Cococubed.com Thermodynamics
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Modifying the Free Energy in Skye (2022) The Skye Equation of State (EOS) is built on top of automatic differentiation machinery. This makes it straightforward to modify, because all the requisite partial derivatives of the Helmholtz free energy are computed automatically via operator overloading. In this research note we demonstrate this advantage by implementing a new prescription for the anharmonic free energy of a crystalline one-component plasma. Apart from boilerplate and variable declarations, only 7 new lines of code were needed. Difference in the adiabatic temperature gradient ∇ad computed with the Skye EOS using the Baiko & Chugunov (2022, BC22) and Potekhin & Chabrier (2010, PC10) prescriptions for the anharmonic Helmholtz free energy is shown for an even-mass mixture of 12C and 16O as a function of density and temperature. Skye: A Differentiable Equation of State (2021) Stellar evolution and numerical hydrodynamics simulations depend critically on access to fast, accurate, thermodynamically consistent equations of state. In this article we present Skye, a new equation of state for fully-ionized matter. Skye includes the effects of positrons, relativity, electron degeneracy, Coulomb interactions, non-linear mixing effects, and quantum corrections. Skye determines the point of Coulomb crystallization in a self-consistent manner, accounting for mixing and composition effects automatically. A defining feature of this equation of state is that it uses analytic free energy terms and provides thermodynamic quantities using automatic differentiation machinery. Because of this, Skye is easily extended to include new effects by simply writing new terms in the free energy. We also introduce a novel thermodynamic extrapolation scheme for extending analytic fits to the free energy beyond the range of the fitting data while preserving desirable properties like positive entropy and sound speed. We demonstrate Skye in action in the MESA stellar evolution software instrument by computing white dwarf cooling curves. The Skye EOS = an improved Helmholtz EOS for the non-interacting parts + an improved Potekhin & Chabrier EOS for the Coulomb plama parts + auto-differentiation. Its the bees knees for ionized plasmas as of 2021. Skye is avaliable at https://github.com/adamjermyn/Skye. Coverage of the Skye EOS Adiabatic temperature gradient Thermodynamic consistency Liquid-solid free energy difference and phase Latent heat cooling delays The Accuracy, Consistency, and Speed Of An Electron–Positron Equation Of State Based On Table Interpolation Of The Helmholtz Free Energy (2000) In this article, an electron-positron equation of state based on table interpolation of the Helmholtz free energy is developed and analyzed. The interpolation scheme guarantees perfect thermodynamic consistency, independent of the interpolating function. The choice of a biquintic Hermite polynomial as the interpolating function results in accurately reproducing the underlying Helmholtz free energy data in the table, and yields derivatives of the pressure, specific entropy and specific internal energy which are smooth and continuous. The execution speed – evaluated across several different machine architectures, compiler options, and mode of operation – suggest that the Helmholtz equation of state routine is faster than any of the five equation of state routines surveyed by Timmes & Arnett (1999). When an optimal balance of accuracy, thermodynamic consistency, and speed is desirable, then the tabular Helmholtz equation of state is an excellent choice, particularly for multidimensional models of stellar phenomena. Pressure Specific internal energy Entropy The Accuracy, Consistency, and Speed of Five Equations of State for Stellar Hydrodynamics (1999) In this article, we compare the thermodynamic properties and execution speed of five independent equations of state. A wide range of temperatures, densities, and compositions are considerebdconditions appropriate for modeling the collapse of a cloud of hydrogen gas (or an exploding supernova) to the outer layers of a neutron star. The pressures and specific thermal energies calculated by each equation-of-state routine are reasonably accurate (typically 0.1% error or less) and agree remarkably well with each other, despite the different approaches and approximations used in each routine. The derivatives of the pressure and specific thermal energies with respect to the temperature and density generally show less accuracy (typically 1% error or less) and more disagreement with one another. Thermodynamic consistency, as measured by deviations from the appropriate Maxwell relations, shows that the Timmes equation of state and the Nadyozhin equation of state achieve thermodynamic consistency to a high degree of precision. The execution speed of the five equation-of-state routinebsevaluated across several different machine architectures, compiler options, and modes of operatiobndiffer dramatically. The Arnett equation of state is the fastest of the five routines, with the Nadyozhin equation of state close behind. Equation of State Software Instruments Open-source codes are avaliable from this link and this link. There are times when a simpler cold fermi gas EOS is a wonderful thing. Such an EOS is in cold_fermi_gas.tbz. One can see this equation of state in action on this cold white dwarf page.