Irreversible Work and Inner Friction in Quantum Thermodynamic Processes

We discuss the thermodynamics of closed quantum systems driven out of equilibrium by a change in a control parameter and undergoing a unitary process. We compare the work actually done on the system with the one that would be performed along ideal adiabatic and isothermal transformations. The comparison with the latter leads to the introduction of irreversible work, while that with the former leads to the introduction of inner friction. We show that these two quantities can be treated on an equal footing, as both can be linked with the heat exchanged in thermalization processes and both can be expressed as relative entropies. Furthermore, we show that a specific fluctuation relation for the entropy production associated with the inner friction exists, which allows the inner friction to be written in terms of its cumulants.
All the authors acknowledge support from COST MP1209 Action. F. P., G. F., and N. L. G. acknowledge insightful discussions with Michele Campisi, John Goold, and Mauro Paternostro. T. J. G. A. is supported by the European Commission, the European Social Fund, and the Region Calabria through the program POR Calabria FSE 2007-2013-Asse IV Capitale Umano-Obiettivo Operativo M2. F. G. and R. Z. acknowledge MINECO, CSIC, the EU commission, UIB and FEDER funding under Grants FIS2011-23526 (TIQS), postdoctoral JAE program (ESF) and Invited professors program.