Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics¶
Gibbs, J. W. (1902). Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics. Yale University Press.
Cited by¶
6 citations across 5 artifacts.
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Primes¶
- Degrees of Freedom
- In statistical mechanics, the Gibbs ensemble
This sourceFounds equilibrium statistical mechanics on ensembles in phase space (microcanonical, canonical, grand-canonical); for N DOFs phase space has 2N dimensions, and macroscopic thermodynamics emerges as ensemble averages over phase-space configurations.
SupportedVerified against the work's full text
Gibbs represents the phase of an n-degree-of-freedom system (n coordinates, n momenta) as a point in 2n-dimensional space and builds ensembles with probability exponential in energy.
“If we regard a phase as represented by a point in space of 2 n dimen- sions”
- In statistical mechanics, the Gibbs ensemble
- Ensemble
- In statistical mechanics, the ensemble concept became foundational through the work of J.W. Gibbs
This sourceThe founding systematic treatment of equilibrium ensembles (microcanonical, canonical, grand-canonical) as collections of microstates weighted by thermal probability, with ensemble averages recovering thermodynamic observables.
Supported in partVerified against the work's full text
Gibbs's 1902 book builds statistical mechanics on ensembles, naming the canonical distribution and treating microcanonical and grand ensembles; it cannot attest its later influence.
“We consider especially ensembles of systems in which the index (or logarithm) of probability of phase is a linear function of the energy. This distribution, on account of its unique importance in the theory of statisti- cal equilibrium, I have ventured to call canonical”
- Population: all microstates of a system in thermal contact with a heat bath at temperature T. Generating mechanism: microstate enumeration (or sampling) weighted by the Boltzmann factor exp(−E/kT), justified by Gibbs's principle
This sourceEnsembles of systems in thermal equilibrium with a heat bath at temperature T obey the Boltzmann distribution exp(−E/kT) over microstates, and the weighted ensemble average of an observable equals the thermodynamic observable.
SupportedVerified against the work's full text
Gibbs shows that a body placed in a large bath is described by the canonical ensemble, whose probability index is linear in energy with a modulus that plays the role of temperature.
“The ensemble of phases of the body, and of the thermometer, approach a standard form as the quantity of the bath is in- definitely increased. This limiting form is easily shown to be what we have described as the canonical distribution.”
- In statistical mechanics, the ensemble concept became foundational through the work of J.W. Gibbs
- Entropy (Thermodynamic Sense)
- Every thermodynamic-entropy articulation specifies (1) the macroscopic system and its thermodynamic variables (energy, volume, particle number, etc.); (2) the choice of statistical ensemble
This sourceFounds modern statistical mechanics; coins and develops the microcanonical, canonical, and grand-canonical ensembles and the ensemble (Gibbs) entropy S = −k Σ p ln p.
Supported in partVerified against the work's full text
Gibbs defines his entropy analogue as an average over a specified (canonical) ensemble, showing that statistical entropy is tied to an ensemble choice, but he does not state the universal checklist.
“has been completely defined as the average value in a canonical ensemble of systems of the logarithm of the coefficient of probability of phase”
- Every thermodynamic-entropy articulation specifies (1) the macroscopic system and its thermodynamic variables (energy, volume, particle number, etc.); (2) the choice of statistical ensemble
- Phase Space
- Phase space appears in classical mechanics (Hamiltonian formulation; the fundamental setting for advanced mechanics and celestial mechanics); in statistical mechanics (ensemble theory — the distribution of system states in phase space, Liouville's theorem, ergodic theory);
This sourceProvides unified statistical-mechanical framework for equilibrium ensembles: microcanonical, canonical, and grand-canonical; shows how ensemble distributions generate equilibrium thermodynamics and how equilibrium states emerge as macroscopic consequences of ensemble averaging.
Supported in partVerified against the work's full text
Gibbs supplies Hamiltonian equations, ensembles distributed in phase, and the conservation of density-in-phase; he does not name Liouville, ergodic theory, or celestial mechanics.
“the condition which the distribution of the systems in phase must satisfy in order that the distribution shall be permanent.”
- Phase space appears in classical mechanics (Hamiltonian formulation; the fundamental setting for advanced mechanics and celestial mechanics); in statistical mechanics (ensemble theory — the distribution of system states in phase space, Liouville's theorem, ergodic theory);
- Thermodynamic Equilibrium
- in statistical mechanics (the Boltzmann-Gibbs distribution characterizes equilibrium ensembles)
This sourceProvides unified statistical-mechanical framework for equilibrium ensembles: microcanonical, canonical, and grand-canonical; shows how ensemble distributions generate equilibrium thermodynamics and how equilibrium states emerge as macroscopic consequences of ensemble averaging.
Supported in partVerified against the work's full text
Gibbs shows that a body in a large heat bath takes the canonical distribution with modulus as temperature; he does not discuss detailed balance or microscopic reversibility.
“approach a standard form as the quantity of the bath is in- definitely increased. This limiting form is easily shown to be what we have described as the canonical distribution.”
- in statistical mechanics (the Boltzmann-Gibbs distribution characterizes equilibrium ensembles)
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