Book contents
- Frontmatter
- Contents
- Preface
- 1 Outline
- 2 Pair correlation function and structure factor of ions
- 3 Thermodynamics
- 4 Electron screening and effective ion-ion interactions
- 5 Interionic forces and structural theories
- 6 Statistical mechanics of inhomogeneous systems and freezing theory
- 7 Electronic and atomic transport
- 8 Hydrodynamic limits of correlation functions and neutron scattering
- 9 Critical behaviour
- 10 Electron states, including critical region
- 11 Magnetism of normal and especially of expanded liquid metals
- 12 Liquid-vapour surface
- 13 Binary liquid-metal alloys
- 14 Two-component theory of pure liquid metals
- 15 Shock-wave studies
- 16 Liquid hydrogen plasmas and constitution of Jupiter
- Appendices
- 2.1 Fluctuation theory derivation of S(0) in terms of compressibility
- 3.1 Percus-Yevick hard sphere solution for direct correlation function
- 3.2 Weeks-Chandler-Andersen (WCA) approximation to structure factor
- 5.1 Pressure dependence of pair function related to three-particle correlations
- 5.2 Conditions to be satisfied by thermodynamically consistent structural theories
- 5.3 Gaussian core model and Kirkwood decoupling of triplet correlations
- 5.4 Specific heats of liquids in terms of higher-order correlation functions
- 5.5 Inversion of measured structure, constrained by pseudopotential theory, to extract ion-ion interaction
- 6.1 Vacancy formation energy evaluated in a hot (model) crystal
- 6.2 Vacancy formation energy related to Debye temperature
- 7.1 Inverse transport theory for noninteracting electrons
- 8.1 Method of fluctuating hydrodynamics
- 8.2 Asymptotic behaviour of other Green-Kubo time correlation functions
- 8.3 Dynamics of S(k, ω) included through self-function Ss(k, ω)
- 8.4 Fourth moment theorem for dynamical structure factor
- 8.5 One-dimensional barrier crossing: Kramers' theory
- 8.6 Mode-coupling and velocity field methods
- 9.1 Ornstein-Zernike treatment of critical correlations
- 9.2 Homogeneity, scaling, and an introduction to renormalization group method
- 9.3 Compressibility ratios and thermal pressure coefficients of simple monatomic liquids from model equations of state
- 9.4 Mode coupling applied to critical behaviour
- 9.5 Proof of Wiedemann-Franz law up to metal-insulator transition for Fermi liquid model
- 10.1 Plasmon properties as function of phenomenological relaxation time
- 11.1 Heavy Fermion theory
- 13.1 Conformal solution theory: thermodynamics and structure
- 13.2 Results for concentration fluctuations from quasi-chemical approximation
- 13.3 Density profiles, direct correlation functions, and surface tension of liquid mixtures
- 13.4 Relation of surface segregation phenomenology to first-principles statistical mechanics
- 13.5 Long-time behaviour of correlation functions in binary alloys
- 13.6 Hydrodynamic correlation functions in a binary alloy
- 13.7 Metallic binary liquid-glass transition
- 13.8 Haeffner effect, electromigration, and thermal transport
- 13.9 Theory of disorder localization of noninteracting electrons
- 14.1 Phonon-plasmon model
- 14.2 Response functions for mass densities
- 14.3 Quantum hydrodynamic limit of two-component theory
- 14.4 Evaluation of transport coefficients
- 14.5 Electron-ion structure factor in a nonequilibrium situation
- 14.6 Relations between long-wavelength limit structure factors in binary metallic alloys
- 16.1 Integral equations for correlations in liquid metals, especially hydrogen
- 16.2 Quantum Monte Carlo calculations of ground state of solid hydrogen
- References
- Index
2.1 - Fluctuation theory derivation of S(0) in terms of compressibility
Published online by Cambridge University Press: 19 January 2010
Book contents
- Frontmatter
- Contents
- Preface
- 1 Outline
- 2 Pair correlation function and structure factor of ions
- 3 Thermodynamics
- 4 Electron screening and effective ion-ion interactions
- 5 Interionic forces and structural theories
- 6 Statistical mechanics of inhomogeneous systems and freezing theory
- 7 Electronic and atomic transport
- 8 Hydrodynamic limits of correlation functions and neutron scattering
- 9 Critical behaviour
- 10 Electron states, including critical region
- 11 Magnetism of normal and especially of expanded liquid metals
- 12 Liquid-vapour surface
- 13 Binary liquid-metal alloys
- 14 Two-component theory of pure liquid metals
- 15 Shock-wave studies
- 16 Liquid hydrogen plasmas and constitution of Jupiter
- Appendices
- 2.1 Fluctuation theory derivation of S(0) in terms of compressibility
- 3.1 Percus-Yevick hard sphere solution for direct correlation function
- 3.2 Weeks-Chandler-Andersen (WCA) approximation to structure factor
- 5.1 Pressure dependence of pair function related to three-particle correlations
- 5.2 Conditions to be satisfied by thermodynamically consistent structural theories
- 5.3 Gaussian core model and Kirkwood decoupling of triplet correlations
- 5.4 Specific heats of liquids in terms of higher-order correlation functions
- 5.5 Inversion of measured structure, constrained by pseudopotential theory, to extract ion-ion interaction
- 6.1 Vacancy formation energy evaluated in a hot (model) crystal
- 6.2 Vacancy formation energy related to Debye temperature
- 7.1 Inverse transport theory for noninteracting electrons
- 8.1 Method of fluctuating hydrodynamics
- 8.2 Asymptotic behaviour of other Green-Kubo time correlation functions
- 8.3 Dynamics of S(k, ω) included through self-function Ss(k, ω)
- 8.4 Fourth moment theorem for dynamical structure factor
- 8.5 One-dimensional barrier crossing: Kramers' theory
- 8.6 Mode-coupling and velocity field methods
- 9.1 Ornstein-Zernike treatment of critical correlations
- 9.2 Homogeneity, scaling, and an introduction to renormalization group method
- 9.3 Compressibility ratios and thermal pressure coefficients of simple monatomic liquids from model equations of state
- 9.4 Mode coupling applied to critical behaviour
- 9.5 Proof of Wiedemann-Franz law up to metal-insulator transition for Fermi liquid model
- 10.1 Plasmon properties as function of phenomenological relaxation time
- 11.1 Heavy Fermion theory
- 13.1 Conformal solution theory: thermodynamics and structure
- 13.2 Results for concentration fluctuations from quasi-chemical approximation
- 13.3 Density profiles, direct correlation functions, and surface tension of liquid mixtures
- 13.4 Relation of surface segregation phenomenology to first-principles statistical mechanics
- 13.5 Long-time behaviour of correlation functions in binary alloys
- 13.6 Hydrodynamic correlation functions in a binary alloy
- 13.7 Metallic binary liquid-glass transition
- 13.8 Haeffner effect, electromigration, and thermal transport
- 13.9 Theory of disorder localization of noninteracting electrons
- 14.1 Phonon-plasmon model
- 14.2 Response functions for mass densities
- 14.3 Quantum hydrodynamic limit of two-component theory
- 14.4 Evaluation of transport coefficients
- 14.5 Electron-ion structure factor in a nonequilibrium situation
- 14.6 Relations between long-wavelength limit structure factors in binary metallic alloys
- 16.1 Integral equations for correlations in liquid metals, especially hydrogen
- 16.2 Quantum Monte Carlo calculations of ground state of solid hydrogen
- References
- Index
Summary
Let us consider an open region, i.e., one in which particles can come and go freely, drawn in a system of infinite extent. What will now be shown is that the fluctuation in the number of particles in this region is given by the volume integral of g(r) – 1, which is specifically the isothermal compressibility of the liquid. Another interesting example of such a relation between fluctuations and thermodynamic quantities yields the specific heat cv; this is discussed in Appendix A5.4.
One reason for the interest in the above relation between the volume integral of the radial distribution function—or, equivalently, from (2.4), the long wavelength limit of the structure factor S(k)—and the compressibility (first derived by Ornstein and Zernike) is because of the difficulty of extending diffraction experiments to very small scattering angles.
Let us consider a member of the grand canonical ensemble in which the open region, of volume V, contains exactly N particles.
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- Information
- Liquid MetalsConcepts and Theory, pp. 332 - 334Publisher: Cambridge University PressPrint publication year: 1990