*John von Neumann*

*Nicholas A. Wheeler (ed.)*

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780691178561
- eISBN:
- 9781400889921
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691178561.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter provides the fundamental basis of the statistical theory, building on the formula introduced in the previous chapter, before elaborating proofs of the statistical formulas. From these, ...
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This chapter provides the fundamental basis of the statistical theory, building on the formula introduced in the previous chapter, before elaborating proofs of the statistical formulas. From these, the chapter shows that the most general statistical ensemble which is compatible with the chapter's qualitative basic assumptions is characterized, according to 𝗧𝗿, by a definite operator 𝗨. Furthermore, those particular ensembles which have been called “homogeneous” were characterized by 𝗨 = 𝙋subscript [φ] (∥φ∥ = 1), and since these are the actual states of the systems 𝗦 (i.e., not capable of further resolution) they can also be called states (specifically, 𝗨 = 𝙋subscript [φ] is the state φ).Less

This chapter provides the fundamental basis of the statistical theory, building on the formula introduced in the previous chapter, before elaborating proofs of the statistical formulas. From these, the chapter shows that the most general statistical ensemble which is compatible with the chapter's qualitative basic assumptions is characterized, according to 𝗧𝗿, by a definite operator 𝗨. Furthermore, those particular ensembles which have been called “homogeneous” were characterized by 𝗨 = 𝙋subscript [φ] (∥φ∥ = 1), and since these are the actual states of the systems 𝗦 (*i.e.*, not capable of further resolution) they can also be called states (specifically, 𝗨 = 𝙋subscript [φ] is the state φ).

*Hans-Peter Eckle*

- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780199678839
- eISBN:
- 9780191878589
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199678839.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

Chapter 4 reviews the basic notions of equilibrium statistical mechanics and begins with its fundamental postulate and outlines the structure of the theory using the most important of the various ...
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Chapter 4 reviews the basic notions of equilibrium statistical mechanics and begins with its fundamental postulate and outlines the structure of the theory using the most important of the various statistical ensembles, the microcanonical, the canonical, and the grand canonical ensemble and their corresponding thermodynamic potential, the internal energy, the Helmholtz free energy, and the grand canonical potential. The notions of temperature, pressure, and chemical potential are obtained and it introduces the laws of thermodynamics, the Gibbs entropy, and the concept of the partition function. It also discusses quantum statistical mechanics using the density matrix and as applied to non-interacting Bosonic and Fermionic quantum gases, the former showing Bose–Einstein condensation. The mean-field theory of interacting magnetic moments and the transfer matrix to exactly solve the Ising model in one dimension serve as applications.Less

Chapter 4 reviews the basic notions of equilibrium statistical mechanics and begins with its fundamental postulate and outlines the structure of the theory using the most important of the various statistical ensembles, the microcanonical, the canonical, and the grand canonical ensemble and their corresponding thermodynamic potential, the internal energy, the Helmholtz free energy, and the grand canonical potential. The notions of temperature, pressure, and chemical potential are obtained and it introduces the laws of thermodynamics, the Gibbs entropy, and the concept of the partition function. It also discusses quantum statistical mechanics using the density matrix and as applied to non-interacting Bosonic and Fermionic quantum gases, the former showing Bose–Einstein condensation. The mean-field theory of interacting magnetic moments and the transfer matrix to exactly solve the Ising model in one dimension serve as applications.

*A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo*

- Published in print:
- 2014
- Published Online:
- October 2014
- ISBN:
- 9780199581931
- eISBN:
- 9780191787140
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199581931.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter begins with a rapid introduction to the definition and construction of the statistical ensembles, following the lines that are usually offered in the basics courses on statistical ...
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This chapter begins with a rapid introduction to the definition and construction of the statistical ensembles, following the lines that are usually offered in the basics courses on statistical mechanics. The reader is assumed to be already acquainted with thermodynamics and basic statistical mechanics, but the introduction is self-contained. It is then shown that with short-range interactions the ensembles are physically equivalent, even in the presence of phase transitions. This is done by introducing the notions of concave functions, of stable and tempered potentials, and the Legendre-Fenchel transform of thermodynamic functions. The concavity of the entropy is at the basis of ensemble equivalence. Finally, the concepts of microstate and macrostate are analysed; they are very important for the study of ensemble equivalence or inequivalence in long-range systems.Less

This chapter begins with a rapid introduction to the definition and construction of the statistical ensembles, following the lines that are usually offered in the basics courses on statistical mechanics. The reader is assumed to be already acquainted with thermodynamics and basic statistical mechanics, but the introduction is self-contained. It is then shown that with short-range interactions the ensembles are physically equivalent, even in the presence of phase transitions. This is done by introducing the notions of concave functions, of stable and tempered potentials, and the Legendre-Fenchel transform of thermodynamic functions. The concavity of the entropy is at the basis of ensemble equivalence. Finally, the concepts of microstate and macrostate are analysed; they are very important for the study of ensemble equivalence or inequivalence in long-range systems.