Virial Expansion Research Articles

Estimating metastable thermodynamic properties by isochoric extrapolation from stable states.

The description of metastable fluids, those in local but not global equilibrium, remains an important problem of thermodynamics, and it is crucial for many industrial applications and all first order phase transitions. One way to estimate their properties is by extrapolation from nearby stable states. This is often done isothermally, in terms of a virial expansion for gases or a Taylor expansion in density for liquids. This work presents evidence that an isochoric expansion of pressure at a given temperature is superior to an isothermal density expansion. Two different isochoric extrapolation strategies are evaluated, one best suited for vapors and one for liquids. Both are exact for important model systems, including the van der Waals equation of state. Moreover, we present a simple method to evaluate all the coefficients of the isochoric expansion directly from a simulation in the canonical ensemble. Using only the properties of stable states, the isochoric extrapolation methods reproduce simulation results with Lennard-Jones potentials, mostly within their uncertainties. The isochoric extrapolation methods are able to predict deeply metastable pressures accurately even from temperatures well above the critical. Isochoric extrapolation also predicts a mechanical stability limit, i.e., the thermodynamic spinodal. For water, the liquid spinodal pressure is predicted to be monotonically decreasing with decreasing temperature, in contrast to the re-entrant behavior predicted by the direct extension of the reference equation of state.

Speed of sound measurements and derived third and fourth acoustic virial coefficients of supercritical neon

We report comprehensive and accurate measurements of the speed of sound in neon. These measurements were carried out by a double-path-length pulse-echo technique and cover the temperature range between 200 K and 420 K with pressures up to 100 MPa. The standard uncertainties are 1.9 mK in temperature, 22 parts in 106 in pressure and 35 parts in 106 in speed of sound. The third and fourth acoustic virial coefficients of neon were derived from the speed of sound data in the temperature range of the measurements by fitting a fourth-order acoustic virial expansion in pressure with the second acoustic virial coefficient constrained from first-principles calculations. To support our claimed uncertainty, we determined the ratio M/γ0 between the molar mass M and the ideal-gas heat capacity ratio γ0 of the neon sample with a relative standard uncertainty of 7.7 parts in 106 by additional speed of sound measurements using a spherical resonator at 273.16 K.

Thermodynamics of spin-1/2 fermions on coarse temporal lattices using automated algebra.

Recent advances in automated algebra for dilute Fermi gases in the virial expansion, where coarse temporal lattices were found advantageous, motivate the study of more general computational schemes that could be applied to arbitrary densities, beyond the dilute limit where the virial expansion is physically reasonable. We propose here such an approach by developing what we call the Quantum Thermodynamics Computational Engine (QTCE). In QTCE, the imaginary-time direction is discretized and the interaction is accounted for via a quantum cumulant expansion, where the coefficients are expressed in terms of non-interacting expectation values. The aim of QTCE is to enable the systematic resolution of interaction effects at fixed temporal discretization, as in lattice Monte Carlo calculations, but here in an algebraic rather than numerical fashion. Using this approach, in combination with numerical integration techniques (both known and alternative ones proposed here), we explore the thermodynamics of spin-1/2 fermions across spatial dimensions, focusing on the unitary limit. We find that, remarkably, extremely coarse temporal lattices, when suitably renormalized using known results from the virial expansion, yield stable partial sums for QTCE's cumulant expansion that are qualitatively and quantitatively correct in wide regions (when compared with known experimental results). This article is part of the theme issue 'The liminal position of Nuclear Physics: from hadrons to neutron stars'.

On the equation of state of the hard disk system: the fluid-hexatic phase transition

The accuracy of four approximations proposed along time as equations of state for a homogeneous system of hard disks is compared against molecular dynamics results of 10 5 disks in equilibrium [L. Mier-y-Terán, J. Munguía and J. Antonio Moreno-Razo, Mol. Phys. (2023). doi:10.1080/00268976.2023.2288702]. Three of the equations studied here are Padé approximants. Only one of the approximations studied, based on the virial expansion of the compressibility factor of the homogeneous fluid, is considered to have sound physical grounds. Assuming that the total pressure predicted generates the positional part of the Helmholtz free energy, only, the four approximations, combined with the molecular dynamics simulation results referred above, allow identifying four linear regions, where different orientational processes can be associated with the well known steps of the melting transition. Results for a finite system are consistent with the KTHNY theory. In particular, under this assumption, the four approximations, with small quantitative differences, show the ordering effects present in the fluid at the low-density-end of the fluid-hexatic first-order phase transition. It is shown that the latter region continues to exist under the assumption proposed. In addition, a linear combination of the virial and compressibility coefficients is used to obtain a more accurate version of the Carnahan-Starling type approximation for the hard-disk system.

Thermodynamic properties of quasi-one-dimensional fluids.

We calculate thermodynamic and structural quantities of a fluid of hard spheres of diameter σ in a quasi-one-dimensional pore with accessible pore width W smaller than σ by applying a perturbative method worked out earlier for a confined fluid in a slit pore [Franosch et al. Phys. Rev. Lett. 109, 240601 (2012)]. In a first step, we prove that the thermodynamic and a certain class of structural quantities of the hard-sphere fluid in the pore can be obtained from a purely one-dimensional fluid of rods of length σ with a central hard core of size σW=σ2-W2 and a soft part at both ends of length (σ - σW)/2. These rods interact via effective k-body potentials veff(k) (k ≥ 2). The two- and the three-body potential will be calculated explicitly. In a second step, the free energy of this effective one-dimensional fluid is calculated up to leading order in (W/σ)2. Explicit results for, e.g., the perpendicular pressure, surface tension, and the density profile as a function of density, temperature, and pore width are presented and partly compared with results from Monte-Carlo simulations and standard virial expansions. Despite the perturbative character of our approach, it encompasses the singularity of the thermodynamic quantities at the jamming transition point.

Structure Factors for Hot Neutron Matter from AbInitio Lattice Simulations with High-Fidelity Chiral Interactions.

We present the first abinitio lattice calculations of spin and density correlations in hot neutron matter using high-fidelity interactions at next-to-next-to-next-to-leading order in chiral effective field theory. These correlations have a large impact on neutrino heating and shock revival in core-collapse supernovae and are encapsulated in functions called structure factors. Unfortunately, calculations of structure factors using high-fidelity chiral interactions were well out of reach using existing computational methods. In this Letter, we solve the problem using a computational approach called the rank-one operator (RO) method. The RO method is a general technique with broad applications to simulations of fermionic many-body systems. It solves the problem of exponential scaling of computational effort when using perturbation theory for higher-body operators and higher-order corrections. Using the RO method, we compute the vector and axial static structure factors for hot neutron matter as a function of temperature and density. The abinitio lattice results are in good agreement with virial expansion calculations at low densities but are more reliable at higher densities. Random phase approximation codes used to estimate neutrino opacity in core-collapse supernovae simulations can now be calibrated with abinitio lattice calculations.

Electrical conductivity of hydrogen plasmas: Low-density benchmarks and virial expansion including e–e collisions

An improved virial expansion for the low-density limit of the electrical conductivity σ(T,n) of hydrogen as the simplest ionic plasma is presented. Quantum statistical methods provide exact values for the lowest virial coefficients, which serve as a benchmark for analytical approaches to electrical conductivity as well as for numerical results from density functional theory-based molecular dynamics simulations (DFT-MD) or path-integral Monte Carlo simulations. The correction factor introduced by Reinholz et al. [Phys. Rev. E 91, 043105 (2015)] is applied to describe the inclusion of electron–electron collisions in DFT-based calculations of transport coefficients. As a benchmark, the first virial coefficient is correctly described with this approach. The value of the second virial coefficient is discussed, and questions about its value according to DFT-MD simulations are addressed.

Virial coefficients of the uniform electron gas from path-integral Monte Carlo simulations.

The properties of plasmas in the low-density limit are described by virial expansions. Analytical expressions are known from Green's function approaches only for the first three virial coefficients. Accurate path-integral Monte Carlo (PIMC) simulations have recently been performed for the uniform electron gas, allowing the virial expansions to be analyzed and interpolation formulas to be derived. The exact expression for the second virial coefficient is used to test the accuracy of the PIMC simulations and the range of validity of the interpolation formula of Groth etal. [Phys. Rev. Lett. 119, 135001 (2017)0031-900710.1103/PhysRevLett.119.135001], and we discuss the fourth virial coefficient, which is not exactly known yet. Combining PIMC simulations with benchmarks from exact virial expansion results would allow us to obtain more accurate representations of the equationof state for parameter ranges of conditions which are of interest, e.g., for helioseismology.

Ab initio study of nuclear clustering in hot dilute nuclear matter

We present a systematic ab initio study of clustering in hot dilute nuclear matter using nuclear lattice effective field theory with an SU(4)-symmetric interaction. We introduce a method called light-cluster distillation to determine the abundances of dimers, trimers, and alpha clusters as a function of density and temperature. Our lattice results are compared with an ideal gas model composed of free nucleons and clusters. Excellent agreement is found at very low density, while deviations from ideal gas abundances appear at increasing density due to cluster-nucleon and cluster-cluster interactions. In addition to determining the composition of hot dilute nuclear matter as a function of density and temperature, the lattice calculations also serve as benchmarks for virial expansion calculations, statistical models, and transport models of fragmentation and clustering in nucleus-nucleus collisions.

Calculating the classical virial expansion using automated algebra.

Using schematic model potentials, we calculate exactly the virial coefficients of a classical gas up to sixth order and use them to calculate the virial expansion of basic thermodynamic quantities such as pressure, density, and compressibility. At sufficiently strong couplings, as expected, the virial expansion fails to converge. However, at least for the interactions and parameter ranges we explored, we find that Padé-Borel resummation methods are very effective in improving the convergence of the expansion.

Perturbation theory in a microcanonical ensemble

The microcanonical ensemble is a natural starting point of statistical mechanics. However, when it comes to perturbation theory in statistical mechanics, mainly the canonical and grand canonical ensembles have been used. In this article we show how the microcanonical ensemble can be directly used to carry out perturbation theory for both non-interacting and interacting systems. We obtain the first non-trivial order answers for the specific heat of anharmonic oscillators and for the virial expansion in real gases. They are in exact agreement with the results obtained from the canonical ensemble. In addition, we show how crossover functions for the specific heat of anharmonic oscillators can be constructed using a microcanonical ensemble and also how the subsequent terms of the virial expansion can be obtained. We find that if we consider quantum free particles in a one-dimensional box of extension L, then both the ensembles give an unusual result for the first correction to the specific heat in the high temperature limit.

On Virial Expansions of Correlation Functions. Canonical Ensemble

On Virial Expansions of Correlation Functions. Canonical Ensemble

Exact Quantum Virial Expansion for the Optical Response of Doped Two-Dimensional Semiconductors.

We introduce a quantum virial expansion for the optical response of a doped two-dimensional semiconductor. As we show, this constitutes a perturbatively exact theory in the high-temperature or low-doping regime, where the electrons' thermal wavelength is smaller than their interparticle spacing. We obtain exact analytic expressions for the photoluminescence and we predict new features such as a nontrivial shape of the attractive branch peak related to universal resonant exciton-electron scattering and an associated energy shift from the trion energy. Our theory furthermore allows us to formally unify the two distinct theoretical pictures that have been applied to this system, where we reveal that the predictions of the conventional trion picture correspond to a high-temperature and weak-interaction limit of Fermi-polaron theory. Our results are in excellent agreement with recent experiments on doped monolayer MoSe_{2} and they provide the foundation for modeling a range of emerging optically active materials such as van der Waals heterostructures.

A General Model for Thermodynamic Properties of Fluid Mixtures Based on Helmholtz Energy Formulations for the Components. Virial Expansion and Reduction to van der Waals Mixing Rules

Over the recent decades, Helmholtz energy formulations became available for a broad range of fluids. These multiparameter equations of state (R. Span, Springer 2000) allow computation of thermodynamic properties essentially within the experimental errorbars. Corresponding states-based model by Lemmon and Tillner-Roth (Fluid Phase Equilib 165:1, 1999) enabled construction of Helmholtz energy formulations for mixtures. However, we show that this model generates a non-physical dependence of virial coefficients on composition, which can be strong when the components are dissimilar. We propose a new mixture model that overcomes this deficiency. It has two main ingredients: (i) Quadratic mixing of “Helmholtz volumities”. This quantity with units of molar volume is introduced as a ratio of the molar residual Helmholtz energy to a product of gas constant, thermodynamic temperature, and molar density. It reduces to the second virial coefficient in the zero-density limit. Helmholtz volumities are considered for components and “cross-components”, hypothetical fluids representing the binary interactions. (ii) Replacing the variables—reduced reciprocal temperatures and reduced densities—with temperature and density scaling functions. Different scaling functions can be used for different components and cross-components, thus providing a highly flexible framework for representing the properties of mixtures. The scaling functions must be expandable into Taylor series in terms of molar concentrations in the zero-density limit. For the proposed mixture model, we develop formulas for computing virial coefficients up to the fourth order. Furthermore, we show that when the proposed mixture model is applied to a cubic equation of state, the conventional van der Waals mixing rules can be retrieved. These findings allow to consider the new model as a viable alternative to the corresponding states method of modeling thermodynamic properties of fluid mixtures.

Phase Behavior of Ternary Polymer Mixtures in a Common Solvent.

The Edmond-Ogston model for phase separation is extended to ternary polymer mixtures in a common solvent (de facto a quaternary mixture). The model assumes a truncated virial expansion of the Helmholtz free energy up to the second-order terms in the concentration of the polymers, and the second virial coefficients (B11, B22, B33, B12, B13, B23) are the six parameters of the model. New results from this model are presented in relation to earlier work on binary mixtures: a necessary condition for the virial coefficients for the occurrence of phase separation in two or three phases, an analysis of the different regions of (local) thermodynamic instability using the Descartes sign rule, an expression for the critical curves, a relation between the tangents in points along the critical curve, a relation between the concentration of components in the different phases according to the so-called Lambert-W function, and a consistency check for the composition of coexisting phases in ternary mixtures. The obtained results are evaluated in the maximally symmetric version of the model, where (B11, B22, B33) are equal and (B12, B13, B23) are equal, which leads to two remarkable observations: the concentration range over which two phases are formed is relatively narrow; not all phase separation occurs within a Gibbs triangle, but also, "out-of-Gibbs-triangle" binodals are observed. These results lead to a deeper insight into the phase behavior of ternary mixtures and show promise as a stepping stone toward modeling phase separation in mixtures with many components.

Temperature-Dependent Contact of Weakly Interacting Single-Component Fermi Gases and Loss Rate of Degenerate Polar Molecules.

Motivated by the experimental realization of single-component degenerate Fermi gases of polar ground state KRb molecules with intrinsic two-body losses [L. De Marco et al., A degenerate Fermi gas of polar molecules, Science 363, 853 (2019).SCIEAS0036-807510.1126/science.aau7230], this work studies the finite-temperature loss rate of single-component Fermi gases with weak interactions. First, we establish a relationship between the two-body loss rate and the p-wave contact. Second, we evaluate the contact of the homogeneous system in the low-temperature regime using p-wave Fermi liquid theory and in the high-temperature regime using the second-order virial expansion. Third, conjecturing that there are no phase transitions between the two temperature regimes, we smoothly interpolate the results to intermediate temperatures. It is found that the contact is constant at temperatures close to zero and increases first quadratically with increasing temperature and finally-in agreement with the Bethe-Wigner threshold law-linearly at high temperatures. Fourth, applying the local-density approximation, we obtain the loss-rate coefficient for the harmonically trapped system, reproducing the experimental KRb loss measurements within a unified theoretical framework over a wide temperature regime without fitting parameters. Our results for the contact are not only applicable to molecular p-wave gases but also to atomic single-component Fermi gases, such as ^{40}K and ^{6}Li.

Virial expansions for correlation functions in canonical ensemble

Virial expansions for correlation functions in canonical ensemble

Quench dynamics of thermal Bose gases across wide and narrow Feshbach resonances

Using high-temperature virial expansion, we study the quench dynamics of the thermal Bose gases near wide, narrow, and intermediate Feshbach resonances. Our results show that the shallow bound state near Feshbach resonances leads to oscillation of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{\mathbf{k}}$. Near the wide Feshbach resonance, the long-time momentum distribution ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{\mathbf{k}}$ oscillates when the scattering length ${a}_{s}$ is quenched to large but with finite positive values. The oscillation frequency is determined by the bound-state energy. When ${a}_{s}$ is quenched to infinity or negative value, the oscillation vanishes. Near the narrow Feshbach resonance, when the background scattering length ${a}_{\mathrm{bg}}$ is larger than de Broglie wave length $\ensuremath{\lambda}$, there is an oscillation, and the frequency is determined by the energy of the shallow bound state in the open channel. When ${a}_{\mathrm{bg}}<0$ or $0<{a}_{\mathrm{bg}}\ensuremath{\ll}\ensuremath{\lambda}$, there is no shallow bound state in the open channel, and hence no oscillation. We check our conclusion using some realistic systems, and the results are consistent with our conclusion.

Physically based equation of state for Mie ν-6 fluids.

We develop a physically based equation of state that describes Mie ν-6 fluids with an accuracy comparable to that of state-of-the-art empirical models. The equation of state is developed within the framework of the uv-theory [T. van Westen and J. Gross, J. Chem. Phys. 155, 244501 (2021)], which is modified by incorporating the third virial coefficient B3 in the low-density description of the model. The new model interpolates between a first-order Weeks-Chandler-Andersen (WCA) perturbation theory at high densities and a modified first-order WCA theory that recovers the virial expansion up to B3 at low densities. A new algebraic equation for the third virial coefficient of Mie ν-6 fluids is developed-other inputs are taken from previous work. Predicted thermodynamic properties and phase equilibria are compared to a comprehensive database of molecular simulation results from the literature, including Mie fluids of repulsive exponents 9 ≤ ν ≤ 48. The new equation of state is applicable to states with densities up to ρ*(T*)⪅1.1+0.12T* and temperatures T* > 0.3. For the Lennard-Jones fluid (ν = 12), the performance of the model is comparable to that of the best empirical equations of state available. As compared to empirical models, the physical basis of the new model provides several advantages, however: (1) the new model is applicable to Mie fluids of repulsive exponents 9 ≤ ν ≤ 48 instead of only ν = 12, (2) the model leads to a better description of the meta-stable and unstable region (which is important for describing interfacial properties by classical density functional theory), and (3) being a first-order perturbation theory, the new model (potentially) allows an easier and more rigorous extension to non-spherical (chain) fluids and mixtures.

Aqueous biphasic systems containing sodium L-tartrate dihydrate and PEG, experimental and application of extended thermodynamic models

In the current study, liquid-liquid equilibrium data of aqueous two-phase systems (ATPS) containing polyethylene glycol at the different molecular weights (1000, 2000, 6000, and 8000) - sodium L-tartrate dihydrate - water at 298.15 K were obtained. Binodal curves of mentioned systems have been determined and the effect of the molecular weight of polymer on binodal curves, tie-lines, and two-phase region sizes was studied. The results showed that increasing the molecular weight of the polymer enhances the biphasic area of the system. Also, an extended UNIQUAC equation and an extended form of Virial expansion models as new models of activity coefficient were examined to predict the phase equilibria of mentioned systems. The fitted binary interaction parameters of the model were reported. Both investigated models were capable to correlate the VLE and LLE data of the pertinent binary and ternary systems successfully. It could be observed that the extended Virial model leads to a better presentation of the data in comparison to the extended UNIQUAC model.