Intuition behind classical virial theorem

The point of using expressions such as $$langle x^i frac{partial H}{partial x^j} rangle $$ is not to necessarily obtain information about any general system, but to obtain a tool to study specific systems or at least classes of systems case by case.

For instance, in Newtonian/Galilean dynamics, most systems of interest will be possible to express in coordinates such that their Hamiltonian has the separable form $$H = T(p_i) + V(x^i) $$ Now let us consider the vicinity of a non-degenerate equilibrium, that is a point $x^i_0$ such that $partial V/partial x^j (x^i_0)$, but the all the eigenvalues of the matrix $V_{ij} equiv partial^2 V/ partial x^j partial x^k (x^i_0)$ are positive (i.e. the second-derivative matrix is non-degenerate and positive-definite). Then we can make a transform to $delta x^i = x^i-x^i_0$ and the Hamiltonian can be reexpressed in the form $$H = T(p_i) + frac{1}{2}V_{ij}delta x^i delta x^j + mathcal{O}(delta x^3)$$(Assuming Einstein summation convention.) In that case we can say that near equilibrium $x^i_0$ $$delta x^kfrac{partial H}{partial x_l} = delta x^k V_{lj}delta x^j$$ (Automatically dropping $mathcal{O}(delta x^3)$ from now on.) If you put $k=l$ and sum over the indices you get $$delta x^lfrac{partial H}{partial x_l}(x^i) = delta x^l V_{lj}delta x^j approx 2 (V(x^i) - V(x^i_0))$$ In other words, $delta x^lfrac{partial H}{partial x_l}$ has the meaning of roughly twice the difference of potential energy as compared to equilibrium.

In principle, if you know the matrix $V_{ij}$, knowing each $delta x^k partial H/partial x^l$ also allows you to figure out the approximate distance $delta x^i$ of the system away from the equilibrium of the potential. You can also use some dimensional analysis for a rule of thumb estimate of the complete release of the system from equilibrium. Assume that from the physics of the system you understand that the potential is associated with a binding energy scale $E_{rm b}$ and that the second-derivative matrix goes as $V_{ij} sim E_{rm b}/L_{rm V}^2$ where $L_{rm V}^2$ is a variability length. You can then estimate that the system stays bound as long as $$delta x^lfrac{partial H}{partial x_l}(x^i) lesssim E_{rm b}$$ We also see that and equivalent condition is $|delta x| lesssim L_{rm V} $, which is also the condition for the small-$delta x$ expansions above to be valid.

So far I have just discussed classical mechanics, no statistical physics involved. Let us now, for simplicity, shift our coordinate system so that $x^i_0 = 0$ and then $$langle x^k frac{partial H}{partial x^l}rangle approx 2 langle V_{lj}x^j x^k rangle$$ The statement that for $kneq l$ this is zero means simply that fluctuations in "energy-orthogonal" directions are uncorrelated. This can be understood particularly well if you rotate into a basis where $x^i$ are eigenvectors of $V_{ij}$ (i.e. a basis where $V_{ij}$ is diagonal). The $k=l$ case (without Einstein summation!) gives you the correlation of "energy-related" fluctuations about the potential equilibrium.

For example, once using the virial theorem and the rule of thumb estimate for bound systems near equilibrium, we get a condition for the system staying bound as $$langle x^l frac{partial H}{partial x^l}rangle approx 2nlangle V - V_{rm eq}rangle = n k_{rm B} T lesssim E_{rm B}$$ (here $n$ is the number of degrees of freedom.) I.e. $langle x^l frac{partial H}{partial x^l}rangle$ allows you to analyse in detail whether the system stays in equilibrium, possibly reaches other local equilibria etc. etc.

Of course, this is just an example of a class of systems. There are systems with degenerate equilibria for which the discussion is changed in a few details but the general meaning of the term $langle x partial H/partial xrangle$ is similar. In quantum mechanics one actually has to use a similar analysis to answer whether the temperature is sufficient to excite a degree of freedom at least by a single quantum jump and whether it thus has to be included in the state sum. In astrophysics one also often discusses systems where the virial theorem is very important but the gravitational potential is $sim 1/|x - x'|$ between every two particles. However, the meaning of the term $langle x partial H/partial xrangle$ is not quite universal and becomes particularly murky in relativistic physics. So it is as I said in the beginning, the virial theorem provides a useful tool for specific classes of systems but perhaps not all systems.