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## On the validity of the Boltzmann equation for short range potentials (2013)

Citations: | 8 - 2 self |

### Citations

527 |
Statistical Mechanics. Rigorous results
- Ruelle
- 1999
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Citation Context ...e further assume 52 1) either Φ ∈ C2(R3), or Φ ∈ C2(R3 \ {0}) and Φ(q)→ +∞ as q → 0; 2) Φ is stable. In what follows we will use the usual notational inconsistency Φ(r) = Φ||q|=r. We remind (see e.g. =-=[14]-=-) that an interaction is stable if it fulfills the following condition: U(q1, · · · , qj) = ∑ i<h Φ(|qi − qh|) ≥ −Bj (8.1) for some constant B > 0 . In particular, Φ is positive (possibly diverging) a... |

469 |
Large Scale Dynamics of Interacting Particles
- Spohn
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Citation Context ...f argument. In [10], although all the main ideas, as well as the strategy of the proof, were clearly discussed, the details were missing. The complete proof was presented later on in [9], [16], [19], =-=[18]-=- and [3]. We mention also that the ideas of Lanford can be applied to derive the Boltzmann equation globally in time, in the special case of an expanding cloud of a rare gas in the vacuum [7, 8]. Shor... |

357 |
The Mathematical Theory of Dilute Gases
- Cercignani, Illner, et al.
- 1994
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Citation Context ...t. In [10], although all the main ideas, as well as the strategy of the proof, were clearly discussed, the details were missing. The complete proof was presented later on in [9], [16], [19], [18] and =-=[3]-=-. We mention also that the ideas of Lanford can be applied to derive the Boltzmann equation globally in time, in the special case of an expanding cloud of a rare gas in the vacuum [7, 8]. Shortly afte... |

119 |
On the Dynamical Theory of Gases
- Maxwell
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Citation Context ...potentials we have considered are fairly general, but the basic hypothesis is the short-range assumption. From the very beginning of the Kinetic Theory, Boltzmann himself (see [1]), following Maxwell =-=[12, 13]-=-, considered only inverse power law potentials, besides the hard–sphere system, originally investigated in deriving his famous equation. This is probably due to the good scaling properties of such pot... |

111 |
Time Evolution of Large Classical Systems
- Lanford
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Citation Context ...ε−x`|>ε}(ν) ·(vj+1 − vi) · νfNj+1(zj, xi + νε, vj+1, t) , = ε2(N − j)Cεj+1fNj+1(zj, t) , (3.18) where (3.18) defines Cεj+1, which is the same collision operator appearing in the hard–sphere case, see =-=[10]-=-. Actually it is clear that, in the BG limit, this term is the only O(1) term in the sum in the right hand side of (3.16). Indeed, for m > 0 and fixed j, the size of Aεj+1+m will be O(Nm+1ε2ε3m) , (3.... |

90 |
Principles of the kinetic theory of gases
- Grad
- 1958
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Citation Context ...e path and will be fixed to one for notational simplicity. The scaling (2.6) is usually called low–density limit and it is equivalent to the BG limit originally introduced for the hard–sphere system, =-=[6]-=-. If we want to picture the dynamics in macroscopic variables, we can say that a triple collision - namely a situation in which three or more particles are simultaneously interacting - will be very un... |

22 |
Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum. Erratum and improved result: “Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum
- Illner, Pulvirenti
(Show Context)
Citation Context ..., [19], [18] and [3]. We mention also that the ideas of Lanford can be applied to derive the Boltzmann equation globally in time, in the special case of an expanding cloud of a rare gas in the vacuum =-=[7, 8]-=-. Shortly after the appearance of the Lanford’s paper, F. King in his unpublished thesis [9] approached the same validity problem for a particle system interacting by means of a positive, smooth and s... |

19 |
Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum
- Illner, Pulvirenti
- 1986
(Show Context)
Citation Context ..., [19], [18] and [3]. We mention also that the ideas of Lanford can be applied to derive the Boltzmann equation globally in time, in the special case of an expanding cloud of a rare gas in the vacuum =-=[7, 8]-=-. Shortly after the appearance of the Lanford’s paper, F. King in his unpublished thesis [9] approached the same validity problem for a particle system interacting by means of a positive, smooth and s... |

13 | From Newton to Boltzmann: hard spheres and short–range potentials - Gallagher, Raymond, et al. |

10 | The linear Boltzmann equation for long-range forces: a derivation from particle systems
- Desvillettes, Pulvirenti
- 1999
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Citation Context ...es so that we consider it as an interesting, open problem. For an analysis concerning the much easier problem of the validity of the linear Boltzmann equation for long range potentials, see reference =-=[4]-=-. 2. In the present paper we give an explicit estimate of the error in case of a completely repulsive potential (Theorem 2), while, for stable potentials, we only show the convergence. It would be int... |

9 |
The Boltzmann-Grad Limit and Cauchy-Kovalevskaya Theorem
- Ukai
(Show Context)
Citation Context ...mit (BG limit in the sequel). The restriction to short times is due to the fact that the two series have been proven to converge absolutely only for a small time interval. Actually it was remarked in =-=[20]-=- that the Lanford’s approach is a Cauchy-Kowalevski kind of argument. In [10], although all the main ideas, as well as the strategy of the proof, were clearly discussed, the details were missing. The ... |

7 | From Newton to Boltzmann: the case of short-range potentials
- Gallagher, Saint-Raymond, et al.
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Citation Context ... Boltzmann equation. This is a revisitation and an extension of the thesis of King [9] (appeared after the well known result of Lanford [10] for hard spheres) and of a recent paper by Gallagher et al =-=[5]-=-. Our analysis applies to any stable and smooth potential. In the case of repulsive potentials (with no attractive parts), we estimate explicitly the rate of convergence. KEYWORDS. Kinetic Theory, sca... |

7 |
BBGKY hierarchy for positive potentials
- King
- 1975
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Citation Context ... that, in the low–density (Boltzmann–Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann equation. This is a revisitation and an extension of the thesis of King =-=[9]-=- (appeared after the well known result of Lanford [10] for hard spheres) and of a recent paper by Gallagher et al [5]. Our analysis applies to any stable and smooth potential. In the case of repulsive... |

7 |
Derivation of the Boltzmann equation from particle dynamics
- Uchiyama
- 1988
(Show Context)
Citation Context ...kind of argument. In [10], although all the main ideas, as well as the strategy of the proof, were clearly discussed, the details were missing. The complete proof was presented later on in [9], [16], =-=[19]-=-, [18] and [3]. We mention also that the ideas of Lanford can be applied to derive the Boltzmann equation globally in time, in the special case of an expanding cloud of a rare gas in the vacuum [7, 8]... |

6 |
On the Boltzmann equation for rigid spheres. Transport Theory Statist. Phys
- Cercignani
- 1972
(Show Context)
Citation Context ...series expansion describing the time evolution of the statistical states of a hard–sphere system. This series is the solution of a hierarchy of equations formally established by C. Cercignani in 1972 =-=[2]-=-, following previous ideas due to H. Grad [6]. The main idea of Lanford is to compare such a series expansion with the one arising from the solution of the Boltzmann equation, claiming the term by ter... |

6 | On the Integrated Form of the BBGKY Hierarchy for Hard Spheres. arXiv: 0605068v1 [math-ph
- Spohn
- 2006
(Show Context)
Citation Context ...l distribution (see Remark on page 6). However, by using a density approximation, (3.1.3) can be proven to hold for a general class of initial measures. The argument can be found in [19] page 281, or =-=[17]-=- page 18 for cases of hard–sphere dynamics, and it can be applied also to general smooth potentials. We list the main steps in what follows. A different approach based on a weak formulation may be fou... |

5 |
On the dynamics of particles in a bounded region: A measure theoretical approach
- Marchioro, Pellegrinotti, et al.
- 1976
(Show Context)
Citation Context ...we have to specify the boundary conditions. Assuming specular reflections, there are additional difficulties which we have to overcome. First the dynamical flow is only almost everywhere defined (see =-=[11]-=-), but this (as for the hard– sphere systems) does not create real difficulties. However the analysis of the recollisions requires some extra geometrical arguments. 59 A Appendix (on the cross–section... |

4 |
Lectures on gas theory. English edition annotated by
- Boltzmann
- 1964
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Citation Context ..., v′1, t)f(x, v ′, t)− f(x, v1, t)f(x, v, t) } (1.1) where S2+ = {ν ∈ S2| (v − v1) · ν ≥ 0}, S2 is the unit sphere in R3, (v, v1) is a pair of velocities in incoming collision configuration –see also =-=[1]-=-– and (v′, v′1) is the corresponding 1 pair of outgoing velocities defined byv′ = v − ω[ω · (v − v1)]v′1 = v1 + ω[ω · (v − v1)] . (1.2) Here ω = ω(ν, V ) is the unit vector bisecting the angle betw... |

3 |
Boltzmann equation and Boltzmann hierarchy
- Spohn
- 1984
(Show Context)
Citation Context ...evski kind of argument. In [10], although all the main ideas, as well as the strategy of the proof, were clearly discussed, the details were missing. The complete proof was presented later on in [9], =-=[16]-=-, [19], [18] and [3]. We mention also that the ideas of Lanford can be applied to derive the Boltzmann equation globally in time, in the special case of an expanding cloud of a rare gas in the vacuum ... |

2 |
The Scientific Letters and Papers of James Clerk Maxwell
- Maxwell
- 1995
(Show Context)
Citation Context ...potentials we have considered are fairly general, but the basic hypothesis is the short-range assumption. From the very beginning of the Kinetic Theory, Boltzmann himself (see [1]), following Maxwell =-=[12, 13]-=-, considered only inverse power law potentials, besides the hard–sphere system, originally investigated in deriving his famous equation. This is probably due to the good scaling properties of such pot... |

1 |
Molecular Gas Dynamics. Asakura-Shoten
- Sone, Aoki
- 1994
(Show Context)
Citation Context ...owing assertion: In the considered class of potentials, if for all q with |q| ∈ (0, 1) |q|Φ′′(|q|) + 2Φ′(|q|) ≥ 0 , (A.8) then dΘ dρ > 0 for all ρ ∈ (0, 1), V 2 > 0. This condition is derived also in =-=[15]-=-. Condition (A.8) can be easily checked for a large subset of potentials. For instance any potential of the form Φ(q) = ( 1 |q|k − 1 ) δ|q|<1, k ≥ 1, satisfies the condition, hence has 61 strictly mon... |