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A model detector undergoing constant, infinite-duration acceleration converges to an equilibrium state described by the Hawking-Unruh temperature $T_a=(a/2π)(\hbar/c)$. To relate this prediction to experimental observables, a point-like charged particle, such as an electron, is considered in place of the model detector. Instead of the detector's internal degree of freedom, the electron's low-momentum fluctuations in the plane transverse to the acceleration provide a degree of freedom and observables which are compatible with the symmetry and thermalize by interaction with the radiation field. General arguments in the accelerated frame suggest thermalization and a fluctuation-dissipation relation but leave underdetermined the magnitude of either the fluctuation or the dissipation. Lab frame analysis reproduces the radiation losses, described by the classical Lorentz-Abraham-Dirac equation, and reveals a classical stochastic force. We derive the fluctuation-dissipation relation between the radiation losses and stochastic force as well as equipartitation $\langle p_\perp^2\rangle = 2mT_a$ from classical electrodynamics alone. The derivation uses only straightforward statistical definitions to obtain the dissipation and fluctuation dynamics. Since high accelerations are necessary for these dynamics to become important, we compare classical results for the relaxation and diffusion times to strong-field quantum electrodynamics results. We find that experimental realization will require development of more precise observables. Even wakefield accelerators, which offer the largest linear accelerations available in the lab, will require improvement over current technology as well as high statistics to distinguish an effect.