Statistical Mechanics of Monatomic Systems in an External Periodic Potential Field. II. Distribution Function Theory for Fluids

1961 ◽  
Vol 34 (5) ◽  
pp. 1543-1553 ◽  
Author(s):  
Terrell L. Hill ◽  
Nobuhiko Saitô
Keyword(s):  
Distribution Function ◽  
Statistical Mechanics ◽  
Potential Field ◽  
Periodic Potential ◽  
Function Theory

scholarly journals On hydrodynamics in the presense of strong external potential field

2021 ◽  
Vol 29 (1) ◽  
pp. 21-28
Author(s):  
A. I. Sokolovsky ◽  
S. A. Sokolovsky
Keyword(s):  
Distribution Function ◽  
Kinetic Equation ◽  
Potential Field ◽  
Chemical Potential ◽  
Local Equilibrium ◽  
Practical Importance ◽  
External Potential ◽  
Equilibrium Distribution Function ◽  
Kinetic Coefficients ◽  
Functional Hypothesis

On the base of the Boltzmann kinetic equation, hydrodynamics of a dilute gas in the presence of the strong external potential field is investigated. First of all, a gravitational field is meant, because the consistent development of hydrodynamics in this environment is of great practical importance. In the present paper it is assumed that it is possible to neglect the influence of the field on the particle collisions. The study is based on the Chapman–Enskog method in a Bogolyubov’s formulation, which uses the idea of the functional hypothesis. Consideration is limited to steady gas states, which are subjected to a simpler experimental study. Chemical potential μ0 of the gas at the point where the external field has zero value and its temperature T are selected as the reduced description parameters of the system. In equilibrium, in the presence of the field, these values do not depend on the coordinates. It is assumed that in thehydrodynamic states T and μ0 are weakly dependent on the coordinates and therefore their gradients, considered on the scale of the free path length of the gas, are small. The kinetic equation, accounting for the functional hypothesis, gives an integro-differential equation for a gas distribution function at the hydrodynamic stage of evolution. This equation is solved in perturbation theory in gradients of T and μ0. The main approximation is analyzed for possibility of the system to be in a local equilibrium by means of comparing it with an equilibrium distribution function. Next, the distribution function is calculated in the first approximation in gradients and it is expressed in terms of solutions Ap , Bp of some first kind integral Fredholm equations. An approach to the approximate solution of these equations is discussed. The found distribution function is used to calculate the fluxes of the number of gas particles and their energy in the first order in gradients T and μ0 . Kinetic coefficients, which describe the structure of these fluxes, are introduced. Matrix elements of the operator of the linearized collision integral (integral brackets) are used for their research. It is a question of validity of the principle of symmetry of kinetic coefficients and definition of their signs.

BET Adsorption Isotherm and Surface Heterogeneity

1992 ◽  
Vol 57 (6) ◽  
pp. 1201-1209
Author(s):  
Lydia Ethel Cascarini de Torre ◽  
Eduardo Jorge Bottani
Keyword(s):  
Distribution Function ◽  
Adsorption Isotherm ◽  
Statistical Mechanics ◽  
Adsorption Isotherms ◽  
Surface Heterogeneity ◽  
Adsorption Energies ◽  
Multilayer Formation ◽  
Effect Of Surface

The BET adsorption isotherm is modified in order to take account of surface heterogeneity. The adsorption isotherm is obtained following the statistical mechanics formalism, proposed by Steele, and the effect of surface heterogeneity is limited to the first layer. A Gaussian adsorption energies distribution function is used to describe surface heterogeneity. The variations of the C parameter, multilayer formation and the inversion of adsorption isotherms are analysed.

scholarly journals On One Application of Infinite Systems of Functional Equations in Function Theory

2019 ◽  
Vol 74 (1) ◽  
pp. 117-144 ◽  
Author(s):  
Symon Serbenyuk
Keyword(s):  
Distribution Function ◽  
Unique Solution ◽  
Local Structure ◽  
Functional Equations ◽  
Function Theory ◽  
Shift Operator ◽  
Random Variable ◽  
System Of Functional Equations ◽  
Infinite Systems

Abstract The paper presents the investigation of applications of infinite systems of functional equations for modeling functions with complicated local structure that are defined in terms of the nega-˜Q-representation. The infinite systems of functional equations f\left( {{{\hat \varphi }^k}(x)} \right) = \tilde \beta {i_{k + 1}},k + 1 + \tilde p{i_{k + 1}},k + 1f\left( {{{\hat \varphi }^{k + 1}}(x)} \right), where x = \Delta _{{i_1}(x){i_2}(x) \ldots {i_n}(x) \ldots }^{ - \tilde Q} , and φ ̑ is the shift operator of the Q̃-expansion, are investigated. It is proved that the system has a unique solution in the class of determined and bounded on [0, 1] functions. Its analytical presentation is founded. The continuity of the solution is studied. Conditions of its monotonicity and nonmonotonicity, differential, and integral properties are studied. Conditions under which the solution of the system of functional equations is a distribution function of the random variable \eta = \Delta _{{\xi _1}\,\xi 2 \ldots {\xi _n} \ldots }^{\tilde Q} with independent Q̃-symbols are founded.

IMPLICATIONS OF INTERPOLATIVE QUANTUM STATISTICS — A REVIEW

1995 ◽  
Vol 09 (03n04) ◽  
pp. 135-143 ◽  
Author(s):  
R. RAMANATHAN
Keyword(s):  
Distribution Function ◽  
Statistical Mechanics ◽  
Distribution Functions ◽  
Quantum Statistics ◽  
Physical Processes ◽  
Positivity Condition ◽  
Statistical Interaction

A review of the interpolative quantum statistics is presented. It is shown, by imposing a positivity condition on the distribution function, that this statistics subsumes Gentile's intermediate statistics and is far richer than Gentile's statistics, apart from being a genuine quantum statistics. It is also capable of accounting for the possible very small violations of statistics in physical processes. It is finally pointed out that this statistics offers an elegant and useful parametrization of Haldane's 'statistical interaction', while retaining all the interesting features of the latter. Some of the statistical mechanics that these distribution functions lead to are also examined.

A variational approach to distribution function theory

1990 ◽  
Vol 61 (1-2) ◽  
pp. 143-160 ◽  
Author(s):  
Antoine G. Schlijper ◽  
Ryoichi Kikuchi
Keyword(s):  
Distribution Function ◽  
Variational Approach ◽  
Function Theory

Analyzing of the Image Characteristic Base on Potential Field Distraction Function

2013 ◽  
Vol 339 ◽  
pp. 389-395
Author(s):  
Ze Biao Shan ◽  
Yao Wu Shi ◽  
Hong Cheng Li ◽  
Hong Wei Shi
Keyword(s):  
Distribution Function ◽  
Field Distribution ◽  
Potential Field ◽  
Transition Point ◽  
Amplitude Distribution ◽  
Threshold Value ◽  
Object Distance ◽  
Image Characteristics ◽  
Image Characteristic ◽  
Definition Of

A new definition of potential field distribution function of image was proposed, and the characteristics of the distribution function were analyzed. Distribution function of potential field represents whole fluctuant information of image. So it can be used to analyze the image characteristics. Potential field of image background is weak and correspond to Gaussian distribution. Low stage of potential field distribution function can be used to evaluate noise level of image. Inflection point of the amplitude distribution function of potential field represented the transition point between the background and the actual edge in potential field image. So value of this point can be used as the threshold value to identify edge. When object distance of microscope changes, the edge comes more and more clear, and the edge area judged by the method above will be bigger and bigger. This character can be used to instruct auto-focus process.

Sign in / Sign up

Export Citation Format

Share Document