Local temperatures out of equilibrium

Abstract

The temperature of a physical system is operationally defined in physics as "that quantity which is measured by a thermometer" weakly coupled to, and at equilibrium with the system. This definition is unique only at global equilibrium in view of the zeroth law of thermodynamics: when the system and the thermometer have reached equilibrium, the "thermometer degrees of freedom" can be traced out and the temperature read by the thermometer can be uniquely assigned to the system. Unfortunately, such a procedure cannot be straightforwardly extended to a system out of equilibrium, where local excitations may be spatially inhomogeneous and the zeroth law of thermodynamics does not hold. With the advent of several experimental techniques that attempt to extract a single parameter characterizing the degree of local excitations of a (mesoscopic or nanoscale) system out of equilibrium, this issue is making a strong comeback to the forefront of research. In this paper, we will review the difficulties to define a unique temperature out of equilibrium, the majority of definitions that have been proposed so far, and discuss both their advantages and limitations. We will then examine a variety of experimental techniques developed for measuring the non-equilibrium local temperatures under various conditions. Finally we will discuss the physical implications of the notion of local temperature, and present the practical applications of such a concept in a variety of nanosystems out of equilibrium. Figure 4: (a) The product of the density of states η(E) times the global distribution function ϕ|ρ|ϕ forms a strongly pronounced peak at the expectation value of the global system energyĒ. (b) The logarithm of the local distribution function a|ρ|a (solid line) and the logarithm of a canonical distribution (dashed line) with the same local temperature for a harmonic chain. (a) and (b) are reprinted with permission from [74].

Effect of quantum resonances on local temperature in nonequilibrium open systems

“…54 Therefore, the energy transport process in a nanojunction is significantly different from that in a bulk system, and consequently the classical Fourier's law is strongly violated. 1,[55][56][57][58] Such quantum oscillations have also been observed in some other systems, such as the conjugated organic molecules 59 and graphene nanostructures. 60 The ZCC-based definition has the advantage of yielding a unique value of local temperature (T * ,ZCC ) for any nonequilibrium system.…”

“…6(a), when the terminal impurities are coupled strongly to the leads, both the ZCC and the LMPC predict the T * i vary almost linearly with i, i.e., the distribution of local temperature along the chain obeys the classical Fourier's law. 1 Note that here the restoration of the Fourier's law is not because of disorder 82 or dephasing caused by an external source, 83 which are absent from the AIM under study. Instead, the linear profile of T * i is associated with the substantial broadening of the spectral peaks in A(ω).…”

“…Ideally, the definition of local temperature should be universal (so that it can be applied to as many nonequilibrium situations as possible), unique (so that it yields one and only one value of temperature), operationally feasible (so that it can be measured experimentally), and has the correct asymptotic limit (so that it retrieves the thermodynamic temperature as the system approaches towards an equilibrium state). 1 For instance, Engquist and Anderson defined the local temperature based on a local equilibrium condition 42 referred to as zero-current condition (ZCC) in this paper. In this definition, an ideal potentiometer/thermometer (the probe) is weakly coupled to the nonequilibrium system of interest.…”

“…61 However, its experimental re-alization is rather challenging because of the difficulty in measuring the heat current through a nano-sized sample without knowing its priori local temperature. 1 Moreover, the effectiveness of the ZCC-based definition is questionable if the energy level of the system is in quantum resonance with the states of the environment. This is because if the electron or energy transport is dominated by quantum resonant states, the excitations pertinent to the nonequilibrium processes are highly nonlocal, and hence the local equilibrium approximation which forms the basis of T * ,ZCC may no longer be valid.…”

Evaluation of Different Planting Technique in Ratoon Sugarcane under Semi-Arid Conditions

“…Main reason for low cane yield in Pakistan is low soil fertility, suboptimal plant population density, poor management and inappropriate plating method for efficient utilization of natural resources (Anjum et al, 2015). It is well established that fertilization has a key role in sustaining crop productivity (Ullah et al, 2019;Zhang et al, 2019;Ullah et al, 2020a, b) however appropriate plant method also played an important role particularly in sugarcane. Conventional method of planting sugarcane in 60 cm spaced single row limits domestic cane production to considerable extent (Shafi et al, 1990).…”