Refining the Definition of Entropy

2019 ◽  
pp. 266-270
Author(s):  
Robert H. Swendsen
Keyword(s):  
Energy Distribution ◽  
Energy Dependence ◽  
Thermal Contact ◽  
Delta Function ◽  
Ideal Gas ◽  
Square Root ◽  
Macroscopic System ◽  
Densities Of States ◽  
Definition Of ◽  
Grand Canonical

If a macroscopic system as ever been in thermal contact with another macroscopic system, the width of the energy distribution is not zero. This is in contrast to the approximation made in Chapter 7 that the energy dependence of the entropy is given by a delta function. The width is very narrow (proportional to the inverse square root of the number of particles), but this leads to small errors in the predictions of the entropy. Massieu functions are used to derive the canonical entropy because they allow the extension to non-monotonic densities of states, which will be needed in later chapters. The grand canonical entropy is defined similarly. The canonical entropy and the grand canonical entropy of the classical ideal gas are calculated as examples.

The Classical Ideal Gas: Energy Dependence of Entropy

2019 ◽  
pp. 70-80
Author(s):  
Robert H. Swendsen
Keyword(s):  
Energy Distribution ◽  
Energy Dependence ◽  
Configurational Entropy ◽  
Ideal Gas ◽  
Dimensional Sphere ◽  
Total Entropy ◽  
Energy Dependent ◽  
True Energy ◽  
Area And Volume

The energy-dependence of the entropy of the configurational contributions is derived by considering the exchange if energy is exchanged between two or more systems. The argument is analogous to that given in Chapter 5 for the configurational contributions to the entropy. The derivation requires evaluating the area and volume of an $n$-dimensional sphere, which is carried out explicitly. The entropy is calculated within the approximation that the width of the energy distribution is zero. The total entropy is just the sum of the configurational entropy and the energy-dependent terms, as discussed in Section 4.1. The significance of the non-zero width of the true energy distribution will be addressed in Chapter 21.

scholarly journals On definition of the ideal gas

1910 ◽  
Vol 6 (3) ◽  
pp. 409
Author(s):  
E. Buckingham
Keyword(s):  
Ideal Gas ◽  
Definition Of ◽  
The Ideal

Continuous Random Numbers

2019 ◽  
pp. 54-69
Author(s):  
Robert H. Swendsen
Keyword(s):  
Probability Distributions ◽  
Random Variables ◽  
Delta Function ◽  
Ideal Gas ◽  
Dirac Delta Function ◽  
Bayesian Probability ◽  
Random Numbers ◽  
Continuous Variables ◽  
Dirac Delta ◽  
Discrete Random Variables

The theory of probability developed in Chapter 3 for discrete random variables is extended to probability distributions, in order to treat the continuous momentum variables. The Dirac delta function is introduced as a convenient tool to transform continuous random variables, in analogy with the use of the Kronecker delta for discrete random variables. The properties of the Dirac delta function that are needed in statistical mechanics are presented and explained. The addition of two continuous random numbers is given as a simple example. An application of Bayesian probability is given to illustrate its significance. However, the components of the momenta of the particles in an ideal gas are continuous variables.

A quantitative measure of nonlinearity

1993 ◽  
Vol 39 (5) ◽  
pp. 766-772 ◽  
Author(s):  
K Emancipator ◽  
M H Kroll
Keyword(s):  
Analytical Method ◽  
Response Curve ◽  
Polynomial Regression ◽  
Quantitative Measure ◽  
Square Root ◽  
Straight Line ◽  
The Mean ◽  
The Difference ◽  
Definition Of ◽  
Fitting In

Abstract Quantitative measures of the nonlinearity of an analytical method are defined as follows: the "(dimensional) nonlinearity" of a method is the square root of the mean of the square of the deviation of the response curve from a straight line, where the straight line is chosen to minimize the nonlinearity. The "relative nonlinearity" is defined as the dimensional nonlinearity divided by the difference between the maximum and minimum assayed values. These definitions may be used to develop practical criteria for linearity that are still objective. Calculation of the nonlinearity requires a method of curve-fitting. In this article, we use polynomial regression to demonstrate calculations, but the definition of nonlinearity also accommodates alternative nonlinear regression procedures.

Capillary Absorption Dynamics for Cementitious Material Considering Water Evaporation and Tortuosity of Capillary Pores

2013 ◽  
Vol 821-822 ◽  
pp. 1213-1218 ◽  
Author(s):  
Xue Wu Liu ◽  
Jin Bo Yang ◽  
Kai Quan Xia ◽  
Peng Zhang ◽  
Zhan Guo Li
Keyword(s):  
Capillary Flow ◽  
Fractal Theory ◽  
Testing Procedure ◽  
Cementitious Material ◽  
Initial Time ◽  
Water Evaporation ◽  
Capillary Absorption ◽  
Square Root ◽  
Absorption Test ◽  
Definition Of

This paper presents the theoretical analysis of capillary absorption dynamics for cemementitious material. Fractal theory is applied to analyse tortuosity of capillary pores in cementitious material and a definition of tortuosity is given. The dynamic equation of capillary absorption considering water evaporation and tortuosity of capillary pores is derived. Based on the dynamic model, the capillary coefficient and sorptivity of concrete are explained theoretically. In absorption test, water evaporation is one of the main reasons caused variations from linearity between water absorption height and the square root of time, or between water amount absorbed and the square root of time. In cementitious material, the evaporation rate is very small compare to capillary flow velocity at the initial time of absorption test. For simplification of testing procedure, there is no meaning to modify absorption test.

White noise delta functions and infinite-dimensional Laplacians

2020 ◽  
pp. 2050028
Author(s):  
Luigi Accardi ◽  
Ai Hasegawa ◽  
Un Cig Ji ◽  
Kimiaki Saitô
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Time Derivative ◽  
Delta Function ◽  
Itô Formula ◽  
Ito Formula ◽  
Infinite Dimensional ◽  
Delta Functions ◽  
Definition Of ◽  
White Noise Distribution

In this paper, we introduce a new white noise delta function based on the Kubo–Yokoi delta function and an infinite-dimensional Brownian motion. We also give a white noise differential equation induced by the delta function through the Itô formula introducing a differential operator directed by the time derivative of the infinite-dimensional Brownian motion and an extension of the definition of the Volterra Laplacian. Moreover, we give an extension of the Itô formula for the white noise distribution of the infinite-dimensional Brownian motion.

Yang–Yang equilibrium statistical mechanics: A brilliant method

2016 ◽  
Vol 30 (09) ◽  
pp. 1630008
Author(s):  
Xi-Wen Guan ◽  
Yang-Yang Chen
Keyword(s):  
Statistical Mechanics ◽  
Weak Interactions ◽  
Delta Function ◽  
Ideal Gas ◽  
Baxter Equation ◽  
Dimensional System ◽  
Equilibrium Statistical Mechanics ◽  
Rigorous Approach ◽  
Wide Range ◽  
The One

Yang and Yang in 1969 [J. Math. Phys. 10, 1115 (1969)] for the first time proposed a rigorous approach to the thermodynamics of the one-dimensional system of bosons with a delta-function interaction. This paper was a breakthrough in exact statistical mechanics, after Yang [Phys. Rev. Lett. 19, 1312 (1967)] published his seminal work on the discovery of the Yang–Baxter equation in 1967. Yang and Yang’s brilliant method yields significant applications in a wide range of fields of physics. In this paper, we briefly introduce the method of the Yang–Yang equilibrium statistical mechanics and demonstrate a fundamental application of the Yang–Yang method for the study of thermodynamics of the Lieb–Liniger model with strong and weak interactions in a whole temperature regime. We also consider the equivalence between the Yang–Yang’s thermodynamic Bethe ansatz equation and the thermodynamics of the ideal gas with the Haldane’s generalized exclusion statistics.

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