  We examine the thermal fluctuations of the local electric field $E_k^{\rm
loc}$ and the dipole moment $\mu_k$ 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
$|\mu_k|E_0\sim 30 k_{\rm B}T$, $\mu_k$ 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 $\mu_k(t)$ 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.

Soft Condensed Matter;Statistical Mechanics