Current popular anomaly detection algorithms are capable of detecting global anomalies but oftentimes fail to distinguish local anomalies from normal instances. This paper aims to improve unsupervised anomaly detection via the exploration of physics-based diffusion space. Building upon the embedding manifold derived from diffusion maps, we devise Local Anomaly Descriptor (LAD) whose originality results from faithfully preserving intrinsic and informative density-relevant neighborhood information. This robust and effective algorithm is designed with a weighted umbrella Laplacian operator to bridge global and local properties. To further enhance the efficacy of our proposed algorithm, we explore the utility of anisotropic Gaussian kernel (AGK) which can offer better manifold-aware affinity information. Comprehensive experiments on both synthetic and UCI real datasets verify that our LAD outperforms existing anomaly detection algorithms.