Time-Reversal Symmetry in Open Classical and Quantum Systems

Deriving an arrow of time from time-reversal symmetric microscopic dynamics is a fundamental open problem in physics. Here we focus on several derivations of dissipative dynamics and the thermodynamic arrow of time to study precisely how time-reversal symmetry is broken in open classical and quantum systems. These derivations all involve the Markov approximation applied to a system interacting with an infinite heat bath. We find that the Markov approximation does not imply a violation of time-reversal symmetry. Our results show instead that the time-reversal symmetry is maintained in standard dissipative equations of motion, such as the Langevin equation and the Fokker-Planck equation in open classical dynamics, and the Brownian motion, the Lindblad and the Pauli master equations in open quantum dynamics. In all cases, the resulting equations of motion describe thermalisation that occurs into the future as well as into the past. As a consequence, we argue that the resulting dynamics are better described by a definition of Markovianity that is symmetric with respect to the future and the past.Comment: 14 pages, 3 figure