We investigate Brownian motions of a particle coupled to vacuum fluctuationsof a quantum field. The Unruh effect predicts that an observer in anaccelerated motion sees the Minkowski vacuum as thermally excited. Thisaddresses the problem of whether a thermal property appears in a perturbativerandom motion of a particle in an accelerated motion due to the coupling ornot. We revisit this problem by solving the equation of motion of a particlecoupled to vacuum fluctuations including the radiation reaction force. Wecompute a Fourier integral for the variance of the random velocity in arigorous manner. Similarly, we consider a particle coupled to vacuumfluctuations in de Sitter spacetime motivated by the argument that an observerin de Sitter spacetime sees the Bunch-Davies vacuum as a thermally excitedstate with the Gibbons-Hawking temperature. Our investigation clarifies thecondition that the energy equipartition relation arises in the Brownian motionsof a particle.
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