Cosmic rays (CRs) have critical impacts in the multiphase interstellar medium (ISM), driving dynamical motions in low-density plasma and modifying the ionization state, temperature, and chemical composition of higher-density atomic and molecular gas. We present a study of CR propagation between the ionized ISM and a neutral cloud. Using one-dimensional magnetohydrodynamic particle-in-cell simulations which include ion-neutral drag to damp Alfv$acute{text{e}}$n waves in the cloud, we self-consistently evolve the kinetic physics of CRs and fluid dynamics of the multiphase gas. By introducing the cloud in our periodic domain, our simulations break translational symmetry and allow the emergence of spatial structure in the CR distribution function. A negative spatial gradient forms across the fully-ionized ISM region while a positive gradient forms across the neutral cloud. We connect our results with CR hydrodynamics formulations by computing the wave-particle scattering rates as predicted by quasilinear, fluid, and Fokker-Planck theory. For momenta where the mean free path is short relative to the box size, we find excellent agreement among all scattering rates. By exploring different cloud sizes and ion-neutral collision rates, we show that our results are robust. Our work provides a first-principles verification of CR hydrodynamics when particles stream down their pressure gradient, and opens a pathway toward comprehensive calibrations of transport coefficients from self-generated Alfv$acute{text{e}}$n wave scattering with CRs.
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