Repeated Interaction Models in Open Systems

• Repeated interaction models are frameworks describing sequential interactions between a primary system and probe subsystems that lead to relaxation, dissipation, and decoherence.
• They employ unitary (quantum) or Hamiltonian (classical) dynamics to generate discrete-time evolutions, often modeled by CPTP maps or Markov processes.
• These models are pivotal in quantum thermodynamics and control, offering concrete insights into nonequilibrium steady states, ergodic behavior, and spectral properties.

A repeated interaction model describes the sequential interactions of a primary dynamical system with an ordered collection of secondary subsystems ("probes" or "ancillas"), leading to emergent system dynamics that can exhibit relaxation, dissipation, decoherence, steady-state formation, and stochastic behavior. This structure has become foundational for the analysis of both quantum and classical open systems, underpinning modern developments in statistical mechanics, quantum information, ergodic theory, and the mathematics of dynamical semigroups.

1. General Structure and Paradigms

A repeated interaction (RI) system consists of a primary system $\mathcal{S}$ (finite-dimensional in most rigorous studies) and a sequence of subsystems $\mathcal{P}_1, \mathcal{P}_2, \ldots$ (e.g., "probes," "pieces," or "environmental ancillas"), which are initialized independently and interact with $\mathcal{S}$ for a fixed or variable duration. Each interaction is governed by a unitary evolution (quantum case) or a symplectic/Hamiltonian transformation (classical case), usually under a Hamiltonian of the form

$$H_{\text{tot}} = H_\mathcal{S} \otimes 1_\mathcal{P} + 1_\mathcal{S} \otimes H_\mathcal{P} + V,$$

where $V$ is the interaction term, applied for a time $\tau$:

$U(\tau) = \exp(-i\tau H_{\text{tot}}).$

At each step, $\mathcal{S}$ couples to a "fresh" probe, and after interaction the probe is discarded (traced out) and the system potentially interacts with the next probe. The environmental probes may be chosen i.i.d., via a Markov process, or with stronger temporal correlations depending on the modeling scenario (Ekblad et al., 16 Jun 2024).

This construction yields a discrete-time dynamical process or a product of dynamical maps, for instance,

$$\rho_\mathcal{S}^{(n+1)} = \operatorname{Tr}_\mathcal{P}\left[ U(\tau)\left(\rho_\mathcal{S}^{(n)} \otimes \rho_\mathcal{P}\right)U(\tau)^*\right].$$

In classical mechanics, analogous constructions are used, yielding Markov chains on the system's phase space; in appropriate limits, diffusion-type stochastic differential equations (e.g., Langevin equations) can arise (Deschamps, 2011).

2. Mathematical Frameworks: Quantum and Classical

Several mathematical frameworks have been developed for RI models:

• Discrete Quantum Case: The state of $\mathcal{S}$ evolves under a composition of completely positive, trace-preserving (CPTP) maps, with each map induced by tracing out the probe after each interaction. For instance, with identical probes in state $\xi$, each interaction produces a reduced map

$$\mathcal{L}(\rho) = \operatorname{Tr}_\mathcal{P} \left[ U(\tau)(\rho \otimes \xi) U(\tau)^* \right].$$

The $n$-fold evolution is then $\mathcal{L}^n(\rho_0)$.

• Random/Correlated Environments: For i.i.d. or correlated probe preparation (e.g., Markov or arbitrary stationary process), the total evolution becomes a product $$\mathcal{L}_{\omega_n} \circ \cdots \circ \mathcal{L}_{\omega_1}$$ driven by a stochastic process $\{\omega_k\}$; the large $n$ limit is analyzed via ergodic and reducibility theory (Ekblad et al., 16 Jun 2024).
• Continuous-Time Limit: As the interaction duration tends to zero and frequency increases, one can formally derive a Lindblad master equation or stochastic differential equation describing continuous-time effective open-system evolution (Deschamps, 2011, Andreys, 2019).
• Classical Hamiltonian Case: Discrete maps derived from classical Hamiltonians are composed, possibly yielding in the limit an SDE of the form

$dX_t = b(X_t)dt + \sigma(X_t)dW_t$

where $W_t$ is a Wiener process and the drift and diffusion coefficients are determined by properties of the interaction (Deschamps, 2011).

3. Large-Time Asymptotics and Steady-State Behavior

A central goal is to characterize the long-term (steady-state) behavior of $\mathcal{S}$. Under typical conditions (e.g., irreducibility, ergodicity of the environment sequence), one can prove:

• Quantum Case: There exists a unique stationary state $\rho_\text{eq}$ such that for any initial state,

$$\lim_{n \to \infty} \mathcal{L}^n(\rho_0) = \rho_\text{eq},$$

and for random environments, convergence holds in expectation or almost surely (Ekblad et al., 16 Jun 2024, Bougron et al., 2022). If $\mathcal{L}$ is irreducible, this stationary state is unique and the peripheral spectrum lies inside the unit disk except for the eigenvalue $1$.

• Reducibility: When irreducibility fails, the reducibility theory provides a projection-valued decomposition—minimal reducing projections corresponding to invariant subspaces—so that the asymptotic dynamics splits accordingly (Ekblad et al., 16 Jun 2024).
• Classical Case: For appropriate conditions on the Hamiltonian and interaction, the system's distribution converges to a unique invariant measure, which may be Gibbsian or non-equilibrium depending on parameters (Deschamps, 2011, Andreys, 2019).
• Nonequilibrium Steady States (NESS): In many physically realistic models, the asymptotic state differs from the thermal state of the probes due to non-equilibrium driving, lack of detailed balance, or interaction asymmetries (Prositto et al., 9 Jan 2025). For example, with Heisenberg-type interactions, the diagonal population asymptotes to an explicit function of coupling parameters and collision frequency rather than the probe's Boltzmann distribution.

4. Ergodic Theory, Reducibility, and Spectral Properties

A key technical advance for RI models is the adaptation of ergodic theorems, Perron–Frobenius theory, and spectral analysis to dynamical semigroups in both deterministic and random settings:

• Stationary Random Environments: Using the transfer operator $\mathcal{L}$ associated with the stationary process (possibly with arbitrary temporal correlations), the expectation of observables evaluated along system trajectories converges almost surely:

$$\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N \langle \Phi_n(\rho_0), X \circ \theta^n \rangle = \mathbb{E}[\langle\rho_{\mathrm{eq}}, X\rangle].$$

This Cesàro convergence is guaranteed under a minimal irreducibility condition—i.e., existence of a unique minimal reducing projection that is mapped into itself by the dynamics (Ekblad et al., 16 Jun 2024).

• Reducing Projections: In absence of irreducibility, the repeated interaction model exhibits several invariant or recurrent subspaces. Each minimal reducing projection corresponds to a possibly random ergodic component, with the full system's asymptotic behavior decomposing into these invariant sectors.
• Spectral Techniques: For constant or i.i.d. maps, standard results ensure convergence at exponential rates governed by the spectral gap (distance of the subleading eigenvalue to $1$) (Bruneau et al., 2013, Bougron et al., 2022). For Markovian (non-i.i.d.) driving, convergence properties depend on the spectrum of the associated random operator.

5. Distinction from Thermalization and Nonequilibrium Steady-States

A generic feature of deterministic RI protocols is the failure of the system to strictly thermalize (i.e., approach the probe's Boltzmann state), except in special operational regimes—for example, when the interaction conserves energy ($J_{xx} = J_{yy}$ in Heisenberg interactions) or when a protocol involving sufficiently long, randomized collisions is adopted (Prositto et al., 9 Jan 2025). Otherwise, the system typically relaxes to a parameter-dependent NESS, which may be explicitly computed as a function of interaction strengths, detunings, and timings. This NESS can substantially differ from the probe's equilibrium state, reflecting the granular and driven nature of repeated interactions.

6. Quantitative Convergence, Energetic Costs, and Operational Regimes

The quantitative rate of relaxation and the resource requirements of reaching steady or target states are determined by the contraction factors that emerge from the dynamical updating:

$$|\rho^{(n)} - \rho^{(\infty)}| \leq |\eta|^n |\rho^{(0)} - \rho^{(\infty)}|,$$

with $\eta$ computed from system and interaction parameters (Prositto et al., 9 Jan 2025). The number of interaction steps needed to achieve a given precision $\epsilon$ scales logarithmically in $\epsilon$ and inversely with the contraction parameter.

Moreover, the energetic cost of the repeated interaction protocol may be quantified as the accumulated work required to switch the interaction Hamiltonian on and off at each collision. This cost is nonzero except in energy-conserving protocols or in the “thermalizing with randomized collisions” regime.

7. Extensions and Applications

Repeated interaction models provide foundational tools for:

• Quantum Statistical Mechanics and Thermodynamics: Modeling driven and open quantum systems, calculating steady-state transport properties, analyzing entropy productions, and formulating and proving fluctuation theorems (Hanson et al., 2015, Hanson et al., 2017, Bougron et al., 2022).
• Classical and Quantum Control: Describing stochastic control, stabilization, and the engineering of desired asymptotic states (Andreys, 2019, Prositto et al., 9 Jan 2025).
• Quantum Information and Computation: Implementing dissipative state-preparation protocols, quantum trajectories, and controlling decoherence or error-correction dynamics (Dey et al., 2011, Platini et al., 2018, Ekblad et al., 16 Jun 2024).
• Algorithmic and Simulative Protocols: Using RI as a basis for ergodic averaging, as in the design of collision models and the paper of open-system quantum chaos (Platini et al., 2018).

Applications extend beyond physics to models of reciprocation and social networks, repeated games, and beyond, with modifications of the basic repeated interaction formalism used to describe reciprocal decision-making, collective activation, and cooperation (Polevoy et al., 2016, Hoang et al., 2015, Piedrahita et al., 2017, Chen et al., 2022).

Summary Table—Key Features of Repeated Interaction Models

Aspect	Quantum/General Setting	Classical Setting
System Update	CPTP maps, driven by sequence	Hamiltonian maps, Markov chains
Environment	Probes (identical/i.i.d./Markov)	Environment pieces (e.g., field samples)
Steady state	Unique/ergodic under irreducibility; NESS possible	Unique/Gibbs under suitable dissipation
Convergence Rate	Set by spectral gap/contraction	Set by interaction strength and rates
Energetic Cost	Accumulated work per collision	Work input in switching interactions
Applications	Open quantum systems, control, thermodynamics	Open systems, SDE limits, statistical physics

The modern theory of repeated interaction models, with its rigorous analysis of stationary, random, and correlated probe sequences, provides a unifying mathematical and physical framework for the paper of open-system evolution, steady-state emergence, and resource requirements in both quantum and classical domains (Ekblad et al., 16 Jun 2024, Prositto et al., 9 Jan 2025, Bougron et al., 2022, Deschamps, 2011, Andreys, 2019).