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A statistical thermodynamics variational criterion is propounded to study thermal hysteresis in reversible clustering of gold (Au) nanoparticles. Experimentally, a transient equilibrium mapping analysis is employed to characterize it thermodynamically, further measurements being performed at the nanostructural and electrochemical levels (UV-Vis-NIR spectra, SLS/SAXS, zeta potential). Theoretically, it is successfully interpreted as a thermodynamic cycle, prompting that nanoclusters has potential to produce useful work from heat and paving the way to nanoclustering heat engines. By taking into account the virial expansion of hysteretic pressure, an entropy measure is deduced for a dilute system with given virial coefficients. This allows us to figure out the role of relevant interparticle potential parameters (i.e. surface potential, nanoparticle size, Debye's length, Hamaker energy) in both isothermal and isochoric variations at the onset of hysteresis. Application to spherical Au nanoparticles in watery salt solution (NaCl) is developed when an ad-hoc (DLVO) pairwise potential governs the second virial coefficient at the nanoscale. In particular, the variational criterion predicts a pressure drop between heating and cooling paths which is likely at the base of some energy redistribution (e.g. ordering/restructuring of electric double layers). We found an integrating factor that is able to numerically predict the existence of a critical value for the initial salt concentration maximizing the hysteretic area, and the effect of nanoparticle size on the cycle extent., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2022 The Author(s). Published by Elsevier Inc. All rights reserved.)

Differential geometry offers a powerful framework for optimising and characterising finite-time thermodynamic processes, both classical and quantum. Here, we start by a pedagogical introduction to the notion of thermodynamic length. We review and connect different frameworks where it emerges in the quantum regime: adiabatically driven closed systems, time-dependent Lindblad master equations, and discrete processes. A geometric lower bound on entropy production in finite-time is then presented, which represents a quantum generalisation of the original classical bound. Following this, we review and develop some general principles for the optimisation of thermodynamic processes in the linear-response regime. These include constant speed of control variation according to the thermodynamic metric, absence of quantum coherence, and optimality of small cycles around the point of maximal ratio between heat capacity and relaxation time for Carnot engines.

In this paper, we analyze the total work extracted and the efficiency of the magnetic Otto cycle in its classic and quantum versions. As a general result, we found that the work and efficiency of the classical engine is always greater than or equal to its quantum counterpart, independent of the working substance. In the classical case, this is due to the fact that the working substance is always in thermodynamic equilibrium at each point of the cycle, maximizing the energy extracted in the adiabatic paths. We apply this analysis to the case of a two-level system, finding that the work and efficiency in both the Otto's quantum and classical cycles are identical, regardless of the working substance, and we obtain similar results for a multilevel system where a linear relationship between the spectrum of energies of the working substance and the external magnetic field is fulfilled. Finally, we show an example of a three-level system in which we compare two zones in the entropy diagram as a function of temperature and magnetic field to find which is the most efficient region when performing a thermodynamic cycle. This work provides a practical way to look for temperature and magnetic field zones in the entropy diagram that can maximize the power extracted from an Otto magnetic engine.

The trade-off between large power output, high efficiency and small fluctuations in the operation of heat engines has recently received interest in the context of thermodynamic uncertainty relations (TURs). Here we provide a concrete illustration of this trade-off by theoretically investigating the operation of a quantum point contact (QPC) with an energy-dependent transmission function as a steady-state thermoelectric heat engine. As a starting point, we review and extend previous analysis of the power production and efficiency. Thereafter the power fluctuations and the bound jointly imposed on the power, efficiency, and fluctuations by the TURs are analyzed as additional performance quantifiers. We allow for arbitrary smoothness of the transmission probability of the QPC, which exhibits a close to step-like dependence in energy, and consider both the linear and the non-linear regime of operation. It is found that for a broad range of parameters, the power production reaches nearly its theoretical maximum value, with efficiencies more than half of the Carnot efficiency and at the same time with rather small fluctuations. Moreover, we show that by demanding a non-zero power production, in the linear regime a stronger TUR can be formulated in terms of the thermoelectric figure of merit. Interestingly, this bound holds also in a wide parameter regime beyond linear response for our QPC device.

The goal of this comment note is to express my concerns about the recent paper by Tian Zhao et al. ( Entropy 2018 , 20 , 542). It is foreseen that this comment will stimulate a fruitful discussion of the issues involved. The principle of the least thermodynamic action is applicable for the analysis of the Carnot cycle using the entropy (not heat) generation extrema theorem. The transversality conditions of the variational problem provide the rectangular shape of the ST diagram for the Carnot cycle.