Violation of Boltzmann'sHTheorem in Real Gases
Abstract
The well-known variational (maximum-entropy) property of the Maxwellian velocity distribution is used to shed some light on the range of validity of the Boltzmann transport equation. It permits a characterization of the initial states for which the Boltzmann $H$ theorem is violated. In particular, it is shown that: (a) Any monatomic system for which the equilibrium potential energy exceeds the minimum possible value possesses a continuum of initial states for which the approach to equilibrium takes place through an increase, rather than a decrease, in Boltzmann's $H$. (b) If the initial distribution of particles is spatially homogeneous and Maxwellian, the approach to equilibrium will take place through an increase (decrease) in the Boltzmann $H$, according as the initial potential energy is less (greater) than the equilibrium value. (c) A necessary condition for the $H$-theorem-violating phenomenon is that the approach to equilibrium takes place through a conversion of kinetic energy into potential energy; a sufficient condition requires also that the initial velocity distribution be sufficiently close to Maxwellian. (d) These $H$-theorem-violating conditions are readily attained experimentally; for example, the free expansion of oxygen gas at 160 \ifmmode^\circ\else\textdegree\fi{}K and 45-atm pressure produces an experimentally realizable violation of the Boltzmann $H$ theorem.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Attempts to remedy the use of S_G through coarse-grained ensembles may be useful mathematically, but conceptually they are just variations on the Boltzmann entropy.
“Attempts to remedy this through the use of coarse grained ensembles may be useful mathematically, but conceptually they are just variations on the Boltzmann entropy [19].”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [19]
For open systems in contact with thermal reservoirs, the time evolution of the microstate X is no longer given by a Hamiltonian since the system is not isolated, and S_G(ρ_t) need no longer be constant in time.
“The situation is different when one considers open systems, e.g. systems in contact with thermal reservoirs [19, 20]. The time evolution of the microstate X of such a system is then no longer given by a Hamiltonian since the system is not isolated and S G ( ρ t ) need no longer be constant in time.”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [19]
The connection between the steady states of systems evolving under the invented thermostated dynamics and those obtained from more realistic models for which the stationary ̄ρ is not singular is still not entirely clear; the same is true for the correct identification of entropy in such systems.
“The connection between the steady states of systems evolving under the invented thermostated dynamics and those obtained from more realistic models for which the stationary ¯ ρ is not singular, is still not entirely clear. The same is true for the correct identification of entropy in such systems [17-23].”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [19]
For a closed system evolving under deterministic non-Hamiltonian thermostated dynamics, starting with an initial density ρ(X,0) uniform (or absolutely continuous) with respect to the appropriate Lebesgue measure, the dynamics leads to ρ(X,t)→ ̄ρ as t→∞, with ̄ρ singular with respect to Lebesgue measure (generally an SRB measure), and S_G(ρ)→-∞ decreasing with time.
“Starting with an initial density ρ ( X, 0), uniform (or absolutely continuous) with respect to the appropriate Lebesgue measure, the dynamics leads to ρ ( X,t ) → ¯ ρ , as t →∞ , with ¯ ρ singular with respect to Lebesgue measure, (¯ ρ is generally an SRB measure), and S G ( ρ ) →-∞ decreasing with time”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [19]
The σ in the thermostated dynamics is frequently interpreted as an 'entropy production' and in some cases has a form similar to the hydrodynamic entropy production in a stationary open system with currents.
“This σ is frequently interpreted as an 'entropy production' and indeed has, in some cases, a form similar to the hydrodynamic entropy production in a stationary open system in which there are currents.”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [19]
Cited by
Sorted by citations
On the (Boltzmann) entropy of non-equilibrium systems
Goldstein, S., & Lebowitz, J. L. (2004) · Physica D: Nonlinear Phenomena · doi:10.1016/j.physd.2004.01.008 · Cited by 240 · see citing works
“. (40) simplifies for a spatially uniform system for which u= 0, ∂n ∂t = 0 and TK is independent of x. If, furthermore, ft happens to be a Maxwellian with a temperature TK (a case considered by Jaynes [16]), then (at that instant) dS(ft,E)/dt = dK dt [T−1 K −T −1 Φ tot. ]. (42) 25 where K = E − Φtot. is the total kinet…”
BackgroundResult
- Authors
- Jaynes, E. T.
- Affiliations
- Washington University in St. Louis (US)
- Corresponding author
- Jaynes, E. T.
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 1971-08-01
- Volume
- 4
- Issue
- 2
- Pages
- 747-750
- ISSN
- 0556-2791 (ISSN-L)
- OpenAlex
- W2333430510
- Cited by
- 47
- References
- 0
- Citations by year
- 2026: 1 · 2024: 1 · 2023: 1 · 2022: 2 · 2020: 1 · 2018: 3 · 2017: 2 · 2015: 2 · 2013: 2 · 2012: 2
- FWCI
- 0.00
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review A (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Gas Dynamics and Kinetic Theory
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Statistical Mechanics and Entropy
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Keywords
- H-theorem, Boltzmann equation, Maxwell-Boltzmann distribution, variational principle, convergence to equilibrium, real gases, monatomic gases, potential energy, kinetic energy conversion, free expansion, oxygen gas, initial velocity distribution
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
