Thermal Equilibrium Solutions Research Articles

Gravothermal catastrophe and critical dimension in a D -dimensional asymptotically AdS spacetime

We investigate the structure and stability of the thermal equilibrium states of a spherically symmetric self-gravitating system in a $D$-dimensional asymptotically Anti-de Sitter(AdS) spacetime. The system satisfies the Einstein-Vlasov equations with a negative cosmological constant. Due to the confined structure of the AdS potential, we can construct thermal equilibrium states without any artificial wall in the asymptotically AdS spacetime. Accordingly, the AdS radius can be regarded as the typical size of the system. Then the system can be characterized by the gravothermal energy and AdS radius normalized by the total particle number. We investigate the catastrophic instability of the system in a $D$-dimensional spacetime by using the turning point method. As a result, we find that the curve has a double spiral structure for $4\le D\le 10$ while it does not have any spiral structures for $D\ge11$ as in the asymptotically flat case confined by an adiabatic wall. Irrespective of the existence of the spiral structure, there exist upper and lower bounds for the value of the gravothermal energy. This fact indicates that there is no thermal equilibrium solution outside the allowed region of the gravothermal energy. This property is also similar to the asymptotically flat case.

Thermal Instability in Radiation Hydrodynamics: Instability Mechanisms, Position-dependent S-curves, and Attenuation Curves

Local thermal instability can plausibly explain the formation of multiphase gas in many different astrophysical environments, but the theory of local TI is only well-understood in the optically thin limit of the equations of radiation hydrodynamics (RHD). Here, we lay groundwork for transitioning from this limit to a full RHD treatment assuming a gray opacity formalism. We consider a situation where the gas becomes thermally unstable due to the hardening of the radiation field when the main radiative processes are free–free cooling and Compton heating. We identify two ways in which this can happen: (i) when the Compton temperature increases with time, through a rise in either the intensity or energy of a hard X-ray component; and (ii) when attenuation reduces the flux of the thermal component such that the Compton temperature increases with depth through the slab. Both ways likely occur in the broad-line region of active galactic nuclei where columns of gas can be ionization-bounded. In such instances where attenuation is significant, thermal equilibrium solution curves become position-dependent and it no longer suffices to assess the stability of an irradiated column of gas at all depths using a single equilibrium curve. We demonstrate how to analyze a new equilibrium curve—the attenuation curve—for this purpose, and we show that, by Field’s instability criterion, a negative slope along this curve indicates that constant-density slabs are thermally unstable whenever the gas temperature increases with depth.

Thermal Equilibrium Solutions of Black Hole Accretion Flows: Outflows versus Advection

Observations and numerical simulations have shown that outflows generally exist in the accretion process. We revisit the thermal equilibrium solutions of black hole accretion flows by including the role of outflows. Our study focuses on the comparison of the cooling rate of outflows with that of advection. Our results show that, except for the inner region, outflows can dominate over advection in a wide range of flows, which is in good agreement with previous numerical simulations. We argue that an advection-dominated inner region together with an outflow-dominated outer region should be a general radial distribution for both super-Eddington accretion flows and optically thin flows with low accretion rates.

Impact of convection and resistivity on angular momentum transport in dwarf novae

The eruptive cycles of dwarf novae are thought to be due to a thermal-viscous instability in the accretion disk surrounding the white dwarf. This model has long been known to imply enhanced angular momentum transport in the accretion disk during outburst. This is measured by the stress to pressure ratio α, with α ≈ 0.1 required in outburst compared to α ≈ 0.01 in quiescence. Such an enhancement in α has recently been observed in simulations of turbulent transport driven by the magneto-rotational instability (MRI) when convection is present, without requiring a net magnetic flux. We independently recover this result by carrying out PLUTO magnetohydrodynamic (MHD) simulations of vertically stratified, radiative, shearing boxes with the thermodynamics and opacities appropriate to dwarf novae. The results are robust against the choice of vertical boundary conditions. The thermal equilibrium solutions found by the simulations trace the well-known S-curve in the density-temperature plane that constitutes the core of the disk thermal-viscous instability model. We confirm that the high values of α ≈ 0.1 occur near the tip of the hot branch of the S-curve, where convection is active. However, we also present thermally stable simulations at lower temperatures that have standard values of α ≈ 0.03 despite the presence of vigorous convection. We find no simple relationship between α and the strength of the convection, as measured by the ratio of convective to radiative flux. The cold branch is only very weakly ionized so, in the second part of this work, we studied the impact of non-ideal MHD effects on transport. Ohmic dissipation is the dominant effect in the conditions of quiescent dwarf novae. We include resistivity in the simulations and find that the MRI-driven transport is quenched (α ≈ 0) below the critical density at which the magnetic Reynolds number Rm ≤ 104. This is problematic because the X-ray emission observed in quiescent systems requires ongoing accretion onto the white dwarf. We verify that these X-rays cannot self-sustain MRI-driven turbulence by photo-ionizing the disk and discuss possible solutions to the issue of accretion in quiescence.

Thermal equilibrium solution to new model of bipolar hybrid quantum hydrodynamics

In this paper we study the hybrid quantum hydrodynamic model for nano-sized bipolar semiconductor devices in thermal equilibrium. By introducing a hybrid version of the Bhom potential, we derive a bipolar hybrid quantum hydrodynamic model, which is able to account for quantum effects in a localized region of the device for both electrons and holes. Coupled with Poisson equation for the electric potential, the steady-state system is regionally degenerate in its ellipticity, due to the quantum effect only in part of the device. This regional degeneracy of ellipticity makes the study more challenging. The main purpose of the paper is to investigate the existence and uniqueness of the weak solutions to this new type of equations. We first establish the uniform boundedness of the smooth solutions to the modified bipolar quantum hydrodynamic model by the variational method, then we use the compactness technique to prove the existence of weak solutions to the original hybrid system by taking hybrid limit. In particular, we account for two different kinds of hybrid behaviour. We perform the first hybrid limit when both electrons and holes behave quantum in a given region of the device, and the second one when only one carrier exhibits hybrid behaviour, whereas the other one is presented classically in the whole domain. The semi-classical limit results are also obtained. Finally, the theoretical results are tested numerically on a simple toy model.

Critical luminosity of Super-Dddington accretion disks

Slim disk is a kind of optically thick, advection dominated accretion flow, in which the disk height can be comparable with the disk radius. The Slim disk has been widely used to explain the observation phenomena of various Super Eddington accretion disks. Outflows are always present in the numerical simulations of Super Eddington accretion disks, which implies that only a small fraction of gas in the disk is finally swallowed by the black hole. In the study of the Slim disk, we adopt an explicit vertical gravitational force in the thermal equilibrium solutions. The results of our calculation show that there is a maximum for the gravitational force of a black hole in the vertical direction of the accretion disk, and there exists a maximally possible accretion rate for each radius in the outer region of the slim disk. The existence of a maximally possible accretion rate in the case of Super-Eddington accretion implies that only the inner region of these flows can possibly take the form of Slim disk, and strong outflows from the outer region are required to reduce the accretion rate in order for Slim disks to be realized. This result is in agreement with the numerical simulation result. Finally, we fit the curve of the disk luminosity L disk/ L Edd with the dimensionless accretion rate m ˙ , and derive an analytic relation between the total luminosity and the mass accretion rate.

A possible origin of the rapid variability of gamma-ray bursts due to convective energy transfer in hyperaccretion discs

We investigate the effects of the convection in the hyperaccretion disc around a stellar-mass black hole, which is considered to be the central engine of gamma-ray bursts (GRBs), with simple analytical calculations. If the convective energy transfer in the vertical direction becomes efficient compared with the inward advective energy transport, the hyperaccretion disc is expected to be hotter and the neutrino emission due to the electron–positron annihilation becomes the most efficient cooling process. We find that the sequence of the thermal equilibrium solutions for the convective hyperaccretion disc would have a viscously unstable branch, especially when the viscosity parameter is relatively small (α≲ 0.01). This means that the sporadical mass accretion on to a black hole would occur in this disc. We propose that this process can be the origin of the highly variable light curves observed in the prompt emissions of GRBs.

Black funnels and droplets in thermal equilibrium

It has recently been proposed that the strong coupling behaviour of quantum field theories on a non-dynamical black hole background can be described, in the context of the AdS/CFT correspondence, by a competition between two gravity duals: a black funnel and a black droplet. We present here thermal equilibrium solutions which represent such spacetimes, providing the first example where the thermal competition between the gravity duals can be studied. The solutions correspond to a special family of charged AdS C-metrics. We compute the corresponding Euclidean actions and find that the black funnel always dominates the canonical ensemble in our example, meaning that the field theory does not undergo a phase transition.

THERMAL EQUILIBRIA OF OPTICALLY THIN, MAGNETICALLY SUPPORTED, TWO-TEMPERATURE, BLACK HOLE ACCRETION DISKS

We obtained thermal equilibrium solutions for optically thin, two-temperature black hole accretion disks incorporating magnetic fields. The main objective of this study is to explain the bright/hard state observed during the bright/slow transition of galactic black hole candidates. We assume that the energy transfer from ions to electrons occurs via Coulomb collisions. Bremsstrahlung, synchrotron, and inverse Compton scattering are considered as the radiative cooling processes. In order to complete the set of basic equations, we specify the magnetic flux advection rate. We find magnetically supported (low-beta), thermally stable solutions. In these solutions, the total amount of the heating via the dissipation of turbulent magnetic fields goes into electrons and balances the radiative cooling. The low-$\beta$ solutions extend to high mass accretion rates and the electron temperature is moderately cool. High luminosities and moderately high energy cutoffs in the X-ray spectrum observed in the bright/hard state can be explained by the low-beta solutions.

THERMAL EQUILIBRIA OF MAGNETICALLY SUPPORTED BLACK HOLE ACCRETION DISKS

We present new thermal equilibrium solutions for optically thin and optically thick disks incorporating magnetic fields. The purpose of this paper is to explain the bright hard state and the bright/slow transition observed in the rising phases of outbursts in black hole candidates. On the basis of the results of three-dimensional magnetohydrodynamic simulations, we assume that magnetic fields inside the disk are turbulent and dominated by the azimuthal component and that the azimuthally averaged Maxwell stress is proportional to the total (gas, radiation, and magnetic) pressure. We prescribe the magnetic flux advection rate to determine the azimuthal magnetic flux at a given radius. Local thermal equilibrium solutions are obtained by equating the heating, radiative cooling, and heat advection terms. We find magnetically supported (β = (pgas + prad)/pmag < 1), thermally stable solutions for both optically thin disks and optically thick disks, in which the heating enhanced by the strong magnetic field balances the radiative cooling. The temperature in a low-β disk (T ∼ 107–1011K) is lower than that in an advection-dominated accretion flow (or radiatively inefficient accretion flow) but higher than that in a standard disk. We also study the radial dependence of the thermal equilibrium solutions. The optically thin, low-β branch extends to , where is the mass accretion rate and is the Eddington mass accretion rate, in which the temperature anticorrelates with the mass accretion rate. Thus, optically thin low-β disks can explain the bright hard state. Optically thick, low-β disks have the radial dependence of the effective temperature Teff ∝ ϖ−3/4. Such disks will be observed as staying in a high/soft state. Furthermore, limit cycle oscillations between an optically thick low-β disk and a slim disk will occur because the optically thick low-β branch intersects with the radiation pressure dominated standard disk branch. These limit cycle oscillations will show a smaller luminosity variation than that between a standard disk and a slim disk.

Constraints on Slim Accretion Discs

We show that when the gravitational force in the vertical direction is correctly calculated, the well-known S- shaped sequence of thermal equilibrium solutions can be constructed only for small radii of black hole accretion flows, such that slim accretion discs can possibly exist only in the inner regions of these flows.

A Note on the Slim Accretion Disk Model

We show that when the gravitational force is correctly calculated in dealing with the vertical hydrostatic equilibrium of black hole accretion disks, the relationship that is valid for geometrically thin disks, i.e., $c_s/\Omega_K H =$ constant, where $c_s$ is the sound speed, $\Omega_K$ is the Keplerian angular velocity, and $H$ is the half-thickness of the disk, does not hold for slim disks. More importantly, by adopting the correct vertical gravitational force in studies of thermal equilibrium solutions, we find that there exists a maximally possible accretion rate for each radius in the outer region of optically thick accretion flows, so that only the inner region of these flows can possibly take the form of slim disks, and strong outflows from the outer region are required to reduce the accretion rate in order for slim disks to be realized.

On the bipolar multidimensional quantum Euler–Poisson system: The thermal equilibrium solution and semiclassical limit

In this paper we establish the existence of a unique thermal equilibrium solution of the bipolar multidimensional quantum Euler–Poisson system over the whole space R 3 and obtain the relevant semiclassical limit and a combined Planck–Debye length limit through using energy estimating methods and a fixed point theory.

A Unified Perspective on the Dynamics of Axisymmetric Hurricanes and Monsoons

Abstract This paper provides a unified perspective on the dynamics of hurricane- and monsoonlike vortices by identifying them as specific limiting cases of a more general flow system. This more general system is defined as stationary axisymmetric balanced flow of a stably stratified non-Boussinesq atmosphere on the f plane. The model is based on the primitive equations assuming gradient wind balance in the radial momentum equation. The flow is forced by heating in the vortex center, which is implemented as relaxation toward a specified equilibrium temperature Te. The flow is dissipated through surface friction, and it is assumed to be almost inviscid in the interior. The heating is assumed supercritical, which means that Te does not allow a regular thermal equilibrium solution with zero surface wind, and which gives rise to a cross-vortex secondary circulation. Numerical solutions are obtained using time stepping to a steady state, where at each step the Eliassen secondary circulation is diagnosed as part of the solution strategy. Reality and regularity of the solution is discussed, putting this work in relation to previous work. Scaling analysis suggests that for a given geometry, essential vortex properties are controlled by the ratio ℱ = αT/cD, where αT is the rate of thermal relaxation and cD quantifies the strength of surface friction for a given surface wind. For large ℱ, the temperature is close to Te and the vortex shows properties that can be associated with a hurricane including strong cyclonic surface winds. On the other hand, for small ℱ, the vortex shows properties that can be associated with a monsoon; that is, the surface winds are small and the secondary circulation keeps the temperature significantly away from Te. The scaling analysis is verified by numerical solutions spanning a wide range of the parameter space. It is shown how the two limiting cases correspond with the respective approximate semianalytical theories presented previously. The results imply an important role of αT for hurricane formation.

Analysis and Synthesis IV: Limits and Supplements

Analysis and Synthesis IV: Limits and Supplements

Nonlinear time evolution of thermal structures

The nonlinear stability and time evolution of thermal structures constituted by optically thin plasmas are analyzed. The structure has been assumed to be heated at a rate ∼Tm, cooled by the standard cooling function for plasmas with solar abundances and with an anisotropic thermal conduction coefficient. Second-order analytical results are obtained for the thermal equilibrium solutions. For nonhomogeneous solutions and strong disturbances, a numerical analysis is carried out. The angle between the temperature gradient and the magnetic field is a crucial parameter in determining the stability of structures close to the marginal state. A central overheating may occur for large enough amplitudes of the initial disturbance imposed on stable steady-state thermal structures. Implications of the above results in the formation of cool structures in the solar atmosphere and in the interstellar medium are outlined.

Einstein-Fokker-Planck Equation for a Collapsing Perfect Fluid Star in a Thermal Noise Bath: Static Thermal Black Hole as the Equilibrium Solution

This paper considers sphericalOppenheimer-Snyder gravitational collapse of dust orperfect fluid “stars” immersed within aspacetime containing a thermal bath of (Gaussian) whitenoise at a temperature T, obeying the autocorrelations of thefluctation-dissipation theorem. Candidates for theresulting non-linear Einstein-Langevin (EL) stochasticdifferential field equations are developed. A collapsing fluid or dust star coupled to the stochastic,external thermal bath of fluctuations is theninterpreted as an example of a non-linear, noisy system,somewhat analogous to a non-linear Brownian motion in a viscous, thermal bath at temperature T. AnEinstein-Fokker-Planck (EFP) hydrodynamical continuityequation, describing the collapse as a probability flowwith respect to the exterior standard time ts outside the collapsing body, is developed. Thethermal equilibrium or stationary solution can bederived in the infinite standard time relaxation limit.This limit (ts → ∞) only exists for a static, external observer within thenoise bath viewing the collapsing sphere such that R→ 1 (the event horizon) with unit probability asts → ∞. The stationary or thermalequilibrium solution of the efp equations therefore seemsto correspond to a static black hole in a Hartle-Hawkingstate at the Hawking temperature tH. The OSmodel first predicted event horizons and singularities. It is interesting that through a simplestochastic extension of the model, one can conclude thatthe final collapsed, static, equilibrium state of thebody must be a thermal black hole at the Hawkingtemperature.

Thermally Forced Stationary Axisymmetric Flow on thefPlane in a Nearly Frictionless Atmosphere

Abstract This paper investigates stationary axisymmetric balanced flow of a stably stratified dry non-Boussinesq atmosphere on the f plane. The circulation is forced in the troposphere through thermal relaxation toward a specified equilibrium temperature and is damped through Rayleigh friction in the interior of the domain. Surface friction is sufficiently strong to ensure weak surface winds. As in the analogous zonally symmetric problem studied by Plumb and Hou there is threshold behavior in the frictionless limit with a thermal equilibrium solution for subcritical forcing and a highly nonlinear so-called angular momentum conserving (AMC) solution for supercritical forcing. The latter is characterized by a sharp outward edge of the vortex circulation and a nonvanishing secondary cross-vortex circulation. In the frictionless limit, the secondary circulation does not reach above the region of the thermal forcing. Noticeable differences of the current problem with respect to the zonally symmetric problem ar...

Light scattering properties of paramagnetic particles

We present an experimental study of light scattering properties of paramagnetic particles. To account for the magnetic dipole radiation and the Brownian motion of the particles in a thermal equilibrium solution, we calculate the scattering intensity and its auto-correlation function g(t). Experimentally, we examine the scattering properties of the paramagnetic particles and compare the results with those from isotropic and anisotropic dielectric particles. The experiment verifies the calculation and reveals that the magnetic dipole radiation of the paramagnetic particles is unusually large and equals to approximately 1/3 of the electric dipole radiation of the particles. Dynamic light scattering measurements show that the measured g(t) for the depolarized scattering is strongly influenced by the size distribution of the particles. This is because the large paramagnetic particles tend to have more magnetite content and hence weigh more in the depolarized scattering. With a simple sedimentation method, we are able to separate the particles of different sizes and obtain relatively monodispersed scattering samples. These samples give the expected translational and rotational diffusion constants of the particles.

The thermal equilibrium state of semiconductor devices

The thermal equilibrium state of two oppositely charged gases confined to a bounded domain ▪, m = 1,2 or m = 3, is entirely described by the gases' particle densities p, n minimizing the total energy ε(p, n). it is shown that for given P, N > 0 the energy functional ε admits a unique minimizer in {(p, n) ϵ L2(Ω) x L2(Ω) : p, n ≥ 0, ʃΩp = P, ʃΩn = N} and that p, n ϵ C(Ω) ∩ L∞(Ω).The analysis is applied to the hydrodynamic semiconductor device equations. These equations in general possess more than one thermal equilibrium solution, but only the unique solution of the corresponding variational problem minimizes the total energy. It is equivalent to prescribe boundary data for electrostatic potential and particle densities satisfying the usual compatibility relations and to prescribe Ve and P, N for the variational problem.