All examples seen in the preceding chapters have dealt with a single particle. In this chapter, the theory is expanded to systems with several identical particles. Here, ‘many’ in practice means two. However, this does allow the introduction of several central aspects. Perhaps the most important one is spin, which is the topic of the first part. Central elements in this context are the Stern–Gerlach experiment and the Pauli matrices. The characteristics of these matrices are studied in some detail as they play crucial roles in the remainder of the book. The concept of entanglement in quantum physics is introduced – exemplified using both the two-particle spin wave function and the combined spin–space wave function for a single particle. Due to the Pauli principle, the importance of spin and exchange symmetry in a many-body context is hard to underestimate. The fact that identical particles are indistinguishable has implications for the symmetry of the wave function. This, in turn, has significant consequences for the structure of the system – including its ground state. This is investigated by performing calculations of energy estimates. Most of these apply the variational principle, but also the notion of self-consistent field and the Hartree–Fock method are introduced.

The multivariate fluctuation relation is established for the full counting statistics of the energy and particle fluxes across an open quantum system in contact with several reservoirs on the basis of microreversibility The quantum version of the nonequilibrium work fluctuation relation is recovered in the presence of a single reservoir. In the long-time limit, the time-reversal symmetry relation is expressed in terms of the cumulative generating function for the full counting statistics. In systems with independent particles, the symmetry relation can be obtained in the scattering approach for the transport of bosons and fermions. The temporal disorder and its time asymmetry can be characterized by the quantum version of the entropy and coentropy per unit time. Their difference gives the thermodynamic entropy production rate. Furthermore, the stochastic approach is also considered for electron transport in quantum dots, quantum point contacts, and single-electron transistors.