Extended Mean-Field Theory: Reversible and Irreversible Quantum Evolution of Trapped Gases

R. Walser, J. Cooper, Murray Holland

We have formulated a kinetic theory for a condensed atomic gas in a trap, i.e., a generalized Gross-Pitaevskii equation, as well as a quantum-Boltzmann equation for the normal and anomalous fluctuations [R. Walser et al., Phys. Rev. A 59, 3878 (1999)]. In this paper, the theory is applied to the case of an isotropic configuration and we present numerical and analytical results for the reversible real-time propagation, as well as irreversible evolution towards equilibrium.

Henk T. C. Stoof

Dieter Schuch, Kyu -Myung Chung

Abstract The aim of our work was to find an unambiguous connection between irreversible macroscopic dynamics and reversible microdynamics that makes it possible to manifest irreversibility on a submacroscopic level without the use of coarse graining arguments. On this level the state of a physical system can be described by a field amplitude Ψ and the time evolution of this system is determined by a field equation for Ψ. For conservative systems, this field equation is formally identical with the linear Schrödinger equation, which can be constructed with the help of the classic Hamiltonian function for the corresponding problem. Regarding irreversible phenomena like damping due to a frictional force, for those dissipative systems no classic Hamiltonian function exists. Therefore the corresponding field equation cannot be obtained in the usual way. Nevertheless, also for dissipative systems it is possible to obtain a field equation in an unambiguous way using only Newton's form of classic mechanics. The result of our method is a nonlinear Schrödinger‐type field equation with a logarithmic nonlinearity. We discuss in more detail the properties of our logarithmic nonlinearity that corresponds to a macroscopic frictional force in a unique way. A figurative interpretation in terms of environment and interaction can be given. From a more formal point of view, the compatibility of our nonlinear operator with principles known from the theory of linear operators is investigated. One of the surprising results is the fact that although our nonlinear Hamiltonian HNL is “Hermitean” in the usual sense, in contrast to the linear theory an operator exp(i · HNL) is not unitary. Furthermore, in our theory the time‐derivative of the mean value of an operator is not only essentially determined by (the mean value of) its commutator with the Hamiltonian. There also occurs an additional term that causes irreversibility of the evolution and is connected with the feature of our theory that (in general) time derivative and construction of the mean value are no longer commuting operations. This fact shows some similarity with coarse graining theories, but in our theory the reason can be traced back unambiguously to an irreversible physical phenomenon.

R. Jackiw

Lyoubi-Idrissi, Abderrahim

This paper introduces the Entropic I-Field Theory, a fundamental framework that establishes irreversibility as a physical principle rather than an emergent property. We propose a universal scalar field—the I-field—that couples dissipatively to all matter and radiation. By breaking time-reversal symmetry at the fundamental level through an extended Euler-Lagrange-Rayleigh variational principle, the theory preserves causality and conservation laws while introducing intrinsic directionality to physical evolution.The framework replaces purely unitary quantum evolution with non-conservative dynamics in which the I-field coupling introduces dissipative terms into the equations of motion. Through numerical validation across quantum, chaotic, and thermodynamic systems, we demonstrate that the theory provides unified solutions to foundational paradoxes: the quantum measurement problem is resolved through objective I-field-mediated collapse, thermodynamic irreversibility arises from fundamental symmetry breaking rather than statistical coarse-graining, and the arrow of time emerges from field dynamics rather than initial conditions. The theory makes testable predictions accessible with current technology: temperatureindependent decoherence at ultralow temperatures, universal noise floors violatingfluctuation-dissipation relations, modified atomic energy levels proportional to I-field strength, and measurable time variation of fundamental constants. These predictionsdistinguish the framework from standard physics and provide experimental pathways for falsification. If confirmed, this would establish irreversibility as a fundamental interaction encoded in the Lagrangian rather than an emergent statistical phenomenon.

László Erdős, Benjamin Schlein