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Bulletin of the American Physical Society

Convex Decompositions of Thermal Equilibrium for Non-interacting Non-relativistic Particles

Aurélia Chenu, Agata M. Brańczyk, John E. Sipe

Published May 26, 2016• 0 citations
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Abstract

We provide convex decompositions of thermal equilibrium for non-interacting non-relativistic particles in terms of localized wave packets. These quantum representations offer new calculation tools and insights as they help relate to the classical picture.

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Convex Decompositions of Thermal Equilibrium for Non-interacting Non-relativistic Particles

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Journal of Physics A Mathematical and Theoretical•Apr 23, 2010•14 citations
The classical and quantum representations of thermal equilibrium are strikingly different, even for free, non-interacting particles. While the first involves particles with well-defined positions and momenta, the second usually involves energy eigenstates that are delocalized over a confining volume. In this paper, we derive convex decompositions of the density operator for non-interacting, non-relativistic particles in thermal equilibrium that allow for a connection between these two descriptions. Associated with each element of the decomposition of the N-particle thermal state is an N-body wave function, described as a set of wave packets; the distribution of the average positions and momenta of the wave packets can be linked to the classical description of thermal equilibrium, while the different amplitudes in the wave function capture the statistics relevant for fermions or bosons.
Within the framework of relativistic quantum field theory, a novel method is established which allows to distinguish non-equilibrium states admitting locally a thermodynamic interpretation. The basic idea is to compare these states with global equilibrium states (KMS states) by means of local thermal observables. With the help of such observables, the states can be ordered into classes of increasing local thermal stability. Moreover, it is possible to identify states exhibiting certain specific thermal properties of interest, such as a definite local temperature or entropy density. The method is illustrated in a simple model describing the spatio-temporal evolution of a “big heat bang”. 1
Non-equilibrium evolution at absolute temperature T and approach to equilibrium of statistical systems in long-time (t) approximations, using both hierarchies and functional integrals, are reviewed. A classical non-relativistic particle in one spatial dimension, subject to a potential and a heat bath (hb), is described by the non-equilibrium reversible Liouville distribution (W) and equation, with a suitable initial condition. The Boltzmann equilibrium distribution Weq generates orthogonal (Hermite) polynomials Hn in momenta. Suitable moments Wn of W (using the Hn’s) yield a non-equilibrium three-term hierarchy (different from the standard Bogoliubov–Born–Green–Kirkwood–Yvon one), solved through operator continued fractions. After a long-t approximation, the Wn’s yield irreversibly approach to equilibrium. The approach is extended (without hb) to: (i) a non-equilibrium system of N classical non-relativistic particles interacting through repulsive short range potentials and (ii) a classical ϕ4 field theory (without hb). The extension to one non-relativistic quantum particle (with hb) employs the non-equilibrium Wigner function (WQ): difficulties related to non-positivity of WQ are bypassed so as to formulate approximately approach to equilibrium. A non-equilibrium quantum anharmonic oscillator is analyzed differently, through functional integral methods. The latter allows an extension to relativistic quantum ϕ4 field theory (a meson gas off-equilibrium, without hb), facing ultraviolet divergences and renormalization. Genuine simplifications of quantum ϕ4 theory at high T and large distances and long t occur; then, through a new argument for the field-theoretic case, the theory can be approximated by a classical ϕ4 one, yielding an approach to equilibrium.
A general representation of the thermal quantum state is given for any interacting quantum system such as fermi, bose and quantum spin systems. A fundamental equation of quantum non-equilibrium systems is obtained on the basis of the time-dependent thermal quantum state ∣Ψ(t)⟩|\varPsi(t)\rangle∣Ψ(t)⟩.
Numerical computations (Valentini and Westman 2005 Proc. R. Soc. A 461 253–72) demonstrate that an initially arbitrary particle density, stirred by the field of de Broglie velocity associated with the Schrödinger wavefunction for a sum of energy eigenstates, relaxes to the quantum thermal equilibrium that is the Born probability density, provided the particle density is coarse grained. The results are explained here by a Lagrangian or trajectory analysis, in terms of the relative dispersion of passive particles in a turbulent fluid. The analysis assumes that the turbulence statistics are stationary and isotropic, although these assumptions may be weakened. The relaxation to equilibrium is not reversible, owing to the coarse graining of the particle density and to the statistical inevitability of particle separation. There is no effective stirring toward equilibrium in very simple quantum systems such as a Gaussian wave packet or an energy eigenstate. However, it is argued that relaxation takes place during the emission of the packet or the establishment of the eigenstate, owing to stirring by the transients in the wavefunction for the entire system. The Lagrangian analysis is readily extended to nonrelativistic many-particle systems and to relativistic single-particle systems.