Continuous Particle Exchange Thermal Machines

• Continuous particle exchange thermal machines are quantum devices that maintain steady-state flows of energy and particles between reservoirs via static, energy-selective elements.
• They exploit differences in temperature and chemical potential to autonomously perform tasks like heat engines, refrigerators, or heat pumps, enabling versatile multitasking operations.
• Device architectures such as quantum dots, molecular junctions, and multiterminal systems provide precise energy filtering, high efficiency near Carnot limits, and observable quantum self-oscillations.

Continuous particle exchange thermal machines are quantum thermodynamic devices in which steady-state flows of energy and particles are maintained between multiple reservoirs via a static, energy-selective structure—typically a quantum dot, molecule, or array thereof—without time-dependent driving or mechanically moving parts. They operate by exploiting differences in chemical potentials and temperatures between reservoirs, leading to the autonomous transport of electrons (or other particles), which in turn can perform work, mediate cooling, or transfer heat between various environments. Fundamental to their operation is the role of energy filtering, microscopic reversibility, and the interplay between elastic and inelastic (e.g., phonon-assisted) transport processes. These properties enable the design and engineering of multitasking (hybrid) machines that can perform combinations of thermodynamic tasks, such as acting simultaneously as a heat engine and a refrigerator, all within the framework permitted by quantum stochastic thermodynamics.

1. Fundamental Principles and Theoretical Framework

Continuous particle exchange thermal machines function by mediating electron (or carrier) transport through static, energy-selective elements that couple two or more reservoirs held at different chemical potentials ($\mu_\alpha$) and temperatures ($T_\alpha$). The transport occurs continuously in steady state, without explicit time-dependent driving. The system is generally described by a Hamiltonian of the form: $$H = H_\text{sys} + H_\text{leads} + H_\text{tun} + H_\text{ph}$$ where $H_\text{sys}$ may be a quantum dot, molecule, or set of levels; $H_\text{leads}$ models the electronic reservoirs; $H_\text{tun}$ covers the tunneling between system and leads; and $H_\text{ph}$ describes phonon or bosonic baths in multiterminal setups.

Key features:

• Conserved Quantities: Both energy and particle number are independently exchanged. The local equilibrium of each reservoir is characterized by a generalized Gibbs state $$\rho_\alpha \propto \exp[-\beta_\alpha (H_\alpha - \mu_\alpha N_\alpha)]$$.
• Steady-State Master Equation: In the weak coupling regime, electronic occupation probabilities evolve via

$$\dot P_n = \sum_m W_{n \leftarrow m} P_m - \left(\sum_l W_{l \leftarrow n}\right) P_n$$

with transition rates $W_{n \leftarrow m}$ satisfying detailed balance with respect to their respective reservoirs.

• Entropy Production and Constraints: Total entropy production rate is given by

$$\dot S_\text{tot} = -\sum_i \beta_i \dot Q_i = J_E F_E + J_N F_N \geq 0$$

where $J_E$, $J_N$ are net energy and particle currents, and $F_E$, $F_N$ are the thermodynamic affinities.

• Flux-Force Linear Response: In linear regime, coupled Onsager relations hold:

$$\begin{pmatrix} J_E \ J_N \end{pmatrix} = \begin{pmatrix} L_{EE} & L_{EN} \ L_{NE} & L_{NN} \end{pmatrix} \begin{pmatrix} F_E \ F_N \end{pmatrix}$$

with positivity and symmetry relations constrained by the second law and, in absence of magnetic fields, Onsager reciprocity.

2. Device Architectures and Experimental Realizations

Implementations predominantly utilize gate-tunable quantum dots, molecular junctions, or multiterminal nanostructures, each providing distinct energy filtering properties. Three canonical regimes have been established:

• Single Quantum Dot Machines: The simplest realization consists of a quantum dot with level energy $\varepsilon$ connecting two reservoirs ("hot" and "cold") at $T_H$ and $T_C$, respectively. The system functions as a steady-state heat engine, refrigerator, or heat pump depending on applied bias and thermal gradients (Pyurbeeva et al., 6 Nov 2025).
• Molecular Machines: Molecular-scale devices, leveraging strong electron correlations (e.g., Kondo resonance), achieve energy filtering with additional tunability via gate voltages and molecular design. Such single-molecule quantum engines have demonstrated high efficiency at low temperatures, reaching up to 53% of the Curzon-Ahlborn bound (Volosheniuk et al., 23 Aug 2025).
• Phonon-Assisted/Multiterminal Machines: By adding third or fourth terminals coupled via bosonic baths (e.g., phononic), inelastic transport is enabled, with phonon exchange assisting electron hopping. These setups—such as three-terminal double-quantum-dot (DQD) or four-terminal four-QD systems—allow for parallel or hybrid operation as engine/refrigerator/pump (Lu et al., 2022).

Device Hamiltonians for multiterminal examples take forms such as: $$H_\text{DQD} = \sum_{i={L,R}} E_i b_i^\dagger b_i + t (b_L^\dagger b_R + \text{h.c.}) + \sum_q \lambda_q b_L^\dagger b_R (d_q + d_q^\dagger) + \sum_q \omega_q d_q^\dagger d_q + \text{lead/tunneling terms}$$ with phonon coupling providing inelastic channels critical for multitasking operation.

3. Transport, Efficiency, and Performance Metrics

Transport observables include electrical current ($I$), heat current ($J_Q$), and output power ($P = I \cdot V_b$). For energy-selective transport through a sharp level, currents are given by Landauer-type or master-equation formulas: $$I = (e/h) \int d\varepsilon \, \tau(\varepsilon)[f_H(\varepsilon) - f_C(\varepsilon)]$$

$$J_Q^H = (1/h) \int d\varepsilon (\varepsilon - \mu_H) \tau(\varepsilon)[f_H(\varepsilon) - f_C(\varepsilon)]$$

where $\tau(\varepsilon)$ is the transmission function.

Efficiencies follow:

• Heat Engine: $\eta = P / J_Q^H = e V_b / (\varepsilon - \mu_H)$; bounded by Carnot ($\eta_C = 1 - T_C/T_H$).
• Efficiency at Maximum Power (Linear Response): $\eta(P_\text{max}) = \eta_C / 2$ for optimally matched conductance/loading.
• Exergy Efficiency for Multi-Tasking: For multi-terminal machines, efficiencies generalize to "exergy efficiencies"

$$\phi = -(\text{sum of negative } T \sigma \text{ terms})/(\text{sum of positive } T \sigma \text{ terms}) \leq 1$$

encoding combinations of work, refrigeration, and heating contributions.

In multiterminal hybrids, coefficients of performance (COP) and hybrid efficiencies (e.g., for engine+refrigerator) are computed by partitioning entropy production according to thermodynamic affinities.

Maximum power, efficiency at maximum power, and noise ("constancy") trade off against each other. In single-level quantum dot models with minimized parasitic heat flow, Carnot efficiency can be approached arbitrarily closely, while the experimentally attainable $\eta(P_\text{max})$ has reached up to $0.7\,\eta_C$ (Pyurbeeva et al., 6 Nov 2025).

4. Multitasking, Cooperative Effects, and Functionality Maps

Adding additional terminals, especially those coupled via inelastic (phonon or bosonic) processes, enables multitask operation:

• Hybrid Modes: By exploiting control over reservoir temperatures and chemical potentials, continuous particle-exchange thermal machines can act as heat engines, refrigerators, heat pumps, or any two (or three) simultaneously. For instance, a three-reservoir quantum dot system with a "gate" heat bath can access engine+refrigerator, engine+heater, refrigerator+heater, and even triple-task modes depending on the reference temperature (Manzano et al., 2020).
• Cooperative Thermoelectric Effects: In multiterminal systems (e.g., DQD, four-QD), performance is characterized by Onsager matrices of dimension $3\times 3$ or $4\times 4$, respectively. The existence of both longitudinal and transverse thermopower coefficients enables geometric tuning of thermodynamic affinities. By combining longitudinal and transverse effects, one can optimize both the figure of merit and power factor, often exceeding those attainable in purely two-terminal elastic machines (Lu et al., 2022).

Functionality diagrams partition parameter space (e.g., $\Delta \mu$ vs. gate voltages) into regions of distinct hybrid operation. Transitions between conduction regimes (e.g., from engine-only to simultaneous engine+refrigerator) occur at characteristic cutoff voltages or affinities set by the level energies and bath temperatures.

Quantitatively, multiterminal machines using realistic parameters can reach exergy efficiencies in the range of $50{-}80\%$ of Carnot, with additional terminals and nondegenerate temperature configurations boosting performance by $\sim 20{-}30\%$ over their degenerate counterparts.

5. Noise, Stability, and Engineering Guidelines

Practical operation demands not only high efficiency and power but also low noise, i.e., stable, predictable output. Particle-exchange machines allow calculation of the zero-frequency current noise,

$$S_I = 2 \int_{-\infty}^\infty dt \langle \delta I(t) \delta I(0) \rangle$$

with analytic expressions available in the four-rate quantum dot model (Pyurbeeva et al., 6 Nov 2025). Output fluctuations scale with tunneling asymmetry and detailed-balance breaking effects.

Guidelines for optimization include:

• Enhancing Conductance Scale: Maximize tunneling rates' harmonic mean $\Gamma = \Gamma_H \Gamma_C / (\Gamma_H + \Gamma_C)$.
• Engineering Asymmetry: Tune entropy difference (degeneracy), coupling asymmetry, and, if present, detailed-balance violation to achieve desired trade-off among power, efficiency, and noise.
• Suppressing Passive Heat Leaks: Minimize parasitic phononic, cotunnel, and non-sharp-level contributions to enforce energy selectivity and optimal performance.
• Gate-Tuning: Precisely gate $\varepsilon$ to maximize required performance metric (e.g., efficiency at maximum power or maximal cooling current).
• Selecting Load: Choose load resistance to select between regimes of high power (at $G R = 1$), high efficiency ($G R \gg 1$), or minimal noise.
• Leveraging Many-Body Effects: For example, Kondo correlations in molecular systems enhance energy filtering sharpness, boosting both output power and efficiency (Volosheniuk et al., 23 Aug 2025).

6. Advanced Modes: Inelastic, Autonomous, and Self-Oscillating Machines

Inclusion of strong inelastic processes (phonon-assisted electron hopping or coupling to mechanical resonators) introduces new operational regimes:

• Phonon-Assisted and Multiterminal Devices: Transport is enabled by absorption or emission of bosons, expanding the operational diagram and enabling hybrid functionality (simultaneous multitasking) not accessible in purely elastic transport (Lu et al., 2022).
• Quantum Self-Oscillations: When a quantum dot is strongly coupled to a quantum mechanical resonator (QMR), part of the electronic transport energy can be converted into mechanical self-oscillation. The onset of self-oscillations is tightly linked to the thermal machine operating in "heater" mode, with genuine limit-cycle motion witnessed by phase-space measures such as "torotropy." The efficiency of current-to-oscillation conversion is maximized at moderate coupling strengths; strong coupling suppresses transport and distorts the limit-cycle structure (Sevitz et al., 22 Aug 2025).
• Experimentally Observable Signatures: The occurrence of self-oscillations is detectable via nonlinearities or kinks in current-bias characteristics and can be quantified directly in physical observables.

7. Applications, Performance Limits, and Outlook

Continuous particle exchange thermal machines are established platforms for exploring quantum thermodynamics, non-equilibrium statistical mechanics, and the design of highly miniaturized, autonomously operating engines, refrigerators, and hybrid thermal multitaskers. Applications include:

• Ultra-compact engines for low-temperature electronics.
• Energy harvesting, on-chip cooling, and quantum information thermal control tasks in the few-nanometre regime.
• Prototypes for quantum thermodynamic resource manipulation, including implementation of generalized Gibbs states and multitasking protocols (Manzano et al., 2020).

Performance boundaries are dictated by the universal Carnot limit, details of energy filtering, and the suppression of parasitic channels. By fine-tuning system asymmetries, many-body correlations, and environmental couplings, quantum dot and molecular thermal machines can approach theoretical efficiency optima under realistic operating conditions. Scalable implementations in mesoscopic and molecular systems, with feedback from ongoing experimental studies, continue to set benchmarks for quantum thermodynamic devices.