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1

Ergodicity of non-Hamiltonian Equilibrium Systems

Evans D. J., Williams S. R., Rondoni L., Searles D. J.

Computational Methods in Science and Technology

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2017

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Vol. 23, No. 3

175--184

EN

It is well known that ergodic theory can be used to formally prove a form of relaxation to microcanonical equilibrium for finite, mixing Hamiltonian systems. In this manuscript we substantially modify this proof using an approach similar to that used in umbrella sampling, and use this approach to consider relaxation in both Hamiltonian and non- Hamiltonian systems. In doing so, we demonstrate the need for a form of ergodic consistency of the initial and final distribution. The approach only applies to relaxation of averages of physical properties and low order probability distribution functions. It does not provide any information about whether the full 6N-dimensional phase space distribution relaxes towards the equilibrium distribution or how long the relaxation of physical averages takes.

2

Teoria dyfuzji wieloskładnikowej w ośrodkach płynnych

Burghardt A.

Prace Naukowe Instytutu Inżynierii Chemicznej Polskiej Akademii Nauk

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2016

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nr 20

17--61

PL

Celem niniejszego opracowania jest przedstawienie szczegółowych równań dyfuzji wieloskładnikowej, wyprowadzonych w oparciu o mechaniczną teorię dyfuzji wieloskładnikowej. Porównano różne postacie równań definiujących siły napędowe dyfuzji wieloskładnikowej (uogólnione równania Maxwella-Stefana i Ficka-Onsagera). Przedstawiono również inne teorie dyfuzji (termodynamika nierównowagowa, mechanika statystyczna).

EN

The aim of the study is to present in detail the constitutive equations of multicomponent diffusion derived on the basis of the mechanical theory of multicomponent diffusion. Various formulations of the diffusion driving force have been compared (generalized Maxwell-Stefan, generalized Fick-Onsager equations). Other theories of the multicomponent diffusion (nonequilibrium thermodynamics, statistical mechanics) have been also discussed.

3

Why must we work in the phase space?

Sławianowski J. J., Schroeck Jr F. E., Martens A.

IPPT Reports on Fundamental Technological Research

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2016

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nr 1

1--162

EN

We are going to prove that the phase-space description is fundamental both in the classical and quantum physics. It is shown that many problems in statistical mechanics, quantum mechanics, quasi-classical theory and in the theory of integrable systems may be well-formulated only in the phase-space language. There are some misunderstandings and confusions concerning the concept of induced probability and entropy on the submanifolds of the phase space. First of all, they are restricted only to hypersurfaces in the phase space, i.e., to the manifolds of the defect of dimension equal to one. But what is more important, it was assumed there that the phase-space geometry was metrical-Euclidean and the resulting metric geometry of the microcanonical ensemble was obtained by the reduction of the primary Euclidean geometry to the corresponding submanifold. But it is well-known that the phase-space manifold has no natural metric geometry and that all concepts to be used must be of symplectic origin. Otherwise they are just accidental or artificial. So, instead we show that even if the configuration space is endowed with some metric, then in general the true geometry of submanifolds in the corresponding cotangent bundle (phase-space) is of different origin which has nothing to do with the mentioned configuration space Riemannian geometry, instead it is of purely symplectic origin. And this is sufficient to constructing microcanonical ensemble and entropy concepts. In any case, the purely symplectic phase-space geometry is sufficient to obtain everything within the completely metric-free language.

PL

Chcemy wykazać, że opis zjawisk mechanicznych oparty na pojęciu przestrzeni fazowej jest fundamentalny zarówno z klasycznego jak i kwantowego punktu widzenia. Pokazujemy, że liczne problemy mechaniki statystycznej, teorii kwantów i mechaniki quasiklasycznej oraz teorii układów całkowalnych mogą być dobrze sformułowane wyłącznie w języku symplektycznej przestrzeni fazowej. Istnieje mnóstwo nieporozumień czy wręcz błędów dotyczących pojęcia prawdopodobieństwa warunkowego i entropii w przypadku podrozmaitości przestrzeni fazowej. Przede wszystkim są one zazwyczaj definiowane dla przypadku powierzchni o defekcie wymiaru jeden. Co jednak dużo ważniejsze, zwykle zakłada się, że przestrzeń fazowa ma jednocześnie metryczną geometrię Euklidesową. Geometria metryczna podrozmaitości, używana w konstrukcji zespołu mikrokanonicznego, jest otrzymywana jako redukcja, ograniczenie pierwotnej geometrii Euklidesowej. Wiadomo jednak, że rozmaitość przestrzeni fazowej nie ma żadnej „wrodzonej” geometrii metrycznej i że wszystkie podstawowe pojęcia, wyjąwszy dynamikę konkretnych modeli, muszą mieć czysto symplektyczną genezę. W przeciwnym wypadku są one przypadkowe lub wręcz sztuczne. Zatem, nawet jeśli wyjściowa przestrzeń konfiguracyjna ma zadaną geometrię typu metrycznego, to na ogół właściwa geometria podrozmaitości w wiązce ko-stycznej, przynajmniej ta istotna dla pojęć statystycznych, nie jest związana z metryką konfiguracyjną i ma czysto symplektyczną genezę. I to wystarcza dla skonstruowania pojęcia zespołu mikrokanonicznego i entropii. W każdym razie, czysto symplektyczna geometria przestrzeni fazowej wystarcza do otrzymania pojęć mechaniki statystycznej w obrębie języka całkowicie niemetrycznego. W przypadku, gdy przestrzeń konfiguracyjna jest Euklidesowa, implikowane przez metrykę pojęcia statystyczne pokrywają się z symplektycznymi. W ogólnym wypadku nie musi tak być. Pokazujemy, że pojęcia te dadzą się wprowadzić w języku czysto symplektycznym, niezależnym od metryki konfiguracyjnej. Dotyczy to także uogólnionych rozkładów mikrokanonicznych.

4

Physical interpretation of the Cosserat mechanics for a collection of atoms

Sikoń M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2016

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Vol. 64, nr 2

333--338

EN

In this work, the Cosserat medium is analyzes as a set of atoms. These atoms are under the action of a mechanical load. The statistical analysis is preceded by a description of a single atom using classical mechanics and quantum mechanics. The behavior of the atoms in the field generated by mechanical change of the interatomic distance is shown as a phenomenon which can explain the Cosserat mechanics in a continuum.

5

Statistical mechanics in earth physics and natural hazards

Vallianatos F., Telesca L.

Acta Geophysica

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2012

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Vol. 60, no. 3

499-501

6

Earthquakes, model systems and connections to q-statistics

Celikoglu A., Tirnakli U.

Acta Geophysica

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2012

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Vol. 60, no. 3

535-546

EN

In this work, we make an attempt to review some of the recent studies on earthquakes using either real catalogs or synthetic data coming from some model systems. A common feature of all these works is the use of q -statistics as a tool.

7

On Noise Analysis of Oscillators Based on Statistical Mechanics

Thiessen T., Mathis W.

International Journal of Electronics and Telecommunications

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2010

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Vol. 56, No. 4

357-366

EN

In this paper a new approach of thermal noise analysis of electronic oscillators is presented. Although nonlinear electronic oscillators are one of the most essential subcircuits in electronic systems typical design concepts for these oscillators are based on ideas of linear circuits. Because the functionality of oscillators depends on nonlinearities, advanced design methods are developed where nonlinearities are an integral part. Since low voltage oscillator concepts have to be developed in modern IC technologies there is a need to include at least thermal noise aspects into the design flow. For this reason we developed new physical descriptions of thermal noise in electronic oscillators where we use ideas from nonequilibrium statistical mechanics as well as the Langevin approach. We illustrate our concepts by some examples.

8

Non-equilibrium statistical mechanics formulation of thermal transport properties for binary and ternary gas mixtures

Avsec J., Naterer G., Marcic M.

Archives of Thermodynamics

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2008

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Vol. 29, no. 2

21-46

EN

The paper develops a new statistical formulation to calculate the thermal diffusivity, binary diffusion coefficient, thermal diffusion factor, viscosity and thermal conductivity of gas mixtures with non-equilibrium statistical mechanics. For the analytical calculation of transport properties, the models of Kihara and Chapman-Cowling (up to the third order) have been used. Thermal transport properties for mixtures involving carbon monoxide, helium, argon, xenon and krypton have been computed with the new formulation in this paper. New mixing rules for the calculation of transport properties for mixtures are developed. Close agreement is obtained between the analytical results (based on statistical mechanics) and experimental data. The results exhibit comparable or better accuracy than previous methods, while providing new insight regarding the detailed statistical mechanisms of intermolecular interactions, as they contribute to the transport property variations with temperature.

9

Calculation of equilibrium and nonequilibrium thermophysical properties by means of statistical mechanics

Avsec J.

Journal of Technical Physics

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2003

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Vol. 44, no 2

163-179

EN

The paper features the mathematical model of computing equilibrium and nonequilibrium ther-mophysical properties of state in the liquid and gas domain for pure refrigerants and mixtures with the help of classical thermodynamics and statistical thermodynamics. The paper features all important contributions (translation, rotation, internal rotation, vibration, intermolecular potential energy and influence of electron and nuclei excitation). To calculate the thermodynamic properties of real fluid, the models on the basis of Lennard-Jones intermolecular potential were applied. To calculate the thermodynamic properties of real fluid by means of classical thermodynamics, we used the Tillner-Roth-Watanabe-Wagner (TRWW) equation on the base of Helmholtz type. We have developed a mathematical model for the calculation of all equilibrium and nonequilibrium thermodynamic functions of state for pure refrigerants. The analytical results obtained by statistical thermodynamics are compared with the TRWW model and show a relatively good agreement.

10

Comparison of thermodynamic functions calculated by means of various methods

Avsec J., Marcic M.

Journal of Technical Physics

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2002

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Vol. 43, no 3

309-321

EN

The paper deals with the Lennard-Jones fluid and presents the mathematical model of computating thermodynamic functions of state in the liquid and gas domain by means of statistical thermodynamics. To calculate the thermodynamic properties of a real fluid, we used the Johnson-Zollweg-Gubbins model based on the modified Benedict-Webb-Rubin equation of state, the Chunxi-Yigui-Jiufang equation of state based on the simple perturbation theory, and the complex Tang-Tong-Lu model based on the solution of the Ornstein-Zernike equation obtained by means of the perturbation theory. The analytical results are compared with the thermodynamical data, and with the results obtained from classical thermodynamics.

11

Computation of transport and equilibrium properties of molecular systems with quantum mechanical calculation of interaction potentials

Arefiev K. M., Shevkunov S. V.

International Journal of Applied Mechanics and Engineering

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2002

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Vol. 7, no 4

1095-1123

EN

The computation of transport coefficients in gases on the basis of molecular theory requires the determination of interaction potentials. As a rule, the dependencies of interaction energy on the distances between the molecules extracted from experimental data on different measurable characteristics are used. At the same time, direct calculation of interaction potentials on the basis of approximate solution of Schroedinger equation is possible for a number of relatively simple in their electron structure, but important for applications systems. Mixtures of the vapors of the atoms of metals with noble gases represent a typical example of such systems. In this paper, a comparison between experimental and calculated diffusion coefficients of the vapors of metals in the first and the second groups of Periodic Table dissolved in noble gases is presented. A sufficient for practical needs convergence of numerical results is demonstrated. The interaction potentials obtained can be used in the calculations of other transport coefficients, such as viscosity and thermal conductivity, in the mixtures of the vapors of metals with gases. Along with the traditional approach based on Schroedinger formalism, modern alternative methods of quantum mechanics and quantum statistics are presented. One example is the Path Integral Monte Carlo method based on Feynman representation of quantum mechanics. This formalism makes it possible to solve quantum statistical problems for thermally excited electron states and in this way to simulate numerically equilibrium properties of dense non-ideal plasma. Exchange and all correlation effects can be described in this formalism in an explicit way. Another modern approach aimed at stochastic simulations of electron quantum states is the so-called Diffusion Method, representing a solution of Schroedinger equation in imaginary time. Applications of stochastic methods in the problems of thermodynamics and plasma physics are presented. The perspectives and possible directions of development of new methods in the statistical description of condensed matter is briefly discussed.