Convection is a key transport phenomenon important in many different areas, from hydrodynamics and ocean circulation to planetary atmospheres or stellar physics. However its microscopic understanding still remains challenging. Here we numerically investigate the onset of convective flow in a compressible (non-Oberbeck-Boussinesq) hard disk fluid under a temperature gradient in a gravitational field. We uncover a surprising two-step transition scenario with two different critical temperatures. When the bottom plate temperature reaches a first threshold, convection kicks in (as shown by a structured velocity field) but gravity results in hindered heat transport as compared to the gravity-free case. It is at a second (higher) temperature that a percolation transition of advection zones connecting the hot and cold plates triggers efficient convective heat transport. Interestingly, this novel picture for the convection instability opens the door to unknown piecewise-continuous solutions to the Navier-Stokes equations.
Hard particle systems are among the most successful, inspiring and prolific models of physics. They contain the essential ingredients to understand a large class of complex phenomena, from phase transitions to glassy dynamics, jamming, or the physics of liquid crystals and granular materials, to mention just a few. As we discuss in this paper, their study also provides crucial insights on the problem of transport out of equilibrium. A main tool in this endeavour are computer simulations of hard particles. Here we review some of our work in this direction, focusing on the hard disks fluid as a model system. In this quest we will address, using extensive numerical simulations, some of the key open problems in the physics of transport, ranging from local equilibrium and Fourier’s law to the transition to convective flow in the presence of gravity, the efficiency of boundary dissipation, or the universality of anomalous transport in low dimensions. In particular, we probe numerically the macroscopic local equilibrium hypothesis, which allows to measure the fluid’s equation of state in nonequilibrium simulations, uncovering along the way subtle nonlocal corrections to local equilibrium and a remarkable bulk-boundary decoupling phenomenon in fluids out of equilibrium. We further show that the the hydrodynamic profiles that a system develops when driven out of equilibrium by an arbitrary temperature gradient obey universal scaling laws, a result that allows the determination of transport coefficients with unprecedented precision and proves that Fourier’s law remains valid in highly nonlinear regimes. Switching on a gravity field against the temperature gradient, we investigate numerically the transition to convective flow. We uncover a surprising two-step transition scenario with two different critical thresholds for the hot bath temperature, a first one where convection kicks but gravity hinders heat transport, and a second critical temperature where a percolation transition of streamlines connecting the hot and cold baths triggers efficient convective heat transport. We also address numerically the efficiency of boundary heat baths to dissipate the energy provided by a bulk driving mechanism. As a bonus track, we depart from the hard disks model to study anomalous transport in a related hard-particle system, the $1d$ diatomic hard-point gas. We show unambiguously that the universality conjectured for anomalous transport in $1d$ breaks down for this model, calling into question recent theoretical predictions and offering a new perspective on anomalous transport in low dimensions. Our results show how carefully-crafted numerical simulations of simple hard particle systems can lead to unexpected discoveries in the physics of transport, paving the way to further advances in nonequilibrium physics.
We introduce a general class of stochastic lattice gas models, and derive their fluctuating hydrodynamics description in the large size limit under a local equilibrium hypothesis. The model consists in energetic particles on a lattice subject to exclusion interactions, which move and collide stochastically with energy-dependent rates. The resulting fluctuating hydrodynamics equations exhibit nonlinear coupled particle and energy transport, including particle currents due to temperature gradients (Soret effect) and energy flow due to concentration gradients (Dufour effect). The microscopic dynamical complexity is condensed in just two matrices of transport coefficients: the diffusivity matrix (or equivalently the Onsager matrix) generalizing Fick-Fourier’s law, and the mobility matrix controlling current fluctuations, which are coupled via a fluctuation-dissipation theorem. Interestingly, the positivity of entropy production in the system then leads to detailed constraints on the microscopic dynamics. We further demonstrate the Gaussian character of the noise terms affecting the local currents. The so-called kinetic exclusion process has as limiting cases two of the most paradigmatic models of nonequilibrium physics, namely the symmetric simple exclusion process of particle diffusion and the Kipnis-Marchioro-Presutti model of heat flow, making it the ideal testbed where to further develop modern theories of nonequilibrium behavior.
We use extensive computer simulations to probe local thermodynamic equilibrium (LTE) in a quintessential model fluid, the two-dimensional hard-disks system. We show that macroscopic LTE is a property much stronger than previously anticipated, even in the presence of important finite size effects, revealing a remarkable bulk-boundary decoupling phenomenon in fluids out of equilibrium. This allows us to measure the fluid’s equation of state in simulations far from equilibrium, with an excellent accuracy comparable to the best equilibrium simulations. Subtle corrections to LTE are found in the fluctuations of the total energy which strongly point out to the nonlocality of the nonequilibrium potential governing the fluid’s macroscopic behavior out of equilibrium.
When driven out of equilibrium by a temperature gradient, fluids respond by developing a nontrivial, inhomogeneous structure according to the governing macroscopic laws. Here we show that such structure obeys strikingly simple scaling laws arbitrarily far from equilibrium, provided that both macroscopic local equilibrium and Fourier’s law hold. Extensive simulations of hard disk fluids confirm the scaling laws even under strong temperature gradients, implying that Fourier’s law remains valid in this highly nonlinear regime, with putative corrections absorbed into a nonlinear conductivity functional. In addition, our results show that the scaling laws are robust in the presence of strong finite-size effects, hinting at a subtle bulk-boundary decoupling mechanism which enforces the macroscopic laws on the bulk of the finite-sized fluid. This allows to measure for the first time the marginal anomaly of the heat conductivity predicted for hard disks.
Additional material: video demonstrating the scaling procedure (credit: J. del Pozo 2014)