Quantum and classical thermodynamics, work-extraction, heat engines and cooling

The aim of this research to understand whether quantum mechanics offers any definite opportunity in making more efficient heat engines and -- more generally -- whether there are specifically quantum limits to the laws of thermodynamics, mostly the second law. Since recently, I also started to work on hydrodynamic coolers and work-extractors: hydrodynamics is a specific theory that does impose its limits on energy conversion and cooling. This research also closely connected to foundations of thermodynamics, e.g. what is the physically correct definition of the mean temperature. 

I provide below a sample of research results with pdf version of papers and comments that do not repeat the abstracts. 

-- A.E. Allahverdyan et al., Thermodynamic definition of mean temperature, Phys. Rev. E, 108, 004100 (2023).

Defining mean values of physical quantities is not trivial, but sometimes we are so familiar with concrete definitions that we never ask where they come from. The most widespread definition of the mean is the mean arithmetic, but there are certainly more (also practical, but non-equivalent) definitions including mean geometric, mean harmonic, etc.  For additive quantities (length, volume, energy) the arithmetic mean is physical because it is the only one consistent with the operational procedure of putting the systems together. However, the situation is altogether different for temperature, which is not an additive quantity. Moreover, it is defined for an arbitrary valid thermometer. In principle, all such thermometers are equally good. The temperature scales we use in everyday life (Celsius, Fahrenheit, and Kelvin) are conventional. Thus, the definition of the mean temperature is an open and non-trivial problem, and it does not reduce to applying a mathematical definition. A deeper inquiry into the thermodynamic nature of temperature is needed to understand the definition of its mean. This definition is urgently needed for climate science, atmospheric thermodynamics, and hydrodynamics. In this manuscript, an interdisciplinary team of authors (two physicists and two environmental scientists) addressed the problem starting from the laws of thermodynamics. They implemented the operational procedure of defining the mean by putting the systems together, which in the present context means that the definition of the mean temperature is sought by looking at jointly equilibrium states for the systems. Such states are not unique, because they depend on how precisely (i.e. under which external conditions) the equilibration was achieved. There is however a general way of characterizing the domain of such states, which is based on the notion of the extracted work, a central quantity in thermodynamics. This notion allows us to propose upper and lower bounds for the mean temperature. In the next step, the authors applied the laws of thermodynamics -- together with a natural assumption on the relevance of energy and entropy as the main thermodynamic state functions -- to deduce a unique expression of the mean temperature. This definition of the mean fully respects the conventionality of the temperature definition. Importantly, the mean temperature for two systems at different temperatures depends not on these two temperatures only, but on the thermodynamic states of both systems. For example, according to this definition, two pieces of iron at temperatures 0 C and 50 C, respectively, will have a different mean temperature compared with two pieces of wood having the same temperature 0 C and 50 C. However, things simplify considerably for ideal and weakly-non ideal (van der Waals) gasses, as always in thermodynamics. Fortunately, this is the most relevant case in atmospheric thermodynamics. Now only the concentrations and heat-capacities of the gases enter into the mean temperature, which is expressed by a simple analytic formula.

-- Modeling gasodynamic vortex cooling (Ranque effect)

A. E. Allahverdyan and S. Fauve, Phys. Rev. Fluids, 2, 084102 (2017). 

The Ranque, or vortex cooling effect occupies a special place in hydrodynamics. It is straightforward to demonstrate it even in a simple laboratory experiment: pressurised air is injected tangentially such that a swirling flow is generated in a tube. Air escaping from the periphery of the tube is hot whereas a cold stream is collected from the vicinity of the tube axis. But confusing theories have been proposed for the relevant mechanism describing this refrigeration process, leading people to question accepted links between thermodynamics and hydrodynamics. Confusion started since its inception in 1930's, when contemporaries of Ranque refused to believe in the effect. We took seriously the task of working out the minimal hydrodynamic model that consistently reproduces the main aspects of the vortex cooling effect in its simplest (uniflow) configuration. We noted that models based on several simplified assumptions (e.g. incompressibility or adiabaticity) failed to describe the effect, e.g.

because they could produce arbitrary low temperatures. Compressibility, viscosity,heat-conductivity, and also appropriate boundary conditions should be taken into account for describing cooling. Turbulence is not necessary for cooling but should be

taken into account, at least using turbulent viscosity, in most experimental regimes of operation. Importantly, we predicted a cooling effect with efficiency larger than 1.This does not contradict the second law, but systematic observations of such an effect are lacking.

-- Adaptive heat engine.

A.E. Allahverdyan, S.G. Babajanyan,  N.H. Martirosyan, and A.V. Melkikh, Physical Review Letters,  117, 030601 (2016). 

A major limitations for many heat engines is that their functioning demands on-line control, and/or a fine-tuning between environmental parameters (temperatures of thermal baths) and internal parameters of the engine. This has serious ecological consequences, e.g. car engines abandon the fuel that is depleted only partially. In this paper we propose an adaptive heat engine, where the engine's structure adapts to given environment. Hence no on-line control and no fine-tuning is needed. The engine can employ unknown resources; it can also adapt to results of its own functioning. Our approach is motivated  by photosynthesis: the major heat engine of life that operates between the hot Sun temperature and the low-temperature Earth environment. It does have adaptive features that allow its functioning under decreased hot temperature (shadowing) or increased cold temperature (hot whether). 

The model we studied consists of to two thermal baths (hot and cold) that interact with the three-level engine model; please see the picture. The source of work -- schematically represented as the shining lamp on the picture -- couples with the energy levels 1 and 2. The controller of energy levels adjusts the energy levels to environmental changes, but it does not couple with the environment directly, because for the studied class of models this is useless. 

-- Non-equilibrium quantum fluctuations of work. 

A.E. Allahverdyan, Physical Review E 90, 032137 (2014)                                                                                                           

This paper emerged after some 10-year effort to understand fluctuations of work in quantum statistical mechanics. One side of the problem is well-known in quantum mechanics and amounts to the fact that joint probabilities of non-commuting variables are not unique and are not well-defined (e.g. probabilities are sometimes negative etc). The context of statistical mechanics adds to this problem a specific physical meaning: what is work, how to define the second law on the microscopic level or for non-equilibrium states. It also forces us to re-think the relation between Heisenberg and Schroedinger representations of quantum mechanics. 

-- Carnot Cycle at Finite Power: Attainability of Maximal Efficiency

A. E. Allahverdyan, K. V. Hovhannisyan, A. V. Melkikh and S. G. Gevorkian, Phys. Rev. Lett. 109, 248903 (2013)

This paper emerged from the desire to approach the maximal (Carnot) efficiency as close as possible without decreasing the power of heat engine (i.e. without making it useless). The real heat engines known so far do have this feature: whenever they get rather efficient, they simultaneously decrease their power, so that the purpose was to understand this feature deeper and (if possible) to surmount it. It appeared that this feature closely relates to two known issues: computational complexity of searching and Levinthal paradox in protein physics. The way real proteins overcome this paradox also suggested a scenario by which it was possible to obtain efficient (but so far not arbitrary close to the maximal efficiency) and powerful heat engines.