Boltzmann Statistical Mechanics Research Articles

Invariant Model of Boltzmann Statistical Mechanics and its Implications to Hydrodynamic Model of Electromagnetism, Physical Foundations of Quantum Mechanics, Relativity, Quantum Gravity and Quantum Cosmology

Invariant Model of Boltzmann Statistical Mechanics and its Implications to Hydrodynamic Model of Electromagnetism, Physical Foundations of Quantum Mechanics, Relativity, Quantum Gravity and Quantum Cosmology

Scale-Invariant Model of Boltzmann Statistical Mechanics and Generalized Thermodynamics

Some implications of a scale-invariant model of Boltzmann statistical mechanics to the laws of generalized thermodynamics are investigated. Through definition of stochastic Planck and Boltzmann universal constants, dimension of Kelvin absolute temperature T (degrees kelvin) is identified as a length (meters) associated with Wien wavelength Tβ = λwβ of particle thermal oscillations. Hence, thermodynamic temperature and atomic mass of the field 𝔽β at scale β provide internal measures of (extension, duration) of background space 𝕊β+1 = 𝔽β needed to define external space and time coordinates and atomic-mass-unit of 𝔽β+1. Introduction of invariant internal thermodynamic spacetime and Boltzmann factor are in harmony with modern concepts of quantum gravity as deterministic dissipative dynamic system [73]. The connections between de Pretto number 8338 and Joule-Mayer mechanical equivalent of heat Jc = 4.169 kJ / kcal and universal gas constant Ro = 8338 J / kcal.m are identified leading to the modified mechanical equivalent of heat J = 2Jc = 8338 J / kcal . It is shown that with work defined as Helmholtz free energy W = F = U – TS, Helmholtz decomposition of total thermal energy into free heat U and latent heat pV results in modified form of the first law of thermodynamics Q = H = U- W = U + pV. Finally, by application of Boltzmann combinatorics method, entropy of ideal gas is expressed in terms of the number of Heisenberg-Kramers virtual oscillators as S = 4Nk in exact agreement with predicted entropy of black hole by Major and Setter [159].

Quantum theory of multiscale coarse-graining.

Coarse-grained (CG) models serve as a powerful tool to simulate molecular systems at much longer temporal and spatial scales. Previously, CG models and methods have been built upon classical statistical mechanics. The present paper develops a theory and numerical methodology for coarse-graining in quantum statistical mechanics, by generalizing the multiscale coarse-graining (MS-CG) method to quantum Boltzmann statistics. A rigorous derivation of the sufficient thermodynamic consistency condition is first presented via imaginary time Feynman path integrals. It identifies the optimal choice of CG action functional and effective quantum CG (qCG) force field to generate a quantum MS-CG (qMS-CG) description of the equilibrium system that is consistent with the quantum fine-grained model projected onto the CG variables. A variational principle then provides a class of algorithms for optimally approximating the qMS-CG force fields. Specifically, a variational method based on force matching, which was also adopted in the classical MS-CG theory, is generalized to quantum Boltzmann statistics. The qMS-CG numerical algorithms and practical issues in implementing this variational minimization procedure are also discussed. Then, two numerical examples are presented to demonstrate the method. Finally, as an alternative strategy, a quasi-classical approximation for the thermal density matrix expressed in the CG variables is derived. This approach provides an interesting physical picture for coarse-graining in quantum Boltzmann statistical mechanics in which the consistency with the quantum particle delocalization is obviously manifest, and it opens up an avenue for using path integral centroid-based effective classical force fields in a coarse-graining methodology.

Notes on Contributors

Notes on Contributors

Re-visiting the O/Cu(111) system--when metastable surface oxides could become an issue!

Surface oxidation processes are crucial for the functionality of Cu-based catalytic systems used for methanol synthesis, partial oxidation of methanol or the water-gas shift reaction. We assess the stability and population of the "8"-structure, a [formula, see text:] oxide phase, on the Cu(111) surface. This structure has been observed in X-ray photoelectron spectroscopy and low-energy electron diffraction experiments as a Cu(111) surface reconstruction that can be induced by a hyperthermal oxygen molecular beam. Using density-functional theory calculations in combination with ab initio atomistic thermodynamics and Boltzmann statistical mechanics, we find that the proposed oxide superstructure is indeed metastable and that the population of the "8"-structure is competitive with the known "29" and "44" oxide film structures on Cu(111). We show that the configuration of O and Cu atoms in the first and second layers of the "8"-structure closely resembles the arrangement of atoms in the first two layers of Cu2O(110), where the atoms in the "8"-structure are more constricted. Cu2O(110) has been suggested in the literature as the most active low index facet for reactions such as water splitting under light illumination. If the "8"-structure were to form during a catalytic process, it is therefore likely to be one of the reactive phases.

DSM-V and the Future of Suicidology

Classifications are an evolutionistic answer to individual generalizations. They are conceived to establish common grounds and shared languages, and to fight “natural” chaos. Especially from Rudolf Clausius (entropy) and Ludwig Boltzmann (statistical mechanics) have we learned that chaos can be triggered – in fact always is triggered – by strictly deterministic causes (Cardwell, 1971). We have also learned that, from a deterministic point of view, we can’t go on for more than three levels of physical interventions; after that we should be prepared to face the chaos. Incidentally, Boltzmann was a bipolar sufferer and a believer in a truly Darwinistic perspective for cultural evolution. He hanged himself in 1906, during a summer holiday near Trieste (which at the time belonged to the Austro-Hungarian empire). Suicidologists, neuroscientists, epidemiologists, mentalhealth professionals, and many more (not the least patients) will soon be challenged in their nosographic habits (or destinies) by the new versions of the International Classification of Diseases (ICD-11, to be released in 2015) and the Diagnostic and Statistical Manual of Mental Disorders (DSM-V, expected to be released in early 2013). The latter constitutes the focus of this editorial.

The Statistical Interpretation of Entropy: An Activity

The second law of thermodynamics, which states that the entropy of an isolated macroscopic system can increase but will not decrease, is a cornerstone of modern physics. Ludwig Boltzmann argued that the second law arises from the motion of the atoms that compose the system. Boltzmann's statistical mechanics provides deep insight into the functioning of the second law and also provided evidence for the existence of atoms at a time when many scientists (like Ernst Mach and Wilhelm Ostwald) were skeptical.1

Is galaxy distribution non-extensive and non-Gaussian?

Self-gravitating systems in the Universe are generally thought to be non-extensive, and often show long-tails in various distribution functions. In principle, these non-Boltzmann properties are naturally expected from the peculiar property of gravity, long-range and unshielded. Therefore, the ordinary Boltzmann statistical mechanics would not be applicable for these self-gravitating systems in its naive form. In order to step further, we quantitatively investigate the above two properties, non-extensivity and long-tails, by explicitly introducing various models of statistical mechanics. We use the data of CfA II South redshift survey and apply the count-in-cell method. We study four statistical mechanics: (1) Boltzmann statistical mechanics, (2) Fractal statistical mechanics, (3) Rényi-entropy-based (REB) statistical mechanics, and (4) Tsallis statistical mechanics, and use Akaike information criteria (AIC) for the fair comparison. We found that both Rényi-entropy-based statistical model and Tsallis statistical model are far better than the other two models. Therefore, the long-tail in the distribution function turns out to be essential.

Lesche stability of κ-entropy

The Lesche stability condition for the Shannon entropy (J. Stat. Phys. 27 (1982) 419), represents a fundamental test, for its experimental robustness, for systems obeying the Maxwell–Boltzmann statistical mechanics. Of course, this stability condition must be satisfied by any entropic functional candidate to generate non-conventional statistical mechanics. In the present effort we show that the κ-entropy, recently introduced in literature (Phys. Rev. E 66 (2002) 056125), satisfies the Lesche stability condition.

Ludwig Boltzmann: The Man Who Trusted Atoms

Foreword Preface Introduction 1. short biography of Ludwig Boltzmann 2. Physics before Boltzmann 3. Kinetic theory before Boltzmann 4. The Boltzmann equation 5. Time irreversibility and the H-theorem 6. Boltzmann's relation and the statistical interpretation of entropy 7. Boltzmann, Gibbs and equilibrium statistical mechanics 8. The problem of polyatomic molecules 9. Boltzmann's contributions to other branches of physics 10. Boltzmann as a philosopher 11. Boltzmann and his contemporaries 12. The influence of Boltzmann's ideas on the science and technology of the twentieth century Epilogue Chronologys A German professor's journey to Eldorado Appendices

Application of Boltzmann Statistical Mechanics in the Validation of the Gaussian Summit-Height Distribution in Rough Surfaces

In theoretical modeling of contact mechanics, a homogeneously, isotropically rough surface is usually assumed to be a flat plane covered with asperities of a Gaussian summit-height distribution. This assumption yields satisfactory results between theoretical predictions and experimental measurements of the physical characteristics, such as thermal/electrical contact conductance and friction coefficient. However, lack of theoretical basis of this assumption motivates further study in surface modeling. This paper presents a theoretical investigation by statistical mechanics to determine surface roughness in terms of the most probable distribution of surface asperities. Based upon the surface roughness measurements as statistical constraints, the Boltzmann statistical model derives a distribution equivalent to Gaussian. The Boltzmann statistical mechanics derivation in this paper provides a rigorous validation of the Gaussian summit-height assumption presently in use for study of rough surfaces.

Pseudopotentials for centroid molecular dynamics: Application to self-diffusion in liquid para-hydrogen

The path integral centroid formulation of quantum Boltzmann statistical mechanics is employed to obtain temperature dependent pairwise effective pseudopotentials for use in centroid molecular dynamics (CMD) simulations of liquid para-hydrogen at 25 and 14 K. The self-diffusion constants calculated from these simulations are in good agreement with experiment, and with the results of more time consuming calculations based on the same theory. The CMD with pairwise centroid pseudopotential approach is therefore demonstrated to provide an extremely efficient and straightforward method for including quantum dynamical effects within a classical simulation framework.

Georgescu-Roegen's defense of classical thermodynamics revisited

Nicholas Georgescu-Roegen's criticisms, added to those of physicists and philosophers, result in a definitive refutation of Boltzmann's claim that his H-theorem derives the Second Law of Thermodynamics solely from Newtonian Mechanics. Some authors using Boltzmann's Statistical Mechanics go so far as to claim that perpetual motion machines are feasible in principle; this is also incorrect. Modern approaches by Ilya Prigogine and by the “Astrophysical School” to build thermodynamics on a basis that avoids Georgescu-Roegen's objections are surveyed. These conclusions have both epistemological and technological importance for economics; this paper emphasizes the former. The epistemological argument is: (1) The entropy law is an evolutionary law. (2) Modern neoclassical economics is arithmomorphic. (3) Arithmomorphic laws cannot capture evolutionary changes. (4) Hence neoclassical economics cannot incorporate the entropy law's implications for the economic process. (5) If one can show that modern physicists use evolutionary laws despite their non-arithmomorphic nature, then maybe (6) economists will become accepting of evolutionary laws. (7) This would be good because the entropy law, which is evolutionary (point 1), has important implications for economics. This paper primarily treats the most unfamiliar point for non-physicists, point 5.

Poincaré versus Boltzmann in Šil'nikov phenomena

By suitable adjustment of the control parameters in a CO 2 laser with feedback we show experimental evidence of Šil'nikov chaos, characterized by intensity pulses almost equal in shape, but irregularly separated in time. The times of return to a Poincaré section are statistically spread, however their iteration map is one-dimensional and in close agreement with that arising from Šil'nikov theory. Thus, the iteration map of the time intervals becomes the most appropriate indicator of this chaos. The residual width of the experimentally measured maps is due to a transient fluctuation enhancement peculiar to macroscopic systems, which is absent in low-dimensional chaotic dynamics. This appears as an inestricable mixture of deterministic unpredictability as introduced by Poincaré and stochastic fluctuations as considered in Boltzmann statistical mechanics.

A stochastic dynamics approach to quantum Boltzmann statistical mechanics

An alternative to the standard Metropolis-based approach for evaluating quantum statistical mechanical ensemble averages is introduced. The method is based on the close similarity between the Bloch equation and the Smoluchowski diffusion equation. Stochastic ‘‘Bloch trajectories’’ are constructed from local solutions to the Bloch equation for the density matrix and used to evaluate Boltzmann ensemble averages and partition functions. This is analogous to the use of Brownian dynamics trajectories in diffusion calculations. Sample results for selected one-dimensional systems are presented and the extension to multidimensional problems is discussed.

A Monte Carlo method for quantum Boltzmann statistical mechanics using Fourier representations of path integrals

By expanding Feynman path integrals in a Fourier series a practical Monte Carlo method is developed to calculate the thermodynamic properties of interacting systems obeying quantum Boltzmann statistical mechanics. Working expressions are developed to calculate internal energies, heat capacities, and quantum corrections to free energies. The method is applied to the harmonic oscillator, a double-well potential, and clusters of Lennard-Jones atoms parametrized to mimic the behavior of argon. The expansion of the path integrals in a Fourier series is found to be rapidly convergent and the computational effort for quantum calculations is found to be within an order of magnitude of the corresponding classical calculations. Unlike other related methods no special techniques are required to handle systems with strong short-range repulsive forces.

A Monte Carlo method for quantum Boltzmann statistical mechanics

An alternate approach to quantum Boltzmann statistical mechanics in calculations on interacting many-body systems is presented in the form of the Monte-Carlo method. (AIP)

Multiphonon Processes in the Nonradiative Decay of Large Molecules

Radiationless transition rates for polyatomic molecules are investigated in the simplest case of the statistical limit, i.e., large energy gaps between the two electronic states. By analogy with the theory of optical line shapes in solids, the “golden rule” rate expression for the nonradiative decay, which is usually written as a double sum over initial and final vibronic states, is equivalently represented in closed form as a single Fourier integral. For the case in which the vibrations are assumed to be harmonic, but may have different frequencies and equilibrium positions in the two electronic states, general closed-form analytic expressions, which include all of the vibrational modes, are obtained for the transition rates. The energy gap law is again obtained in the weak coupling statistical limit along with a proper description of the propensity rules for the promoting mode. For the case of the aromatic hydrobarbons, the effective energy gap is found to be in agreement with Siebrand's analysis, and explicit account is taken of the competition between the C–H modes and the C–C skeletal modes (which have large oscillator displacements) or the out-of-plane modes (which have large frequency changes) for the electronic energy which is involved in the relaxation process. The deuterium isotope effect is again found to be dominated by the modes of highest frequency (C–H or C–D), but corrections due to the C–C modes, which may be measurable, are explicitly included. The theory is also extended to include the role of the host medium as an inert heat bath by considering the intermolecular phonons as potential accepting modes. The approximations employed are compared with those usually made in Boltzmann statistical mechanics, thereby providing greater understanding as to the role of the Franck–Condon principle in the determination of the nonradiative decay rates.