# Velocity correlations of different particles

In section 13.3.2 of Statistical Mechanics: Theory and Molecular Simulation by Mark E. Tuckerman, the author derives the Green-Kubo relations for the diffusion constant. In the derivation, he makes the following claim:

Recall that in equilibrium, the velocity (momentum) distribution is a product of independent Gaussian distributions. Hence, $$\langle \dot{x}_i\dot{x}_j \rangle$$ is $$0$$, and moreover, all cross correlations $$\langle \dot{x}_i (0) \dot{x}_j (t) \rangle$$ vanish when $$i\neq j$$.

However, I have doubts regarding the second claim ($$\langle \dot{x}_i (0) \dot{x}_j (t) \rangle = 0$$ when $$i\neq j$$). I am able to prove it for non-interacting particles using the Langevin equation (with the assumption that the white noise terms corresponding to different particles are uncorrelated). However, I am not sure that this statement is true for a system of interacting particles. Is there a rigorous proof that the velocity cross-correlations vanish for different particles even in the presence of interactions?

In case of velocity-independent interactions, the Hamiltonian is something like $$H(x_1, p_1;..;x_N, p_N)=\sum_{i=1}^N\frac{p_i^2}{2m} + \sum_{i=1}^{N}\sum_{j=1}^{i-1}U(|x_i-x_j|).$$ In this case we automatically have that equilibrium, the velocity (momentum) distribution is a product of independent Gaussian distributions: $$\rho(x_1, p_1;..;x_N, p_N)\propto e^{-\beta H(x_1, p_1;..;x_N, p_N)}.$$
• I thought that too, but it seems that the author made no such assumptions. He begins his derivation from the general Hamiltonian $\mathcal{H}^{\prime}=\sum_{i=1}^{N}\frac{\mathbf{p}_{i}^{2}}{2m_{i}}+U\left(\mathbf{r}_{1},\dots,\mathbf{r}_{N}\right)-f\sum_{i=1}^{N}x_{i}$ (the last term describes diffusion). Regarding your remark - how does the factorization of the velocity distribution into independent Gaussian distributions imply $\langle \dot{x}_i (0) \dot{x}_j (t) \rangle=0$ for $i \neq j$? After all, $\rho$ doesn't even contain time. Thank you!
• @drnk factorization into Gaussian distributions in equilibrium does not imply absence of the correlation between velocities at different times. In fact, I don't see the flaw with my argument that the particle velocities become correlated after a collision, and that BBGKY hierarchy does not stop at the first equation confirms it - it is the collision integral that couples lower probability functions to higher ones: $w(x_1,p_1)$ to $w(x_1,p_1;x_2,p_2)$, etc. That $w(x_1,p_1;x_2,p_2)$ doe snot factorize means that velocities are correlated. Anyhow, I would look for an exact answer in BBGKY. Sep 15, 2023 at 9:26