In a similar way, we can prove that only the real parts of $\Delta_3$ and $\Delta_4$ contribute to (129) and that

$$ \Re(\Delta_3) = -4\pi^3 \tilde{\mathcal{C}}_{VV}(\mathbf{k}, \omega) \frac{\hat{W}(\mathbf{k})^2}{|\epsilon(\mathbf{k}, \omega)|^2} \mathbf{k} \cdot \frac{\partial f}{\partial \mathbf{v}_1} f(\mathbf{v}_2) \delta(\omega - \mathbf{k} \cdot \mathbf{v}_1) \delta(\omega - \mathbf{k} \cdot \mathbf{v}_2) \int d\mathbf{v}' \mathbf{k} \cdot \frac{\partial f}{\partial \mathbf{v}'} \delta(\omega - \mathbf{k} \cdot \mathbf{v}'), $$

and

$$ \Re(\Delta_4) = -4\pi^3 \tilde{\mathcal{C}}_{VV}(\mathbf{k}, \omega) \frac{\hat{W}(\mathbf{k})^2}{|\epsilon(\mathbf{k}, \omega)|^2} \mathbf{k} \cdot \frac{\partial f}{\partial \mathbf{v}_2} f(\mathbf{v}_1) \delta(\omega - \mathbf{k} \cdot \mathbf{v}_1) \delta(\omega - \mathbf{k} \cdot \mathbf{v}_2) \int d\mathbf{v}' \mathbf{k} \cdot \frac{\partial f}{\partial \mathbf{v}'} \delta(\omega - \mathbf{k} \cdot \mathbf{v}'). $$

To compute the contribution of $\Delta_5$, let use this simple identity

$$ \frac{1}{\epsilon^2} = \frac{1}{\epsilon^2} - \frac{1}{|\epsilon|^2} + \frac{1}{|\epsilon|^2}. $$

And in a similar way that we did for the other terms, we notice that

$$ \frac{1}{\epsilon^2} - \frac{1}{|\epsilon|^2} = \frac{1}{\epsilon|\epsilon|^2} \{\bar{\epsilon} - \epsilon\} = \frac{-2i\Im(\epsilon)}{\epsilon|\epsilon|^2}. $$

Then, the fifth term $\Delta_5$ reads

$$ \Delta_5 = \frac{2i\Im(\epsilon)}{\epsilon|\epsilon|^2} (2\pi\hat{W}(\mathbf{k}))^2 f(\mathbf{v}_1) f(\mathbf{v}_2) \delta(\mathbf{v}_1 - \mathbf{k}\cdot\mathbf{v}_1) \delta(\mathbf{v}_2 - \mathbf{k}\cdot\mathbf{v}_2) \\ - \left( \frac{2\pi\hat{W}(\mathbf{k})}{|\epsilon(\mathbf{k}, \omega)|^2} \right)^2 f(\mathbf{v}_1) f(\mathbf{v}_2) \delta(\mathbf{v}_1 - \mathbf{k}\cdot\mathbf{v}_1) \delta(\mathbf{v}_2 - \mathbf{k}\cdot\mathbf{v}_2). $$

Once again, with parity arguments we can show that only the real part of $\Delta_5$ contributes to (129), that is to say

$$ \Re(\Delta_5) = \frac{2i\Im(\epsilon)^2}{|\epsilon|^4} (2\pi\hat{W}(\mathbf{k}))^2 f(\mathbf{v}_1) f(\mathbf{v}_2) \delta(\mathbf{v}_1 - \mathbf{k}\cdot\mathbf{v}_1) \delta(\mathbf{v}_2 - \mathbf{k}\cdot\mathbf{v}_2) \\ - \left( \frac{2\pi\hat{W}(\mathbf{k})}{|\epsilon(\mathbf{k}, \omega)|^2} \right)^2 f(\mathbf{v}_1) f(\mathbf{v}_2) \delta(\mathbf{v}_1 - \mathbf{k}\cdot\mathbf{v}_1) \delta(\mathbf{v}_2 - \mathbf{k}\cdot\mathbf{v}_2). $$

Thanks to this analysis, we can compute C as following

$$ C = \frac{1}{2(2\pi)} \sum_{\mathbf{k}} \int_{\Gamma} d\omega \mathbf{k} \mathbf{k} \Re\{\Delta_1 + \Delta_2 + \Delta_3 + \Delta_4 + \Delta_5\}. \quad (130) $$

Furthermore, the quadratic term of the Hamiltonian $H^{(2)}$ is linked to this cumulant via the following formula

$$ H^{(2)} = \int \mathrm{d}\mathbf{r}\mathbf{d}\mathbf{v}_1 \mathbf{v}_2 \frac{\partial p}{\partial \mathbf{v}_1} \frac{\partial p}{\partial \mathbf{v}_2} \mathbf{C}(\mathbf{v}_1, \mathbf{v}_2). \quad (131) $$

Using equations (123, 85, 130, 131), we can show that

$$ H^{(2)} = \int \mathrm{d}\mathbf{r}\mathbf{d}\mathbf{v}_1 \frac{\partial p}{\partial \mathbf{v}_1} \frac{\partial p}{\partial \mathbf{v}_1} : \mathbf{D}(\mathbf{v}_1) f(\mathbf{v}_1) \\ - \int \mathrm{d}\mathbf{r}\mathbf{d}\mathbf{v}_1 \mathbf{v}_2 \frac{\partial p}{\partial \mathbf{v}_1} \frac{\partial p}{\partial \mathbf{v}_2} : \mathbf{B}[f](\mathbf{v}_1, \mathbf{v}_2) f(\mathbf{v}_1) f(\mathbf{v}_2) \\ + \int \mathrm{d}\mathbf{r}\mathbf{d}\mathbf{v}_1 \mathbf{v}_2 \mathbf{v}_3 \mathbf{v}_4 \frac{\partial p}{\partial \mathbf{v}_1} \frac{\partial p}{\partial \mathbf{v}_2} \mathbf{B}^{(2)}(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4) \left\{ f(\mathbf{v}_1) f(\mathbf{v}_2) \frac{\partial f}{\partial \mathbf{v}_3} \frac{\partial f}{\partial \mathbf{v}_4} \right. \\ \left. - 2f(\mathbf{v}_1) \frac{\partial f}{\partial \mathbf{v}_2} f(\mathbf{v}_3) \frac{\partial f}{\partial \mathbf{v}_4} + \frac{\partial f}{\partial \mathbf{v}_1} \frac{\partial f}{\partial \mathbf{v}_2} f(\mathbf{v}_3) f(\mathbf{v}_4) \right\}, \quad (132) $$

with

$$ \mathbf{B}^{(2)}(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4) = 2\pi^3 \left(\frac{\lambda_D}{L}\right)^3 \sum_{\mathbf{k}} \int d\omega \mathbf{k} \mathbf{k} \mathbf{k} \mathbf{k} \frac{\hat{W}(\mathbf{k})^4}{|\epsilon(\mathbf{k}, \omega)|^4} \prod_{i=1}^4 \delta(\omega - \mathbf{k} \cdot \mathbf{v}_i), \quad (133) $$

being a fully symmetric order-4 tensor.
## G Expression of the fourth cumulant

In this appendix, we report the result of the computation of $H^{(4)}$ the contribution of the fourth cumulant to the large deviation Hamiltonian:

$$ H^{(4)} = \int \mathrm{d}\mathbf{r} \mathrm{d}\mathbf{v}_1 \cdots \mathrm{d}\mathbf{v}_4 \left\{ \left\{ \frac{\partial p}{\partial \mathbf{v}_1} \frac{\partial p}{\partial \mathbf{v}_1} - \frac{\partial p}{\partial \mathbf{v}_2} \frac{\partial p}{\partial \mathbf{v}_1} \right\} \left\{ \frac{\partial p}{\partial \mathbf{v}_3} \frac{\partial p}{\partial \mathbf{v}_3} - \frac{\partial p}{\partial \mathbf{v}_4} \frac{\partial p}{\partial \mathbf{v}_3} \right\} \mathbf{B}^{(2)} f(\mathbf{v}_1) f(\mathbf{v}_2) f(\mathbf{v}_3) f(\mathbf{v}_4) \right. \\ + 3 \int \mathrm{d}\mathbf{r} \mathrm{d}\mathbf{v}_1 \cdots \mathrm{d}\mathbf{v}_6 \left\{ \frac{\partial p}{\partial \mathbf{v}_1} \frac{\partial p}{\partial \mathbf{v}_2} \frac{\partial p}{\partial \mathbf{v}_3} \left\{ \frac{\partial p}{\partial \mathbf{v}_3} - \frac{\partial p}{\partial \mathbf{v}_4} \right\} \mathbf{B}^{(3)} f(\mathbf{v}_3) f(\mathbf{v}_4) \left\{ f(\mathbf{v}_1) f(\mathbf{v}_2) \frac{\partial f}{\partial \mathbf{v}_5} \frac{\partial f}{\partial \mathbf{v}_6} \right. \\ \left. \qquad - 2 \frac{\partial f}{\partial \mathbf{v}_1} f(\mathbf{v}_2) f(\mathbf{v}_5) \frac{\partial f}{\partial \mathbf{v}_6} + \frac{\partial f}{\partial \mathbf{v}_1} \frac{\partial f}{\partial \mathbf{v}_2} f(\mathbf{v}_5) f(\mathbf{v}_6) \right\} \\ + \int \mathrm{d}\mathbf{r} \mathrm{d}\mathbf{v}_1 \cdots \mathrm{d}\mathbf{v}_8 \left\{ \frac{\partial p}{\partial \mathbf{v}_1} \frac{\partial p}{\partial \mathbf{v}_2} \frac{\partial p}{\partial \mathbf{v}_3} \frac{\partial p}{\partial \mathbf{v}_4} \mathbf{B}^{(4)} \left\{ f(\mathbf{v}_1) f(\mathbf{v}_2) f(\mathbf{v}_3) f(\mathbf{v}_4) \frac{\partial f}{\partial \mathbf{v}_5} \frac{\partial f}{\partial \mathbf{v}_6} \frac{\partial f}{\partial \mathbf{v}_7} \frac{\partial f}{\partial \mathbf{v}_8} \right. \right. \\ \left. - 4 \frac{\partial f}{\partial \mathbf{v}_1} f(\mathbf{v}_2) f(\mathbf{v}_3) f(\mathbf{v}_4) f(\mathbf{v}_5) \frac{\partial f}{\partial \mathbf{v}_6} \frac{\partial f}{\partial \mathbf{v}_7} \frac{\partial f}{\partial \mathbf{v}_8} + 6 \frac{\partial f}{\partial \mathbf{v}_1} \frac{\partial f}{\partial \mathbf{v}_2} f(\mathbf{v}_3) f(\mathbf{v}_4) f(\mathbf{v}_5) f(\mathbf{v}_6) \frac{\partial f}{\partial \mathbf{v}_7} \frac{\partial f}{\partial \mathbf{v}_8} \right. \\ \left. - 4 \frac{\partial f}{\partial \mathbf{v}_1} \frac{\partial f}{\partial \mathbf{v}_2} \frac{\partial f}{\partial \mathbf{v}_3} f(\mathbf{v}_4) f(\mathbf{v}_5) f(\mathbf{v}_5) f(\mathbf{v}_6) f(\mathbf{v}_7) \frac{\partial f}{\partial \mathbf{v}_8} + \frac{\partial f}{\partial \mathbf{v}_1} \frac{\partial f}{\partial \mathbf{v}_2} \frac{\partial f}{\partial \mathbf{v}_3} \frac{\partial f}{\partial \mathbf{v}_4} f(\mathbf{v}_5) f(\mathbf{v}_6) f(\mathbf{v}_7) f(\mathbf{v}_8) \right\}. \quad (134) $$

**Acknowledgements** We thank J. Barré and G. Eyink for pointing us to relevant literature and for interesting comments on this manuscript. The research leading to this work was supported by a Subagreement from the Johns Hopkins University with funds provided by Grant No. 663054 from Simons Foundation. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of Simons Foundation or the Johns Hopkins University. We thank the two anonymous reviewers for their useful recommendations, which helped us to improve our first version of our manuscript.

## References

1. Akhiezer, A.I., Akhiezer, I., Polovin, R., Sitenko, A., Stepanov, K.: Plasma electrodynamics. volume 1-linear theory. volume 2-non-linear theory and fluctuations. OISNP 1 (1975)

2. Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctuation theory. Reviews of Modern Physics 87(2), 593 (2015)

3. Bodineau, T., Gallagher, I., Saint-Raymond, L., Simonella, S.: Fluctuation theory in the boltzmann-grad limit. Journal of Statistical Physics pp. 1–23 (2020)

4. Bouchet, F.: Is the boltzmann equation reversible? a large deviation perspective on the irreversibility paradox. Journal of Statistical Physics 181(2), 515–550 (2020). DOI 10.1007/s10955-020-02588-y. URL https://doi.org/10.1007/s10955-020-02588-y

5. Bouchet, F., Grafke, T., Tangarife, T., Vanden-Eijnden, E.: Large deviations in fast-slow systems. Journal of Statistical Physics pp. 1–20 (2015)

6. Braun, W., Hepp, K.: The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Commun. Math. Phys. 56, 101–113 (1977)

7. Carrillo, J.A., Delgadino, M.G., Desvillettes, L., Wu, J.: The Landau equation as a Gradient Flow. arXiv e-prints arXiv:2007.08591 (2020)

8. Derrida, B., Lebowitz, J.L., Speer, E.R.: Exact Free Energy Functional for a Driven Diffusive Open Stationary Nonequilibrium System. Physical Review Letters 89(3), 030601+ (2002). DOI 10.1103/PhysRevLett.89.030601

9. Duerinckx, M.: On the size of chaos via glauber calculus in the classical mean-field dynamics. Communications in Mathematical Physics 382(1), 613–653 (2021)

10. Duerinckx, M., Saint-Raymond, L.: Lenard-balescu correction to mean-field theory. Probability and Mathematical Physics 2(1), 27–69 (2021)

11. Feng, J., Kurtz, T.G.: Large deviations for stochastic processes. 131. American Mathematical Soc. (2006)
12. Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer-Verlag, 3dr ed. New York (2012)

13. Golse, F.: On the dynamics of large particle systems in the mean field limit. In: Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity, pp. 1–144. Springer (2016)

14. Kifer, Y.: Averaging principle for fully coupled dynamical systems and large deviations. Ergodic Theory and Dynamical Systems **24**(03), 847–871 (2004)

15. Lancellotti, C.: Time-asymptotic evolution of spatially uniform gaussian vlasov fluctuation fields. Journal of Statistical Physics **163**(4), 868–886 (2016)

16. Leblé, T., Serfaty, S.: Large deviation principle for empirical fields of log and riesz gases. Inventiones mathematicae **210**(3), 645–757 (2017)

17. Leblé, T., Serfaty, S.: Fluctuations of two dimensional coulomb gases. Geometric and Functional Analysis **28**(2), 443–508 (2018)

18. Lifshitz, E.M., Pitaevskii, L.P.: Physical kinetics. Course of theoretical physics, Oxford: Pergamon Press, 1981 (1981)

19. Mielke, A., Peletier, M.A., Renger, D.M.: On the relation between gradient flows and the large-deviation principle, with applications to markov chains and diffusion. Potential Analysis **41**(4), 1293–1327 (2014)

20. Morrison, P., Shadwick, B.: On the fluctuation spectrum of plasma. Communications in Nonlinear Science and Numerical Simulation **13**(1), 130–140 (2008)

21. Mouhot, C., Villani, C.: On Landau damping. Acta Mathematica **207**, 29–201 (2011)

22. Nicholson, D.: Introduction to plasma theory. Wiley, New-York (1983)

23. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation (2001)

24. Padilla-Garza, D.: Large Deviation Principle for local empirical measure of Coulomb gases at intermediate temperature regime. arXiv e-prints arXiv:2011.00480 (2020)

25. Paul, T., Pulvirenti, M., Simonella, S.: On the size of chaos in the mean field dynamics. Archive for Rational Mechanics and Analysis **231**(1), 285–317 (2019)

26. Rezakhanlou, F.: Large deviations from a kinetic limit. Ann. Probab. **26**(3), 1259–1340 (1998). DOI 10.1214/aop/1022855753. URL https://doi.org/10.1214/aop/1022855753

27. Schram, P.P.: Kinetic theory of gases and plasmas, vol. 46. Springer Science & Business Media (2012)

28. Serfaty, S.: Microscopic description of Log and Coulomb gases. arXiv e-prints arXiv:1709.04089 (2017)

29. Serfaty, S.: Gaussian Fluctuations and Free Energy Expansion for 2D and 3D Coulomb Gases at Any Temperature. arXiv e-prints arXiv:2003.11704 (2020)

30. Varadhan, S.S.: Large deviations and applications. SIAM (1984)

31. Velázquez, J.J., Winter, R.: The two-particle correlation function for systems with long-range interactions. Journal of Statistical Physics **173**(1), 1–41 (2018)

32. Veretennikov, A.Y.: On Large Deviations in the Averaging Principle for SDEs with a "Full Dependence". Annals of Probab. **27**, 284 (1999)

33. Villani, C.: Optimal transport: old and new, vol. 338. Springer Science & Business Media (2008)

34. Yamaguchi, Y.Y., Barré, J., Bouchet, F., Dauxois, T., Ruffo, S.: Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model. Physica A Statistical Mechanics and its Applications **337**, 36–66 (2004). DOI 10.1016/j.physa.2004.01.041