PHYSICAL REVIEW E 96, 032143 (2017) Macroscopic violation of the law of heat conduction Michael M. Cândido * and Welles A. M. Morgado Department of Physics, PUC-Rio, Rua Marquês de São Vicente 225, 22453-900 Rio de Janeiro–RJ, Brazil Sílvio M. Duarte Queirós Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud, 150, 22290-180 Rio de Janeiro–RJ, Brazil (Received 2 May 2017; revised manuscript received 17 August 2017; published 29 September 2017) We analyze a model describing an anharmonic macroscopic chain in contact with general reservoirs that follow the Lévy-Itô theorem on the Gaussian-Poissonian decomposition of the measure. We do so by considering a perturbative approach to compute the heat flux and the (canonical) temperature profile when the system reaches the steady state. This approach allows observing a macroscopic violation of the law of the heat conduction equivalent to that found for small (N = 2) systems in contact with general reservoirs, which conveys the ascendency of the nature of the reservoirs over the size of the system. DOI: 10.1103/PhysRevE.96.032143 I. INTRODUCTION From the ancient Egyptians—to whom the first accounts on the nature of heat are accredited—continuing with pre-Socratic philosophers and later on with eminent figures in the History of Science, it took 3000-off years and Joule’s experimental work to reach a proper definition of heat as the amount of energy that is transferred between a system and its surroundings but in the form of whatever kind of work (mechanical, chemical, etc.) [1]. Two decades earlier than “The Mechanical Equivalent of Heat” experiment [2]—and still within the caloric theory—Fourier had established his law stating that the (local) heat flux density,  h, is equal to the product of thermal conductivity, κ , by the negative (local) gradient of the temperature T ,  h ≡ −κ  ∇T , which has been proved thermodynamically correct. With the advent of statistical mechanics—namely, kinetic theory—it was possible to connect mechanical microscopic mechanisms with Fourier’s law [3]. In due course, the same microscopical effort was made aiming to figure out the phenomenon of heat transport in crystals in contact with reservoirs at different temperatures, T C and T H (T C <T H ) [4,5]. Soon, it was realized that because of the ballistic character of the transmission of the energy, by the harmonic lattice, models of harmonic coupled oscillators are unable to retrieve Fourier’s macroscopic behavior; actually, they yield infinite heat conductivity with subsequent dynamically based studies showing that favorable mixing properties assure normal heat transport properties to a system [6] while ergodicity apparently plays a secondary role [7]. Along with the microscopic mechanical features of the system through which heat is transferred, it must be recalled that the thorough characterization of this nonequilibrium prob- lem must take the reservoirs into account. Markovian matters apart, heat reservoirs are assumed as thermal baths—described by either deterministic or stochastic analytical formulations, each presenting its pros and cons [8,9]—yielding Gaussian fluctuations, i.e., presenting a purely continuous Lévy-Itô measure [10] with a single source of stochasticity: the variance. * Present address: Colégio Técnico, Universidade Federal Rural do Rio de Janeiro, Rodovia BR 465–km 8 s/n, 23890-000 Seropédica– RJ, Brazil. The most typical instances are the Nosé-Hoover thermostat for the former and the Langevin thermostat for the latter. With respect to a stochastic approach to the reservoirs, they allow the employing of a quite useful arsenal of techniques and simplifications in the treatment of Gaussian variables that are provided by both stochastic calculus and probability theory. Nevertheless, the concept of reservoir goes beyond the thermal (heat) classification: in several physical and biological processes we have mechanical(-like) systems in contact with sources of energy which do not abide by the canonical conditions of thermodynamics to be classified as thermal—and thus they are called athermal reservoirs—different from other types of sources that act upon the system by performing pure work or exchanging information [11,12]. By reason of their statistical features, athermal reservoirs have been analytically represented by processes other than Gaussian and Brownian. For instance, the shot-noise Pois- sonian process can be used to represent athermal reservoirs which interact with the system at a rate λ, and effective force of magnitude (t ). When 〈(t )〉 = 0 these reservoirs can be understood as work performing reservoirs, whereas when 〈(t )〉= 0 they only change the average energy of the system by stochasticity (variance and higher-order cumulants) and therefore they are viewed as heat sources. The former can be depicted by some types of molecular motors or experimental implementations of ratchets [13,14], whereas the latter can be represented by a (little dense) granular gas [15] or bacterial colonies [16] as well as problems described by generalizations of the Onsager-Machlup fluctuation theory of the second order in time [17]. It is worth mentioning that Poissonian noise has been attracting the attention of the physical community due its applications in a wide set of phe- nomena, such as (i) solid-state problems wherein shot (singular measure) noise is related to the quantization of the charge [18]; (ii) resistor-inductor-capacitor circuits with injection of power at some rate resembling heat pumps [19]; (iii) surface diffusion and low vibrational motion with adsorbates, e.g., Na/Cu(001) compounds [20]; (iv) biological motors in which shot noise mimics the nonequilibrium stochastic hydrolysis of adenosine triphosphate [21–23]; (v) molecular dynamics when the Andersen thermostat is applied [24]; (vi) the use of detectors based on Josephson junctions in order to probe higher-order cumulants in fluctuating currents [25,26]; and 2470-0045/2017/96(3)/032143(11) 032143-1 ©2017 American Physical Society