The DOI matches its Crossref record; 4 of the 4 references it cites were found in a registry or an index.
Time evolution of large classical systems
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In the citing paper
“One starts with a nonequilibrium system with energy E in which the kinetic energy K is larger than what it would be for an equilibrium system with the same energy E . The system will evolve towards equilibrium with a Maxwellian velocity distribution at temperature T ∞ = 2 3 k K E where K E is the equilibrium kinetic energy corresponding to E .”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [16]
In the citing paper
“However, since the microstates corresponding to the initial situation just described are not typical for the macrostates Γ f 0 , this example does not contradict the typical second law behavior of S ( f t ).”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [16]
In the citing paper
“Assuming that f 0 is a Maxwellian with temperature T 0 we will have T ∞ < T 0 and so S gas ( f ∞ ) < S gas ( f 0 ).”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [16]
Citations
Cites
Mathematical Methods in Kinetic Theory
Cercignani, C. (1990) · Springer eBooks · doi:10.1007/978-1-4899-7291-0 · Cited by 601
Mathematical Methods In Kinetic Theory
Cercignani, C., & Street, R. E. (1970) · Physics Today · doi:10.1063/1.3021788 · Cited by 373
Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas
McKean, H. P. (1966) · Archive for Rational Mechanics and Analysis · doi:10.1007/bf00264463 · Cited by 286
On the Boltzmann equation for rigid spheres
Cercignani, C. (1972) · Transport Theory and Statistical Physics · doi:10.1080/00411457208232538 · Cited by 103
Cited by
Sorted by citations
Items 501–552 of 552
Рязанов, В. В., & Ryazanov, V. V. (2018) · Теоретическая и математическая физика · doi:10.4213/tmf9333
H Theorem and Trend to Equilibrium
Villani, C. (2002) · Handbook of mathematical fluid dynamics · doi:10.1016/s1874-5792(02)80007-6
Coagulation and diffusion: a probabilistic perspective on the Smoluchowski PDEPreprint
Hammond, A. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1401.0829
On the particle approximation to stationary solutions of the Boltzmann equation
Pulvirenti, M., mathematics, †. T. C. I. P. O. T. S. I. N. M. I. A., Coclite, G. E. G. M., & Link: (2019) · Mathematics in Engineering · doi:10.3934/mine.2019.4.699
Villani, C. (2002) · Handbook of mathematical fluid dynamics · doi:10.1016/s1874-5792(02)80008-8
Villani, C. (2002) · Handbook of mathematical fluid dynamics · doi:10.1016/s1874-5792(02)80005-2
Statistical and probabilistic modeling of a cloud of particles coupled with a turbulent fluid
Goudenège, L., Larat, A., Llobell, J., Massot, M., Mercier, D., Thomine, O., & Vié, A. (2019) · Springer Link (Chiba Institute of Technology) · doi:10.1051/proc/201965401/pdf
About the Derivation of the Quasilinear Approximation in Plasma Physics
Bardos, C., & Besse, N. (2023) · Applied and numerical harmonic analysis · doi:10.1007/978-3-030-45847-8_14
Dynamics of correlations in a system of hard spheresPreprint
Gerasimenko, V. I., & Gapyak, I. V. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2012.03001
Beikirch, M., Cramer, S., Frank, M., Otte, P., Pabich, E., & Trimborn, T. (2020) · Advances in Complex Systems · doi:10.1142/s0219525920500174
The Banach manifold of measures and the Lagrange multipliers of statistical mechanics
Dostoglou, S., & Hughes, A. M. (2021) · Information Geometry · doi:10.1007/s41884-021-00057-3
Statistical Properties of Turbulence
Gallavotti, G. (2002) · Springer eBooks · doi:10.1007/978-3-662-04670-8_7
Subdynamics of fluctuations in a many-particle quantum system
Los, V. F. (2022) · Journal of Statistical Mechanics Theory and Experiment · doi:10.1088/1742-5468/ac47ff
Influence of surface properties on the dynamics of fluid flow
Istenič, E., Šajn, V., & Brojan, M. (2022) · Physics of Fluids · doi:10.1063/5.0078745
Recollisional effects on dissipation of granular gases driven by Gaussian thermostat
Yano, R. (2022) · The European Physical Journal Plus · doi:10.1140/epjp/s13360-022-02436-w
Gallavotti, G. (2014) · Theoretical and mathematical physics · doi:10.1007/978-3-319-06758-2_5
(1984) · North-Holland mathematics studies · doi:10.1016/s0304-0208(08)70666-2
Cercignani, C., Illner, R., & Pulvirenti, M. (1994) · Applied mathematical sciences · doi:10.1007/978-1-4419-8524-8_2
Dürr, D., & Teufel, S. (2009) · Springer eBooks · doi:10.1007/b99978_4
Chapter 6. Topics in Differential Operators
(1986) · Pure and applied mathematics · doi:10.1016/s0079-8169(08)60061-6
(2018) · Elsevier eBooks · doi:10.1016/b978-1-78548-282-3.50008-7
Scaling Limits of Large Systems of Non-linear Partial Differential Equations
Benedetto, D., & Pulvirenti, M. (2009) · Encyclopedia of Complexity and Systems Science · doi:10.1007/978-0-387-30440-3_463
Calzetta, E., & Hu, B. L. (2022) · Cambridge University Press eBooks · doi:10.1017/9781009291248.022
Bihan, C. L. (2022) · arXiv (Cornell University) · doi:10.48550/arxiv.2212.04392
On cluster dynamics of infinite Newtonian dynamics
Zessin, H. (2023) · Journal of Mathematical Physics · doi:10.1063/5.0128104
Boltzmann Equation and the H-Theorem
Lazarovici, D. (2023) · New directions in the philosophy of science · doi:10.1007/978-3-031-33448-1_10
Pickl, P., & Schaal, A. (2024) · Fundamental theories of physics · doi:10.1007/978-3-031-45434-9_19
Recent Progress on the Mathematical Theory of Wave Turbulence
Deng, Y. (2024) · Trends in mathematics · doi:10.1007/978-3-031-48579-4_10
Boltzmann-type kinetic equations and discrete models
Bobylev, A. V. (2024) · Успехи математических наук · doi:10.4213/rm10161
The Emergence of Order in Many Element Systems
Einav, A. (2024) · Journal of Statistical Physics · doi:10.1007/s10955-024-03307-7
On an L2 critical Boltzmann equation
Chen, T., Denlinger, R., & Pavlović, N. (2024) · Nonlinear Analysis · doi:10.1016/j.na.2024.113609
Boltzmann-type kinetic equations and discrete models
Bobylev, A. V. (2024) · Russian Mathematical Surveys · doi:10.4213/rm10161e
Han, B. (2025) · Journal of Differential Equations · doi:10.1016/j.jde.2025.01.048
Variational interacting particle systems and Vlasov equations
Gladbach, P., & Kepka, B. (2025) · Communications in Partial Differential Equations · doi:10.1080/03605302.2025.2460098
Conjectures and Suggested Applications
Gallavotti, G. (2025) · Lecture notes in physics · doi:10.1007/978-3-031-85497-2_5
Gallavotti, G. (2025) · Lecture notes in physics · doi:10.1007/978-3-031-85497-2_1
Bihan, C. L. (2025) · Archive for Rational Mechanics and Analysis · doi:10.1007/s00205-025-02105-z
Scheibe, E. (2025) · Fundamental theories of physics · doi:10.1007/978-3-662-70298-7_3
Penrose, O. (1996)
Processus Limites et Irréversibilité en Mécanique Statistique Hors d’Équilibre
Jancel, R. (1995) · Springer eBooks · doi:10.1007/978-94-015-8543-9_26
A Stochastic Particle System Modelling the Broadwell Equaton
Pulvirenti, M. (1991) · Springer eBooks · doi:10.1007/978-94-009-1908-2_13
Dolmaire, T., Miele, N., & Nota, A. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6135432
Fluctuations and Moderate Deviations for a Binary Collision Model
Gao, F., & Xia, X. (2026) · Journal of Statistical Physics · doi:10.1007/s10955-026-03570-w
Cui, W. (2026) · European journal of applied sciences · doi:10.14738/ejas.1403.1267
Analysis of the kinetic equation arising from the one-dimensional FPUT chain
Menegaki, A. (2026) · Journées Équations aux dérivées partielles · doi:10.5802/jedp.696
Hilbert’s Sixth Problem: Particles and Waves
Deng, Y. (2026) · doi:10.1137/25m1804406
Trivial resonances for a system of Klein-Gordon equations and statistical applications
de Suzzoni, A.-S. (2026) · Séminaire Laurent Schwartz — EDP et applications · doi:10.5802/slsedp.188
Metadata
- Authors
- Lanford, O. E.
- Affiliations
- University of California System (US)
- Corresponding author
- Lanford, Oscar E.
- Venue
- Lecture Notes in Physics
- Publisher
- Springer Berlin Heidelberg
- Published
- 1975
- Pages
- 1-111
- ISSN
- 0075-8450 (ISSN-L)
- 1616-6361
- OpenAlex
- W1568056205
- Cited by
- 557
- References
- 4
- Citations by year
- 2026: 7 · 2025: 18 · 2024: 19 · 2023: 17 · 2022: 22 · 2021: 14 · 2020: 17 · 2019: 16 · 2018: 21 · 2017: 14 · 2016: 22 · 2015: 22 · 2014: 35 · 2013: 15 · 2012: 20
- FWCI
- 32.95
- Type
- book chapter
- Language
- en
- Open access
- closed
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Lecture notes in physics (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Statistical Mechanics and Entropy
- Physical Sciences › Mathematics › Mathematical Physics › Mathematical Dynamics and Fractals
- Keywords
- nonequilibrium statistical mechanics, microcanonical ensemble, ergodic theory, time evolution, large-scale systems
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