On the emergence of classical stochasticity

TL;DR

The work probes how classical stochasticity can emerge from quantum dynamics governed by a Pauli-type master equation. It shows that decoherence alone does not guarantee definite intermediate states, but in the ultradecoherence limit coherences can be adiabatically eliminated to yield a symmetric, Markovian classical master equation for probabilities, with transition rates W_{μν} that can reproduce both Fermi Golden Rule and quantum Zeno regimes. The analysis extends to bosons and fermions, deriving their respective master equations with bosonic stimulation and Pauli exclusion, and includes pump/loss processes. In non-equilibrium transport, the framework reveals condensation-like behavior for bosons and blocking for fermions, and provides explicit mean-field, two-point, and first-arrival time results, including Monte-Carlo verifications. Overall, the paper clarifies the logical conditions under which classical stochastic reasoning applies to quantum dynamics and connects these results to decoherence theory and quantum-time problems, while outlining avenues for future work on memory effects and incomplete decoherence.

Abstract

We examine the logical structure of the emergence of classical stochasticity for a quantum system governed by a Pauli-type master equation. It is well-known that while such equations describe the evolution of probabilities, they do not automatically justify classical reasoning based on the assumption that the system exists in a definite state at intermediate times. On the other hand, we show that this assumption is crucial for the standard calculation of stochastic times such as the persistent time and the time of first arrivals. We then consider examples of single particles, bosons, and fermions in the so-called ultradecoherence limit to illustrate how classical stochasticity may emerge from quantum mechanics.

Figures (3)

• Figure 1: Schematic description of the logical structure of the emergence of the classical stochasticity. The decoherence theory for the quantum-to-classical transition indicates that the Pauli master equation does not imply the notion of system existing in definite states, leaving the picture of classical stochasticity incomplete. On the other hand, assuming that the system accommodates definite states, the Pauli master equation can be obtained by the application of the Fermi Golden Rule. Generally, the standard derivations of stochastic times assume not only the Pauli master equation, but also that system accommodates definite states at intermediate time points.
• Figure 2: Sketch of the one-dimensional lattice we consider with $L+1$ sites. Particles are injected with rate $\eta$ at the first site $\mu=0$ and absorbed with rate $\theta$ at the last site $\mu = L$. The hopping rate $\Gamma$ is symmetric and uniform along the lattice.
• Figure 3: (a) Mean occupation numbers of particles at each lattice site from Monte Carlo simulations (points) and analytical results (lines). Parameters: $L+1=10$ sites, $\Gamma=1$; $\eta=\theta=0.2$ for fermions (red); $\eta=0.01$ and $\theta =0.02$ for bosons (blue), as an extremely low gain is required for a stationary state to exist, see eq. \ref{['eqbosonlim']}. In simulation, the population is counted in the stationary regime and averaged over $\approx 10^5$ iterations. (b) Results of Monte-Carlo simulations of the probability distribution of arrival times for fermions (red) and bosons (blue) in the low-gain regime $\eta/\Gamma=0.01$. The analytical expression is plotted as a green dashed line. (c,d) Results of Monte-Carlo simulations of the probability distribution of arrival times with different gain values ($\eta/\Gamma=0.01,0.1,0.5,1,2$) for fermions (c), bosons (d). Note that the scales vary between panels.