Hydrodynamic Equations Research Articles

One-dimensional Carrollian fluids. Part I. Carroll-Galilei duality

Galilean and Carrollian algebras acting on two-dimensional Newton-Cartan and Carrollian manifolds are isomorphic. A consequence of this property is a duality correspondence between one-dimensional Galilean and Carrollian fluids. We describe the dynamics of these systems as they emerge from the relevant limits of Lorentzian hydrodynamics, and explore the advertised duality relationship. This interchanges longitudinal and transverse directions with respect to the flow velocity, and permutes equilibrium and out-of-equilibrium observables, unveiling specific features of Carrollian physics. We investigate the action of local hydrodynamic-frame transformations in the Galilean and Carrollian configurations, i.e. dual Galilean and Carrollian local boosts, and comment on their potential breaking. Emphasis is laid on the additional geometric elements that are necessary to attain complete systems of hydrodynamic equations in Newton-Cartan and Carroll spacetimes. Our analysis is conducted in general Cartan frames as well as in more explicit coordinates, specifically suited to Galilean or Carrollian use.

Thermodynamics and generalized hydrodynamics of simple integrable QFT in finite volume

Abstract We derive thermodynamic (TBA) and general hydrodynamic (GHD) equations corrected by virtual processes for integrable QFT on large but finite size space circle. Obtained TBA’s are solved numerically for the sinh-Gordon model. Complicated Euler scale GHD equations are expanded explicitly for small occupation ratio of virtual quasiparticles. The spectrum of velocities for the linear approximation to GHD is numerically calculated.

Theory of Turbulent Equilibrium Spheres with Power-law Linewidth–Size Relation

Dense cores inherit turbulent motions from the interstellar medium in which they form. As a tool for comparison to both simulations and observations, it is valuable to construct theoretical core models that can relate their internal density and velocity structure while predicting their stability to gravitational collapse. To this end, we solve the angle-averaged equations of hydrodynamics under two assumptions: (1) the system is in a quasi-steady equilibrium; (2) the velocity field consists of radial bulk motion plus isotropic turbulence, with turbulent dispersion increasing as a power law in the radius. The resulting turbulent equilibrium sphere (TES) solutions form a two-parameter family, characterized by the sonic radius r s and the power-law index p. The TES is equivalent to a Bonnor–Ebert (BE) sphere when r s → ∞ . The density profile in the outer regions of the TES is slightly shallower than the BE sphere, but is steeper than the logotropic model. Stability analysis shows that the TESs with size exceeding a certain critical radius are unstable to radial perturbations. The center-to-edge density contrast, mass, and radius of the marginally stable TES all increase with increasing average velocity dispersion. The FWHM of the column density profile is always smaller than the critical radius, by a larger factor at higher velocity dispersion, suggesting that observations need to probe beyond the FWHM to capture the full extent of turbulent cores. When applied to the highly turbulent regime typical of cluster-forming clumps, the critical mass and radius of the TES intriguingly resembles the typical mass and radius of observed star clusters.

Multiple solutions of nonlinear coupled constitutive relation model and its rectification in non-equilibrium flow computation

In this study, the multiple solutions of Nonlinear Coupled Constitutive Relation (NCCR) model are observed and a way for identifying the physical solution is proposed. The NCCR model proposed by Myong is constructed from the generalized hydrodynamic equations of Eu, and aims to describe rarefied flows. In the non-equilibrium regime, the NCCR equations are more reliable than the Navier-Stokes equations. And the NCCR equations have an advantage of efficiency over the discrete velocity methods and stochastic particle methods. However, the NCCR model is a complicated nonlinear system. Many assumptions have been used in the schemes for solving the NCCR equations. The corresponding numerical methods may be associated with unphysical solution and instability. At the same time, it is hard to analyze the physical accuracy and stability of NCCR model due to the uncertainties in the numerical discretization. In this study, a new numerical method for solving NCCR equations is proposed and used to analyze the properties of NCCR equations. More specifically, the nonlinear equations are converted into the solutions of an objective function of a single variable. Under this formulation, the multiple solutions of the NCCR system are identified and the criteria for picking up the physical solution are proposed. Therefore, a numerical scheme for solving NCCR equations is constructed. A series of flow problems in the near continuum and low transition regimes with a large variation of Mach numbers are conducted to validate the numerical performance of proposed method and the physical accuracy of NCCR model.

Hydrodynamic equations for a U(N) invariant superfluid

Abstract In this paper, we develop an appropriate set of hydrodynamic equations for a U(N) invariant superfluid that couple the dynamics of superflow and magnetization. In the special case when both the superfluid and normal velocities are zero, the hydrodynamic equations reduce to a generalized version of Landau-Lifshitz equation for ferromagnetism with U(N) symmetry. When both velocities are non-zero, there appears couplings between the superflow and magnetization dynamics, and the superfluid velocity no longer satisfies the irrotational condition. On the other hand, the magnitude of magnetization is no longer a constant of motion as was the case for the standard Landau-Lifshitz theory. In comparison with the simple superfluid, the first and second sounds are modified by a non-zero magnetization through various thermodynamic functions. For U(2) invariant superfluid, we get both (zero-) sound wave and a spin wave at zero temperature. It is found that the dispersion of spin wave is always quadratic, which is consistent with microscopic analysis. In the Appendix, we
show that the hydrodynamic equation for a U(N) invariant superfluid can be obtained from the general hydrodynamic equation with arbitrary internal symmetries.

Structure, maximum mass, and stability of compact stars in f(Q,T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(\\mathcal {Q,T})$$\\end{document} gravity

We investigate the properties of compact objects in the f(Q, T) theory, where Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {Q}$$\\end{document} is the non-metricity scalar and T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${ \\mathcal {T}}$$\\end{document} is the trace of the energy–momentum tensor. We derive an interior analytical solution for anisotropic perfect-fluid spheres in hydrostatic equilibrium using the linear form of f(Q,T)=Q+ψT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(\\mathcal {Q}, { \\mathcal {T}})=\\mathcal {Q}+\\psi { \\mathcal {T}}$$\\end{document}, where ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} represents a dimensional parameter. Based on the observational constraints related to the mass and radius of the pulsar SAX J1748.9-2021, ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} is set to a maximum negative value of ψ1=ψ/κ2=-0.04\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _1=\\psi / \\kappa ^2=-0.04$$\\end{document}, where κ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa ^2$$\\end{document} is the gravitational coupling constant. The solution results in a stable compact object, which does not violate the speed of sound condition cs2≤c23\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_s^2\\le \\frac{c^2}{3}$$\\end{document}. The effective equation of state is similar to the quark matter equation of state, and involves the presence of an effective bag constant. When ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} is negative, the star has a slightly larger size as compared to GR stars with the same mass. The difference in the predicted star size between the theory with a negative ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} and GR for the same mass is attributed to an additional force appearing in the hydrodynamic equilibrium equation. The maximum compactness allowed by the strong energy condition for f(Q,T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(\\mathcal {Q}, { \\mathcal {T}})$$\\end{document} theory and for GR is C=0.514\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C = 0.514$$\\end{document} and 0.419, respectively, with the f(Q,T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(\\mathcal {Q}, { \\mathcal {T}})$$\\end{document} prediction about 10%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10\\%$$\\end{document} higher than the GR one. Assuming a surface density at saturation nuclear density of ρnuc=4×1014g/cm3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _{\ ext {nuc}} = 4\ imes 10^{14}~\\hbox {g}/\\hbox {cm}^3$$\\end{document}, the maximum mass of the star is 4.66M⊙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4.66 M_\\odot $$\\end{document}, with a radius of 14.9 km.

Simulating Nonlinear Radiation Diffusion Through Quantum Computing

The recent success of the fusion ignition experiment at Lawrence-Livermore National Laboratory has renewed excitement for inertial confinement fusion. Designing such experiments relies on computational modeling using radiation hydrodynamic codes run on massively parallel supercomputers which simulate hydrodynamic flows, radiation diffusion, and thermonuclear burn. Constructing a quantum algorithm for radiation hydrodynamics that provides a quantum speedup is of great interest. A recent quantum algorithm that solves the Navier-Stokes equations of hydrodynamics with a quadratic speedup marks a first step towards this goal. Here we take the next step and present a quantum algorithm for nonlinear radiation diffusion that also has a quadratic speedup. To verify the algorithm we consider radiation striking a cold, optically thick target that generates a Marshak wave in the target. Results of numerical simulation of the quantum algorithm are compared with those of a standard partial differential equation solver and excellent agreement is found.

Lattice Boltzmann Model for Rarefied Gaseous Mixture Flows in Three-Dimensional Porous Media Including Knudsen Diffusion

Numerical modeling of gas flows in rarefied regimes is crucial in understanding fluid behavior in microscale applications. Rarefied regimes are characterized by a decrease in molecular collisions, and they lead to unusual phenomena such as gas phase separation, which is not acknowledged in hydrodynamic equations. In this work, numerical investigation of miscible gaseous mixtures in the rarefied regime is performed using a modified lattice Boltzmann model. Slip boundary conditions are adapted to arbitrary geometries. A ray-tracing algorithm-based wall function is implemented to model the non-equilibrium effects in the transition flow regime. The molecular free flow defined by the Knudsen diffusion coefficient is integrated through an effective and asymmetrical binary diffusion coefficient. The numerical model is validated with mass flow measurements through microchannels of different cross-section shapes from the near-continuum to the transition regimes, and gas phase separation is studied within a staggered arrangement of spheres. The influence of porosity and mixture composition on the gas separation effect are analyzed. Numerical results highlight the increase in the degree of gas phase separation with the rarefaction rate and the molecular mass ratio. The various simulations also indicate that geometrical features in porous media have a greater impact on gaseous mixtures’ effective permeability at highly rarefied regimes. Finally, a permeability enhancement factor based on the lightest species of the gaseous mixture is derived.

A multi-scale IMEX second order Runge-Kutta method for 3D hydrodynamic ocean models

Understanding complex physical phenomena often involves dealing with partial differential equations (PDEs) where different phenomena exhibit distinct timescales. Fast terms, associated with short characteristic times, coexist with slower ones requiring relatively longer time steps for resolution. The challenge becomes more manageable when, despite the varying characteristic times of fast and slow terms, the computational cost associated with faster terms is significantly lower than that of slower terms. Additionally, slower terms can also exhibit two distinct longer characteristic times, adding complexity to the system and resulting in a total of three characteristic timescales. In this paper, an innovative split second-order IMEX (IMplicit-EXplicit) temporal scheme is introduced to address this temporal complexity. It is used to solve the primitive equation ocean model. Extremely short times are handled explicitly with small time steps, while longer timescales are managed explicitly and semi-implicitly using larger time steps. The decision to solve a portion of the slower terms semi-implicitly is due to the fact that it does not significantly increase the total computational cost, allowing for greater flexibility in the time step without imposing a substantial burden on the overall computational efficiency. This strategy enables efficient management of the various temporal scales present in the equations, thereby optimizing computational resources. The proposed scheme is applied to solve 3D hydrodynamics equations encompassing three time scale: fast terms representing wave phenomena, slow terms describing horizontal aspects and stiff terms for vertical ones. Furthermore, the scheme is designed to respect crucial physical properties, namely global and local conservation. The obtained results on different test cases demonstrate the robustness and efficiency of the IMEX approach in simulating these complex systems.

On the expansion of a flow into vacuum for spherically symmetric relativistic hydrodynamic equations

Fluid expansion into a vacuum is an interesting and important problem in continuum physics as well as in mathematics. This paper studies the relativistic Euler equations with an initial value that is a combination of a radial flow in a spherical domain and vacuum in the remaining domain. We prove that this expansion into vacuum problem admits a global-in-time classical solution, provided that the initial mass-energy density in the spherical domain is sufficiently small.

Description of the emission of cumulative secondary particles in collisions of heavy ions of intermediate energies based on the non-equilibrium hydrodynamic approach

We have analyzed the double-differential cross sections for the emission of cumulative protons, pions, kaons, and antiprotons in collisions of carbon nuclei on a fixed target at an energy of 19.6 GeV/nucleon obtained in the IHEP experiment at the U-70 accelerator. When describing these spectra, the nonequilibrium approach was taken into account as a result of the joint solution of the kinetic equation with the equations of hydrodynamics. Comparisons with other approaches are made.

Nonlinear Structures of Dispersive Electrostatic Solitary Waves in a Multi‐Ion Partially Ionized Plasma

ABSTRACTThe nonlinear structure and dynamics of dispersive solitons and breather waves described by Korteweg‐de Vries and nonlinear Schrödinger equations are studied. The theoretical and numerical study of the generalized hydrodynamic equations, accounting for wave dissipation and particle production‐loss mechanism, are considered. The reductive expansion method has been used in the context of the instability problem of multi‐fluid dynamics, applied to the study of electrostatic solitons and ion‐acoustic waves. A nonlocal model of interacting solitary‐breather waves has been presented. Applications of the theory, concerning the ion streaming instability in the framework of plasma physics, are presented.

Boundary Value and Control Problems for the Stationary Magnetic Hydrodynamic Equations of Heat Conducting Fluid with Variable Coefficients

Boundary Value and Control Problems for the Stationary Magnetic Hydrodynamic Equations of Heat Conducting Fluid with Variable Coefficients

Klein–Gordon Equations, Equations of Relativistic Quantum Hydrodynamics, and Quantum Shocks in Describing Collisions of Atomic Nuclei

Klein–Gordon Equations, Equations of Relativistic Quantum Hydrodynamics, and Quantum Shocks in Describing Collisions of Atomic Nuclei

A linearized hydrodynamic code for laser ablation and its application to laser pulse shaping for direct-drive fusion

One of the most harmful processes in inertial confinement fusion is Rayleigh–Taylor instability (RTI), and an efficient way to mitigate it is pulse shaping. However, because shaped laser pulses lead to unsteady ablation, it is insufficient to evaluate RTI based solely on the instability growth rate. Here, for better prediction of RTI during linear growth, hydrodynamic equations for laser ablation (including both balance and linearized perturbation equations) are solved numerically and used to optimize the laser pulse shape for direct-drive inertial confinement fusion. For given target conditions and laser energy, simulations show that a picket pulse before the main laser pulse can reduce RTI significantly, and it is clear that the reduction comes from two aspects: (i) the lower RTI seed due to rarefaction at the descending edge of the picket in the imprint stage and (ii) the smaller growth rate due to enhanced ablation velocity at the main pulse in the acceleration stage. It is found that the perturbed laser deposition in an underdense plasma also has a profound influence on RTI seeds in the imprint stage.

A nonequilibrium statistical mechanics derivation of the hydrodynamic equations of simple fluids using a noncanonical form of the Poisson bracket

The continuum level hydrodynamic equations of a simple fluid were derived by Irving and Kirkwood directly from discrete particle dynamics using statistical mechanics almost 75 years ago. Their elegant derivation demonstrated the fundamental molecular basis of macroscopic fluid flow and culminated in molecular expressions for the stress tensor and heat current density that have since been employed in countless molecular simulations to date. In this article, an alternative derivation is presented, which leads to more general expressions for the fundamental transport relationships and which arrives at them in a more straightforward chain of consistency that ensues directly from the Principle of Least Action. The main point of departure from the Irving–Kirkwood derivation is the application of a transformation mapping of the total momentum of each individual particle onto the sum of its peculiar momentum and its momentum relative to the local velocity field. This mapping provides a phase-space distribution function applicable in the space of particle positions and peculiar momentum, from which a noncanonical Poisson bracket can be derived in terms of the same set of microscopic variables. For a given dynamic variable, expressed in terms of particle positions and peculiar momenta, the expectation value of the noncanonical Poisson bracket of the dynamic variable is shown to correspond to the evolution equation of the expectation value of the dynamic variable. This allows for a direct derivation of all macroscopic density evolution equations (mass, momentum, and energy density fields) using a systematic procedure free of assumptions concerning the macroscopic state of the system. Furthermore, an explicit expression of the time evolution of the entropy density at the hydrodynamic level is derived following the same procedure. Finally, in the limit of short-range interparticle interactions, a molecular-based expression for the local stress tensor as properly defined from continuum mechanics is developed at the hydrodynamic level that elucidates the continuum mechanics connection of the general stress expression of Irving and Kirkwood.

An analytical solution to the soil’s dynamic response induced by wave interaction with a marine current characterized by a non-uniform vertical profile

In this work, we obtain an analytical solution to the soil’s dynamic response induced by wave interaction with a marine current with a prescribed non-uniform vertical profile. The control volume used to analyze the present problem is divided into a water region and another composed of a poroelastic soil region. The governing equations for the water hydrodynamics are presented in its dimensionless version and they are simplified for the limit of linear water waves. The velocity field in the water region is determined from the continuity equation and the “inviscid Orr-Sommerfeld linear equation”. We obtain a dispersion equation for wave-current interaction. For modeling the soil’s dynamics response, we adopt the u-p approximation. Considering a characteristic criterion for soil liquefaction, the maximum liquefaction depth is obtained as an eigenvalue problem. The analytical estimations, valid for the limit of linear water waves, provided by the present analysis show that, marine currents traveling in the same direction to the wave propagation induces larger values of the liquefaction depth than those obtained for currents opposite to wave propagation. As a first approximation, the present analysis may help understand the dynamic response of poroelastic soil induced by the wave-non-uniform marine current interaction.

Role of Cairns–Gurevich ion distribution on nonlinear wave propagation in negatively charged dusty plasma

The study of dust-acoustic (DA) nonlinear electrostatic waves in dusty plasma is crucial for understanding plasma behavior in both astrophysical and laboratory environments. Dusty plasmas, characterized by the presence of negatively charged dust particles, exhibit complex dynamics influenced by various factors, including nonthermal electron and ion populations following Cairns–Gurevich distributions. This study addresses the problem of understanding wave propagation under these specific conditions by employing a set of fluid hydrodynamic equations for dust fluid, alongside appropriate electron and Cairns–Gurevich ion distributions, to model the dynamics and propagation of DA nonlinear electrostatic waves under specific plasma conditions with negatively charged dust particles. We derived a nonlinear Zakharov–Kuznetsov (ZK) equation to model the evolution of these waves and further transformed it into the nonlinear Schrödinger (NLS) equation to analyze rogue wave formation and identify unstable and stable zones within the plasma. We focused our analysis on the effects of key parameters, such as the nonthermal index, magnetic field strength, negative dust temperature ratio, polarization force, and density ratio, on the behavior of dust-acoustic waves (DAWs). The results demonstrated that soliton waves, explosive waves, and kink waves exhibit distinct behaviors depending on these parameters and the spatial context of Earth's magnetosphere. These findings provide new insights compared to previous studies into the conditions under which these waveforms emerge and evolve, especially in magnetized environments where earlier models were less effective at accurately characterizing or predicting wave behavior. By applying two different analytical methods, a direct integration approach and a generalized Kudryashov method, we expanded our understanding of nonlinear wave phenomena in dusty plasmas. Our results not only advance theoretical knowledge but also have practical implications for interpreting space plasma observations and guiding experimental design in plasma physics research. This study thus contributes to a more comprehensive understanding of plasma dynamics, with potential applications to astrophysical observation and laboratory plasma experimentation.

Generalized hydrodynamics for the volterra lattice: ballistic and non-ballistic behavior of correlation functions

Abstract In recent years, a lot of effort has been put in describing the hydrodynamic behavior of in- tegrable systems. In this paper, we describe such picture for the Volterra lattice. Specifically, we are able to explicitly compute the susceptibility matrix and the current-field correlation matrix in terms of the density of states of the Volterra lattice endowed with a Generalized Gibbs ensemble. Furthermore, we apply the theory of linear Generalized Hydrodynamics to describe the Euler scale behavior of the correlation functions. We anticipate that the solution to the Generalized Hydrodynamics equations develops shocks at ξ0=x/t ; so this linear approximation does not fully describe the behavior of correlation functions. Intrigued but this fact, we performed several numerical investigations which show that, exactly when the solution to the hydrodynamic equations develops shock, the correlation functions show an highly oscillatory behavior. In view of this empirical observation, we believe that at this point ξ0 the diffusive contribution are not sub-leading corrections to the ballistic transport, but they are of the same order.

Temperature effects on the electrical conductivity of K‐feldspar

AbstractK‐feldspar, which constitutes about 60 of the Earth's crust, is crucial for understanding electrical conductivity in porous rocks. Its electrical properties are vital for applications in ceramics, electrical insulation and conductive polymers. In this work, we study the time evolution of electrical conductivity of K‐feldspar‐rich rocks with varying temperatures, at high and low pH, which has been studied through simulation using time domain random walk. Random walkers, mimicking ions in transport, move in accordance with appropriate hydrodynamic equations, dissolution and precipitation kinetics. Electrical conductivity has been calculated considering variations in the parameters of temperature, fluid pH and the abundance of K‐feldspar in rocks. Electrical conductivity is found to increase with temperature up to a critical value, after which it decreases. The sharpness of the rise and fall in electrical conductivity is quantified through a measure defined as the conductivity quality factor . We find that increases with a decrease in the availability of K‐feldspar mineral. Our simulated results of electrical conductivity show a good match with the experimental trends reported.