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An Introduction to Statistical Mechanics and Thermodynamics最新文献

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Continuous Random Numbers 连续随机数
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2012-03-01 DOI: 10.1093/acprof:oso/9780199646944.003.0005
R. Swendsen
{"title":"Continuous Random Numbers","authors":"R. Swendsen","doi":"10.1093/acprof:oso/9780199646944.003.0005","DOIUrl":"https://doi.org/10.1093/acprof:oso/9780199646944.003.0005","url":null,"abstract":"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.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"104 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":"116101430","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 Harmonic Solid 谐波固体
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2012-03-01 DOI: 10.1093/ACPROF:OSO/9780199646944.003.0025
R. Swendsen
{"title":"The Harmonic Solid","authors":"R. Swendsen","doi":"10.1093/ACPROF:OSO/9780199646944.003.0025","DOIUrl":"https://doi.org/10.1093/ACPROF:OSO/9780199646944.003.0025","url":null,"abstract":"In Chapter 26 we return to calculating the contributions to the specific heat of a crystal from the vibrations of the atoms. The vibrations of a model of a solid, for which the interactions are quadratic in form, is investigated. Calculations are restricted to one dimension for simplicity in the derivations of the Fourier modes and the equations of motion. Both pinned and periodic boundary conditions are discussed. The representation of the Hamiltonian in terms of normal modes and the solution in terms of the equations of motion are derived. The Debye approximation is then introduced for three-dimensional systems.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"49 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":"130639472","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 经典理想气体
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2012-03-01 DOI: 10.1093/acprof:oso/9780199646944.003.0002
R. Swendsen
{"title":"The Classical Ideal Gas","authors":"R. Swendsen","doi":"10.1093/acprof:oso/9780199646944.003.0002","DOIUrl":"https://doi.org/10.1093/acprof:oso/9780199646944.003.0002","url":null,"abstract":"This chapter provides an introduction to an important model system in classical statistical mechanics. The classical ideal gas is defined, and it is explained why it is useful as an example in an introduction to the concept of entropy. What distinguishes an ‘ideal’ gas from a ‘real’ gas is the absence of interactions between the particles. Although an ideal gas might seem to be an unrealistic model, its properties are experimentally accessible by studying real gases at low densities. Since even the molecules in the air you are breathing are separated by an average distance of about ten times their diameter, nearly ideal gases are easy to find.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"43 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":"133676243","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
Classical Gases: Ideal and Otherwise 经典气体:理想气体和非理想气体
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2012-03-01 DOI: 10.1093/ACPROF:OSO/9780199646944.003.0007
R. Swendsen
{"title":"Classical Gases: Ideal and Otherwise","authors":"R. Swendsen","doi":"10.1093/ACPROF:OSO/9780199646944.003.0007","DOIUrl":"https://doi.org/10.1093/ACPROF:OSO/9780199646944.003.0007","url":null,"abstract":"As preparation for the derivation of the entropy for systems with interacting particles, the position and momentum variables are treated simultaneously, in this chapter, for the ideal gas. Releasing a constraint on the exchange of volume between two systems leads to an entropy maximum, just as the release of an energy- or particle-number constraint. This same principle is shown to be true for asymmetric pistons, which allow the total volume to change. The entropy of systems with interacting particles is then derived. The Second Law of Thermodynamics is established for general systems. Finally, the Zeroth Law of Thermodynamics is derived.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"12 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":"115757642","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
Temperature, Pressure, Chemical Potential, and All That 温度，压强，化学势等等
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2012-03-01 DOI: 10.1093/acprof:oso/9780199646944.003.0008
R. Swendsen
{"title":"Temperature, Pressure, Chemical Potential, and All That","authors":"R. Swendsen","doi":"10.1093/acprof:oso/9780199646944.003.0008","DOIUrl":"https://doi.org/10.1093/acprof:oso/9780199646944.003.0008","url":null,"abstract":"The Maxwell–Boltzmann distribution of momentum is obtained from statistical mechanics. Expressions for the temperature, pressure, and chemical potential are formulated as partial derivatives of the entropy with respect to energy, volume, and particle-number. The temperature scale is derived from comparison with the ideal gas law. The concept of the fundamental relation is defined as an expression that contains all thermodynamic information about the system of interest. Its differential form is introduced. Equations of state contain partial information about the thermal properties of a system and can be expressed as partial derivatives of the fundamental relation. The function of thermometers, pressure gauges, and thermal reservoirs are derived from these principles.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"5 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":"132826972","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
Black-Body Radiation 黑体辐射
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2012-03-01 DOI: 10.1093/ACPROF:OSO/9780199646944.003.0024
R. Swendsen
{"title":"Black-Body Radiation","authors":"R. Swendsen","doi":"10.1093/ACPROF:OSO/9780199646944.003.0024","DOIUrl":"https://doi.org/10.1093/ACPROF:OSO/9780199646944.003.0024","url":null,"abstract":"A black body is a perfect absorber of electromagnetic radiation. The energy spectrum was correctly calculated by Max Planck under the assumption that the energy of light waves only came in discrete multiples of a constant (called Planck’s constant) times the frequency. This was perhaps the first achievement of quantum mechanics. The derivation is presented here. The purpose of the current chapter is to calculate the spectrum of radiation emanating from a black body. The calculation was originally carried out by Max Planck in 1900 and published the following year. This was before quantum mechanics had been invented, or perhaps it could be regarded the first step in its invention.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"1 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":"130762351","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
Ensembles in Classical Statistical Mechanics 经典统计力学中的系综
An Introduction to Statistical Mechanics and Thermodynamics Pub Date : 2012-03-01 DOI: 10.1093/ACPROF:OSO/9780199646944.003.0019
R. Swendsen
{"title":"Ensembles in Classical Statistical Mechanics","authors":"R. Swendsen","doi":"10.1093/ACPROF:OSO/9780199646944.003.0019","DOIUrl":"https://doi.org/10.1093/ACPROF:OSO/9780199646944.003.0019","url":null,"abstract":"This chapter explores more powerful methods of calculation than were seen previously. Among them are Molecular Dynamics (MD) and Monte Carlo (MC) computer simulations. Another is the canonical partition function, which is related to the Helmholtz free energy. The derivation of thermodynamic identities within statistical mechanics is illustrated by the relationship between the specific heat and the fluctuations of the energy. It is shown how the canonical ensemble allows us to integrate out the momentum variables for many classical models. The factorization of the partition function is presented as the best trick in statistical mechanics, because of its central role in solving problems. Finally, the problem of many simple harmonic oscillators is solved, both for its importance and as an illustration of the best trick.","PeriodicalId":102491,"journal":{"name":"An Introduction to Statistical Mechanics and Thermodynamics","volume":"49 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":"121815804","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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