Non-coherent evolution of closed weakly interacting system leads to equidistribution of probabilities of microstates

TL;DR

The paper addresses how irreversibility and the arrow of time emerge in a closed quantum system by introducing non-coherent evolution, which converts unitary dynamics into a Markovian process via phase averaging from a finite spectral width. The resulting framework yields a symmetric generator $\mathcal{Q}$ for a continuous-time Markov chain, with the probability vector evolving as $\mathcal{P}(t) = \exp(\mathcal{Q} t)\mathcal{P}(0)$ and transition rates given by $Q_{kl} = \frac{2\pi}{\hbar} |V_{kl}|^2 \delta(E_k - E_l)$, leading to irreversibility and equiprobability of microstates in the long-time limit. By projecting to one-particle distributions, the theory derives the Boltzmann collision integral for two- and three-phonon processes and shows that stationary points and phase averaging yield Fermi-Dirac or Bose-Einstein statistics, respectively. Overall, the approach provides a quantum-mechanical route to Gibbs equilibrium and kinetic theory without appealing to an external heat bath, with potential tests in optical and nanoscale transport systems.

Abstract

We introduce a concept of non-coherent evolution of macroscopic quantum systems. We show that for weakly interacting systems such evolution is a Markovian stochastic process. The transition rates between system states, which characterize the process, are determined by Fermi's golden rule. Such evolution is time-irreversible and leads to the equidistribution of probabilities across every state of the system. Furthermore, we investigate the time dependence of the mean numbers of particles in single-particle states and find that, under the given assumptions, it is governed by the Boltzmann collision integral. The proposed mechanism that transforms time-reversible unitary evolution into time-irreversible stochastic evolution is non-coherence. In the presented theory, the non-coherence is not associated with interaction with a heat bath, but rather with the finite spectral width of quantum states. This understanding of non-coherence is analogous to the one used in wave optics. Thus, we present a novel approach to the famous arrow of time problem.

Figures (5)

• Figure 1: Two beams of light are incident on the interface and are transformed either coherently or non-coherently, depending on the conditions. The interface can be, for example, between air and optical glass (shown in light blue). The gray rectangles represent mirrors. In subfigures (a) and (b), two beams are emitted by the same source and separated by the interface. The initial state of the system is defined by the amplitudes after separation. The column of initial amplitudes is denoted as $\mathcal{A}_i$, and the column of initial intensities as $\mathcal{P}_i$. If the difference between the optical paths of the two split beams is smaller than the coherence length of the light, $\Delta l < l_{\text{coh}}$, a coherent transformation occurs according to formula \ref{['Gen1']}, as depicted in subfigure (a). This transformation is reversible, and with appropriate phase differences between the beams, they can recombine into a single beam, meaning the splitting transformation has been reverted. If the difference between the optical paths exceeds the coherence length of the light, $\Delta l > l_{\text{coh}}$, a non-coherent transformation occurs according to formula \ref{['T1']}. The resulting intensities no longer depend on the phase difference and are strictly greater than zero, which means the split beams cannot merge back into a single beam.
• Figure 2: As in Fig. \ref{['non-coherence-fig1']}, two beams of light are incident on the interface and are transformed either coherently or non-coherently, depending on the conditions. We present a qualitative scheme for interpreting the experiment in terms of photon packets. Subfigure (a) shows a beam of light divided into segments of equal length—the coherence length $l_{\text{coh}}$. Each segment roughly corresponds to a single packet of photons, and the segments are numbered sequentially from left to right. In subfigure (b), the left end of packet number one is shown crossing the mirror. Since the difference between the optical paths of the two split parts of the beam is smaller than the coherence length, $\Delta l < l_{\text{coh}}$, the state interacts with itself, and the interaction is coherent. In particular, it is packet number one interacting with itself. In subfigure (c), the difference between the optical paths exceeds the coherence length, $\Delta l > l_{\text{coh}}$, and the state interacts with another state, resulting in non-coherent behavior. Specifically, it is state number $n$ interacting with state number $m$, with $n \neq m$.
• Figure 3: The diagram illustrates the characteristic scales of the system. The coherence length $l_{\text{coh}}$ is much smaller than the size of the system. For example, we depict the interaction of two bosonic particles resulting in their merging into a single bosonic particle. The regions of the system from which the two particles originate are separated by a distance much greater than the coherence length, and their interaction is therefore non-coherent.
• Figure 4: The diagram illustrates the characteristic scales of transitions in a non-coherent regime. To make the diagram easier to understand, quantities that are an order of magnitude greater than others are shown as only slightly greater. Both the time scale and energy scale are depicted, as some proportions are more intuitive on the timescale, while others are more intuitive on the energy scale. The quantities on the time scale and energy scale are related by: $T_k = \frac{\hbar}{E_k}$, $t_{\text{coh}} = \frac{1}{\omega_{\text{idth}}}$, $\Delta t = \frac{\pi\hbar}{\Delta E}$, $T_A = \frac{\hbar}{V_M}$. (a) shows a microstate $|k\rangle$ and the probability amplitude of the system to be in such a microstate, $A$. It demonstrates that the characteristic scale of the amplitude evolution, $T_A$, is much higher than the working scale, $\Delta t$, which is much higher than the wave period of a state, $T_k$. Therefore, we have: $T_k \ll \Delta t \ll T_A$. (b) depicts a sheaf of states (represented as an oval), with a frequency width $\omega_{\text{idth}}$, containing a single state $|k\rangle$ in the first sheaf and a group of states in the second sheaf, indexed by $l$, such that the energy difference between them and the initial state is less than $\Delta E$. The width of the sheaf is much greater than the working broadening, $\Delta E$. It is evident that the number of states to which a transition can occur from the initial state $|k\rangle$ is proportional to the broadening $\Delta E$.
• Figure 5: The diagram shows how the Boltzmann equation and the Gibbs microcanonical ensemble are deduced from the equation of non-coherent evolution (Eq. \ref{['TransitionRates']}). In rectangles, the names of equations are indicated, and arrows represent the mathematical procedures used to derive a formula from a more general one. $t \to \infty$ denotes taking the stationary point of the equation, and $\mathbb{P} \to \mathbb{E}$ indicates averaging -- that is, the transition from probabilities to average (expected) values. The diagram is commutative in the sense that taking the stationary point of Eq. \ref{['TransitionRates']} and then averaging, or averaging first and then taking the stationary point, both lead to well-known single-particle distribution functions (Fermi-Dirac or Bose-Einstein distributions).