Feynman rules in statistical mechanics

Paolo Amore, Francisco Marcelo Fernández

Yoel Tikochinsky

John Edward Gough

We introduce a Feynman diagram calculus for quantum stochastic fields and show that from the weak coupling limit of a Bose quantum field reservoir in a Gaussian state one obtains a quantum stochastic field process which has extended Wigner statistics. That is, we calculate explicitly the moments using the diagram rules and show that only moments corresponding to non-crossing diagrams are non-trivial. We further give the physical mechanism underlying this fact; namely the combination of momentum conservation (arising from the correct treatment of the full responsive interaction) and the mass-shell energy conservation for the virtual (stochastic) reservoir quanta. In our analysis we introduce a volume cut-off initially for the system particle to discretize its energy spectrum and then take the weak coupling limit. The individual noise fields associated with energy state transitions were Gaussian however, when the volume cut-off is then removed and we recover the noise correlations. Thus for this problem the removal of the cut-off and the weak coupling limit are in fact interchangeable.

T. Kita

In this article, we present a concise and self-contained introduction to nonequilibrium statistical mechanics with quantum field theory by considering an ensemble of interacting identical bosons or fermions as an example. Readers are assumed to be familiar with the Matsubara formalism of equilibrium statistical mechanics such as Feynman diagrams, the proper self-energy, and Dyson's equation. The aims are threefold: (i) to explain the fundamentals of nonequilibrium quantum field theory as simple as possible on the basis of the knowledge of the equilibrium counterpart; (ii) to elucidate the hierarchy in describing nonequilibrium systems from Dyson's equation on the Keldysh contour to the Navier-Stokes equation in fluid mechanics via quantum transport equations and the Boltzmann equation; (iii) to derive an expression of nonequilibrium entropy that evolves with time. In stage (i), we introduce nonequilibrium Green's function and the self-energy uniquely on the round-trip Keldysh contour, thereby avoiding possible confusions that may arise from defining multiple Green's functions at the very beginning. We try to present the Feynman rules for the perturbation expansion as simple as possible. In particular, we focus on the self-consistent perturbation expansion with the Luttinger-Ward thermodynamic functional, i.e., Baym's Phi-derivable approximation, which has a crucial property for nonequilibrium systems of obeying various conservation laws automatically. We also show how the two-particle correlations can be calculated within the Phi-derivable approximation, i.e., an issue of how to handle the "Bogoliubov-Born-Green-Kirkwood-Yvons (BBGKY) hierarchy".

Sergei Vladimirovich Tyablikov

G. E. Uhlenbeck