Zeroth Law Research Articles

A Four Laws Structure for Looking at Economics through the Eyes of Thermodynamics

Thermodynamics is based on its Four Laws structure. Based on Thermodynamics and its Four Laws structure, this work presents and explores its economic counterpart, proposing an economic Four Laws structure to look at Economics through the eyes of Thermodynamics. The Economic Zeroth Law is based on the merchandise economic thermal equilibrium, thus defining the economic temperature. The Economic First Law is based on balance equations for the units of merchandise and monetary species. The economic Second Law is based on the observation that the traded merchandise units are transferred in the increasing unit price direction (in the decreasing economic temperature direction). This allows economic entropy to be defined through a differential equation, as well as the definition of economic irreversibility and economic entropy generation. The economic Third Law states that the merchandise economic entropy of an economic system at the null economic temperature (at an infinite merchandise unit price) is null. Conjugation of the economic Second and Third Laws sets how to evaluate the absolute value of the economic entropy of an economic system. The presented developments setting the proposed economic Four Laws structure introduce and explore a large set of concepts, variables, and relations that are of major relevance for looking at Economics through the eyes of Thermodynamics.

Black hole zeroth law of the Einstein-complex scalar-Gauss-Bonnet gravity

The zeroth law of black hole mechanics asserts that the stationary black hole has a constant surface gravity on the event horizon. This study investigates the zeroth law within the Einstein-Scalar-Gauss-Bonnet gravity, particularly involving a complex scalar field (ECSGB). Assuming that the Lie derivative of the complex scalar field is proportional to itself other than stationary, and that ECSGB gravity converges to the Einstein's gravity analyticaly as the coupling constant approaches zero, as well as the minimally coupled matter fields satisfy the dominant energy condition, we demonstrate that the surface gravity remains constant over the entire event horizon without reliance on additional symmetries.

Zeroth and second laws of black hole mechanics in Einstein-Maxwell-scalar effective field theory

There has been recent progress in extending the zeroth and second laws of black hole mechanics to gravitational effective field theories (EFTs). We generalize these results to a much larger class of EFTs describing gravity coupled to electromagnetism and a real scalar field. We also show that the zeroth law holds for the EFT of gravity coupled to electromagnetism and a charged scalar field. Published by the American Physical Society 2024

Constraints on Tsallis Cosmology from Big Bang Nucleosynthesis and the Relic Abundance of Cold Dark Matter Particles.

By employing Tsallis' extensive but non-additive δ-entropy, we formulate the first two laws of thermodynamics for gravitating systems. By invoking Carathéodory's principle, we pay particular attention to the integrating factor for the heat one-form. We show that the latter factorizes into the product of thermal and entropic parts, where the entropic part cannot be reduced to a constant, as is the case in conventional thermodynamics, due to the non-additive nature of Sδ. The ensuing two laws of thermodynamics imply a Tsallis cosmology, which is then applied to a radiation-dominated universe to address the Big Bang nucleosynthesis and the relic abundance of cold dark matter particles. It is demonstrated that the Tsallis cosmology with the scaling exponent δ∼1.499 (or equivalently, the anomalous dimension Δ∼0.0013) consistently describes both the abundance of cold dark matter particles and the formation of primordial light elements, such as deuterium 2H and helium 4He. Salient issues, including the zeroth law of thermodynamics for the δ-entropy and the lithium 7Li problem, are also briefly discussed.

Entropy defect: Algebra and thermodynamics

We investigate the way the entropy of a system can be partitioned into the entropies of its constituents in consistency with thermodynamics. This partitioning is described through the concept of an entropy defect, which measures the missing entropy between the sum of entropies of a system's constituents and the entropy of the combined system; this decrease of entropy corresponds to the order induced by the additional long-range correlations developed among the constituents of the combined system. We conclude that the most generalized addition rule is the one characterizing the kappa entropy; when the system resides in stationary states, the kappa entropy becomes the one associated with kappa distributions, while, in general, this entropy applies more broadly, in stationary or nonstationary states. Moreover, we develop the specific algebra of the addition rule with entropy defect. The addition rule forms a mathematical group on the set of any measurable physical-quantity (e.g., entropy). Finally, we use these algebraic properties to restate the generalized zeroth law of thermodynamics so that it is applicable for nonstationary as well as stationary states: If a body C measures the entropies of two other bodies, A and B, then, their combined entropy is measured as the connected A and B entropy, where the entropy defect is involved in all measurements.

Explicit formulae for surface gravities in stationary circular axi-symmetric spacetimes

Using minimalist assumptions we develop a natural functional decomposition for the spacetime metric, and explicit tractable formulae for the surface gravities, in arbitrary stationary circular (PT symmetric) axisymmetric spacetimes. We relate rigidity results, (the existence of a Killing horizon), and the zeroth law to the absence of curvature singularities at the would-be horizons. These observations are of interest to both observational astrophysicists (modelling the cold, dark, heavy objects at the center of most spiral galaxies), and to the analogue spacetime community, (wherein the presence of naked singularities is not necessarily deprecated, and the occurrence of non-Killing horizons is relatively common).

Killing Horizons and Surface Gravities for a Well-Behaved Three-Function Generalization of the Kerr Spacetime

Thanks to the recent advent of the event horizon telescope (EHT), we now have the opportunity to test the physical ramifications of the strong-field near-horizon regime for astrophysical black holes. Herein, emphasizing the trade-off between tractability and generality, the authors discuss a particularly powerful three-function distortion of the Kerr spacetime, depending on three arbitrary functions of the radial coordinate r, which on the one hand can be fit to future observational data, and on the other hand is sufficiently general so as to encompass an extremely wide class of theoretical models. In all of these spacetimes, both the timelike Hamilton–Jacobi (geodesic) and massive Klein–Gordon (wave) equations separate, and the spacetime geometry is asymptotically Kerr; hence, these spacetimes are well-suited to modeling real astrophysical black holes. The authors then prove the existence of Killing horizons for this entire class of spacetimes, and give tractable expressions for the angular velocities, areas, and surface gravities of these horizons. We emphasize the validity of rigidity results and zeroth laws for these horizons.

Approach to Equilibrium in Translation-Invariant Quantum Systems: Some Structural Results

We formulate the problem of approach to equilibrium in algebraic quantum statistical mechanics and study some of its structural aspects, focusing on the relation between the zeroth law of thermodynamics (approach to equilibrium) and the second law (increase in entropy). Our main result is that approach to equilibrium is necessarily accompanied by a strict increase in the specific (mean) energy and entropy. In the course of our analysis, we introduce the concept of quantum weak Gibbs state which is of independent interest.

Null surface thermodynamics in topologically massive gravity

We study three dimensional topologically massive gravity (TMG) in presence of a generic codimension one null boundary. The existence of the boundary is accounted for by enlarging the Hilbert space of the theory by degrees of freedom which only reside at the boundary, the boundary degrees of freedom. The solution phase space of this theory in addition to bulk massive chiral gravitons of the TMG, involves boundary modes which are labeled by surface charges associated with large diffeomorphisms. We show boundary degrees of freedom obey a local thermodynamic description over the solution phase space, null surface thermodynamics, described by a local version of the first law, a local Gibbs–Duhem equation, and local zeroth law. Due to the expansion of the boundary and also the passage of the bulk mode through the boundary, our null surface thermodynamics describes an open boundary system that is generically out of thermal equilibrium.

Kinetic Temperature of Structures for Resilience, Instability, and Failure Analysis of Building Systems

From theory, calibration and application of the equipartition theorem of statistical physics to structural failure and instability analysis, we introduce the kinetic temperature of structures as an order parameter to ascertain equilibrium and out-of-equilibrium states in structural mechanics. Set within the framework of molecular dynamics-based structural mechanics, this is achieved by connecting the set of momentum balance equations to an outside bath reservoir maintained at a reference temperature history through the Nosé-Hoover thermostat. The problem thus comes down to solving the momentum balance equation with a dissipative mass damping term, which evolves in function of the difference in temperature between the structure’s kinetic temperature/energy and the bath temperature. Following the Zeroth Law of Thermodynamics, it is recognized that a structure is in (thermal) equilibrium as long as the structure’s kinetic temperature attains the bath temperature; whereas it is out-of-equilibrium when the open system (structure plus bath) exhibits a sustained temperature difference. In this case, the structure has exhausted its fluctuation-dissipation capacity, which is indicative—for structures—of a progressive failure and instability. The implementation of the kinetic temperature as an order parameter in structural failure and instability analysis is illustrated for a prototype five-storey building subject to excessive wind and fire loads. It is suggested that the proposed order parameter becomes an integral part of the structural engineering toolbox for resilience studies of buildings and structures.

Operational Definition of the Temperature of a Quantum State.

Temperature is usually defined for physical systems at thermal equilibrium. Nevertheless one may wonder if it would be possible to attribute a meaningful notion of temperature to an arbitrary quantum state, beyond simply the thermal (Gibbs) state. In this Letter, we propose such a notion of temperature considering an operational task, inspired by the zeroth law of thermodynamics. Specifically, we define two effective temperatures for quantifying the ability of a quantum system to cool down or heat up a thermal environment. In this way we can associate an operationally meaningful notion of temperature to any quantum density matrix. We provide general expressions for these effective temperatures, for both single- and many-copy systems, establishing connections to concepts previously discussed in the literature. Finally, we consider a more sophisticated scenario where the heat exchange between the system and the thermal environment is assisted by a quantum reference frame. This leads to an effect of "coherent quantum catalysis," where the use of a coherent catalyst allows for exploiting quantum energetic coherences in the system, now leading to much colder or hotter effective temperatures. We demonstrate our findings using a two-level atom coupled to a single mode of the electromagnetic field.

Thermodynamics of stationary states of the ideal gas in a heat flow.

There is a long-standing question as to whether and to what extent it is possible to describe nonequilibrium systems in stationary states in terms of global thermodynamic functions. The positive answers have been obtained only for isothermal systems or systems with small temperature differences. We formulate thermodynamics of the stationary states of the ideal gas subjected to heat flow in the form of the zeroth, first, and second law. Surprisingly, the formal structure of steady state thermodynamics is the same as in equilibrium thermodynamics. We rigorously show that U satisfies the following equation dU = T*dS* - pdV for a constant number of particles, irrespective of the shape of the container, boundary conditions, the size of the system, or the mode of heat transfer into the system. We calculate S* and T* explicitly. The theory selects stable nonequilibrium steady states in a multistable system of ideal gas subjected to volumetric heating. It reduces to equilibrium thermodynamics when heat flux goes to zero.

Gauge invariant derivation of zeroth and first laws of black hole thermodynamics

In gauge invariant theories, like Einstein-Maxwell theory, physical observables should be gauge invariant. In particular, mass, entropy, angular momentum, electric charge and their respective chemical potentials, temperature, horizon angular velocity and electric potential which appear in the laws of black hole thermodynamics should be gauge invariant. In the usual construction of the laws of black hole thermodynamics, gauge invariance of the intensive quantities is subtle; here we remedy this and provide a gauge invariant derivation and the proof of the zeroth and first laws of black hole thermodynamics.

The zeroth law of black hole thermodynamics in arbitrary higher derivative theories of gravity

We consider diffeomorphism invariant theories of gravity with arbitrary higher derivative terms in the Lagrangian as corrections to the leading two derivative theory of Einstein’s general relativity. We construct a proof of the zeroth law of black hole thermo-dynamics in such theories. We assume that a stationary black hole solution in an arbitrary higher derivative theory can be obtained by starting with the corresponding stationary solution in general relativity and correcting it order by order in a perturbative expansion in the coupling constants of the higher derivative Lagrangian. We prove that surface gravity remains constant on its horizon when computed for such stationary black holes, which is the zeroth law. We argue that the constancy of surface gravity on the horizon is related to specific components of the equations of motion in such theories. We further use a specific boost symmetry of the near horizon space-time of the stationary black hole to constrain the off-shell structure of the equations of motion. Our proof for the zeroth law is valid up to arbitrary order in the expansion in the higher derivative couplings.

Interdependence of the First and Second Laws of Thermodynamics

Historically, classical thermodynamics was developed during attempts to improve the efficiency of steam engines. Those investigations led to the well-known zeroth, first, and second laws of thermodynamics. Although these laws are presented independently, linkages between the zeroth and second laws have been pointed out in the physics teaching literature. In our 2020 paper, “What if Energy Flowed from Cold to Hot? Counterfactual Thought Experiments,” we imagined a universe in which energy spontaneously flowed from colder to hotter objects. That paper inspired us to explore the connections between the first law and the Clausius statement of the second law in the present paper.

Discovery of the zeroth law of helicity spectrum in the pre-inertial range of wall turbulence

We report an unprecedented existence of the zeroth law of helicity spectrum (i.e., the helicity spectrum becomes independent of the wavenumber) in the transition from production range to inertial range, herein termed the pre-inertial range, of wall turbulence. The zeroth law is explained by the superposition effect of the forward joint cascade of energy and helicity caused by twisting and stretching of wall-attached superstructures in an equilibrium layer. The phenomenological model perfectly predicts the zeroth law in the pre-inertial range. Experimental data support the existence of the zeroth law.

Self-regularization in turbulence from the Kolmogorov 4/5-law and alignment.

A defining feature of three-dimensional hydrodynamic turbulence is that the rate of energy dissipation is bounded away from zero as viscosity is decreased (Reynolds number increased). This phenomenon-anomalous dissipation-is sometimes called the 'zeroth law of turbulence' as it underpins many celebrated theoretical predictions. Another robust feature observed in turbulence is that velocity structure functions [Formula: see text] exhibit persistent power-law scaling in the inertial range, namely [Formula: see text] for exponents [Formula: see text] over an ever increasing (with Reynolds) range of scales. This behaviour indicates that the velocity field retains some fractional differentiability uniformly in the Reynolds number. The Kolmogorov 1941 theory of turbulence predicts that [Formula: see text] for all [Formula: see text] and Onsager's 1949 theory establishes the requirement that [Formula: see text] for [Formula: see text] for consistency with the zeroth law. Empirically, [Formula: see text] and [Formula: see text], suggesting that turbulent Navier-Stokes solutions approximate dissipative weak solutions of the Euler equations possessing (nearly) the minimal degree of singularity required to sustain anomalous dissipation. In this note, we adopt an experimentally supported hypothesis on the anti-alignment of velocity increments with their separation vectors and demonstrate that the inertial dissipation provides a regularization mechanism via the Kolmogorov 4/5-law. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Scalar-hairy Lovelock gravity respects zeroth law

We study the zeroth law for Killing horizon in scalar-hairy Lovelock gravity, and show that the surface gravity of a general Killing horizon in the scalar-hairy Lovelock gravity is constant, provided that the dominant energy condition is obeyed by the matter field.

How geometrically frustrated systems challenge our notion of thermodynamics

Although Boltzmann’s definition of entropy and temperature are widely accepted, we will show scenarios which apparently are inconsistent with our normal notion of thermodynamics. We focus on generic geometrically frustrated systems (GFSs), which stay at constant negative Boltzmann temperatures, independent from their energetic state. Two weakly coupled GFSs at same temperature exhibit, in accordance with energy conservation, the same probability for all energetic combinations. Heat flow from a hot GFS to a cooler GFS or an ideal gas increases Boltzmann entropy of the combined system, however the maximum is non-local, which, in contrast to conventional thermodynamics, implies that both subsystems maintain different temperatures here. Re-parametrization can transform these non-local into local maxima with corresponding equivalence of re-defined temperatures. However, these temperatures cannot be assigned solely to a subsystem but describe combinations of both. The non-local maxima of entropy restrict the naive application of the zeroth law of thermodynamics. Reformulated this law is still valid with the consequence that a GFS at constant negative temperature can measure positive temperatures. Heat exchange between a GFS and a polarized paramagnetic spin gas, i.e. a system that may achieve besides positive also negative temperatures, drives the combined system to a local-, or non-local maximum of entropy, with equivalent or non-equivalent temperatures here. Energetic constraints determine which scenario results. In case of a local maximum, the spin gas can measure temperature of the GFS like a usual thermometer, however, this reveals no information about the energetic state of the GFS.

Thermalization of small quantum systems: from the zeroth law of thermodynamics

Thermalization of isolated quantum systems has been studied intensively in recent years and significant progresses have been achieved. Here, we study thermalization of small quantum systems that interact with large chaotic environments under the consideration of Schrödinger evolution of composite systems, from the perspective of the zeroth law of thermodynamics. Namely, we consider a small quantum system that is brought into contact with a large environmental system; after they have relaxed, they are separated and their temperatures are studied. Our question is under what conditions the small system may have a detectable temperature that is identical with the environmental temperature. This should be a necessary condition for the small quantum system to be thermalized and to have a well-defined temperature. By using a two-level probe quantum system that plays the role of a thermometer, we find that the zeroth law is applicable to quantum chaotic systems, but not to integrable systems.