Ballistic and diffusive dynamics in a two-dimensional ideal gas of macroscopic chaotic Faraday waves

Kyle J. Welch, Isaac Hastings-Hauss, Raghuveer Parthasarathy, Eric I. Corwin
2014 Physical Review E  
We have constructed a macroscopic driven system of chaotic Faraday waves whose statistical mechanics, we find, are surprisingly simple, mimicking those of a thermal gas. We use real-time tracking of a single floating probe, energy equipartition, and the Stokes-Einstein relation to define and measure a pseudotemperature and diffusion constant and then self-consistently determine a coefficient of viscous friction for a test particle in this pseudothermal gas. Because of its simplicity, this
more » ... plicity, this system can serve as a model for direct experimental investigation of nonequilibrium statistical mechanics, much as the ideal gas epitomizes equilibrium statistical mechanics. Classical kinetic theory requires molecular chaos and homogeneity [1, 2] . Atomic-scale collisions provide the source of homogeneous random motion in thermal systems and form the foundation of classical thermodynamics. In equilibrium thermal systems the crossover from ballistic motion to diffusive motion is a fundamental link between microscopic statistical mechanics and macroscopic thermodynamics [3] . In Einstein's classic thought experiment of a pollen grain in water a thorough study of the grain's ballistic motion would require simultaneous temporal resolution of 10 μs and spatial resolution of 1 nm [4] . Therefore, this crossover from ballistic to diffusive has only recently been experimentally demonstrated in the equilibrium thermal systems of rarefied gases [5,6] and liquids [7, 8] . By contrast, macroscopic systems allow studies of their constituent dynamics that are impossible in the thermal world. Macroscopic systems dilate the characteristic length and time scales but lose the stochastic excitations of thermal systems. This means that any of the random motion necessary for mimicking equilibrium thermal behavior macroscopically must be produced by some stochastic energy input. Because chaotic Faraday waves are very well understood and characterized they present an ideal source of randomness. The Faraday instability is excited on the surface of a fluid subject to vertical oscillations beyond some critical amplitude [9] [10] [11] . Above a second, higher, critical amplitude the surface waves transition from stable ordered waves to spatiotemporal chaos [12] . Surface flows have been measured using fluorescent dyes [13, 14] and tracer particles considerably smaller than the characteristic wave size [13, [15] [16] [17] . A ballistic to diffusive crossover has been observed in chaotic Faraday waves using virtual tracer data drawn from particle image velocimetry measurements [18] . However, for real tracer measurements to date, diffusive motion and fractional Brownian motion have been observed at long and at relatively short time scales, respectively [17, 19] ; the ballistic regime for real particles has not been demonstrated. It remains unproven, therefore, whether a driven (and therefore nonequilibrium) athermal system such as chaotic Faraday waves still exhibits all characteristics of equilibrium statistical mechanics, such as a ballistic-diffusive crossover and a well-defined temperature derived from atomistic chaos, as it is in classical kinetic theory. A variety of attempts to define pseudotemperatures have been proposed for nonequilibrium systems [20] . These have had limited success, describing aspects of the systems' dynamics only over narrow parameter ranges and for few measured properties. A typical approach is to use the Stokes-Einstein relation [21, 22] or fluctuationdissipation theorem [22] to define an effective temperature. While successful in producing a well-defined temperature, these studies do not comment on whether or not their internally determined quantities exhibit behavior consistent with classical kinetic theory. In the present study we achieve random excitation and homogeneity by floating a particle large relative to the characteristic length of the Faraday waves on a chaotic fluid surface (Fig. 1) . Despite the decidedly nonequilibrium nature of chaotic Faraday waves, we show that they drive a buoyant tracer to undergo fully ballistic (short times) and diffusive (long times) Brownian motion, a hallmark of isotropic equilibrium statistical mechanics. This system admits of a well-defined temperature, diffusion constant, and drag coefficient consistent with the system being a nearly ideal gas of excitations. We have created a macroscopic pseudothermal system (which, if treated like a conventional thermal system, would be 10 8 times hotter than the center of the sun). We generate the Faraday waves in a circular aluminum dish brimful with water ( Fig. 1) . We use a circular dish to discourage ordering of the waves [23] . The inner portion, where the water is held, has a radius of 9 cm and a depth of 1.27 cm. A gutter to collect spillover is carved around this inner region. The dish is vertically agitated by a shaker (Vibration Test Systems VTS-100) whose frequency f and peak-peak amplitude A we control independently using a digital function generator (Stanford Research Systems Model DS335) and a voltage amplifier (Behringer Europower EP4000). Mounted on the bottom of the dish is an accelerometer (CTC AC244-2D/010) used to measure stroke amplitude and frequency. The dish is filled to brimful conditions [11] against a knife edge to avoid pinning of the contact angle, which would create a "cold" zone near the edges. However, the fluid position remains pinned to the knife edge, leading to some damping of the waves near the edges. Our floating test particle is a three-dimensional printed section of cone (selectively laser sintered nylon, coated in black spray paint, with a density of approximately 0.7 g/cm 3 ) with a top radius of 0.75 cm, height 0.25 cm, side angle
doi:10.1103/physreve.89.042143 pmid:24827228 fatcat:6vq6ixe53fct5myolnbronx4fq