Stability and causality of Carter's multifluid theory

Abstract

Stability and causality are studied for linear perturbations about equilibrium in Carter's multifluid theory. Our stability analysis is grounded on the requirement that the entropy of the multifluid, plus that of the environment, must be maximised at equilibrium. This allows us to compute a quadratic Lyapunov functional, whose positive definiteness implies stability. Furthermore, we verify explicitly that, also for multifluids, thermodynamic stability implies linear causality. As a notable stability condition, we find that the entrainment matrix must always be positive definite, confirming a widespread intuition. I. INTRODUCTIONCarter's multifluid theory [1][2][3] is the hydrodynamic framework currently adopted for modelling superfluid-normal mixtures in full general relativity [4][5][6]. It extends the notion of perfect fluid to interacting systems in which nondiffusive relative flows can survive over hydrodynamic time-scales. As such, it finds application in neutron star physics [7][8][9]: dense hadronic matter is believed to be a mixture of several chemical components, some of which are superfluid (free to spin at different rates [10][11][12]). Furthermore, the ability to describe non-diffusive out-of-equilibrium fluxes makes Carter's theory well suited for modelling dissipation beyond Fick's law [13][14][15], elevating the formalism to a pillar of Relativistic Extended Irreversible Thermodynamics [16][17][18].Despite the relevance of Carter's multifluid approach, both for non-equilibrium statistical mechanics and for astrophysical modelling, very little is known about its mathematical properties. In particular, to date no systematic study of its stability and causality properties has ever been carried out. In other words, we do not know under which conditions the initial value formulation of Carter's theory is reliable, and produces physically meaningful solutions. Only few specific hydrodynamic models, built using Carter's approach, have been shown to be reliable (or non-reliable [19]). This has always been done by invoking some mathematical correspondence [20] with the Israel-Stewart theory [21], whose stability-causality properties are well known [22]. Unfortunately, such correspondence is limited to theories with only two currents (entropy and particles), and cannot be extended to, e.g., neutron star hydrodynamics (which requires at least three currents: entropy, protons, and neutrons). This is a serious issue, not only because we do not know if the currently adopted multifluid models are reliable, but also because we have no idea of which factors contribute to make a theory stable, and which approximations, instead, may be harmful. The case of Carter's "regular theory" for heat conduction is emblematic: Carter correctly identified the origin of the instability of the theories of Eckart [23] and Landau and Lifshitz [24]; he formulated a new theory with the precise goal of fixing such pathologies [13]; nevertheless, the resulting theory turned out to be still unstable (for some realistic equations of state...

Simulating bulk viscosity in neutron stars I: formalism

“…They adopted the prescription of extended irreversible thermodynamics [50,51] of promoting the dissipative fluxes to new dynamic degrees of freedom. The resulting theory is by construction causal [36], and, as a consequence [52], stable [53,54]. Moreover, since the expansion is performed near local equilibrium, and not for small gradients 2 , Müller-Israel-Stewart theories are able (in some cases [58]) to accurately describe the dynamics of the slowest non-equilibrium degrees of freedom [56,59].…”

“…Recently, Gavassino et al [19] proposed a framework for bulk-viscous fluids that builds on the formalism of Carter [35] and naturally allows for hyperbolic equations and causal solutions [54], which are the requirements needed for numerical applications and for thermodynamic stability [52,61]. As shown in Gavassino et al [19], when the quasi-equilibrium state of a fluid departing from full equilibrium can be parameterized by a number of scalar variables, it can be mapped into a chemically reacting mixture.…”

Local thermodynamical equilibrium and relativistic dissipation

“…The formulas in (25)(26)(27)(28)(29)(30)(31)(32)(33) express the transformation laws of the various thermodynamical variables under a frame change described in ( 24) and there will be used frequently in the following analysis. Under such frame change, they show that most of the thermodynamical variables remain practically frame independent as long as second order and higher order deviations from the state of "local thermodynamical equilibrium" specified by (u µ , s(ρ, n)), are neglected.…”

“…Notice that the assumption ǫ << 1 in (19) implies that the inequality τ C (p) << τ M (p), becomes observer independent in the sense that as long as (u µ , ûµ ) in ( 24) are chosen to lie within the "cone" of the opening pseudo-angle ǫ = O 1 << 1 then time dilatation and length contraction are effects considered as been inessential. Moreover, the "invariance" of the inequality τ C (p) << τ M (p), under the frame change in (24) in combination to (25)(26)(27)(28)(29)(30)(31)(32)(33), permit us to simply refer to a state of "local thermodynamical equilibrium" without any further reference to which particular (u µ , s(ρ, n)) this fictitious state' is associated with.…”

“…[36], [32] the stability properties of transient thermodynamics has been addressed. The stability of equilibrium states and causality properties of Carter's theory of heat conducting relativistic fluids (for an introduction to this theory see [23] [24] [25]) has been the focus of refs [26], [27], [29].…”