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?


1 Answer 1


I believe that the statement specifically refers to non-interacting particles. If any two particles collide, their velocities after the collision are clearly correlated - hence the BBGKY hierarchy of equations for the joint distribution functions.

Alternatively, there might be that the author of the text have made additional assumptions about the interaction/derivation (which perhaps motivated the OP to use the Langevin equations.)

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: $$ \rho(x_1, p_1;..;x_N, p_N)\propto e^{-\beta H(x_1, p_1;..;x_N, p_N)}.$$

  • $\begingroup$ 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! $\endgroup$
    – dnrk
    Sep 15, 2023 at 9:13
  • $\begingroup$ @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. $\endgroup$
    – Roger V.
    Sep 15, 2023 at 9:26
  • $\begingroup$ @dnrk However that the point of Kubo is to reduce everything to the correlations in equilibrium... in which case, the collision integral in BBGKY chain vanishes... you can try to follow the notes linked in this answer $\endgroup$
    – Roger V.
    Sep 15, 2023 at 9:30

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