This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.How to introduce thermodynamics to quantum mechanics? From numerous possibilities of solving this task, the simple choice is here the conventional von Neumann equation deals with a density operator whose probability weights are time-independent. Because there is no reason apart from the reversible quantum mechanics that these weights have to be time-independent, this constraint is waived, which allows one to introduce thermodynamical concepts to quantum mechanics. This procedure is similar to that of Lindblad’s equation, but different in principle. But beyond this simple starting point, the applied thermodynamical concepts of discrete systems may perform a ‘source theory’ for other versions of phenomenological quantum thermodynamics. This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.Even though irreversibility is one of the major hallmarks of any real-life process, an actual understanding of irreversible processes remains still mostly semi-empirical. In this paper, we formulate a thermodynamic uncertainty principle for irreversible heat engines operating with an ideal gas as a working medium. In particular, we show that the time needed to run through such an irreversible cycle multiplied by the irreversible work lost in the cycle is bounded from below by an irreducible and process-dependent constant that has the dimension of an action. The constant in question depends on a typical scale of the process and becomes comparable to Planck’s constant at the length scale of the order Bohr radius, i.e. the scale that corresponds to the smallest distance on which the ideal gas paradigm realistically applies. This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.A comprehensive constitutive theory for the thermo-mechanical behaviour of generalized continua is established within the framework of continuum thermodynamics of irreversible processes. It represents an extension of the class of generalized standard materials to higher order and higher grade continuum theories. It reconciles most existing frameworks and proposes some new extensions for micromorphic and strain gradient media. The special case of strain gradient plasticity is also included as a contribution to the current debate on the consideration of energetic and dissipative mechanisms. Finally, the stress gradient continuum theory emerges as a new research field for which an elastic-viscoplastic theory at finite deformations is provided for the first time. This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.How can we derive the evolution equations of dissipative systems? What is the relation between the different approaches? How much do we understand the fundamental aspects of a second law based framework? Is there a hierarchy of dissipative and ideal theories at all? How far can we reach with the new methods of nonequilibrium thermodynamics? This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.Variational principles play a fundamental role in deriving the evolution equations of physics. They work well in the case of non-dissipative evolution, but for dissipative systems, the variational principles are not unique and not constructive. With the methods of modern nonequilibrium thermodynamics, one can derive evolution equations for dissipative phenomena and, surprisingly, in several cases, one can also reproduce the Euler-Lagrange form and symplectic structure of the evolution equations for non-dissipative processes. In this work, we examine some demonstrative examples and compare thermodynamic and variational techniques. Then, we argue that, instead of searching for variational principles for dissipative systems, there is another viable programme the second law alone can be an effective tool to construct evolution equations for both dissipative and non-dissipative processes. This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.When thermodynamics is understood as the science (or art) of constructing effective models of natural phenomena by choosing a minimal level of description capable of capturing the essential features of the physical reality of interest, the scientific community has identified a set of general rules that the model must incorporate if it aspires to be consistent with the body of known experimental evidence. Some of these rules are believed to be so general that we think of them as laws of Nature, such as the great conservation principles, whose ‘greatness’ derives from their generality, as masterfully explained by Feynman in one of his legendary lectures. The second law of thermodynamics is universally contemplated among the great laws of Nature. In this paper, we show that in the past four decades, an enormous body of scientific research devoted to modelling the essential features of non-equilibrium natural phenomena has converged from many different directions and frameworks towards the general recognition (albeit still expressed in different but equivalent forms and language) that another rule is also indispensable and reveals another great law of Nature that we propose to call the ‘fourth law of thermodynamics’. We state it as follows every non-equilibrium state of a system or local subsystem for which entropy is well defined must be equipped with a metric in state space with respect to which the irreversible component of its time evolution is in the direction of steepest entropy ascent compatible with the conservation constraints. To illustrate the power of the fourth law, we derive (nonlinear) extensions of Onsager reciprocity and fluctuation-dissipation relations to the far-non-equilibrium realm within the framework of the rate-controlled constrained-equilibrium approximation (also known as the quasi-equilibrium approximation). This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.The paper aims to construct a rational extended thermodynamics (RET) theory of dense polyatomic gases by taking into account the experimental evidence that the relaxation time of molecular rotation and that of molecular vibration are quite different from each other. selleckchem For simplicity, we focus on gases with only one dissipative process due to bulk viscosity. In fact, in some polyatomic gases, the effect of bulk viscosity is much larger than that of shear viscosity and heat conductivity. The present theory includes the previous RET theory of dense gases with six fields as a particular case, and it also includes the RET theory of rarefied polyatomic gases with seven fields in the rarefied-gas limit. The closure is carried out by using the universal principles, that is, Galilean invariance and objectivity, entropy principle, and thermodynamic stability (convexity of entropy), where the duality principle connecting rarefied gases to dense gases also plays an important role. A detailed discussion is devoted to the expression of the production terms in the system of balance equations.