Abstract
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.