Title: Time-reversal symmetry relations for currents in nonequilibrium stochastic and quantum systems

Abstract: The purpose of this review is to present the new advances in nonequilibrium
statistical mechanics based on the time-reversal symmetry relations of large-deviation type
for the transport properties flowing through nonequilibrium systems.
Starting from quantum mechanics, master equations are introduced which rule
the stochastic processes taking place in open systems such as mesoscopic electronic circuits.
The counting statistics of quantum electron transport is presented
and the fluctuation theorem is established from microreversibility
for the different fluctuating currents flowing across the open system.
The Onsager reciprocity relations as well as their generalizations to
the nonlinear response properties are deduced from the current fluctuation theorem.
The fluctuation theorem is obtained using symmetry relations of the time
evolution operator.  The link to similar relationships derived in the context
of the quantum scattering approach is discussed.

Keywords: Microreversibility, Liouville equation, master equation,
phase-space probability distribution, second law of thermodynamics, entropy production,
detailed balance, Onsager-Casimir reciprocity relations, nonlinear response theory,
nonequiibrium steady state, affinity, fluctuation theorem for currents,
quantum dot, quantum point contact, counting statistics,
Levitov-Lesovik formula, Landauer-BŁttiker formula

Table of content (tentative):
- Microreversibility without and with an external magnetic field
- Time-dependent quantum systems
- Open quantum systems between several reservoirs
- Equilibrium and nonequilibrium systems: detailed balance and its breaking
- Heat and particle fluctuating current in small systems
- Coupling between several currents and energy transduction
- Transient fluctuation theorem for currents and work relation in quantum systems [1,2]
- From the transient to the steady-state fluctuation theorem for currents in quantum systems [2]
- Full counting statistics and the steady-state fluctuation theorem for currents in quantum systems [2]
- Connection to the scattering approach and the Levitov-Lesovik formula for counting statistics
- Connection to the Keldysh approach
- Connection to the master-equation approach [3]
- The fluctuation theorem for currents and response theory [1,2,3,4]
- Linear and nonlinear response coefficients [2,3,4]
- Generalizations of Green-Kubo and Casimir-Onsager reciprocity relations [2,3,4]
- Application to electronic transport in coherent quantum conductors
- Application to electronic transport in quantum dots and quantum point contacts [5]

[1] D. Andrieux and P. Gaspard, Phys. Rev. Lett. 100 (2008) 230404.
[2] D. Andrieux, P. Gaspard, T. Monnai, and S. Tasaki, New J. Phys. 11 (2009) 043014.
[3] D. Andrieux and P. Gaspard, J. Stat. Mech. (2006) P01011.
[4] D. Andrieux and P. Gaspard, J. Stat. Mech. (2007) P02006.
[5] G. Bulnes Cuetara, M. Esposito, and P. Gaspard, Phys. Rev. B 84 (2011) 165114.