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Fluctuations of local electric field and dipole moments in water between metal walls

We examine the thermal fluctuations of the local electric field $E_k^{\rm loc}$ and the dipole moment $μ_k$ in liquid water at $T=298$ K between metal walls in electric field applied in the perpendicular direction. We use analytic theory and molecular dynamics simulation. In this situation, there is a global electrostatic coupling between the surface charges on the walls and the polarization in the bulk. Then, the correlation function of the polarization density $p_z(r)$ along the applied field contains a homogeneous part inversely proportional to the cell volume $V$. Accounting for the long-range dipolar interaction, we derive the Kirkwood-Fr$\ddot{\rm{o}}$hlich formula for the polarization fluctuations when the specimen volume $v$ is much smaller than $V$. However, for not small $v/V$, the homogeneous part comes into play in dielectric relations. We also calculate the distribution of $E_k^{\rm loc}$ in applied field. As a unique feature of water, its magnitude $|E_k^{\rm loc}|$ obeys a Gaussian distribution with a large mean value $E_0 \cong 17~$V$/$nm, which arises mainly from the surrounding hydrogen-bonded molecules. Since $|μ_k|E_0\sim 30 k_{\rm B}T$, $μ_k$ becomes mostly parallel to $E_k^{\rm loc}$. As a result, the orientation distributions of these two vectors nearly coincide, assuming the classical exponential form. In dynamics, the component of $μ_k(t)$ parallel to $E_k^{\rm loc}(t)$ changes on the timescale of the hydrogen bonds $\sim 5$ ps, while its smaller perpendicular component undergoes librational motions on timescales of 0.01 ps.