This paper presents a file replication scheme with analogy to thermal diffusion for storage load balancing in unstructured peer-to-peer (P2P) file sharing networks. The proposed scheme is intended to balance storage load among peers in a dynamic, distributed, and autonomous manner, as in thermal diffusion. Theoretical analysis results show that the presented scheme actually has a statistical analogy with a thermal diffusion equation. In addition, the proposed scheme includes parameters to widely explore the trade-off between storage load balancing and search performance existing in unstructured P2P file sharing networks. Simulation results show that compared to other replication schemes, the proposed scheme has better ability not only in balancing storage load among peers, which is the primary objective of the proposal, but also in widely exploring the performance trade-off.