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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?

dnrk
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1 Answers1

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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.)

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

Roger V.
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