Scaling laws and bulk-boundary decoupling in heat flow

Abstract

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.PACS numbers: 05.70.Ln, 44.10.+i, The understanding of nonequilibrium behavior remains as one of the major challenges in theoretical physics, even in the simplest situations posed by nonequilibrium steady states (NESSs) [1][2][3][4][5][6][7][8]. The first thing one notices in typical NESSs (as those obtained for fluids under a temperature gradient) is the nontrivial, inhomogeneous structure that the system of interest develops in response to the nonequilibrium driving. This structure, readily measurable in experiments or simulations, carries information on the governing nonequilibrium macroscopic laws (e.g. Fourier's law) which emerge from the myriad of interacting microscopic constituents. It is therefore of paramount importance to understand general properties of these structures, consubstantial to nonequilibrium behavior. With this idea in mind, we derive here a set of simple yet general scaling laws for a broad class of d-dimensional fluids driven far from equilibrium by a temperature gradient. In particular, we show that the fluid's density and temperature profiles follow from two master curves, independent of the driving force and the system parameters, after a simple linear scaling of space. This strong result is based on two mild hypotheses, namely macroscopic local equilibrium and Fourier's law, together with a rather general assumption on the fluid's equation of state.We then proceed to test the emerging picture in a quintessential model, the hard disk fluid. Hard sphere (HS) models and their relatives are among the most successful, inspiring and prolific models of physics, as they contain the essential ingredients to understand a large class of complex phenomena, from phase transitions or heat transport to glassy dynamics, jamming, or the physics of liquid crystals and granular materials, to mention just a few [6,[9][10][11][12][13][14][15][16][17][18][19][20][21][22], turning general results for * jpozo@onsager.ugr.es † garrido@onsager.ugr.es ‡ phurtado@onsager.ugr.es these systems even more appealing. Extensive computer simulations of hard disks under tempe...

A violation of universality in anomalous Fourier’s law

“…3.b strongly support this conclusion. This is a manifestation of the bulk-boundary decoupling phenomenon already reported in hard disks out of equilibrium [35], which enforces the macroscopic laws on the bulk of the finite-sized fluid.…”

“…In any case, in order to test quantitatively this idea, we di- Supplementary Table S1. Small points correspond to the scaling collapse obtained for N ∈ [10 2 + 1, 10 4 + 1], T0 ∈ [2,20], and η ∈ [0.5, 3], while bigger points correspond to additional results obtained from extensive simulations for larger system sizes, namely N = 31623 ( ) and N = 10 5 + 1 (2), with T0 = 20 and η ∈ [0.5, 3]. The line stands for the theoretical prediction, and the master curve for µ = 2.2 has been shifted vertically for better comparison.…”

Probing local equilibrium in nonequilibrium fluids

“…As hard disks exhibit density-temperature separability (i.e. temperature scales out of all thermodynamic relations) [19,46], the associatedQ depends exclusively on density, meaning that a complete collapse is expected for the projection of the EoS surface on theQ − ρ plane, as we indeed observe, see top panel in Fig. 4.…”

“…Strikingly, although density and temperature profiles, as well as pressures, all depend strongly on N , see Figs macroscopically and thus obeys locally the thermodynamic EoS, and the boundary layers near the thermal walls, which sum up all sorts of artificial finite-size and boundary corrections to renormalize the effective boundary conditions on the remaining bulk. This remarkable bulk-boundary decoupling phenomenon, instrumental in the recent discovery of novel scaling laws in nonequilibrium fluids [46,47], is even more surprising at the light of the long range correlations present in nonequilibrium fluids [4,17,18], offering a tantalizing method to obtain macroscopic properties of nonequilibrium fluids without resorting to unreliable finite-size scaling extrapolations [47]. For comparison, we include in Fig.…”

Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state

“…However, they do not solve the problem of establishing theoretical interrelations between different macroscopic quantities. In recent works by del Pozo et al [47,48] computer simulation data for the twodimensional hard disks in the heat-conduction steady states are analyzed in terms of the equilibrium-like equation of state and the local Fourier law. Bulk behaviour of the temperature and particle density profiles are shown to obey specific scaling relations valid even for strong nonequilibrium conditions.…”