Bjarne Andresen

The thermodynamic stability of the evolution of a simple conservatively perturbed chemical equilibrium, that is, a two-step reversible reaction, is investigated using the Lyapunov thermodynamic stability formulation. We show that such systems are asymptotically thermodynamically stable.

The year 1975 marked the beginning of an entirely new direction for thermodynamics with the publication of Curzon and Ahlborn’s innocent-looking short paper “Efficiency of a Carnot engine at maximum power output”

This paper extends the concept of epitropy, as introduced in previous work, to capture the effects of extreme tail behavior arising naturally over very long time and large space scales. Epitropy has some qualities that parallel entropy, although it is not quite the same. Its function is to capture the effects of a probability distribution function (PDF) having only a finite populated domain, which was introduced to eliminate divergent moment integrals. Unlike entropy, it represents hidden properties from the external (not the internal) that influence other regimes. This paper uses kinetic theory methods to show the necessity of epitropy in order to ensure that divergent moment integrals do not diverge. While on laboratory scales, the issues in question are negligible, we arrive at dynamics for the grand regime where the tail-generated epitropy can drive the movement of energy.

Three approaches for determining the thermodynamic stability of irreversible processes are described in generalized formulations. The simplest is the Gibbs–Duhem theory, specialized to irreversible trajectories, which uses the concept of virtual displacement in the reverse direction. Its only drawback is that even a trajectory leading to an explosion is identified as a thermodynamically stable motion. In the second approach, we use a thermodynamic Lyapunov function and its time rate from the Lyapunov thermodynamic stability theory (LTS, previously known as CTTSIP). In doing so, we demonstrate that the second differential of entropy, a frequently used Lyapunov function, is useful only for investigating the stability of equilibrium states. Nonequilibrium steady states do not qualify. Without using explicit perturbation coordinates, we further identify asymptotic thermodynamic stability and thermodynamic stability under constantly acting disturbances of unperturbed trajectories as well as of nonequilibrium steady states. The third approach is also based on the Lyapunov function from LTS, but here we additionally use the rates of perturbation coordinates, based on the Gibbs relations and without using their explicit expressions, to identify not only asymptotic thermodynamic stability but also thermodynamic stability under constantly acting disturbances. Only those trajectories leading to an infinite rate of entropy production (unstable states) are excluded from this conclusion. Finally, we use these findings to formulate the Fourth Law of thermodynamics based on the thermodynamic stability. It is a comprehensive statement covering all nonequilibrium trajectories, close to as well as far from equilibrium. Unlike previous suggested “fourth laws”, this one meets the same level of generality that is associated with the original zeroth to third laws. The above is illustrated using the Schlögl reaction with its multiple steady states in certain regions of operation.

In this work, we lay the foundations for computing the behavior of a quantum heat engine whose working medium consists of an ensemble of non-harmonic quantum oscillators. In order to enable this analysis, we develop a method based on the Schrödinger picture. We investigate different possible choices on the basis of expanding the density operator, as it is crucial to select a basis that will expedite the numerical integration of the time-evolution equation without compromising the accuracy of the computed results. For this purpose, we developed an estimation technique that allows us to quantify the error that is unavoidably introduced when time-evolving the density matrix expansion over a finite-dimensional basis. Using this and other ways of evaluating a specific choice of basis, we arrive at the conclusion that the basis of eigenstates of a harmonic Hamiltonian leads to the best computational performance. Additionally, we present a method to quantify and reduce the error that is introduced when extracting relevant physical information about the ensemble of oscillators. The techniques presented here are specific to quantum heat cycles; the coexistence within a cycle of time-dependent Hamiltonian and coupling with a thermal reservoir are particularly complex to handle for the non-harmonic case. The present investigation is paving the way for numerical analysis of non-harmonic quantum heat machines.

Typical human-scaled considerations of thermodynamic states depend primarily on the core of associated speed or other relevant distributions, because the wings of those distributions are so improbable that they cannot contribute significantly to averages. However, for long timescale regimes (slow time), previous papers have shown otherwise. Fluctuating local equilibrium systems have been proven to have distributions with non-Gaussian tails demanding more careful treatment. That has not been needed in traditional statistical mechanics. The resulting non-Gaussian distributions do not admit notions such as temperature; that is, a global temperature is not defined even if local regimes have meaningful temperatures. A fluctuating local thermodynamic equilibrium implies that any local detector is exposed to sequences of local states which collectively induce the non-Gaussian forms. This paper shows why tail behavior is observationally challenging, how the convolutions that produce non-Gaussian behavior are directly linked to time-coarse graining, how a fluctuating local equilibrium system does not need to have a collective temperature, and how truncating the tails in the convolution probability density function (PDF) produces even more non-Gaussian behaviors.

The dissipation in an irreversible step process is reduced when the number of steps is increased in any refinement of the steps in the process. This is a consequence of the ladder theorem, which states that, for any irreversible process proceeding by a sequence of relaxations, dividing any relaxation step into two will result in a new sequence that is more efficient than the original one. This results in a more-steps-the-better rule, even when the new sequence of steps is not reoptimized. This superiority of many steps is well established empirically in, e.g., insulation and separation applications. In particular, the fact that the division of any step into two steps improves the overall efficiency has interesting implications for biological evolution and emphasizes thermodynamic length as a central measure for dissipation.