Failures of defining Effective temperature in an active colloidal system

Active colloidal systems contain active particles, which convert energy from the environment into directed or persistent motion. Effective temperature, defined from active diffusion, fluctuation, and sedimentation profile, is used sometimes for quantifying the activity of active particles. However, non-thermal fluctuations were found to lead to failures when using effective temperature under some conditions, for examples, at short times and in strong confinement. The reason and conditions under which would the failure appear is not clear even for a single active particle.
We investigate from experimentation and numerical simulation the Effective temperature in a quadratic potential. We found the maximum probability distribution in histograms of position is at a characteristic length away from the center, which can’t be described with T_eff and Boltzmann distribution. The average potential energies were not proportional to the effective temperatures. We also found that the fluctuation power spectral density (PSD) plots have two characteristic frequencies, which can’t be explained with T_eff and fluctuation-dissipation theorem. Our experiments and numerical simulation provide how the effective temperature fails to describe the system.