Existing logit-based knowledge distillation methods typically employ singularly deterministic categorical distributions, which eliminates the inherent uncertainty in network predictions and thereby limiting the effective transfer of knowledge. To address this limitation, we introduce distribution-based probabilistic modeling as a more comprehensive representation of network knowledge. Specifically, we regard the categorical distribution as a random variable and leverage deep neural networks to predict its distribution, representing it as an evidential second-order distribution. Based on the second-oder modeling, we propose Evidential Knowledge Distillation (EKD) which distills both the expectation of the teacher distribution and the distribution itself into the student. The expectation captures the macroscopic characteristics of the distribution, while the distribution itself conveys microscopic information about the classification boundaries. Additionally, we theoretically demonstrate that EKD's distillation objective provides an upper bound on the expected risk of the student when the teacher’s predictions are treated as ground truth labels. Extensive experiments on several standard benchmarks across various teacher-student network pairs highlight the effectiveness and superior performance of EKD. Our code is available in the Supplementary Material.