will be $0$ (because the interaction is infinitely short in time).

In summary : how to understand the fact that we can have at the same time, transfer of energy from a system $1$ to a system $2$ without an interaction between them ?
Can we say that there is indeed an interaction but the energy of interaction associated to it is actually $0$ ? Thus we can have interaction between systems without any energy associated to this interaction.
I just need to fix my ideas...
 A: The correct relation would indeed be
$$E_{tot}=E_S+E_T+E_{S,T}$$
where $E_{S,T}$ is an interaction term. 
We can neglect the interaction term if two conditions are satisfied:


*

*The interaction potential of the atoms/molecules is short-ranged.

*The number of molecules at the interface between the thermostat and the system is negligible with respect to the number of molecules in the bulk.


Condition 2 is very reasonable. For example, if the system has volume $V$ and number density $\rho$, the number of molecules in the bulk will be
$$N_{bulk} = \rho V$$
while the number of molecules at the interface will be approximately
$$N_{interface} \approx \rho V^{2/3} \Delta r$$
where $\Delta r$ is a length scale of the order of the molecular diameter. Therefore
$$\frac{N_{interface}}{N_{bulk}} \approx V^{-1/3} \Delta r = \frac{\Delta r}{L}$$
where we have defined $L=V^{1/3}$. Since (to be generous) $\Delta r \approx \text 10^{-9}$m, this number is very small in practical situation. 
Condition 1 is also reasonable, unless you are considering charged particle. In that case, the interaction energy will decay as
$$E \propto r^{-1}$$ 
and you will have some issues. This is because the interaction of the molecules at the interface with "far away" molecules in the bulk will decay slowly with distance, and therefore their contribution to the total energy will in general be non-negligible even if condition 1 is satisfied. 
I will note that condition 1 is linked to the convergence of the integral
$$I = \int_0^\infty 4 \pi r^2 u(r) g(r) dr$$ 
where $u$ is the pair interaction potential and $g$ is the radial distribution function. If $u(r) \propto r^{-1}$ you will have
$$I \propto \int_0^{\infty} r g(r) dr$$
which is divergent since $\lim_{r \to \infty} g(r) = 1$.
