A Single-granule Stirling Heat Engine

Single-particle heat engines at atomic and colloidal scales obey the universal thermodynamic bounds on work and efficiency. Here, we translate these principles to the macroscale by building an athermal Stirling engine whose working medium is a millimeter-sized, vibrofluidized granule confined in a time-dependent magnetic trap. By embedding a rattler within the granule to inject noise, we engineer overdamped, Brownian-like dynamics in an otherwise inertial particle. This design enables independent control over the granule’s effective temperature and spatial confinement. Our engine quantitatively reproduces the universal power-efficiency trade-offs of finite-time thermodynamics, achieving the Curzon-Ahlborn efficiency at maximum power. Strikingly, we uncover a control parameter-dependent damping that leads to an unexpected dissipation mechanism - the losses in the compression stroke rival or even exceed those during expansion. Our work establishes an accessible experimental platform to study small-system thermodynamics in intrinsically athermal systems.

💡 Research Summary

The authors present a macroscopic realization of a single‑particle heat engine by constructing a Stirling engine whose working medium is a millimetre‑scale vibro‑fluidized granule. The granule (4 mm × 2.1 mm) encloses a weakly ferromagnetic steel ball (“rattler”) of 1 mm or 1.5 mm diameter. The assembly is placed in a tunable magnetic trap that provides a harmonic potential U(x)=½kx², while vertical vibrations at 37 Hz inject energy. By varying the dimensionless acceleration Γ (5.25 ≤ Γ ≤ 5.85) the effective temperature T_eff of the granule can be controlled; by adjusting the current in the electromagnet the trap stiffness k can be varied independently.

When the internal ball is immobilised the granule exhibits under‑damped dynamics, with a position distribution only slightly deviating from Gaussian and a power‑spectral density (PSD) scaling as f⁻⁴ beyond the trap roll‑off frequency f_c. Introducing the rattler creates internal collisions that act as a stochastic “noise source”. The PSD becomes Lorentzian and follows f⁻² over two decades, indicating overdamped Brownian motion, while the position distribution becomes a perfect Gaussian whose width scales with 1/√k. This transformation enables the definition of an effective temperature via the equipartition relation ½k⟨x²⟩ = ½k_B T_eff. Importantly, T_eff varies with Γ but remains essentially independent of k, allowing the construction of a rectangular T–k diagram that mimics the isothermal and isochoric steps of a Stirling cycle.

The authors implement the Stirling cycle by synchronously modulating T_eff (through Γ) and k according to a predefined protocol. Cycle times τ are varied from 2 s to 32 s, spanning frequencies both below and above the trap’s characteristic frequency, yet always remaining well below the overdamped cutoff f_D, ensuring overdamped dynamics throughout. Work per cycle W_cyc is obtained from the Stratonovich integral of x² dk, while heat Q_cyc follows the integral of k x ẋ dt. As expected for stochastic engines, W_cyc displays large cycle‑to‑cycle fluctuations with a Gaussian distribution; the mean work ⟨W⟩ is negative (engine does work on the surroundings).

In the quasistatic limit (large τ) the measured ⟨W⟩ converges to the theoretical overdamped Stirling work W_∞ = k_B(T_c − T_h) ln(k_max/k_min), confirming that the engine reaches the reversible bound. For finite τ, irreversible losses scale as ⟨W_diss⟩ ≈ Σ/τ, where Σ is a protocol‑dependent irreversibility parameter. Fitting yields Σ ≈ 0.07 k_B T_c for the small rattler and Σ ≈ 0.15 k_B T_c for the larger one. Power P = −⟨W⟩/τ and efficiency η = −⟨W⟩/Q_h obey the universal power‑efficiency trade‑off known from finite‑time thermodynamics. As τ → ∞, η approaches the Stirling efficiency η_∞ = η_C