Entropy Inequality Research Articles

Holographic entropy inequalities and the topology of entanglement wedge nesting

We prove two new infinite families of holographic entropy inequalities. A key tool is a graphical arrangement of terms of inequalities that is based on entanglement wedge nesting. It associates the inequalities with tessellations of the torus and the projective plane, which reflect a certain topological aspect of entanglement wedge nesting. The inequalities prove a prior conjecture about the holographic entropy cone. We discuss their relation to black hole physics and differential entropy, and sketch applications to quantum error correction, quantifying randomness of quantum states, and others. Published by the American Physical Society 2024

The Navier-Stokes system with temperature and salinity for free surface flows. Numerical scheme and validation

In this paper, we propose and study a numerical scheme for the Navier-Stokes-Fourier system derived and studied by the authors in [12]. This system models hydrostatic free surface flows with density variations depending on temperature or salinity. We show that the finite volume/finite element scheme presented – based on a layer averaged formulation of the model – is well-balanced with regards to the steady state of the lake at rest and preserves the nonnegativity of the water height. A maximum principle on the density is also proved as well as a discrete entropy inequality (when the thermal and viscous effects are neglected). Some numerical validations are finally shown with comparisons to 3D analytical solutions and experiments.

Semi-discrete entropy-preserving surface reconstruction schemes for the shallow water equations: Analysis of physical structures

We aim to introduce a new variant of hydrostatic reconstruction scheme, which is physical-structure-preserving and semi-discrete entropy-preserving for shallow water equations with a discontinuous bottom topography, based on novel surface reconstructions (NSR). The NSR is used to define approximate Riemann states with respect to the water depth and the velocity to compute consistent numerical fluxes across the cell interface and the upwind interface source term. This work is motivated by path-conservative surface reconstructions (Dong (2023) [19]), in which the conditions to satisfy the entropy inequality are complicated. We propose the NSR scheme to satisfy the entropy inequality without the corresponding complicated conditions. A key advantage of the NSR scheme is to preserve the physical structure of the water surface in the sense that when the water surface is continuous, then the defined approximate Riemann state should inherit this property. The physical-structure-preserving property is indispensable for obtaining a more accurate discretized source term and avoiding artificial “waterfall” effects. We prove that the NSR scheme satisfies a semi-discrete entropy condition and can guarantee the water depth to be nonnegative, and maintain the stationary state. We show several computed results to compare the NSR scheme with the hydrostatic reconstruction scheme based on subcell reconstructions and the path-conservative surface reconstruction scheme. We also extend the NSR scheme for pollutant transport in shallow water based on adaptive moving triangular meshes. Finally, several numerical results obtained by the adaptive NSR scheme for pollutant transport in shallow water are shown.

Interpretation of Second Law of Thermodynamics in Extended Procedures for the Exploitation of the Entropy Inequality: Korteweg Fluids and Strain-Gradient Elasticity as Examples.

In continuum physics the dissipation principle, first proposed by Coleman and Noll in 1963, regards second law of thermodynamics as a unilateral differential constraint on the constitutive equations. In 1996, Muschik and Ehrentraut provided a rigorous proof of such an approach under the assumption that, at an arbitrary instant, t0, in an arbitrary point, P0, of a continuous system, the entropy production is zero if, and only if, P0 is in thermodynamic equilibrium. In 2022, Cimmelli and Rogolino incorporated such an assumption in a more general formulation of the second law of thermodynamics. In this paper, we prove that the same conclusions hold if both the fundamental balance laws and their gradients are substituted into the entropy inequality. Such a methodology is applied to analyze the strain-gradient elasticity.

A geometrically and thermodynamically compatible finite volume scheme for continuum mechanics on unstructured polygonal meshes

We present a novel Finite Volume (FV) scheme on unstructured polygonal meshes that is provably compliant with the Second Law of Thermodynamics and the Geometric Conservation Law (GCL) at the same time. The governing equations are provided by a subset of the class of symmetric and hyperbolic thermodynamically compatible (SHTC) models introduced by Godunov in 1961. Specifically, our numerical method discretizes the equations for the conservation of momentum, total energy, distortion tensor and thermal impulse vector, hence accounting in one single unified mathematical formalism for a wide range of physical phenomena in continuum mechanics, spanning from ideal and viscous fluids to hyperelastic solids. By means of two conservative corrections directly embedded in the definition of the numerical fluxes, the new schemes are proven to satisfy two extra conservation laws, namely an entropy balance law and a geometric equation that links the distortion tensor to the density evolution. As such, the classical mass conservation equation can be discarded. Firstly, the GCL is derived at the continuous level, and subsequently it is satisfied by introducing the new concepts of general potential and generalized Gibbs relation. The new potential is nothing but the determinant of the distortion tensor, and the associated Gibbs relation is derived by introducing a set of dual or thermodynamic variables such that the GCL is retrieved by dot multiplying the original system with the new dual variables. Once compatibility of the GCL is ensured, thermodynamic compatibility is tackled in the same manner, thus achieving the satisfaction of a local cell entropy inequality. The two corrections are orthogonal, meaning that they can coexist simultaneously without interfering with each other. The compatibility of the new FV schemes holds true at the semi-discrete level, and time integration of the governing PDE is carried out relying on Runge-Kutta schemes. A large suite of test cases demonstrates the structure preserving properties of the schemes at the discrete level as well.

Relative entropy inequality for capillary fluids with density dependent viscosity and applications

We derive a relative entropy inequality for capillary compressible fluids with density dependent viscosity. Applications in the context of weak–strong uniqueness analysis, pressureless fluids and high-Mach number flows are presented.

Entropy constraints for ground energy optimization

We study the use of von Neumann entropy constraints for obtaining lower bounds on the ground energy of quantum many-body systems. Known methods for obtaining certificates on the ground energy typically use consistency of local observables and are expressed as semidefinite programming relaxations. The local marginals defined by such a relaxation do not necessarily satisfy entropy inequalities that follow from the existence of a global state. Here, we propose to add such entropy constraints that lead to tighter convex relaxations for the ground energy problem. We give analytical and numerical results illustrating the advantages of such entropy constraints. We also show limitations of the entropy constraints we construct: they are implied by doubling the number of sites in the relaxation and as a result they can at best lead to a quadratic improvement in terms of the matrix sizes of the variables. We explain the relation to a method for approximating the free energy known as the Markov Entropy Decomposition method.

Identification-robust methods for comparing inequality with an application to regional disparities

We propose Fieller-type methods for inference on generalized entropy inequality indices in the context of the two-sample problem which covers testing the statistical significance of the difference in indices, and the construction of a confidence set for this difference. In addition to irregularities arising from thick distributional tails, standard inference procedures are prone to identification problems because of the ratio transformation that defines the considered indices. Simulation results show that our proposed method outperforms existing counterparts including simulation-based permutation methods and results are robust to different assumptions about the shape of the null distributions. Improvements are most notable for indices that put more weight on the right tail of the distribution and for sample sizes that match macroeconomic type inequality analysis. While irregularities arising from the right tail have long been documented, we find that left tail irregularities are equally important in explaining the failure of standard inference methods. We apply our proposed method to analyze income per-capita inequality across U.S. states and non-OECD countries. Empirical results illustrate how Fieller-based confidence sets can: (i) differ consequentially from available ones leading to conflicts in test decisions, and (ii) reveal prohibitive estimation uncertainty in the form of unbounded outcomes which serve as proper warning against flawed interpretations of statistical tests.

Entropy inequalities for random walks and permutations

Entropy inequalities for random walks and permutations

The log-Minkowski inequality of curvature entropy for convex bodies

The log-Minkowski inequality of curvature entropy for convex bodies

Discrete entropy inequalities via an optimization process

The solutions of hyperbolic systems may contain discontinuities. These weak solutions verify not only the original PDEs, but also an entropy inequality that acts as a selection criterion determining whether a discontinuity is physical or not. Obtaining a discrete version of these entropy inequalities when approximating the solutions numerically is crucial to avoid convergence to unphysical solutions or even unstability. However such a task is difficult in general, if not impossible for schemes of order 2 or more. In this paper, we introduce an optimization framework that enables us to quantify a posteriori the decrease or increase of entropy of a given scheme, locally in space and time. We use it to obtain maps of numerical diffusion and to prove that some schemes do not have a discrete entropy inequality. A special attention is devoted to the widely used second order MUSCL scheme for which almost no theoretical results are known.

A note on equivalent conditions for majorization

<abstract><p>In this paper, we introduced novel characterizations of the classical concept of majorization in terms of upper triangular (resp., lower triangular) row-stochastic matrices, and in terms of sequences of linear transforms on vectors. We used our new characterizations of majorization to derive an improved entropy inequality.</p></abstract>

A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.

On weak (measure valued)–strong uniqueness for Navier–Stokes–Fourier system with Dirichlet boundary condition

In this article, our goal is to define a measure valued solution of compressible Navier–Stokes–Fourier system for a heat conducting fluid with Dirichlet boundary condition for temperature in a bounded domain. The definition will be based on the weak formulation of entropy inequality and ballistic energy inequality. Moreover, we obtain the weak (measure valued)–strong uniqueness property of this solution with the help of relative energy.

High order entropy stable discontinuous Galerkin spectral element methods through subcell limiting

Subcell limiting strategies for discontinuous Galerkin spectral element methods do not provably satisfy a semi-discrete cell entropy inequality. In this work, we introduce an extension to the subcell and monolithic convex limiting strategies [12,11,2] that satisfies the semi-discrete cell entropy inequality by formulating the limiting factors as solutions to an optimization problem. The optimization problem is efficiently solved using a deterministic greedy algorithm. We also discuss the extension of the proposed subcell limiting strategy to preserve general convex constraints. Numerical experiments confirm that the proposed limiting strategy preserves high-order accuracy for smooth solutions and satisfies the cell entropy inequality.

Thermomechanics of ferri-antiferromagnetic phase transition in finitely-strained rocks towards paleomagnetism

The thermodynamic model of visco-elastic deformable magnetic materials at finite strains is formulated in a fully Eulerian way in rates with the aim to describe thermoremanent paleomagnetism in crustal rocks. The Landau theory applied to a ferro-to-para-magnetic phase transition, the gradient theory for magnetization (leading to exchange energy) with general mechanically dependent coefficient, hysteresis in magnetization evolution by Gilbert equation involving objective corotational time derivative of magnetization, and demagnetizing field are considered in the model. The Jeffreys viscoelastic rheology is used with temperature-dependent creep to model solidification or melting transition. The model complies with energy conservation and the Clausius-Duhem entropy inequality.

Trace- and Improved Data-Processing Inequalities for von Neumann Algebras

We prove a version of the data-processing inequality for the relative entropy for general von Neumann algebras with an explicit lower bound involving the measured relative entropy. The inequality, which generalizes previous work by Sutter et al. on finite-dimensional density matrices, yields a bound for how well a quantum state can be recovered after it has been passed through a channel. Some natural applications of our results are in quantum field theory where the von Neumann algebras are known to be of type III. Along the way we generalize various multi-trace inequalities to general von Neumann algebras.

Artificial Viscosity to Get Both Robustness and Discrete Entropy Inequalities

Artificial Viscosity to Get Both Robustness and Discrete Entropy Inequalities

Confined run-and-tumble model with boundary aggregation: Long-time behavior and convergence to the confined Fokker–Planck model

The motile micro-organisms such as Escherichia coli, sperm, or some seaweed are usually modeled by self-propelled particles that move with the run-and-tumble process. Individual-based stochastic models are usually employed to model the aggregation phenomenon at the boundary, which is an active research field that has attracted a lot of biologists and biophysicists. Self-propelled particles at the microscale have complex behaviors, while characteristics at the population level are more important for practical applications but rely on individual behaviors. Kinetic PDE models that describe the time evolution of the probability density distribution of the motile micro-organisms are widely used. However, how to impose the appropriate boundary conditions that take into account the boundary aggregation phenomena is rarely studied. In this paper, we propose the boundary conditions for a 2D confined run-and-tumble model (CRTM) for self-propelled particle populations moving between two parallel plates with a run-and-tumble process. The proposed model satisfies the relative entropy inequality and thus long-time convergence. We establish the relation between CRTM and the confined Fokker–Planck model (CFPM) studied in [J. Fu, B. Perthame and M. Tang, Fokker–Plank system for movement of micro-organism population in confined environment, J. Statist. Phys. 184 (2021) 1–25]. We prove theoretically that when the tumble is highly forward peaked and frequent enough, CRTM converges asymptotically to the CFPM. A numerical comparison of the CRTM with aggregation and CFPM is given. The time evolution of both the deterministic PDE model and individual-based stochastic simulations are displayed, which match each other well.

Global Existence of Weak Solutions to 3D Compressible Primitive Equations of Atmospheric Dynamics with Degenerate Viscosity

In this work, we show the existence of global weak solutions to the three-dimensional compressible primitive equations of atmospheric dynamics with degenerate viscosity density-dependent for large initial data. With a pressure law of the form ρ2, we represent the vertical velocity as a function of the density and the horizontal one which will be important in using Faedo-Galerkin method to obtain the global existence of the approximate solutions. In analogy with the cases in [28] and [26, 27], we prove that the weak solutions satisfy the basic energy inequality and the Bresch-Desjardins entropy inequality. Based on these estimates and using compactness arguments, we prove the global existence of weak solutions of (1.1) by vanishing the parameters in our approximate system step by step.