Quantum stochastic dynamics. I.

ADAM W. MAJEWSKI

Yun‐An Yan, Jiushu Shao

Classical Brownian motion has well been investigated since the pioneering work of Einstein, which inspired mathematicians to lay the theoretical foundation of stochastic processes. A stochastic formulation for quantum dynamics of dissipative systems described by the system-plus-bath model has been developed and found many applications in chemical dynamics, spectroscopy, quantum transport, and other fields. This article provides a tutorial review of the stochastic formulation for quantum dissipative dynamics. The key idea is to decouple the interaction between the system and the bath by virtue of the Hubbard-Stratonovich transformation or Itô calculus so that the system and the bath are not directly entangled during evolution, rather they are correlated due to the complex white noises introduced. The influence of the bath on the system is thereby defined by an induced stochastic field, which leads to the stochastic Liouville equation for the system. The exact reduced density matrix can be calculated as the stochastic average in the presence of bath-induced fields. In general, the plain implementation of the stochastic formulation is only useful for short-time dynamics, but not efficient for long-time dynamics as the statistical errors go very fast. For linear and other specific systems, the stochastic Liouville equation is a good starting point to derive the master equation. For general systems with decomposable bath-induced processes, the hierarchical approach in the form of a set of deterministic equations of motion is derived based on the stochastic formulation and provides an effective means for simulating the dissipative dynamics. A combination of the stochastic simulation and the hierarchical approach is suggested to solve the zero-temperature dynamics of the spin-boson model. This scheme correctly describes the coherent-incoherent transition (Toulouse limit) at moderate dissipation and predicts a rate dynamics in the overdamped regime. Challenging problems such as the dynamical description of quantum phase transition (local- ization) and the numerical stability of the trace-conserving, nonlinear stochastic Liouville equation are outlined.

Piotr Garbaczewski

Yuri A. Rylov

The formalism of the particle dynamics in the space-time, where motion of free particles is primordially stochastic, is considered. The conventional dynamic formalism, obtained for the space-time, where the motion of free particles is primordially deterministic, seems to be unsuitable. The statistical ensemble of stochastic (or deterministic) systems is considered to be the main object of dynamics. At such a logical reloading the statistical description becomes a component of dynamics, and capacities of dynamics increase. At such an approach one can describe deterministic, stochastic and quantum particles by means of the uniform technique. The quantum particle is described as a stochastic particle, i.e. without a reference to the quantum principles. Besides, by means of this technique one can describe classical inviscid fluid. There are four different versions of the formalism: (1) description in Euler dynamic variables, (2) description in Lagrange dynamic variables, (3) description in terms of the generalized stream function, (4) description in terms of the wave function. The uniform formalism is purely dynamic. Even describing stochastic systems, it does not refer to probability and probabilistic structures. In relativistic case the uniform formalism can describe pair production and pair annihilation.

A Frigerio

Curtis Wade Vinson Curtis Vinson, Curtis Wade Vinson Curtis Vinson

We propose Stochastic Quantum Geometry (SQG) as a framework for emergent spacetime based on the interplay of quantum noise and discrete geometry. SQG models spacetime as an SU(2)-safe spin network evolving through stochastic continuous-time Markov dynamics, while matter degrees of freedom evolve under a completely-positive, trace-preserving Lindblad equation with rates derived from classical geometry noise. A local, bounded back-reaction couples matter fluctuations to geometry rewiring, producing co-evolutionary dynamics that remain mathematically consistent (CPTP, ergodic, non-explosive). Emergent signatures include modularity plateaus, a robust negative entropy–structure correlation, and discrete curvature–energy slopes resembling Einsteinian balance. A hub-dominated failure mode is corrected by a variant, SQG-H, which suppresses unstable long-range bridges through local harmony couplers. Unlike quantum graphity or adaptive network models, SQG introduces the unique triad of Lindblad-driven open quantum dynamics, SU(2)-safe stochastic geometry evolution, and bounded local back-reaction, a combination not present in prior literature. Keywords: stochastic quantum geometry, open quantum systems, spin networks (SU(2)), emergent spacetime, Lindblad dynamics