Nonequilibrium quantum anharmonic oscillator and scalar field: high temperature approximations

Abstract

We treat a relativistic quantum boson gas, described by a scalar quantum field, with quartic self-interaction (φ 4 ) in three spatial dimensions: we review the known equilibrium case and present new proposals offequilibrium. For high temperature and large spatial scales, the behaviour of the gas at equilibrium simplifies nonperturbatively (equilibrium dimensional reduction or EDR): its thermodynamics is described by classical statistical mechanics with some quantum field effects. By assumption, the initial state of the gas offequilibrium includes interactions and inhomogeneities and is not far from thermal equilibrium. We employ real-time generating functionals and obtain the free nonequilibrium correlators at non-zero temperature. The nonequilibrium quantum gas appears to simplify nonperturbatively in the regime of high temperature and large temporal and spatial scales (nonequilibrium dimensional reduction or NEDR), its dynamics being described by classical statistical mechanics with some quantum field effects. We outline the renormalization of the φ 4 theory, the nonequilibrium statistical mechanics of a quantum anharmonic oscillator and the high temperature simplifications, all of which provide very useful hints for NEDR in the field case. Our main proposals are NEDR and the associated new (renormalized) real-time nonequilibrium generating functionals for the φ 4 theory.

Classical and Quantum Models in Non-Equilibrium Statistical Mechanics: Moment Methods and Long-Time Approximations

“…We shall review in outline items (a)-(c), analyzed in our previous work [14][15][16][17][18][19]. We shall study further convergence aspects for issues (b) and (c), which, even if shortly, will provide some additional partial clarification, so far unpublished.…”

Non-Equilibrium Liouville and Wigner Equations: Moment Methods and Long-Time Approximations

“…Notice the key fact that in Equations (18) and (19), σ only appears as s + (n/σ), so that + (n/σ) > 0 (for s = > 0 and n = 0, 1, 2, ...).…”

“…We shall adopt the following standpoint, certainly different from those in [11][12][13][14] and bearing connections to and, simultaneously, differences from those in [15][16][17][18][19][20][21][22]. As a direct analysis of large systems is very difficult, we shall start with an open small statistical system, subject to an hb at thermal equilibrium at absolute temperature, T ( = 0), but with negligible external dissipation, due to the hb.…”

“…(c) Could it be extended to quantum systems by means of Wigner functions [6], at least approximately or in some regime? The present work, which will extend, non-trivially, [15][16][17][18][19][20][21][22], is devoted to study questions (a), (b) and (c).…”