Statistical complexity, virial expansion, and van der Waals equation

Particle number fluctuations are considered within the van der Waals (VDW) equation, which contains both attractive (mean-field) and repulsive (eigenvolume) interactions.The VDW equation is used to calculate the scaled variance of particle number fluctuations in generic Boltzmann VDW system and in nuclear matter.The strongly intensive measures ∆[E * , N ] and Σ[E * , N ] of the particle number and excitation energy fluctuations are also considered, and, similarly, show singular behavior near the critical point.The ∆[E * , N ] measure is shown to attain both positive and negative values in the vicinity of critical point.Based on universality argument, similar behavior is expected to occur in the vicinity of the QCD critical point.

The solvent around protein molecules in solutions is structured and this structuring introduces a repulsion in the intermolecular interaction potential at intermediate separations. We use Monte Carlo simulations with isotropic, pair-additive systems interacting with such potentials. We test if the liquid-liquid and liquid-solid phase lines in model protein solutions can be predicted from universal curves and a pair of experimentally determined parameters, as done for atomic and colloid materials using several laws of corresponding states. As predictors, we test three properties at the critical point for liquid-liquid separation: temperature, as in the original van der Waals law, the second virial coefficient, and a modified second virial coefficient, all paired with the critical volume fraction. We find that the van der Waals law is best obeyed and appears more general than its original formulation: A single universal curve describes all tested nonconformal isotropic pair-additive systems. Published experimental data for the liquid-liquid equilibrium for several proteins at various conditions follow a single van der Waals curve. For the solid-liquid equilibrium, we find that no single system property serves as its predictor. We go beyond corresponding-states correlations and put forth semiempirical laws, which allow prediction of the critical temperature and volume fraction solely based on the range of attraction of the intermolecular interaction potential.

Virial expansion is widely used in cold atoms to analyze high temperature strongly correlated many-body systems. As the n-th order virial expansion coefficient can be accurately obtained by exactly solving up to n-body problems, the virial expansion offers a few-body approach to study strongly correlated many-body problems. In particular, the virial expansion has successfully been applied to unitary Fermi gas. We review recent progress of the virial expansion studies in the unitary Fermi gas, in particular the fourth order virial coefficient.

Positivity of the virial coefficients in lattice dimer models and upper bounds on the number of matchings on graphs

The study of detonation processes in gaseous media is of interest for the coal mining, aviation, aerospace industries, and hydrogen energy. Under high pressure conditions during the passage of a detonation wave, it is necessary to take into account the equation of state of a real gas, the effects of endothermicity and exothermicity. The paper considers the flow of real gas through a plane detonation wave. A modified Rankine-Hugoniot equation has been obtained to describe the dynamics of changes in the parameters of a van der Waals gas flow when it passes through a detonation wave. The relations that determine the limiting pressure for the existence of detonation in a Van der Waals gas were obtained. The dependence of the limiting pressure on the parameters A and B of the Van der Waals equation of state is shown. An increase in parameter A slows down the growth of pressure in the detonation wave, and an increase in parameter B intensifies it. An equation for determining the velocity of combustion products in a real gas is obtained. Compared with the velocity of combustion products in an ideal gas. For an ideal gas, combustion products flow from the detonation front at a critical (sonic) velocity. For a Van der Waals gas, the velocity of combustion products can be greater than the critical one. Moreover, both parameters characterizing the additional pressure (A) and the additional volume (B) lead to the acceleration of combustion products. The additional pressure transfers its energy to the flow and thus causes acceleration. The additional volume leads to a decrease in the degree of compression in the same way as the limiting compression (28). Therefore, the flow velocity increases.

We characterize the high-temperature thermodynamics of rotating bosons and fermions in two- (2D) and three-dimensional (3D) isotropic harmonic trapping potentials. We begin by calculating analytically the conventional virial coefficients $b_n$ for all $n$ in the noninteracting case, as functions of the trapping and rotational frequencies. We also report on the virial coefficients for the angular momentum and associated moment of inertia. Using the bn coefficients, we analyze the deconfined limit (in which the angular frequency matches the trapping frequency) and derive explicitly the limiting form of the partition function, showing from the thermodynamic standpoint how both the 2D and 3D cases become effectively homogeneous 2D systems. To tackle the virial coefficients in the presence of weak interactions, we implement a coarse temporal lattice approximation and obtain virial coefficients up to third order.

The van der Waals equation is well known for the description of two-dimensional monolayers. The formation of a monolayer is the result of a compromise between the process of self-organization on the surface and the probabilities of spatial configurations of adsorbate molecules near the surface. The main reasons for the geometric heterogeneity of the monolayer are the geometric disorder and the energy inhomogeneity of the surface profile. A monolayer is a statistically related system and its symmetry causes correlations of processes at different spatial scales. The classical van der Waals equation is written for the two-dimensional, completely symmetric Euclidean space. In the general case, the geometry of the monolayer must be defined for the Euclidean space of fractional dimension (fractal space) with symmetry breaking. In this case, the application of the classical van der Waals equation is limited. Considering the fractal nature of the monolayer–solid interface, a quasi-two-dimensional van der Waals equation is developed. The application of the equation to experimental data of an activated carbon is shown.

This study evaluated the monthly average of the enthalpy, geopotential, latent heat content, moist static energy, and dry static energy at several levels of the atmosphere as a Van Der Waals gas from radiosonde data at 00:00 UTC and 12:00 UTC over the metropolitan region of Natal-RN-Brazil. The years 2015 and 2017 were considered because they presented more than 70% of complete data, with days coinciding at both times. To construct the vertical profiles of the variables, the density, specific humidity, and geopotential were calculated at each level through a mathematical approximation based on the Calculus. The Van der Waals-like atmosphere results showed that the latent heat content compensated for any change in enthalpy and potential energy from the surface to the mid-troposphere. Energy was exchanged among enthalpy, geopotential, and latent heat content, showing a roughly constant moist static energy. The moist static energy was a conserved quantity under adiabatic and hydrostatic transformations, although the dry static energy was not conserved. The results of the atmosphere as a Van Der Waals gas and the atmosphere as an ideal gas were compared, proving to be similar.

The gas equation of state (EOS) serves as a critical tool for analyzing the thermal effects within the hydrogen storage tank during refueling processes. It quantifies the dynamic relationships among pressure, temperature and volume, playing a vital role in numerical simulations of hydrogen refueling, the development of refueling protocols, and ensuring refueling safety. This study first establishes a lumped-parameter thermodynamic model for the hydrogen refueling process, which combines a zero-dimensional gas model with a one-dimensional tank wall model (0D1D). The model’s accuracy was validated against experimental data and will be used in combination with different EOSs to simulate hydrogen temperature and pressure. Subsequently, parameter values are derived for the van der Waals EOS and its modified forms—Redlich–Kwong, Soave, and Peng–Robinson. The accuracy of the modified forms is evaluated using the Joule–Thomson inversion curve. A polynomial EOS is formulated, and its parameters are numerically determined. Finally, the hydrogen temperatures and pressures calculated using the van der Waals EOS, Redlich–Kwong EOS, polynomial EOS, and the National Institute of Standards and Technology (NIST) database are compared. Within the initial and boundary conditions set in this study, the results indicate that among the modified forms for van der Waals EOS, the Redlich–Kwong EOS exhibits higher accuracy than the Soave and Peng–Robinson EOSs. Using the NIST-calculated hydrogen pressure as a benchmark, the relative error is 0.30% for the polynomial EOS, 1.83% for the Redlich–Kwong EOS, and 17.90% for the van der Waals EOS. Thus, the polynomial EOS exhibits higher accuracy, followed by the Redlich–Kwong EOS, while the van der Waals EOS demonstrates lower accuracy. This research provides a theoretical basis for selecting an appropriate EOS in numerical simulations of hydrogen refueling processes.

Onsager's prediction of a first-order phase transition in a gas of long thin rods is verified by extensive calculations on a simple model. The molecules are rectangular parallelepipeds, of length l and square cross section d×d. The long molecular axes can point in only three mutually perpendicular directions. Virial coefficients, up to the seventh, are calculated exactly as functions of orientation, in the limit l→ ∞, d→0, and l2d=constant. The transition predicted by Onsager's theory, based on a second virial approximation to the potential of mean force in the space of relative orientations, is observed also when all virial coefficients up to the seventh are included. The virial expansion appears to converge well for the isotropic phase; but the properties of the anisotropic phase depend sensitively on the order of approximation.

In the classical ideal gas there are no interparticular forces. The Dutch scientist Johannes Diderik van der Waals derived a correction leading to the van der Waals equation. The interparticular forces in the van der Waals equation can not be explained by covalent bonds (characterized by the sharing of electron pairs) of the particles (atoms and molecules) in the gas. Those forces can only be explained via weak interactions and can be attractive or repulsive. The weak interactions between atoms and molecules have later been named after Johannes Diderik van der Waals. Weak interactions are not restricted to intermolecular interactions but can also occur intramolecular. In both cases the weak interactions can be responsible for the shape of molecules, supersystems and clusters. A famous example for this is the protein folding in RiboNucleic Acid (RNA) and DeoxyriboNucleic Acid (DNA). The effect of medical compounds often is founded on weak interactions with something in the human body. Unfortunately, the calculation of weak interactions is a challenging task. Very accurate ab initio quantum mechanical methods and a good description of the molecular orbitals are needed. Both requirements lead to high computational demands (calculation time and storage usage for main memory and hard disc). In ab initio quantum chemistry one normally only calculates the (total) energy E of a system. This is done with an accuracy of approximately 99% (when compared to the exact result) if the Hartree-Fock method (HF) is used. For many cases this accuracy is not sufficient to achieve the chemical accuracy of 1 kcal/mol. In contrast to thermochemistry where (reaction) enthalpies are typically calculated from standard enthalpies of formation (which are of the same order of magnitude as the reaction enthalpies and a certain accuracy of the formation enthalpies leads to a result of similar quality of the relative energies) in ab initio quantum chemistry calculations, the calculated total energies are larger by orders of magnitude than the energy of the desired property. For weak interactions the ratio of calculated values and the desired property is even larger and dynamic electron correlation, which is not covered at all by HF, plays a larger role or is sometimes responsible for the complete interaction energy. In particular the dispersion energy is a pure electron correlation effect. Methods that give a correction to HF to include dynamic electron correlation in the calculated energy need larger basis sets for the description of the molecular orbitals than HF itself. In the example of the interaction energy of the benzene dimer, the total electronic energy of the dimer is approximately 300000 kcal/mol, whereas the interaction energy is approximately 3 kcal/mol. This ratio of 105 illustrates why an accuracy of more than 99% and hence methods that cover dynamic electron correlation are necessary. The Coupled Cluster method with single and double excitations and triple excitations at perturbation theory level (CCSD(T)) (see Section 3.10) is considered to be highly accurate for most weak interactions. The convergence of the total energy with the basis set size can be greatly improved by the introduction of explicit correlation. Unfortunately CCSD(T) is rather expensive and systems with more than 30 atoms are very hard to calculate. Many systems of interest with weak interactions are larger, and therefore CCSD(T) is not feasible for their calculation. The MP2 method is much cheaper[c] than the CCSD(T) method, but unfortunately not as accurate. Earlier attempts to improve MP2 like SCS-MP2 have been successful to some extent but were never totally satisfying. Other attempts such as MP2.5 are more expensive and therefore also restricted to smaller systems. The goal of this work was to test the effects of explicit correlation in the CCSD(T) method and to develop a new method for the calculation of weak interactions that is able to improve on MP2 and SCS-MP2 towards CCSD(T). A new method - Dispersion-Weighted-MP2[d] (DW-MP2) has been developed in this work. It combines MP2 and SCS-MP2 and will be shown to achieve the accuracy needed for chemical accuracy.

Zeolitic encapsulation of small non-polar gas molecules under appropriate p–T conditions (Vr > 1) can be conveniently dealt with in terms of van der Waals' equation of state for real gases. Thus, the amount of gas trapped in the zeolitic cages can be predicted if the cage volumes are known; otherwise, void-volume values may be obtained from encapsulation experiments. These void-volume values, as obtained from hydrogen encapsulation in A-type zeolites, are ca. 20% larger than those reported in the literature, but in agreement with calculated void volumes derived from crystallographic data and from a refined structural model of zeolite A. The ‘jump’ in encapsulation capacity at ca. 2.5 univalent blocking cations per unit cell is explained in terms of the percolation theory as transition from β-cage encapsulation to encapsulation in both the α- and β-cages. Based on the van der Waals equation, calculated available void volumes and the percolation theory, a theoretical encapsulation curve is proposed which agrees fairly well with experimental results reported so far for H2 in Cs, Na-A zeolites and for Ar, Kr and CH4 in K, Na-A zeolites.

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.

We theoretically investigate the high-temperature thermodynamics of a\nstrongly interacting trapped Fermi gas near either s-wave or p-wave Feshbach\nresonances, using a second order quantum virial expansion. The second virial\ncoefficient is calculated based on the energy spectrum of two interacting\nfermions in a harmonic trap. We consider both isotropic and anisotropic\nharmonic potentials. For the two-fermion interaction, either s-wave or p-wave,\nwe use a pseudopotential parametrized by a scattering length and an effective\nrange. This turns out to be the simplest way of encoding the energy dependence\nof the low-energy scattering amplitude or phase shift. This treatment of the\npseudopotential can be easily generalized to higher partial-wave interactions.\nWe discuss how the second virial coefficient and thermodynamics are affected by\nthe existence of these finite-range interaction effects. The virial expansion\nresult for a strongly interacting s -wave Fermi gas has already been proved\nvery useful. In the case of p-wave interactions, our results for the\nhigh-temperature equation of state are applicable to future high-precision\nthermodynamic measurements for a spin-polarized Fermi gas near a p-wave\nFeshbach resonance.\n

Thermal gravity convection in a horizontal layer of compressible perfect gas heated from below and a van der Waals gas near the critical state is investigated. The characteristics of the isentropic equilibrium of a compressible medium with a van der Waals equation of state are considered. The known conditions of convection onset in the perfect and van der Waals gases are checked on the basis of a solution of the complete and linearized equations. The restrictions imposed in deriving the known formulas for the adiabatic temperature gradients used in the conditions of absence and onset of convection are discussed. The characteristics of the convective heat transfer are examined, including the causes of the heat-transfer deterioration in the near-critical medium above the hydrostatic equilibrium threshold.