Master equations for Wigner functions with spontaneous collapse and their relation to thermodynamic irreversibility

Abstract

Wigner functions, allowing for a reformulation of quantum mechanics in phase space, are of central importance for the study of the quantum-classical transition. A full understanding of the quantum-classical transition, however, also requires an explanation for the absence of macroscopic superpositions to solve the quantum measurement problem. Stochastic reformulations of quantum mechanics based on spontaneous collapses of the wavefunction are a popular approach to this issue. In this article, we derive the dynamic equations for the four most important spontaneous collapse models—Ghirardi–Rimini–Weber (GRW) theory, continuous spontaneous localization (CSL) model, Diósi-Penrose model, and dissipative GRW model—in the Wigner framework. The resulting master equations are approximated by Fokker–Planck equations. Moreover, we use the phase-space form of GRW theory to test, via molecular dynamics simulations, David Albert’s suggestion that the stochasticity induced by spontaneous collapses is responsible for the emergence of thermodynamic irreversibility. The simulations show that, for initial conditions leading to anti-thermodynamic behavior in the classical case, GRW-type perturbations do not lead to thermodynamic behavior. Consequently, the GRW-based equilibration mechanism proposed by Albert is not observed.

Understanding probability and irreversibility in the Mori-Zwanzig projection operator formalism

“…Albert (1994a,b) argues that the GRW collapses will bring a system starting in an "abnormal" initial state 3 into a state that leads to a normal thermodynamic time evolution. However, te Vrugt et al (2021b) have found no such effect in their simulations.…”

“…Broadly speaking, there are (at least) three ways around this issue. The first one (not considered here, but in a certain variant refuted in te Vrugt et al (2021b)) is to argue that the microscopic dynamics of a closed system is not actually Hamiltonian. The second one is that external perturbations destroy correlations and thereby lead to an equilibrium form for ρ (interventionism, see Ridderbos and Redhead (1998)).…”

“…Evidently, this is also the case for the entropy (28). This view is plausible by the Gibbs paradox, which shows that whether a mixing entropy is to be assigned to a certain mixing process depends on whether we decide to view it as a change of the system's state (see the brief discussion in te Vrugt (2021b)).…”

“…( 24), we integrate from the initial time s forward in time to t, rather than backwards in time -the memory term is supposed to take into account knowledge about the previous states of the system (Grabert, 1982). However, it is certainly possible that a system exhibits anti-thermodynamic behavior (in fact, it can be observed in simulations using Hamiltonian microdynamics (te Vrugt et al, 2021b)), and that such a behavior is never observed requires an explanation that appeals to the initial state of the system. Whether a system exhibits anti-thermodynamic behavior is a feature of this system quite independent of our probabilistic reasoning about it, and we cannot explain the absence of such behavior by any argument that is related to our probabilistic reasoning.…”

Understanding probability and irreversibility in the Mori-Zwanzig projection operator formalism

“…Since GRW theory involves objective stochasticity, one would then have a straightforward explanation for the existence of objective probabilities in statistical mechanics. However, recent computer experiments by te Vrugt et al (2021b) indicate that Albert's suggestion is not successful as an explanation of thermodynamic irreversibility even if GRW theory is true. Albert (1994a, b) argues that the GRW collapses will bring a system starting in an "abnormal" initial state 3 into a state that leads to a normal thermodynamic time evolution.…”