The fourth law of thermodynamics: steepest entropy ascent.

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'.

Time-Energy and Time-Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations.

In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam-Tamm-Messiah time-energy uncertainty relation τ F Δ H ≥ ℏ / 2 provides a general lower bound to the characteristic time τ F = Δ F / | d 〈 F 〉 / d t | with which the mean value of a generic quantum observable F can change with respect to the width Δ F of its uncertainty distribution (square root of F fluctuations). A useful practical consequence is that in unitary dynamics the states with longer lifetimes are those with smaller energy uncertainty Δ H (square root of energy fluctuations). Here we show that when unitary evolution is complemented with a steepest-entropy-ascent model of dissipation, the resulting nonlinear master equation entails that these lower bounds get modified and depend also on the entropy uncertainty Δ S (square root of entropy fluctuations). For example, we obtain the time-energy-and-time-entropy uncertainty relation ( 2 τ F Δ H / ℏ ) 2 + ( τ F Δ S / k B τ ) 2 ≥ 1 where τ is a characteristic dissipation time functional that for each given state defines the strength of the nonunitary, steepest-entropy-ascent part of the assumed master equation. For purely dissipative dynamics this reduces to the time-entropy uncertainty relation τ F Δ S ≥ k B τ , meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty Δ S .

Multi-physics interactions drive VEGFR2 relocation on endothelial cells.

Vascular Endothelial Growth Factor Receptor-2 (VEGFR2) is a pro-angiogenic receptor, expressed on endothelial cells (ECs). Although biochemical pathways that follow the VEGFR2 activation are well established, knowledge about the dynamics of receptors on the plasma membrane remains limited. Ligand stimulation induces the polarization of ECs and the relocation of VEGFR2, either in cell protrusions or in the basal aspect in cells plated on ligand-enriched extracellular matrix (ECM). We develop a mathematical model in order to simulate the relocation of VEGFR2 on the cell membrane during the mechanical adhesion of cells onto a ligand-enriched substrate. Co-designing the in vitro experiments with the simulations allows identifying three phases of the receptor dynamics, which are controlled respectively by the high chemical reaction rate, by the mechanical deformation rate, and by the diffusion of free receptors on the membrane. The identification of the laws that regulate receptor polarization opens new perspectives toward developing innovative anti-angiogenic strategies through the modulation of EC activation.

Essential equivalence of the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) and steepest-entropy-ascent models of dissipation for nonequilibrium thermodynamics.

By reformulating the steepest-entropy-ascent (SEA) dynamical model for nonequilibrium thermodynamics in the mathematical language of differential geometry, we compare it with the primitive formulation of the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) model and discuss the main technical differences of the two approaches. In both dynamical models the description of dissipation is of the "entropy-gradient" type. SEA focuses only on the dissipative, i.e., entropy generating, component of the time evolution, chooses a sub-Riemannian metric tensor as dissipative structure, and uses the local entropy density field as potential. GENERIC emphasizes the coupling between the dissipative and nondissipative components of the time evolution, chooses two compatible degenerate structures (Poisson and degenerate co-Riemannian), and uses the global energy and entropy functionals as potentials. As an illustration, we rewrite the known GENERIC formulation of the Boltzmann equation in terms of the square root of the distribution function adopted by the SEA formulation. We then provide a formal proof that in more general frameworks, whenever all degeneracies in the GENERIC framework are related to conservation laws, the SEA and GENERIC models of the dissipative component of the dynamics are essentially interchangeable, provided of course they assume the same kinematics. As part of the discussion, we note that equipping the dissipative structure of GENERIC with the Leibniz identity makes it automatically SEA on metric leaves.

Steepest entropy ascent model for far-nonequilibrium thermodynamics: unified implementation of the maximum entropy production principle.

By suitable reformulations, we cast the mathematical frameworks of several well-known different approaches to the description of nonequilibrium dynamics into a unified formulation valid in all these contexts, which extends to such frameworks the concept of steepest entropy ascent (SEA) dynamics introduced by the present author in previous works on quantum thermodynamics. Actually, the present formulation constitutes a generalization also for the quantum thermodynamics framework. The analysis emphasizes that in the SEA modeling principle a key role is played by the geometrical metric with respect to which to measure the length of a trajectory in state space. In the near-thermodynamic-equilibrium limit, the metric tensor is directly related to the Onsager's generalized resistivity tensor. Therefore, through the identification of a suitable metric field which generalizes the Onsager generalized resistance to the arbitrarily far-nonequilibrium domain, most of the existing theories of nonequilibrium thermodynamics can be cast in such a way that the state exhibits the spontaneous tendency to evolve in state space along the path of SEA compatible with the conservation constraints and the boundary conditions. The resulting unified family of SEA dynamical models is intrinsically and strongly consistent with the second law of thermodynamics. The non-negativity of the entropy production is a general and readily proved feature of SEA dynamics. In several of the different approaches to nonequilibrium description we consider here, the SEA concept has not been investigated before. We believe it defines the precise meaning and the domain of general validity of the so-called maximum entropy production principle. Therefore, it is hoped that the present unifying approach may prove useful in providing a fresh basis for effective, thermodynamically consistent, numerical models and theoretical treatments of irreversible conservative relaxation towards equilibrium from far nonequilibrium states. The mathematical frameworks we consider are the following: (A) statistical or information-theoretic models of relaxation; (B) small-scale and rarefied gas dynamics (i.e., kinetic models for the Boltzmann equation); (C) rational extended thermodynamics, macroscopic nonequilibrium thermodynamics, and chemical kinetics; (D) mesoscopic nonequilibrium thermodynamics, continuum mechanics with fluctuations; and (E) quantum statistical mechanics, quantum thermodynamics, mesoscopic nonequilibrium quantum thermodynamics, and intrinsic quantum thermodynamics.

Dissolution of a liquid microdroplet in a nonideal liquid-liquid mixture far from thermodynamic equilibrium.

A droplet placed in a liquid-liquid solution is expected to grow, or shrink, in time as approximately t;{1/2}. In this Letter, we report experimental evidence that when the composition in the interface is far from thermodynamic equilibrium due to the nonideality of the mixture, a droplet shrinks as approximately t. This scaling is due to the coupling between mass and momentum transfer known as Korteweg forces as a result of which the droplet self-propels around. The consequent hydrodynamic convection greatly enhances the mass transfer between the droplet and the bulk phase. Thus, the combined effect of nonideality and nonequilibrium modifies the dynamical behavior of the dissolving droplet.

Nonlinear model dynamics for closed-system, constrained, maximal-entropy-generation relaxation by energy redistribution.

We discuss a nonlinear model for relaxation by energy redistribution within an isolated, closed system composed of noninteracting identical particles with energy levels with . The time-dependent occupation probabilities are assumed to obey the nonlinear rate equations where and are functionals of the 's that maintain invariant the mean energy and the normalization condition . The entropy is a nondecreasing function of time until the initially nonzero occupation probabilities reach a Boltzmann-like canonical distribution over the occupied energy eigenstates. Initially zero occupation probabilities, instead, remain zero at all times. The solutions of the rate equations are unique and well defined for arbitrary initial conditions and for all times. The existence and uniqueness both forward and backward in time allows the reconstruction of the ancestral or primordial lowest entropy state. By casting the rate equations in terms not of the 's but of their positive square roots , they unfold from the assumption that time evolution is at all times along the local direction of steepest entropy ascent or, equivalently, of maximal entropy generation. These rate equations have the same mathematical structure and basic features as the nonlinear dynamical equation proposed in a series of papers ending with G. P. Beretta, Found. Phys. 17, 365 (1987) and recently rediscovered by S. Gheorghiu-Svirschevski [Phys. Rev. A 63, 022105 (2001);63, 054102 (2001)]. Numerical results illustrate the features of the dynamics and the differences from the rate equations recently considered for the same problem by M. Lemanska and Z. Jaeger [Physica D 170, 72 (2002)]. We also interpret the functionals and as nonequilibrium generalizations of the thermodynamic-equilibrium Massieu characteristic function and inverse temperature, respectively.

Thermodynamic derivations of conditions for chemical equilibrium and of Onsager reciprocal relations for chemical reactors.

For an isolated chemical reactor, we derive the conditions for chemical equilibrium in terms of either energy, volume, and amounts of constituents or temperature, pressure, and composition, with special emphasis on what is meant by temperature and chemical potentials as the system proceeds through nonequilibrium states towards stable chemical equilibrium. For nonequilibrium states, we give both analytical expressions and pictorial representations of the assumptions and implications underlying chemical dynamics models. In the vicinity of the chemical equilibrium state, we express the affinities of the chemical reactions, the reaction rates, and the rate of entropy generation as functions of the reaction coordinates and derive Onsager reciprocal relations without recourse to statistical fluctuations, time reversal, and the principle of microscopic reversibility.