Nonequilibrium dynamics of moving mirrors in quantum fields: Influence   functional and Langevin equation
  We employ the Schwinger-Keldysh formalism to study the nonequilibrium dynamics of the mirror with perfect reflection moving in a quantum field. Within the regime of linear response in terms of a first order expansion of the mirror's displacement, the coarse-grained effective action is obtained by integrating out quantum fields with the method of influence functional. The semiclassical Langevin equation is derived, and is found to involve two levels of backreaction effects on the dynamics of mirrors: radiation reaction induced by the motion of the mirror and backreaction dissipation arising from fluctuations in quantum fields via a fluctuation-dissipation relation. Although the theorem of fluctuation and dissipation in linear response theory is of model independence, the study from the first principles derivation shows that the obtained theorem is also {\it independent} of the regulators introduced to deal with short-distance divergences from quantum fields. Thus, when the method of regularization is introduced to compute the dissipation and fluctuation effects, this theorem must be fulfilled as the results are obtained by taking the short-distance limit in the end of calculations. The backreaction effects from vacuum fluctuations on moving mirrors are found to be hardly detected while those effects from thermal fluctuations may be detectable. 
Chun-Hsien Wu and Da-Shin Lee