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$$
\kappa  = \frac{V}{k_B T^2} \int_0^\infty \langle J_x(0)  J_x(t) \rangle \, dt
= \frac{V}{3 k_B T^2} \int_0^\infty \langle \mathbf{J}(0) \cdot  \mathbf{J}(t)  \rangle \, dt
$$

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