During nearly 200 years of development in the knowledge of Brownian motion, the Janus sphere, as a typical Brownian particle with special surface properties, has been widely studied in the past few decades. A standard Janus sphere possesses two distinct surfaces. These two surfaces elicit different hydrodynamic interactions with ambient fluids or other interactions in response to environmental stimuli, such as chemical gradients, magnetic fields, and even light. The diffusion of Janus spheres, particularly when controlled by a remotely applied field, has inspired various applications, ranging from the design of micro-swimmers and novel procedures for probing the mechanical properties of suspensions to the fabrication of composites with enhanced performance. In this work, we report a systematic analysis of field-controlled diffusion of Janus spheres. Commencing with stochastic differential equations of motion at the microscale, we derive a coarse-grained Fokker-Planck equation at the macroscale, describing the evolution of the probability distribution function of the Janus sphere in terms of its position and orientation. Leveraging the concept of the hydrodynamic center, we derive, for the first time, explicit generalized Stokes-Einstein relations for long-time effective diffusivity, incorporating the effects of both the surface discontinuity of the Janus sphere and the external fields. The formulae enable predictions of the effective diffusivity as it varies with the slip length and characteristic angle of Janus spheres, and reveal the impact of an aligning potential field on the diffusion coefficients both parallel and perpendicular to the direction of the field. This work not only deepens the understanding of field-controlled diffusion of Janus particles, but also holds a meaningful impact on the future applications in microfluidics and related fields.
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