Active Billiards: Extracting Work from a Nonequilibrium Process

A simulation of active particles bouncing off a ratcheted wheel — a minimal model for nonequilibrium work extraction.

The question

Can a macroscopic object (a rotor) extract net rotational work from a bath of active particles, even when the particles themselves have no preferred direction? This is a question at the heart of nonequilibrium statistical mechanics and active matter physics.

The answer, in the right setup, is yes — and the key ingredient is broken spatial symmetry (the ratchet) combined with the non-thermal statistics of active particles.

The model

The simulation implements a 2D billiards-like setup:

Active particles move in a bounded domain with quadratic drag. Their velocities follow an AR(1) process (a discrete-time autoregressive process), giving them persistent motion with exponentially decaying correlations. This is a minimal model for run-and-tumble bacteria or self-propelled colloids.
A central ratcheted wheel sits at the center of the domain. The wheel has asymmetric teeth — it is easy to rotate in one direction but resists rotation in the other. This breaks the left-right symmetry of the system.
Collisions between particles and the wheel (and between particles) are elastic: momentum is conserved at each collision. The wheel accumulates angular momentum over many collisions, and this net rotation is the extracted work.

Why AR(1)?

A passive particle in thermal equilibrium at temperature \(T\) has velocity fluctuations with flat power spectrum (white noise). An AR(1) process, in contrast, has colored noise — velocity correlations persist over a characteristic time \(\tau\). This mimics the persistence length of biological swimmers and distinguishes active from passive fluctuations.

The AR(1) process is parameterized by its correlation time and noise amplitude:

\[v(t+dt) = (1 - dt/\tau) v(t) + \sigma \sqrt{2dt/\tau} \, \xi(t)\]

where \(\xi(t)\) is white noise. Longer \(\tau\) means more directed (ballistic) motion; \(\tau \to 0\) recovers Brownian motion.

Implementation

C++ simulation for performance (large particle counts, many collisions per second)
Python batch analysis (batch_run.py) for parameter sweeps
High-resolution frame export (up to 7200px, 600 DPI) for publication figures
Velocity time-series logging for computing power spectra and transport coefficients

Physical picture

At equilibrium (passive particles, \(\tau \to 0\)), detailed balance forbids net rotation: the Second Law wins and the ratchet does not turn. With active particles (finite \(\tau\)), detailed balance is broken and the ratchet can sustain rotation. The rotation rate as a function of \(\tau\), particle density, and ratchet geometry characterizes the nonequilibrium work extraction.

This setup is a minimal model for understanding how biological molecular motors might extract work from the active fluctuations of the cytoplasm — a question of both fundamental and practical interest.

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