Dilute Gas Limit Research Articles

Semiclassical approach to ground-state properties of hard-core bosons in two dimensions

Motivated by some inconsistencies in the way quantum fluctuations are included beyond the classical treatment of hard-core bosons on a lattice in the recent literature, we revisit the large-S semi-classical approach to hard-core bosons on the square lattice at T=0. First of all, we show that, if one stays at the purely harmonic level, the only correct way to get the 1/S correction to the density is to extract it from the derivative of the ground state energy with respect to the chemical potential, and that to extract it from a calculation of the ground state expectation value of the particle number operator, it is necessary to include 1/\sqrt{S} corrections to the harmonic ground state. Building on this alternative approach to get 1/S corrections, we provide the first semiclassical derivation of the momentum distribution, and we revisit the calculation of the condensate density. The results of these as well as other physically relevant quantities such as the superfluid density are systematically compared to quantum Monte Carlo simulations. This comparison shows that the logarithmic corrections in the dilute Bose gas limit are only captured by the semi-classical approach if the 1/S corrections are properly calculated, and that the semi-classical approach is able to reproduce the 1/k divergence of the momentum distribution at k=0. Finally, the effect of 1/S^2 corrections is briefly discussed.

Eigenfunctions for Liouville Operators, Classical Collision Operators, and Collision Bracket Integrals in Kinetic Theory Made Amenable to Computer Simulations

In the kinetic theory of dense fluids the many-particle collision bracket integral is given in terms of a classical collision operator defined in the phase space. To find an algorithm to compute the collision bracket integrals, we revisit the eigenvalue problem of the Liouville operator and re-examine the method previously reported [Chem. Phys. 1977, 20, 93]. Then we apply the notion and concept of the eigenfunctions of the Liouville operator and knowledge acquired in the study of the eigenfunctions to cast collision bracket integrals into more convenient and suitable forms for numerical simulations. One of the alternative forms is given in the form of time correlation function. This form, on a further manipulation, assumes a form reminiscent of the ChapmanEnskog collision bracket integrals, but for dense gases and liquids as well as solids. In the dilute gas limit it would give rise precisely to the Chapman-Enskog collision bracket integrals for two-particle collision. The alternative forms obtained are more readily amenable to numerical simulation methods than the collision bracket integrals expressed in terms of a classical collision operator, which requires solution of classical Lippmann-Schwinger integral equations. This way, the aforementioned kinetic theory of dense fluids is made fully accessible by numerical computation/simulation methods, and the transport coefficients thereof are made computationally as accessible as those in the linear response theory.

Universal phase structure of dilute Bose gases with Rashba spin-orbit coupling

A Bose gas subject to a light-induced Rashba spin-orbit coupling possesses a dispersion minimum on a circle in momentum space; we show that kinematic constraints due to this dispersion cause interactions to renormalize to universal, angle-dependent values that govern the phase structure in the dilute-gas limit. We find that, regardless of microscopic interactions, (a) the ground state involves condensation at two opposite momenta and is, in finite systems, a fragmented condensate and and (b) there is a nonzero-temperature instability toward the condensation of pairs of bosons. We discuss how our results can be reconciled with the qualitatively different mean-field phase diagram, which is appropriate for dense gases.

Four-Dimensional Spin Foam Perturbation Theory

We define a four-dimensional spin-foam perturbation theory for the ${\rm BF}$-theory with a $B\wedge B$ potential term defined for a compact semi-simple Lie group $G$ on a compact orientable 4-manifold $M$. This is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. We then regularize the terms in the perturbative series by passing to the category of representations of the quantum group $U_q(\mathfrak{g})$ where $\mathfrak{g}$ is the Lie algebra of $G$ and $q$ is a root of unity. The Chain-Mail formalism can be used to calculate the perturbative terms when the vector space of intertwiners $\Lambda\otimes \Lambda \to A$, where $A$ is the adjoint representation of $\mathfrak{g}$, is 1-dimensional for each irrep $\Lambda$. We calculate the partition function $Z$ in the dilute-gas limit for a special class of triangulations of restricted local complexity, which we conjecture to exist on any 4-manifold $M$. We prove that the first-order perturbative contribution vanishes for finite triangulations, so that we define a dilute-gas limit by using the second-order contribution. We show that $Z$ is an analytic continuation of the Crane-Yetter partition function. Furthermore, we relate $Z$ to the partition function for the $F\wedge F$ theory.

Entropy−Enthalpy Compensation in Chemical Reactions and Adsorption: An Exactly Solvable Model

The free energies of reaction or activation for many systems respond in a common fashion to a perturbing parameter, such as the concentration of an "inert" additive. Arrhenius plots as a function of the perturbing parameter display a "'compensation temperature" at which the free energy appears to be independent of the perturber, an entropy-enthalpy compensation process. Thus, as the perturber's concentration varies, Arrhenius plots of the rate constant or equilibrium constant exhibit a rotation about the fixed compensation temperature. While this (isokinetic/isoequilibrium) component of the phenomenon of entropy-enthalpy compensation appears in a huge number of situations of relevance to chemistry, biology, and materials science, statistical mechanical descriptions have been almost completely lacking. We provide the general statistical mechanical basis for solvent induced isokinetic/isoequilibrium entropy-enthalpy compensation in chemical reactions and adsorption, understanding that can be used to control of rate processes and binding constants in diverse applications. The general behavior is illustrated with an analytical solution for the dilute gas limit.

Impact of amplifying media on the Casimir force

On the basis of macroscopic quantum electrodynamics, a theory of Casimir forces in the presence of linearly amplifying bodies is presented which provides a consistent framework for studying the effect of, e.g., amplifying left-handed metamaterials on dispersion forces. It is shown that the force can be given in terms of the classical Green tensor and that it can be decomposed into a resonant component associated with emission processes and an off-resonant Lifshitz-type component. We explicitly demonstrate that our theory extends additive approaches beyond the dilute-gas limit.

Calculation of the transport and relaxation properties of dilute water vapor

Transport properties of dilute water vapor have been calculated in the rigid-rotor approximation using four different potential energy hypersurfaces and the classical-trajectory method. Results are reported for shear viscosity, self-diffusion, thermal conductivity, and volume viscosity in the dilute-gas limit for the temperature range of 250-2500 K. Of these four surfaces the CC-pol surface of Bukowski et al. [J. Chem. Phys. 128, 094314 (2008)] is in best accord with the available measurements. Very good agreement is found with the most accurate results for viscosity in the whole temperature range of the experiments. For thermal conductivity the deviations of the calculated values from the experimental data increase systematically with increasing temperature to around 5% at 1100 K. For both self-diffusion and volume viscosity, the much more limited number of available measurements are generally consistent with the calculated values, apart from the lower temperature isotopically labeled diffusion measurements.

Mini-Workshop: Mathematical Approaches to Collective Phenomena in Large Quantum Systems

The workshop Mathematical Approaches to Collective Phenomena in Large Quantum Systems , organised by Stefan Adams (Warwick) and Robert Seiringer (Princeton) was held August 31st–September 6th, 2008. This meeting was well attended with 17 participants with broad geographic representation from all continents and with a good mixture of younger Postdocs/researchers (Boland, Bru, Crawford, Schlein, and Starr) and senior researchers. This workshop was a nice blend of researchers with various backgrounds ranging from analysis, probability theory, and functional integration. The fifteen talks are focused along the three general research directions developing mathematical theories of quantum phase transitions: (1) Dilute gas limit and nonlinear effective theories (2) Random walk representations via Feynman–Kac formula (3) Field theory/semiclassical representations via coherent states Area (1) is represented with new developments in talks by Lieb and Solovej focused on the Bogoliubov theory to derive second order correction to the ground state energy and in a talk by Yngvason on rotating trapped Bose gases. New developments for quantum spin systems have been reported by Nachtergaele and Starr. A new probabilistic approach has been introduced in a talk by König which shows a modelling of the free energy in the thermodynamic limit, and future will show if this approach goes beyond the existing theory on finite temperature Bose gases. An interesting area has been summarised by Ueltschi on random permutations. This area is recently under much focus in the probability community. The contribution from the youngest participant, the PhD-student Boland, showed an interesting result concerning the Bose condensate density and the density of so-called long cycles. Long cycles appear if probability mass of finite cycles gets lost in the thermodynamic limit, an area which is studied in the combinatorics community via shape measure analysis. The schedule of the workshop allowed intensive discussion during the afternoons including Friday. The workshop has been closed with an informal discussion/meeting on Friday evening which was attended by the majority of the participants. The workshop showed interesting new developments, and it is the hope of the organisers that there will be more interdisciplinary collaborations, in particular between the analysis and the probability community.

Calculation of the transport and relaxation properties of methane. II. Thermal conductivity, thermomagnetic effects, volume viscosity, and nuclear-spin relaxation

Transport properties of pure methane have been calculated in the rigid-rotor approximation using the recently proposed intermolecular potential energy hypersurface [R. Hellmann et al., J. Chem. Phys. 128, 214303 (2008)] and the classical-trajectory method. Results are reported in the dilute-gas limit for the temperature range of 80-1500 K. The calculated thermal conductivity values are in very good agreement with the measured data and correlations. In the temperature range of 310-480 K the calculated values underestimate the best experimental data by 0.5%-1.0%. We suggest that the calculated values are more accurate, especially at low and high temperatures, than the currently available correlations based on the experimental data. Our results also agree well with measurements of thermal transpiration and of the thermomagnetic coefficients. We have shown that although the dominant contribution to the thermomagnetic coefficients comes from the Wjj polarization in the spherical approximation, the contribution of a second polarization, Wj, cannot be neglected nor can a full description of the Wjj polarization. The majority of the volume viscosity measurements around room temperature are consistent with the calculated values but this is not the case at high and low temperatures. However, for nuclear-spin relaxation the calculated values consistently exceed the measurements, which are mutually consistent within a few percent.

Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity

We formulate the spin foam perturbation theory for three-dimensional Euclidean Quantum Gravity with a cosmological constant. We analyse the perturbative expansion of the partition function in the dilute-gas limit and we argue that the Baez conjecture stating that the number of possible distinct topological classes of perturbative configurations is finite for the set of all triangulations of a manifold, is not true. However, the conjecture is true for a special class of triangulations which are based on subdivisions of certain 3-manifold cubulations. In this case we calculate the partition function and show that the dilute-gas correction vanishes for the simplest choice of the volume operator. By slightly modifying the dilute-gas limit, we obtain a nonvanishing correction which is related to the second order perturbative correction. By assuming that the dilute-gas limit coupling constant is a function of the cosmological constant, we obtain a value for the partition function which is independent of the choice of the volume operator.

Calculation of the transport and relaxation properties of methane. I. Shear viscosity, viscomagnetic effects, and self-diffusion

Transport properties of pure methane gas have been calculated in the rigid-rotor approximation using the recently proposed intermolecular potential energy hypersurface [R. Hellmann et al., J. Chem. Phys. 128, 214303 (2008)] and the classical-trajectory method. Results are reported in the dilute-gas limit for shear viscosity, viscomagnetic coefficients, and self-diffusion in the temperature range of 80-1500 K. Compared with the best measurements, the calculated viscosity values are about 0.5% too high at room temperature, although the temperature dependence of the calculated values is in very good agreement with experiment between 210 and 390 K. For the shear viscosity, the calculations indicate that the corrections in the second-order approximation and those due to the angular-momentum polarization are small, less than 0.7%, in the temperature range considered. The very good agreement of the calculated values with the experimental viscosity data suggests that the rigid-rotor approximation should be very reasonable for the three properties considered. In general, the agreement for the other measured properties is within the experimental error.

Transport Properties of Simple Fluids from the Soft-Core Double Yukawa Potential

The present work is concerned with checking a new and simple pair potential function (soft-core double Yukawa potential) for noble gases by calculation of the transport properties. The viscosity, thermal conductivity and self diffusion coefficient in dilute gas limit in the temperature range of 298-1400 K are calculated and agreement with the measurements is, in general, within the experimental error. A comparison of the calculated and experimental values of the viscosity, thermal conductivity and the diffusion coefficients yields an average absolute deviation of 0.5%, 1.5% and 1.2%, respectively. Also, the calculated transport properties from this potential have been compared with those calculations via the accurate experimental potential and also the corresponding state.

Nonclassical gasdynamic region of selected fluorocarbons

The nonclassical gasdynamic region of fluorinated substances belonging to the PP, FC, and E series is investigated using different thermodynamic models of increasing complexity. Thermodynamic models range from the simple van der Waals equation of state to the more complex Martin-Hou model and include the Redlich-Kwong, Clausius-II, Soave-Redlich-Kwong, and Peng-Robinson equations, under both the polytropic (constant isochoric specific heat in the dilute gas limit) and nonpolytropic approximations. The possibility of observing nonclassical gasdynamic behavior for a given fluid is confirmed to increase with the molecular weight of the substance times the specific heat at constant volume in the dilute gas limit and at the critical temperature, but to be almost insensitive to the nonpolytropic behavior of the fluid; the dependence on the compressibility at the critical point is also weak. A strong dependence on the acentric factor of the substance is revealed, which points to fluids made of nonpolar molecules with nearly spheroidal shapes as the most favorable to exhibit nonclassical gasdynamic behavior. In this respect, the fluorinated cyclic compounds of the PP series are singled out as the most promising candidates for the experimental verification of the existence of nonclassical phenomena in the vapor phase and for nonclassical gasdynamic applications.

Transport properties of H–N2 mixtures

Transport coefficients (shear viscosity, volume viscosity, thermal conductivity, and mass and thermal diffusion coefficients) of H–N2 mixtures in the dilute-gas limit have been calculated from the intermolecular potential in the temperature range 300–2000K using the classical trajectory method. The intermediate results pertaining to H–N2 binary interactions are reported, mainly in terms of cross-section ratios. Cross-sections evaluated with the Mason–Monchick approximation yield very good results for this system, the largest deviations, about 2.5%, being observed for the thermal diffusion coefficient. The accuracy here of this approximation can primarily be attributed to a light H atom and a weakly non-spherical potential resulting in a high rotational collision number. Furthermore, we investigate to which H–N2 cross-sections and their ratios the values of the mixture transport coefficients are most sensitive. Our results indicate that, for some cross-section ratios, reliance on universal correlations at high temperatures, often used in flame codes, can induce sizeable errors in the thermal conductivity coefficient and especially in the thermal diffusion coefficients. We also observed that the volume viscosity is particularly sensitive to the value of the cross-section for internal energy relaxation in H–N2 collisions.

Thermal Conductivity of Carbon Dioxide–Methane Mixtures at Temperatures Between 300 and 425 K and at Pressures up to 12 MPa

The thermal conductivities of carbon dioxide and three mixtures of carbon dioxide and methane at six nominal temperatures between 300 and 425 K have been measured as a function of pressure up to 12 MPa. The measurements were made with a transient hot-wire apparatus. The relative uncertainty of the reported thermal conductivities at a 95% confidence level is estimated to be ±1.2%. Results of the low-density analysis of the obtained data were used to test expressions for predicting the thermal conductivity of nonpolar mixtures in a dilute-gas limit developed by Schreiber, Vesovic, and Wakeham. The scheme was found to underestimate the experimental thermal conductivity with deviations not exceeding 5%. The dependence of the thermal conductivity on density was used to test the predictive scheme for the thermal conductivity of gas mixtures under pressure suggested by Mason et al. and improved by Vesovic and Wakeham. Comparisons reveal a pronounced critical enhancement on isotherms at 300 and 325 K for mixtures with methane mole fractions of 0.25 and 0.50. For other states, comparisons of the experimental and predicted excess thermal conductivity contributions showed a smaller increase of the experimental data with deviations approaching 3% within the examined range of densities.

Calculation of the transport properties of carbon dioxide. III. Volume viscosity, depolarized Rayleigh scattering, and nuclear spin relaxation.

Transport properties of pure carbon dioxide have been calculated from the intermolecular potential using the classical trajectory method. Results are reported in the dilute-gas limit for volume viscosity, depolarized Rayleigh scattering, and nuclear spin relaxation for temperatures ranging from 200 to 1000 K. Three recent carbon dioxide potential energy hypersurfaces have been investigated. Calculated values for the rotational collision number for all three intermolecular surfaces are consistent with the measurements and indicate that the temperature dependence of the Brau-Jonkman correlation is not applicable for carbon dioxide. The results for the depolarized Rayleigh scattering cross section and the nuclear spin relaxation cross section show that calculated values for the generally more successful potentials differ from the observations by 9% at about 290 K, although agreement is obtained for nuclear spin relaxation at about 400 K.

Density-Dependent Isotropic Raman Line Shapes in Compressed Room-Temperature Nitrogen

Using the model developed previously by Everitt and Skinner [J. Chem. Phys. 2001, 115, 8531], we calculate Raman line shapes of nitrogen at room temperature, from the dilute gas limit to over four times the critical density. At low density, it is necessary to use the full Kubo line shape formula, rather than its motionally narrowed limit. Our results for the average frequency shift from the gas phase, the peak frequency shift, and the line width are all in excellent agreement with experiment over the entire density range. The calculations shed some light on the competing physical mechanisms that are responsible for the interesting nonmonotonic density dependences of both the shift and width.

Calculation of the transport properties of carbon dioxide. II. Thermal conductivity and thermomagnetic effects.

The transport properties of pure carbon dioxide have been calculated from the intermolecular potential using the classical trajectory method. Results are reported in the dilute-gas limit for thermal conductivity and thermomagnetic coefficients for temperatures ranging from 200 K to 1000 K. Three recent carbon dioxide potential energy hypersurfaces have been investigated. Since thermal conductivity is influenced by vibrational degrees of freedom, not included in the rigid-rotor classical trajectory calculation, a correction for vibration has also been employed. The calculations indicate that the second-order thermal conductivity corrections due to the angular momentum polarization (< 2%) and velocity polarization (< 1%) are both small. Thermal conductivity values calculated using the potential energy hypersurface by Bukowski et al. (1999) are in good agreement with the available experimental data. They underestimate the best experimental data at room temperature by 1% and in the range up to 470 K by 1%-3%, depending on the data source. Outside this range the calculated values, we believe, may be more reliable than the currently available experimental data. Our results are consistent with measurements of the thermomagnetic effect at 300 K only when the vibrational degrees of freedom are considered fully. This excellent agreement for these properties indicates that particularly the potential surface of Bukowski et al. provides a realistic description of the anisotropy of the surface.

Transport Coefficients of Nuclear Matter at High Temperatures

High temperature coefficients of viscosity and heat conductivity of nuclear matter are calculated on the basis of Chapman-Enskong approximation in the dilute gas limit. It is observed that increasing temperatures produce significant changes in coefficients.

Comments on the U(2) ADHM two-instanton

We find that the U(N) two-instanton solution presented in [Phys. Lett. B 547 (2002) 85] is incorrect. For the U(2) case, we give a very simple form of the general two-instanton solution, so as to clearly show the faultiness of their solution. We also discuss the dilute instanton gas limit and the 't Hooft two-instanton based on our general solution.