Collectively Operating Heat Engine

• Collectively operating heat engines are systems where multiple interacting units extract work from thermal gradients, exhibiting emergent performance and efficiency beyond isolated devices.
• They employ mechanisms such as quantum coherence, synchronization, and neighborhood-dependent interactions to optimize power output and achieve efficiencies near thermodynamic bounds.
• Scaling analyses reveal super-linear power gains and robust phase transitions, with design strategies tuning interaction strengths, stroke durations, and bath engineering.

A collectively operating heat engine is a thermodynamic system where multiple coupled units (quantum or classical) interact during engine operation, thereby exhibiting emergent nonequilibrium phenomena and performance metrics—particularly power output and efficiency—that surpass those achievable by non-interacting or independently operating units. Such engines realize thermodynamic tasks (work extraction from heat flows between baths at different temperatures) where collective interactions, including quantum coherences, synchronization, or neighborhood-dependent transitions, fundamentally alter engine dynamics and thermodynamic bounds.

1. Minimal Collective Heat Engine Models

At the smallest scale, a minimal collective heat engine consists of two coupled nanomachines, each with a two-level system ($\sigma_k=0,1$). The collective system’s Hamiltonian during engine operation is:

$$\tilde\epsilon^{(\nu)}(\sigma_1, \sigma_2) = V_\nu \left[(1-\sigma_1)\sigma_2 + \sigma_1(1-\sigma_2)\right] + \epsilon_\nu (\sigma_1+\sigma_2),$$

providing direct coupling during each stroke $\nu=1,2$. A general cycle is implemented by switching these parameters in time as the device alternately couples to cold ($\beta_1$) and hot ($\beta_2$) reservoirs for durations $\tau_1$, $\tau_2$ respectively, forming collisional or sequential strokes. The stochastic master equation governs population transitions, with local detailed balance fixing transition rates by the reservoir temperatures and work sources.

In the limit of fast switching ($\tau\rightarrow 0$), the system is effectively in simultaneous contact with both baths, yielding a single master equation with rates being the sum over baths—providing direct benchmarking against traditional (single-particle) finite-time engines. The collective topology introduces multiple microscopic currents (e.g., $J_1$, $J_2$: population fluxes between different composite states), enabling richer thermodynamic behavior than a single working medium (Hawthorne et al., 2023).

2. Thermodynamic Analysis: Currents, Power, and Efficiency

The collective regime admits several design variants:

• Energy-stroke engine (no non-conservative force): The thermodynamic power is

$$\mathcal{P} = (\epsilon_1 - \epsilon_2)(J_1-J_2) + (V_1 - V_2)(J_1+J_2),$$

with the efficiency given by

$$\eta = \frac{\mathcal{P}}{\overline{\dot{Q}}_2} = \frac{(\epsilon_1-\epsilon_2)(J_1-J_2)+(V_1-V_2)(J_1+J_2)}{\epsilon_2(J_1-J_2) + V_2(J_1+J_2)}.$$

• Driving-stroke engine (externally biased): Thermodynamic quantities are determined by both the interaction parameters and the applied bias $F_\nu$, displaying similar tunable functional dependence.

Critically, the collective coupling enables crossing of standard finite-time operation bounds: For optimal tuning of $V_1, V_2$ (interactions), the efficiency at maximum power, $\eta_{mP}$, can exceed Curzon–Ahlborn (CA) limits:

$\eta_{CA} = 1 - \sqrt{T_2/T_1},$

and, in the strong-coupling, fast-switching limit, approaches the Carnot bound $\eta_C = 1 - \beta_2/\beta_1$, which is unattainable in non-interacting/sequential protocols (Hawthorne et al., 2023).

3. Scaling Laws and Optimization

Extending to $N$ coupled units enables analysis of scaling:

• Power Scaling: For moderate $N$, output power scales super-linearly, ${\cal P}_N \propto N^\alpha$ with $1 < \alpha < 1.5$, surpassing the linear sum of $N$ independent engines (Souza et al., 2021). For $N \rightarrow \infty$, properly rescaled high-temperature collective coupling yields an extensive limit: ${\cal P}_N/N$ becomes finite and collective gains $$G = \lim_{N\to\infty} {\cal P}_N/(N {\cal P}_1) > 1$$ appear across broad parameter regimes.
• Efficiency: Remains determined predominantly by microscopic energy gaps (or collective order parameters), often independent of $N$ or explicit interaction, e.g., $\eta_N = \Delta E / E_2$ in the collective quantum spin-pair engine (Souza et al., 2021).
• Optimization Strategies: By tuning period $\tau$ (stroke duration), contact time asymmetry $\kappa$, interaction strengths $(V_1, V_2)$, or bias forces, one can maximize power or efficiency, approach tight coupling, and exploit collective ordering near phase transitions (Hawthorne et al., 2023, Filho et al., 2023). Heatmaps reveal broad optima, particularly in highly ordered (synchronized) phases and neighborhood-dependent reservoir-engineered networks (Fiore, 6 Sep 2025, Mamede et al., 8 Aug 2025).

4. Collective Thermodynamic Phenomena and Quantum Effects

Collective operation enables several fundamentally new effects:

• Multiple Currents and Pathways: Coupling enables distinct probability-current pathways through internal states. For instance, in two interacting two-level systems, there exist independent fluxes $J_1, J_2$ allowing current partitioning to optimize engine objectives, unattainable in single-unit operation (Hawthorne et al., 2023).
• Quantum Coherence and Fluctuations: In quantum collective engines, coherence between subsystems fundamentally modifies performance. Power fluctuations can be suppressed below the classical thermodynamic uncertainty relation (TUR) limit—i.e., constancy ${\cal C}_N > 1$—for finite $N$, only reverting to classical TUR saturation in the thermodynamic limit. Such quantum coherence-induced reversibility is a unique hallmark of collective quantum machines (Uzdin, 2015, Souza et al., 2021).
• Synchronization and Phase Transitions: Collective, interacting units can undergo symmetry-breaking transitions, with ordered phases yielding synchronous unit operation. This tight-coupling regime tightens the relationship between heat and work fluxes, ensuring minimal dissipation, vanishing relative fluctuations, and maximal efficiency at finite power (Filho et al., 2023, Mamede et al., 8 Aug 2025).

5. Beyond Minimal Models: Interactions, Topology, and Multireservoir Effects

The diversity of collective engine architectures encompasses:

• Different Interaction Types: All-to-all (mean-field/Ising), Potts ($q>2$), and Blume–Emery–Griffiths (BEG) interactions exhibit rich phase behavior, including continuous, first-order, and tricritical transitions, with Potts models offering higher maximal power scaling with $q$ (Mamede et al., 8 Aug 2025).
• Neighborhood-dependent Reservoirs: Markovian engines with flip rates depending on the local configuration (number of identical neighbors) and assigned to distinct reservoirs demonstrate new nonequilibrium phase transitions and enable tuning of power, dissipation, and efficiency collectively. Lattice topology and network (regular vs random-regular) have quantitative but not qualitative impact on phase behavior and engine performance (Fiore, 6 Sep 2025).
• Experimental Realizations: Microscopic colloidal engines coupled by non-conservative flows and hydrodynamics demonstrate experimentally observable synergy effects beyond the capabilities of two isolated engines. The synergy factor $S = P_{\text{coll}} / P_{\text{sep}}$ can exceed unity for dimer or small arrays at tight spacing, illustrating general principles of collective enhancement already at minimal scales (Krishnamurthy et al., 2021).

Model	Key Collective Mechanism	Scaling/Enhancement
2-level nanomachines (Hawthorne et al., 2023)	Collisional, coupled cycles	$\eta_{mP} \to \eta_C$, max power boost
Spin-pair QHE (Souza et al., 2021)	Coherent collective drive/dissipation	${\cal P}_N\sim N^\alpha$, stable efficiency $\eta$
Quantum Otto (Latune et al., 2020)	Collectively coupled spin ensemble	$O(N^2)$ power scaling near Carnot
Potts/BEG/Ising (Mamede et al., 8 Aug 2025)	All-to-all, multi-state interactions	Linear $q$-scaling, phase transition-tuned
Neighborhood-reservoir (Fiore, 6 Sep 2025)	Local neighborhood-to-bath mapping	Power/efficiency tuning, phase transitions

6. Universal Thermodynamic Relations and Design Principles

Several universal relations emerge for collectively operating engines:

• Steady-state universal relation: For neighborhood-dependent collective models (Fiore, 6 Sep 2025), the key result $\beta_2 {\cal P} \eta_c = - J_4 \eta \sigma$ (for two-bath, $k=4$) formulates a direct balance among power, Carnot efficiency, entropy production, and interaction. This allows performance tuning without increasing energetic cost: power can be increased at fixed efficiency and dissipation via collective ordering.
• Universal efficiency expression: In many interacting spin models, efficiency takes the form $\eta = 1 - V_1/V_2$, independent of bath temperature, state space size $q$, or topology (Mamede et al., 8 Aug 2025).
• Collective symmetry and reversibility: Quantum collective Otto engines utilizing coherence extraction and injection can achieve entropy pollution $EP \sim 1/N$, rendering the collective engine both more powerful and more reversible than any constituent subunit (Uzdin, 2015).

7. Perspectives and Technological Implications

Collectively operating heat engines establish the foundation for scalable, efficient, and robust nanoscale thermodynamic devices with unprecedented performance bounds. They suggest routes to surpass standard power-efficiency tradeoffs via engineering of interaction topologies, bath architectures, and quantum coherence. These principles are applicable in quantum thermodynamic devices, cold-atom and cavity QED experiments, optomechanical devices, and engineered colloidal micro-machines. Control of collective interactions—whether quantum coherent, classical mean-field, or via neighborhood mapping—enables optimization of power output, efficiency at maximum power, and robustness against fluctuations, marking a paradigm shift from independent-engine architectures to functional, interacting thermal machines (Hawthorne et al., 2023, Souza et al., 2021, Uzdin, 2015, Fiore, 6 Sep 2025, Mamede et al., 8 Aug 2025, Krishnamurthy et al., 2021, Macovei, 2022, Filho et al., 2023, Latune et al., 2020).