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An Introduction to Statistical Mechanics and Thermodynamics最新文献

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Thermodynamic Identities 热力学的身份
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2019-12-10 DOI: 10.1093/oso/9780198853237.003.0014
R. Swendsen
{"title":"Thermodynamic Identities","authors":"R. Swendsen","doi":"10.1093/oso/9780198853237.003.0014","DOIUrl":"https://doi.org/10.1093/oso/9780198853237.003.0014","url":null,"abstract":"Many of the calculations in thermodynamics concern the effects of small changes. To carry out such calculations, we often need to evaluate first and second partial derivatives of some thermodynamic quantities with respect to other thermodynamic quantities. Although there are many such partial second derivatives, they are related by thermodynamic identities. This chapter explains the most straightforward way of deriving the needed thermodynamic identities. After explaining the derivation of Maxwell relations and how to find the right one for any given problem, Jacobian methods are introduced, with an accolade to their simplicity and utility. Several examples of the derivation of thermodynamic identities are given, along with a systematic guide for solving general problems.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128856789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stability Conditions 稳定性条件
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2019-12-10 DOI: 10.1093/oso/9780198853237.003.0016
R. Swendsen
{"title":"Stability Conditions","authors":"R. Swendsen","doi":"10.1093/oso/9780198853237.003.0016","DOIUrl":"https://doi.org/10.1093/oso/9780198853237.003.0016","url":null,"abstract":"Stability exists when a thermodynamic phase remains homogenous instead of separating into phases of high and low density (clumping). Certain conditions on the second partial derivatives of extensive variables are necessary for stability, even when the first derivatives do not vanish. These conditions can be expressed in terms of the compressibility and specific heat. Inequalities involving second partial derivatives with respect to intensive variables are derived. We have been assuming that the density of a gas will remain uniform, rather than having most of the particles clump together in one part of the container, leaving the rest of the volume nearly empty.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130796718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Phase Transitions 相变
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2019-12-10 DOI: 10.1093/oso/9780198853237.003.0017
R. Swendsen
{"title":"Phase Transitions","authors":"R. Swendsen","doi":"10.1093/oso/9780198853237.003.0017","DOIUrl":"https://doi.org/10.1093/oso/9780198853237.003.0017","url":null,"abstract":"Phase transitions are introduced using the van der Waals gas as an example. The equations of state are derived from the Helmholtz free energy of the ideal gas. The behavior of this model is analyzed, and an instability leads to a liquid-gas phase transition. The Maxwell construction for the pressure at which a phase transition occurs is derived. The effect of phase transition on the Gibbs free energy and Helmholtz free energy is shown. Latent heat is defined, and the Clausius–Clapeyron equation is derived. Gibbs' phase rule is derived and illustrated.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116516843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Extremum Principles 极值原理
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2019-12-10 DOI: 10.1093/oso/9780198853237.003.0015
R. Swendsen
{"title":"Extremum Principles","authors":"R. Swendsen","doi":"10.1093/oso/9780198853237.003.0015","DOIUrl":"https://doi.org/10.1093/oso/9780198853237.003.0015","url":null,"abstract":"This chapter derives the energy minimum principle from the entropy maximum principle. It postulates and consider the consequences of extensivity. From this are further derived minimum principles for the Helmholtz free energy, enthalpy, and Gibbs free energy. Because of its importance in engineering, exergy is also introduced, and the exergy minimum principle is justified. Analogously to these minimum principles, maximum principles can be derived for the Massieu functions from the entropy maximum principle. For the analysis of the entropy maximum principle, we isolated a composite system and released an internal constraint. Since the composite system was isolated, its total energy remained constant. The composite system went to the most probable macroscopic state after release of the internal constraint, and the total entropy went to its maximum.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126437739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quantum Ensembles
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2019-12-10 DOI: 10.1093/oso/9780198853237.003.0023
R. Swendsen
{"title":"Quantum Ensembles","authors":"R. Swendsen","doi":"10.1093/oso/9780198853237.003.0023","DOIUrl":"https://doi.org/10.1093/oso/9780198853237.003.0023","url":null,"abstract":"The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126914964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Irreversibility 不可逆性
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2019-12-10 DOI: 10.1093/oso/9780198853237.003.0022
R. Swendsen
{"title":"Irreversibility","authors":"R. Swendsen","doi":"10.1093/oso/9780198853237.003.0022","DOIUrl":"https://doi.org/10.1093/oso/9780198853237.003.0022","url":null,"abstract":"The phenomenon of irreversibility is explained on the basis of an analysis by H. L. Frisch. The history of the debate over irreversibility is briefly discussed, including Boltzmann’s H-theorem, Zermelo's Wiederkehreinwand, Poincaré recurrences, Loschmidt's Umkehreinwand and Liouville’s theorem. The derivation of irreversible behavior for the ideal gas position distribution is carried out explicitly. Using this derivation, the Wiederkehreinwand and the Umkehreinwand are revisited and explained. The first thing we must establish is the meaning of the term ‘irreversibility’. This is not quite as trivial as it might seem. The irreversible behavior I will try to explain is that which is observed. Every day we see that time runs in only one direction in the real world,.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131090578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Classical Ideal Gas: Energy Dependence of Entropy 经典理想气体:熵的能量依赖
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2019-12-10 DOI: 10.1093/oso/9780198853237.003.0006
R. Swendsen
{"title":"The Classical Ideal Gas: Energy Dependence of Entropy","authors":"R. Swendsen","doi":"10.1093/oso/9780198853237.003.0006","DOIUrl":"https://doi.org/10.1093/oso/9780198853237.003.0006","url":null,"abstract":"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.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122745397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Thermodynamic Processes 热力学过程
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2019-12-10 DOI: 10.1093/oso/9780198853237.003.0011
R. Swendsen
{"title":"Thermodynamic Processes","authors":"R. Swendsen","doi":"10.1093/oso/9780198853237.003.0011","DOIUrl":"https://doi.org/10.1093/oso/9780198853237.003.0011","url":null,"abstract":"This chapter begins by defining terms critical to understanding thermodynamics: reversible, irreversible, and quasi-static. Because heat engines are central to thermodynamic principles, they are described in detail, along with their operation as refrigerators and heat pumps. Various expressions of efficiency for such engines lead to alternative expressions of the second law of thermodynamics. A Carnot cycle is discussed in detail as an example of an idealized heat engine with optimum efficiency. A special case, called negative temperatures, where temperatures actually exceed infinity, provides further insights. In this chapter we will discuss thermodynamic processes, which concern the consequences of thermodynamics for things that happen in the real world.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117177031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
The Classical Ideal Gas: Configurational Entropy 经典理想气体:构型熵
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2012-03-01 DOI: 10.1093/ACPROF:OSO/9780199646944.003.0004
R. Swendsen
{"title":"The Classical Ideal Gas: Configurational Entropy","authors":"R. Swendsen","doi":"10.1093/ACPROF:OSO/9780199646944.003.0004","DOIUrl":"https://doi.org/10.1093/ACPROF:OSO/9780199646944.003.0004","url":null,"abstract":"This chapter derives the part of the entropy that is generated by the positions of particles, or the configurational entropy. The remaining part of the entropy, which is generated by the momenta of the particles, is derived in Chapter 6. While both derivations are unconventional, they are based directly on an 1877 paper by Boltzmann that discusses the exchange of energy between two or more systems. The dependence of the entropy on the number of particles is derived solely by assuming that the probability of a given particle being in a specified volume is proportional to that volume. No quantum mechanics is required for this derivation, and the result is valid for both distinguishable and indistinguishable particles.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128268644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bose–Einstein Statistics 玻色-爱因斯坦统计
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2012-03-01 DOI: 10.1093/ACPROF:OSO/9780199646944.003.0027
R. Swendsen
{"title":"Bose–Einstein Statistics","authors":"R. Swendsen","doi":"10.1093/ACPROF:OSO/9780199646944.003.0027","DOIUrl":"https://doi.org/10.1093/ACPROF:OSO/9780199646944.003.0027","url":null,"abstract":"The properties of the ideal Bose gas are calculated from the integral equations for the energy and the number of particles as a function of the temperature and chemical potential. It is shown that the integral equations break down below the Einstein temperature that corresponds to the transition to the low-temperature state. The lowest single-particle energy level must be treated explicitly to get the proper equations. With the inclusion of the lowest single-particle energy level, the low-temperature behavior is calculated. The occupation of the lowest level becomes comparable to the total number of particles in the system below the Einstein temperature, and equal to the total number of particles at zero temperature. A numerical solution to the properties of the Bose gas is discussed, and the detailed calculations are assigned to the problems at the end of the chapter.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117123323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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