{
"type": "Article",
"authors": [
{
"type": "Person",
"familyNames": [
"Bodineau"
],
"givenNames": [
"Thierry"
]
},
{
"type": "Person",
"familyNames": [
"Gallagher"
],
"givenNames": [
"Isabelle"
]
},
{
"type": "Person",
"familyNames": [
"Saint-Raymond"
],
"givenNames": [
"Laure"
]
},
{
"type": "Person",
"familyNames": [
"Simonella"
],
"givenNames": [
"Sergio"
]
}
],
"description": [
{
"type": "Paragraph",
"content": [
"The evolution of a gas can be described by different mathematical models depending on the scale of observation.\nA natural question, raised by Hilbert in his sixth problem, is whether these models provide mutually consistent predictions.\nIn particular, for rarefied gases, it is expected that the equations of the kinetic theory of gases can be obtained from molecular dynamics governed by the fundamental principles of mechanics.\nIn the case of hard sphere gases, Lanford (1975) has shown that the Boltzmann equation does indeed appear as a law of large numbers in the low density limit, at least for very short times.\nThe aim of this paper is to present recent advances in the understanding of this limiting process.\n"
]
}
],
"identifiers": [],
"references": [
{
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"id": "bib-bib1",
"authors": [],
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},
{
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"url": "https://arxiv.org/abs/2201.04514"
},
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"url": "https://arxiv.org/abs/2008.10403"
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},
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}
],
"title": "On the dynamics of dilute gases",
"meta": {},
"content": [
{
"type": "Heading",
"id": "S1",
"depth": 1,
"content": [
"1 A statistical approach to dilute gas dynamics"
]
},
{
"type": "Heading",
"id": "S1.SS1",
"depth": 2,
"content": [
"1.1 The physical model: A dilute gas of hard spheres"
]
},
{
"type": "Paragraph",
"content": [
"Although at the time Boltzmann published his famous paper [",
{
"type": "Cite",
"target": "bib-bib8",
"content": [
"8"
]
},
"] the atomic theory was still rejected by some scientists, it was already well established that matter is composed of atoms, which are the elementary constituents of all solids, liquids and gases.\nThe particularity of gases is that the volume occupied by their atoms is negligible as compared to the total volume occupied by the gas, and there are therefore very few constraints on the atoms’ geometric arrangement: they are thus very loosely bound and almost independent.\nNeglecting the internal structure of the atoms, their possible organization into molecules, and the effect of long-range interactions, a gas can be represented as a system formed by a large number of particles that move in a straight line and occasionally collide with each other, resulting in an almost instantaneous scattering.\nThe simplest example of such a model consists in assuming that the particles are small identical spheres, of diameter ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ε≪1",
"meta": {
"altText": "\\varepsilon\\ll 1"
}
},
" and mass 1, interacting only by contact (Figure ",
{
"type": "Cite",
"target": "S1-F1",
"content": [
"1"
]
},
").\nWe refer to this as a ",
{
"type": "Emphasis",
"content": [
"gas of hard spheres"
]
},
".\nThis microscopic description of a gas is explicit, but very difficult to use in practice because the number of particles is extremely large, their size is tiny and their collisions are very sensitive to small shifts (Figure ",
{
"type": "Cite",
"target": "S1-F2",
"content": [
"2"
]
},
").\nThis model is therefore not efficient for making theoretical predictions.\nA natural question is whether one can extract, from such a complex system, less precise but more stable models suitable for applications, such as kinetic or fluid models.\nThis question was formalized by Hilbert at the International Congress of Mathematicians in 1900, in his sixth problem:"
]
},
{
"type": "QuoteBlock",
"content": [
{
"type": "Paragraph",
"content": [
"Boltzmann’s work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua."
]
}
]
},
{
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"id": "S1-F1",
"caption": [
{
"type": "Paragraph",
"content": [
"At time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "t",
"meta": {
"altText": "t"
}
},
", the system of hard spheres is described by the positions ",
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"mathLanguage": "mathml",
"text": "(xkε(t))k≤N",
"meta": {
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}
},
" and the velocities ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "(vkε(t))k≤N",
"meta": {
"altText": "(v^{\\varepsilon}_{k}(t))_{k\\leq N}"
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},
" of the centers of gravity of the particles.\nThe spheres move in a straight line and when two of them touch, they are scattered according to elastic reflection laws."
]
}
],
"content": [
{
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"contentUrl": "mag-124-fig-1.png",
"mediaType": "image/png",
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}
]
},
{
"type": "Figure",
"id": "S1-F2",
"caption": [
{
"type": "Paragraph",
"content": [
"The particles are very small (of diameter ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ε≪1",
"meta": {
"altText": "\\varepsilon\\ll 1"
}
},
") and the dynamics is very sensitive to small spatial shifts.\nIn the first case depicted above, two particles with initial positions ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "x1,x2",
"meta": {
"altText": "x_{1},x_{2}"
}
},
" and velocities ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "v1,v2",
"meta": {
"altText": "v_{1},v_{2}"
}
},
" collide and are scattered.\nIn the second case, after shifting the position of the first particle by a distance ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "O(ε)",
"meta": {
"altText": "O(\\varepsilon)"
}
},
", they no longer collide and each particle keeps moving in a straight line.\nThus, a perturbation of order ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ε",
"meta": {
"altText": "\\varepsilon"
}
},
" of the initial conditions can lead to very different trajectories."
]
}
],
"content": [
{
"type": "ImageObject",
"contentUrl": "mag-124-fig-2.png",
"mediaType": "image/png",
"meta": {
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},
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"id": "S1.SS1.p2",
"content": [
"The Boltzmann equation, mentioned by Hilbert and described in more detail below, expresses that the particle distribution evolves under the combined effect of free transport and collisions.\nFor these two effects to be of the same order of magnitude, a simple calculation shows that, in dimension ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "d≥2",
"meta": {
"altText": "d\\geq 2"
}
},
", the number of particles ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "N",
"meta": {
"altText": "N"
}
},
" and their diameter ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ε",
"meta": {
"altText": "\\varepsilon"
}
},
" must satisfy the scaling relation ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "Nεd−1=O(1)",
"meta": {
"altText": "N\\varepsilon^{d-1}=O(1)"
}
},
", called ",
{
"type": "Emphasis",
"content": [
"low density scaling"
]
},
" [",
{
"type": "Cite",
"target": "bib-bib14",
"content": [
"14"
]
},
"].\nIndeed, the regime described by the Boltzmann equation is such that the mean free path, i.e., the average distance traveled by a particle moving in a straight line between two collisions, is of order 1.\nThus, a typical particle should go through a tube of volume ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "O(εd−1)",
"meta": {
"altText": "O(\\varepsilon^{d-1})"
}
},
" between two collisions, and on average, this tube should cross one of the ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "N−1",
"meta": {
"altText": "N-1"
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},
" other particles.\nNote that, in this regime, the total volume occupied by the particles at a given time is proportional to ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "Nεd",
"meta": {
"altText": "N\\varepsilon^{d}"
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},
" and is therefore negligible compared to the total volume occupied by the gas.\nWe speak then of a ",
{
"type": "Emphasis",
"content": [
"dilute gas"
]
},
"."
]
},
{
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"id": "S1.SS2",
"depth": 2,
"content": [
"1.2 Three levels of averaging"
]
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{
"type": "Paragraph",
"content": [
"Henceforth, it is assumed that the particle system evolves in the unit domain with periodic boundary conditions ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝕋d=[0,1]d",
"meta": {
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}
},
".\nWe consider that the ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "N",
"meta": {
"altText": "N"
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},
" particles are identical: this is the exchangeability assumption.\nThe state of the system can be represented by a measure in the phase space ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝕋d×ℝd",
"meta": {
"altText": "\\mathbb{T}^{d}\\times\\mathbb{R}^{d}"
}
},
" called ",
{
"type": "Emphasis",
"content": [
"empirical measure"
]
},
","
]
},
{
"type": "MathBlock",
"id": "S1.Ex1",
"mathLanguage": "mathml",
"text": "1N∑i=1Nδx−xiδv−vi,",
"meta": {
"altText": "\\frac{1}{N}\\sum_{i=1}^{N}\\delta_{x-x_{i}}\\delta_{v-v_{i}},"
}
},
{
"type": "Paragraph",
"content": [
"where ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "δx",
"meta": {
"altText": "\\delta_{x}"
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},
" is the Dirac mass at ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "x=0",
"meta": {
"altText": "x=0"
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".\nThis measure is completely symmetric (i.e., invariant under permutation of the indices of the particles) because of the exchangeability assumption.\nThis first averaging is however not sufficient to obtain a robust description of the dynamics when ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "N",
"meta": {
"altText": "N"
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" is large, because of the instabilities mentioned in the previous section (Figure ",
{
"type": "Cite",
"target": "S1-F2",
"content": [
"2"
]
},
") which lead to a strong dependence of the particle trajectories on ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ε",
"meta": {
"altText": "\\varepsilon"
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},
".\nWe will therefore introduce a second averaging, with respect to the initial configurations; from a physical point of view, this averaging is natural since only fragmentary information on the initial configuration is available.\nWe therefore assume that the initial data ",
{
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"mathLanguage": "mathml",
"text": "(XN,VN)=(xi,vi)1≤i≤N",
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},
" are independent random variables, identically distributed according to a distribution ",
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"mathLanguage": "mathml",
"text": "f0=f0(x,v)",
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".\nThis assumption must be slightly corrected to account for particle exclusion: ",
{
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"mathLanguage": "mathml",
"text": "\\varepsilon\" display=\"inline\">|xi−xj|>ε",
"meta": {
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},
" for ",
{
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".\nThis statistical framework is called the ",
{
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"content": [
"canonical"
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},
" setting.\nIt is a simple framework allowing us to establish rigorous foundations for the kinetic theory, i.e., to characterize, in the large ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "N",
"meta": {
"altText": "N"
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" asymptotics, the average dynamics and more precisely the evolution equation governing the distribution ",
{
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"mathLanguage": "mathml",
"text": "f(t,x,v)",
"meta": {
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" at time ",
{
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"mathLanguage": "mathml",
"text": "t",
"meta": {
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" of a typical particle.\nIn this paper, our aim is to go beyond this averaged dynamics, and to describe in a precise way the correlations that appear dynamically inside the gas.\nFixing a priori the number ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "N",
"meta": {
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{
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"meta": {
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" is also a random variable, and that only its average ",
{
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"mathLanguage": "mathml",
"text": "με=ε−(d−1)",
"meta": {
"altText": "\\mu_{\\varepsilon}=\\varepsilon^{-(d-1)}"
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" is determined (according to the low density scaling).\nTo define a system of initially independent (modulo exclusion) identically distributed hard spheres according to ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f0",
"meta": {
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", we introduce the ",
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},
" measure as follows: the probability density of finding ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "N",
"meta": {
"altText": "N"
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" particles of coordinates ",
{
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},
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"text": "\\varepsilon}\\quad\\text{for}\\ N=0,1,2,\\dots,\" display=\"block\">1𝒵εμεNN!∏i=1Nf0(xi,vi)∏i≠j𝟏|xi−xj|>εforN=0,1,2,…,",
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}
},
{
"type": "Paragraph",
"content": [
"where the constant ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝒵ε",
"meta": {
"altText": "\\mathcal{Z}^{\\varepsilon}"
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" is the normalization factor of the probability measure.\nWe will assume in the following that the function ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f0",
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{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℙε",
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"altText": "\\mathbb{P}_{\\varepsilon}"
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},
" and ",
{
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"mathLanguage": "mathml",
"text": "𝔼ε",
"meta": {
"altText": "\\mathbb{E}_{\\varepsilon}"
}
},
"."
]
},
{
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"id": "S1.SS3",
"depth": 2,
"content": [
"1.3 A statistical approach"
]
},
{
"type": "Paragraph",
"content": [
"Once the initial random configuration ",
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"mathLanguage": "mathml",
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},
" is chosen, the hard sphere dynamics evolves deterministically (according to the hard sphere equations shown in Figure ",
{
"type": "Cite",
"target": "S1-F1",
"content": [
"1"
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},
"), and we seek to understand the statistical behavior of the empirical measure"
]
},
{
"type": "MathBlock",
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"mathLanguage": "mathml",
"text": "πtε(x,v)≔1με∑i=1Nδx−xiε(t)δv−viε(t)",
"meta": {
"altText": "\\pi^{\\varepsilon}_{t}(x,v)≔\\frac{1}{\\mu_{\\varepsilon}}\\sum_{i=1}^{N}\\delta_{x-x^{\\varepsilon}_{i}(t)}\\delta_{v-v^{\\varepsilon}_{i}(t)}"
}
},
{
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"content": [
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},
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"A law of large numbers"
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"content": [
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{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "με→∞",
"meta": {
"altText": "\\mu_{\\varepsilon}\\to\\infty"
}
},
".\nIn the case of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "N",
"meta": {
"altText": "N"
}
},
" identically distributed independent variables ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "(ηi)1≤i≤N",
"meta": {
"altText": "(\\eta_{i})_{1\\leq i\\leq N}"
}
},
" of expectation ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝔼(η)",
"meta": {
"altText": "\\mathbb{E}(\\eta)"
}
},
", the law of large numbers implies in particular that the mean converges in probability to the expectation:"
]
},
{
"type": "MathBlock",
"id": "S1.Ex2",
"mathLanguage": "mathml",
"text": "1N∑i=1Nηi→N→∞𝔼(η).",
"meta": {
"altText": "\\frac{1}{N}\\sum_{i=1}^{N}\\eta_{i}\\xrightarrow[N\\to\\infty]{}\\mathbb{E}(\\eta)."
}
},
{
"type": "Paragraph",
"content": [
"One can easily show the following convergence in probability:"
]
},
{
"type": "MathBlock",
"id": "S1.Ex3",
"mathLanguage": "mathml",
"text": "⟨π0ε,h⟩≔1με∑i=1Nh(xiε0,viε0)→με→∞∫f0h(x,v)𝑑x𝑑v,",
"meta": {
"altText": "\\langle\\pi^{\\varepsilon}_{0},h\\rangle≔\\frac{1}{\\mu_{\\varepsilon}}\\sum_{i=1}^{N}h(x^{\\varepsilon 0}_{i},v^{\\varepsilon 0}_{i})\\xrightarrow[\\mu_{\\varepsilon}\\to\\infty]{}\\int f^{0}h(x,v)\\,dx\\,dv,"
}
},
{
"type": "Paragraph",
"content": [
"under the grand canonical measure.\nThe difficulty is to understand whether the initial quasi-independence propagates in time so that there exists a function ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f=f(t,x,v)",
"meta": {
"altText": "f=f(t,x,v)"
}
},
" such that the following convergence in probability holds:"
]
},
{
"type": "MathBlock",
"id": "S1.E3X",
"mathLanguage": "mathml",
"text": "⟨πtε,h⟩→με→∞∫f(t,x,v)h(x,v)𝑑x𝑑v",
"meta": {
"altText": "\\displaystyle\\langle\\pi^{\\varepsilon}_{t},h\\rangle\\xrightarrow[\\mu_{\\varepsilon}\\to\\infty]{}\\int f(t,x,v)h(x,v)\\,dx\\,dv"
}
},
{
"type": "Paragraph",
"content": [
"under the grand canonical measure (",
{
"type": "Cite",
"target": "S1-E1",
"content": [
"1.1"
]
},
") over the initial configurations.\nThe most important result proving this convergence was obtained by Lanford [",
{
"type": "Cite",
"target": "bib-bib16",
"content": [
"16"
]
},
"]: he showed that ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f",
"meta": {
"altText": "f"
}
},
" evolves according to a deterministic equation, namely the Boltzmann equation.\nThis result will be explained in Section ",
{
"type": "Cite",
"target": "S2-SS2",
"content": [
"2.2"
]
},
"."
]
},
{
"type": "Heading",
"id": "S1.SS3.SSSx2",
"depth": 3,
"content": [
"A central limit theorem"
]
},
{
"type": "Paragraph",
"content": [
"The approximation (",
{
"type": "Cite",
"target": "S1-E3",
"content": [
"1.3"
]
},
") of the empirical measure neglects two types of errors.\nThe first is the presence of correction terms that converge to 0 when ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "με→+∞",
"meta": {
"altText": "\\mu_{\\varepsilon}\\to+\\infty"
}
},
".\nThe second is related to the probability, which must tend to zero, of configurations for which this convergence does not occur.\nA classical problem in statistical physics is to quantify more precisely these errors, by studying the fluctuations, i.e., the deviations between the empirical measure and its expectation.\nIn the case of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "N",
"meta": {
"altText": "N"
}
},
" independent and identically distributed variables ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "(ηi)1≤i≤N",
"meta": {
"altText": "(\\eta_{i})_{1\\leq i\\leq N}"
}
},
", the central limit theorem implies that the fluctuations are of order ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "O(1/N)",
"meta": {
"altText": "O(1/\\sqrt{N})"
}
},
", and the following convergence in law holds true:"
]
},
{
"type": "MathBlock",
"id": "S1.Ex4",
"mathLanguage": "mathml",
"text": "N(1N∑i=1Nηi−𝔼(η))→N→∞(law)𝒩(0,Var(η)),",
"meta": {
"altText": "\\sqrt{N}\\biggl(\\frac{1}{N}\\sum_{i=1}^{N}\\eta_{i}-\\mathbb{E}(\\eta)\\biggr)\\xrightarrow[N\\to\\infty]{(\\text{law})}\\mathcal{N}(0,\\operatorname{Var}(\\eta)),"
}
},
{
"type": "Paragraph",
"content": [
"where ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝒩(0,Var(η))",
"meta": {
"altText": "\\mathcal{N}(0,\\operatorname{Var}(\\eta))"
}
},
" is the normal law of variance ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "Var(η)=𝔼((η−𝔼(η))2)",
"meta": {
"altText": "\\operatorname{Var}(\\eta)=\\mathbb{E}((\\eta-\\mathbb{E}(\\eta))^{2})"
}
},
".\nIn particular, at this scale, we find some randomness.\nInvestigating the same fluctuation regime for the dynamics of hard sphere gases consists in considering the fluctuation field ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ζtε",
"meta": {
"altText": "\\zeta^{\\varepsilon}_{t}"
}
},
" defined by duality, namely,"
]
},
{
"type": "MathBlock",
"id": "S1.E4",
"mathLanguage": "mathml",
"text": "⟨ζtε,h⟩≔με(⟨πtε,h⟩−𝔼ε(⟨πtε,h⟩)),",
"meta": {
"altText": "\\langle\\zeta^{\\varepsilon}_{t},h\\rangle≔\\sqrt{\\mu_{\\varepsilon}}\\bigl(\\langle\\pi^{\\varepsilon}_{t},h\\rangle-\\mathbb{E}_{\\varepsilon}(\\langle\\pi^{\\varepsilon}_{t},h\\rangle)\\bigr),"
}
},
{
"type": "Paragraph",
"content": [
"where ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "h",
"meta": {
"altText": "h"
}
},
" is a continuous function, and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝔼ε",
"meta": {
"altText": "\\mathbb{E}_{\\varepsilon}"
}
},
" the expectation with respect to the grand canonical measure.\nAt time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "0",
"meta": {
"altText": "0"
}
},
", one can easily show that, under the grand-canonical measure, the fluctuation field ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ζ0ε",
"meta": {
"altText": "\\zeta^{\\varepsilon}_{0}"
}
},
" converges in the low density limit to a Gaussian field ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ζ0",
"meta": {
"altText": "\\zeta_{0}"
}
},
" with covariance"
]
},
{
"type": "MathBlock",
"id": "S1.Ex5",
"mathLanguage": "mathml",
"text": "𝔼(ζ0(h)ζ0(g))=∫f0(z)h(z)g(z)𝑑z.",
"meta": {
"altText": "\\mathbb{E}\\bigl(\\zeta_{0}(h)\\zeta_{0}(g)\\bigr)=\\int f^{0}(z)h(z)g(z)\\,dz."
}
},
{
"type": "Paragraph",
"content": [
"A series of recent works [",
{
"type": "Cite",
"target": "bib-bib4",
"content": [
"4"
]
},
", ",
{
"type": "Cite",
"target": "bib-bib6",
"content": [
"6"
]
},
", ",
{
"type": "Cite",
"target": "bib-bib7",
"content": [
"7"
]
},
", ",
{
"type": "Cite",
"target": "bib-bib5",
"content": [
"5"
]
},
"] has allowed to characterize the fluctuation field (",
{
"type": "Cite",
"target": "S1-E4",
"content": [
"1.4"
]
},
") and to obtain a stochastic evolution equation governing the limit process.\nThese results are presented in Section ",
{
"type": "Cite",
"target": "S3-SS3",
"content": [
"3.3"
]
},
".\n"
]
},
{
"type": "Heading",
"id": "S1.SS3.SSSx3",
"depth": 3,
"content": [
"On large deviations"
]
},
{
"type": "Paragraph",
"content": [
"The last question generally studied in a classical probabilistic approach is that of the quantification of rare events, i.e., the estimation of the probability of observing an atypical behavior (which deviates macroscopically from the mean).\nFor independent and identically distributed random variables, this probability is exponentially small, and it is therefore natural to study the asymptotics"
]
},
{
"type": "MathBlock",
"id": "S1.Ex6",
"mathLanguage": "mathml",
"text": "I(m)≔limδ→0limN→∞−1Nlogℙ(|1N∑i=1Nηi−m|<δ)withm≠𝔼(η).",
"meta": {
"altText": "I(m)≔\\lim_{\\delta\\to 0}\\lim_{N\\to\\infty}-\\frac{1}{N}\\log\\mathbb{P}\\biggl(\\biggl|\\frac{1}{N}\\sum_{i=1}^{N}\\eta_{i}-m\\biggr|<\\delta\\biggr)\\quad\\text{with}\\ m\\neq\\mathbb{E}(\\eta)."
}
},
{
"type": "Paragraph",
"content": [
"The limit ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "I(m)",
"meta": {
"altText": "I(m)"
}
},
" is called the large deviation functional and can be expressed as the Legendre transform of the log-Laplace transform ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℝ∋u↦log𝔼(exp(uη))",
"meta": {
"altText": "\\mathbb{R}\\ni u\\mapsto\\log\\mathbb{E}(\\exp(u\\eta))"
}
},
".\nTo generalize this statement to correlated variables in a gas of hard spheres, it is necessary to compute the log-Laplace transform of the empirical measure on deterministic trajectories, which requires extremely precise control of the dynamical correlations.\nNote that, at time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "0",
"meta": {
"altText": "0"
}
},
", under the grand canonical measure, one can show that, for any ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "0\" display=\"inline\">δ>0",
"meta": {
"altText": "\\delta>0"
}
},
","
]
},
{
"type": "MathBlock",
"id": "S1.Ex7",
"mathLanguage": "mathml",
"text": "limδ→0limμε→∞−1μεlogℙε(d(π0ε,φ0)≤δ)=H(φ0|f0)≔∫(φ0logφ0f0−(φ0−f0))𝑑x𝑑v,",
"meta": {
"altText": "\\begin{split}&\\lim_{\\delta\\to 0}\\lim_{\\mu_{\\varepsilon}\\to\\infty}-\\frac{1}{\\mu_{\\varepsilon}}\\log\\mathbb{P}_{\\varepsilon}\\bigl(d(\\pi^{\\varepsilon}_{0},\\varphi^{0})\\leq\\delta\\bigr)\\\\[-3.0pt]\n&\\qquad=H(\\varphi^{0}|f^{0})≔\\int\\Bigl(\\varphi^{0}\\log\\frac{\\varphi^{0}}{f^{0}}-(\\varphi^{0}-f^{0})\\Bigr)\\,dx\\,dv,\\end{split}"
}
},
{
"type": "Paragraph",
"content": [
"where ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "d",
"meta": {
"altText": "d"
}
},
" is a distance on the space of measures.\nThe dynamical cumulant method introduced in [",
{
"type": "Cite",
"target": "bib-bib4",
"content": [
"4"
]
},
", ",
{
"type": "Cite",
"target": "bib-bib6",
"content": [
"6"
]
},
"] is a key tool for computing the exponential moments of the hard sphere distribution, thus obtaining the dynamical equivalent of this result in short time.\nWe give an overview of these techniques in Section ",
{
"type": "Cite",
"target": "S3",
"content": [
"3"
]
},
"."
]
},
{
"type": "Heading",
"id": "S2",
"depth": 1,
"content": [
"2 Typical behavior: A law of large numbers"
]
},
{
"type": "Heading",
"id": "S2.SS1",
"depth": 2,
"content": [
"2.1 Boltzmann’s amazing intuition"
]
},
{
"type": "Paragraph",
"content": [
"The equation that rules the typical evolution of a gas of hard spheres was heuristically proposed by Boltzmann [",
{
"type": "Cite",
"target": "bib-bib8",
"content": [
"8"
]
},
"] about a century before its rigorous derivation by Lanford [",
{
"type": "Cite",
"target": "bib-bib16",
"content": [
"16"
]
},
"], as the “limit” of the particle system when ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "με→+∞",
"meta": {
"altText": "\\mu_{\\varepsilon}\\to+\\infty"
}
},
".\nBoltzmann’s revolutionary idea was to write an evolution equation for the probability density ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f=f(t,x,v)",
"meta": {
"altText": "f=f(t,x,v)"
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},
" giving the proportion of particles at position ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "x",
"meta": {
"altText": "x"
}
},
" with velocity ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "v",
"meta": {
"altText": "v"
}
},
" at time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "t",
"meta": {
"altText": "t"
}
},
".\nIn the absence of collisions, and in an unbounded domain, this density ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f",
"meta": {
"altText": "f"
}
},
" would be transported along the physical trajectories ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "x(t)=x(0)+vt",
"meta": {
"altText": "x(t)=x(0)+vt"
}
},
", which means that ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f(t,x,v)=f0(x−vt,v)",
"meta": {
"altText": "f(t,x,v)=f^{0}(x-vt,v)"
}
},
".\nThe challenge is to take into account the statistical effect of collisions.\nAs long as the size of the particles is negligible, one can consider that these collisions are pointwise in both ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "t",
"meta": {
"altText": "t"
}
},
" and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "x",
"meta": {
"altText": "x"
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},
".\nBoltzmann proposed a quite intuitive counting:"
]
},
{
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},
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"type": "Paragraph",
"id": "S2.I1.i1.p1",
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"the number of particles of velocity ",
{
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"mathLanguage": "mathml",
"text": "v",
"meta": {
"altText": "v"
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},
" increases when a particle of velocity ",
{
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"meta": {
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"meta": {
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{
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"mathLanguage": "mathml",
"text": "v",
"meta": {
"altText": "v"
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" (Figure ",
{
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"content": [
"1"
]
},
" and (",
{
"type": "Cite",
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"));"
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"meta": {
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{
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{
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}
],
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},
{
"type": "Paragraph",
"content": [
"The probability of these jumps in velocity is described by a transition rate, called the ",
{
"type": "Emphasis",
"content": [
"collision cross section"
]
},
".\nFor interactions between hard spheres, it is given by ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "((v−v1)⋅ω)+",
"meta": {
"altText": "((v-v_{1})\\cdot\\omega)_{+}"
}
},
", where ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "v−v1",
"meta": {
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},
" is the relative velocity of the colliding particles, and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ω",
"meta": {
"altText": "\\omega"
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},
" is the deflection vector, uniformly distributed in the unit sphere ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝕊d−1⊂ℝd",
"meta": {
"altText": "\\mathbb{S}^{d-1}\\subset\\mathbb{R}^{d}"
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},
".\nThe fundamental assumption of Boltzmann’s theory is that, in a rarefied gas, the correlations between two colliding particles must be very small.\nTherefore, the joint probability of having two pre-colliding particles of velocities ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "v",
"meta": {
"altText": "v"
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},
" and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "v1",
"meta": {
"altText": "v_{1}"
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},
" at position ",
{
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"mathLanguage": "mathml",
"text": "x",
"meta": {
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" at time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "t",
"meta": {
"altText": "t"
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" should be well approximated by the product ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f(t,x,v)f(t,x,v1)",
"meta": {
"altText": "f(t,x,v)f(t,x,v_{1})"
}
},
".\nThis independence property is called the ",
{
"type": "Emphasis",
"content": [
"molecular chaos hypothesis"
]
},
".\nThe Boltzmann equation then reads"
]
},
{
"type": "MathBlock",
"id": "S2.E1",
"mathLanguage": "mathml",
"text": "∂tf+v⋅∇xf⏟transport=C(f,f)⏟collision,",
"meta": {
"altText": "\\partial_{t}f+\\underbrace{v\\cdot\\nabla_{x}f}_{\\mathstrut\\text{transport}}=\\underbrace{C(f,f)}_{\\mathstrut\\text{collision}},"
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},
{
"type": "Paragraph",
"content": [
"where"
]
},
{
"type": "MathBlock",
"id": "S2.Ex1",
"mathLanguage": "mathml",
"text": "C(f,f)(t,x,v)=∬[f(t,x,v′)f(t,x,v1′)⏟gain term−f(t,x,v)f(t,x,v1)⏟loss term]×((v−v1)⋅ω)+⏟cross sectiondv1dω,",
"meta": {
"altText": "\\begin{split}C(f,f)(t,x,v)&=\\iint[\\underbrace{f(t,x,v^{\\prime})f(t,x,v^{\\prime}_{1})}_{\\mathstrut\\text{gain term}}-\\underbrace{f(t,x,v)f(t,x,v_{1})}_{\\mathstrut\\text{loss term}}]\\\\[-1.5pt]\n&\\hskip 30.00005pt\\times\\smash[b]{\\underbrace{\\bigl((v-v_{1})\\cdot\\omega\\bigr)_{+}}_{\\text{cross section}}}\\,dv_{1}\\,d\\omega,\\end{split}"
}
},
{
"type": "Paragraph",
"content": [
"with the scattering rules"
]
},
{
"type": "MathBlock",
"id": "S2.E2",
"mathLanguage": "mathml",
"text": "v′=v−((v−v1)⋅ω)ω,v1′=v1+((v−v1)⋅ω)ω",
"meta": {
"altText": "v^{\\prime}=v-\\bigl((v-v_{1})\\cdot\\omega\\bigr)\\omega,\\quad v_{1}^{\\prime}=v_{1}+\\bigl((v-v_{1})\\cdot\\omega\\bigr)\\omega"
}
},
{
"type": "Paragraph",
"content": [
"being analogous to those introduced in Figure ",
{
"type": "Cite",
"target": "S1-F1",
"content": [
"1"
]
},
", with the important difference that ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ω",
"meta": {
"altText": "\\omega"
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},
" is now a random vector chosen uniformly in the unit sphere ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝕊d−1",
"meta": {
"altText": "\\mathbb{S}^{d-1}"
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},
": indeed, the relative position of the colliding particles disappeared in the limit ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ε→0",
"meta": {
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".\nAs a result, the Boltzmann equation is singular because it involves a product of densities at a single point ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "x",
"meta": {
"altText": "x"
}
},
".\nBoltzmann’s idea of reducing the Hamiltonian dynamics describing atomic behavior to a kinetic equation was revolutionary and paved the way to the description of non-equilibrium phenomena by mesoscopic equations.\nHowever, the Boltzmann equation (",
{
"type": "Cite",
"target": "S2-E1",
"content": [
"2.1"
]
},
") was first strongly criticized because it seems to violate some fundamental physical principles.\nIt actually predicts an irreversible evolution in time: it has a Lyapunov functional, called entropy, defined by ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "S(t)≔−∬flogf(t,x,v)𝑑x𝑑v",
"meta": {
"altText": "S(t)≔-\\iint f\\log f(t,x,v)\\,dx\\,dv"
}
},
", such that ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ddtS(t)≥0",
"meta": {
"altText": "\\frac{d}{dt}S(t)\\geq 0"
}
},
", with equality if and only if the gas is in thermodynamic equilibrium.\nThe Boltzmann equation thus provides a quantitative formulation of the second principle of thermodynamics.\nBut at first glance, this irreversibility seems incompatible with the fact that the dynamics of hard spheres is governed by a Hamiltonian system, i.e., a system of ordinary differential equations that is completely reversible in time.\nSoon after Boltzmann postulated his equation, these two different behaviors were considered, by Loschmidt, as a paradox and an obstruction to Boltzmann’s theory.\nA fully satisfactory mathematical explanation of this question remained elusive for almost a century, until the role of probabilities was precisely identified: the underlying dynamics is reversible, but the description that is given of this dynamics is only partial and is therefore not reversible."
]
},
{
"type": "Heading",
"id": "S2.SS2",
"depth": 2,
"content": [
"2.2 Typical behavior: Lanford’s theorem"
]
},
{
"type": "Paragraph",
"id": "S2.SS2.p1",
"content": [
"Lanford’s result [",
{
"type": "Cite",
"target": "bib-bib16",
"content": [
"16"
]
},
"] shows in which sense the Boltzmann equation (",
{
"type": "Cite",
"target": "S2-E1",
"content": [
"2.1"
]
},
") is a good approximation of the hard sphere dynamics.\nIt can be stated as follows (this is not exactly the original formulation; see in particular Section ",
{
"type": "Cite",
"target": "S2-SS4",
"content": [
"2.4"
]
},
" below for comments)."
]
},
{
"type": "Claim",
"id": "S2.Thm124Thm1",
"claimType": "Theorem",
"label": "Theorem 2.1(Lanford).",
"title": [
{
"type": "Strong",
"content": [
"Theorem 2.1"
]
},
{
"type": "Strong",
"content": []
},
"(Lanford)",
{
"type": "Strong",
"content": [
"."
]
}
],
"content": [
{
"type": "Paragraph",
"content": [
{
"type": "Emphasis",
"content": [
"In the low density limit (",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "με→∞",
"meta": {
"altText": "\\mu_{\\varepsilon}\\to\\infty"
}
},
" with ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "μεεd−1=1",
"meta": {
"altText": "\\mu_{\\varepsilon}\\varepsilon^{d-1}=1"
}
},
"), the empirical measure ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "πtε",
"meta": {
"altText": "\\pi^{\\varepsilon}_{t}"
}
},
" defined by (",
{
"type": "Cite",
"target": "S1-E2",
"content": [
"1.2"
]
},
") concentrates on the solution of the Boltzmann equation (",
{
"type": "Cite",
"target": "S2-E1",
"content": [
"2.1"
]
},
"): for any bounded and continuous function ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "h",
"meta": {
"altText": "h"
}
},
","
]
}
]
},
{
"type": "MathBlock",
"id": "S2.Ex2",
"mathLanguage": "mathml",
"text": "0,\\quad\\lim_{\\mu_{\\varepsilon}\\to\\infty}\\mathbb{P}_{\\varepsilon}\\Bigl(\\Bigl|\\langle\\pi^{\\varepsilon}_{t},h\\rangle-\\int f(t,x,v)h(x,v)\\,dx\\,dv\\Bigr|\\geq\\delta\\Bigr)=0,\" display=\"block\">∀δ>0,limμε→∞ℙε(|⟨πtε,h⟩−∫f(t,x,v)h(x,v)𝑑x𝑑v|≥δ)=0,",
"meta": {
"altText": "\\forall\\delta>0,\\quad\\lim_{\\mu_{\\varepsilon}\\to\\infty}\\mathbb{P}_{\\varepsilon}\\Bigl(\\Bigl|\\langle\\pi^{\\varepsilon}_{t},h\\rangle-\\int f(t,x,v)h(x,v)\\,dx\\,dv\\Bigr|\\geq\\delta\\Bigr)=0,"
}
},
{
"type": "Paragraph",
"content": [
{
"type": "Emphasis",
"content": [
"on a time interval ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "[0,TL]",
"meta": {
"altText": "[0,T_{\\mathrm{L}}]"
}
},
" that depends only on the initial distribution ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f0",
"meta": {
"altText": "f^{0}"
}
},
"."
]
}
]
}
]
},
{
"type": "Paragraph",
"id": "S2.SS2.p2",
"content": [
"The time of validity ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "TL",
"meta": {
"altText": "T_{\\mathrm{L}}"
}
},
" of the approximation is found to be a fraction of the average time between two successive collisions for a typical particle.\nThis time is large enough for the microscopic system to undergo a large number of collisions (of the order of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "με",
"meta": {
"altText": "\\mu_{\\varepsilon}"
}
},
"), but (much) too small to see phenomena such as relaxation to (local) thermodynamic equilibrium, and in particular hydrodynamic regimes.\nPhysically, we do not expect this time to be critical, in the sense that the dynamics would change in nature afterwards.\nIn fact, in practice, Boltzmann’s equation is used in many applications (such as spacecraft reentrance calculations) without time restrictions.\nHowever, it is important to note that a time restriction might not be only technical: from a mathematical point of view, one cannot exclude that the Boltzmann equation presents singularities (typically spatial concentrations that would prevent the collision term from making sense, and that would also locally contradict the low density assumption).\nAt present, the problem of extending Lanford’s convergence result to longer times still faces serious obstacles."
]
},
{
"type": "Heading",
"id": "S2.SS3",
"depth": 2,
"content": [
"2.3 Heuristics of Lanford’s proof"
]
},
{
"type": "Paragraph",
"content": [
"Let us informally explain how the Boltzmann equation (",
{
"type": "Cite",
"target": "S2-E1",
"content": [
"2.1"
]
},
") can be predicted from the dynamics of the particles.\nThe goal is to transport via the dynamics the initial grand canonical measure (",
{
"type": "Cite",
"target": "S1-E1",
"content": [
"1.1"
]
},
") and then to project this measure at time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "t",
"meta": {
"altText": "t"
}
},
" onto the 1-particle phase space.\nWe thus define by duality the density ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "F1ε(t,x,v)",
"meta": {
"altText": "F_{1}^{\\varepsilon}(t,x,v)"
}
},
" of a typical particle with respect to a test function ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "h",
"meta": {
"altText": "h"
}
},
" by"
]
},
{
"type": "MathBlock",
"id": "S2.E3",
"mathLanguage": "mathml",
"text": "∫F1ε(t,x,v)h(x,v)𝑑x𝑑v≔𝔼ε(⟨πtε,h⟩).",
"meta": {
"altText": "\\int F_{1}^{\\varepsilon}(t,x,v)h(x,v)\\,dx\\,dv≔\\mathbb{E}_{\\varepsilon}(\\langle\\pi_{t}^{\\varepsilon},h\\rangle)."
}
},
{
"type": "Paragraph",
"content": [
"Theorem ",
{
"type": "Cite",
"target": "S2-Thm124Thm1",
"content": [
"2.1"
]
},
" states that ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "F1ε",
"meta": {
"altText": "F_{1}^{\\varepsilon}"
}
},
" converges to the solution to the Boltzmann equation ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f",
"meta": {
"altText": "f"
}
},
" in the low density limit.\nSo let ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "h",
"meta": {
"altText": "h"
}
},
" be a regular and bounded function on ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝕋d×ℝd",
"meta": {
"altText": "{\\mathbb{T}}^{d}\\times{\\mathbb{R}}^{d}"
}
},
" and consider the evolution of the empirical measure during a short time interval ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "[t,t+δ]",
"meta": {
"altText": "[t,t+\\delta]"
}
},
".\nSeparating the different contributions according to the number of collisions, we can write"
]
},
{
"type": "MathBlock",
"id": "S2.E4",
"mathLanguage": "mathml",
"text": "1δ(𝔼ε[⟨πt+δε,h⟩]−𝔼ε[⟨πtε,h⟩])=1δ𝔼ε[1με∑jno collision(h(zjε(t+δ))−h(zjε(t)))]+1δ𝔼ε[12με∑(i,j)one collision(h(ziε(t+δ))+h(zjε(t+δ))−h(ziε(t))−h(zjε(t)))]+⋯.",
"meta": {
"altText": "\\begin{split}&\\smash[b]{\\frac{1}{\\delta}}(\\mathbb{E}_{\\varepsilon}[\\langle\\pi^{\\varepsilon}_{t+\\delta},h\\rangle]-\\mathbb{E}_{\\varepsilon}[\\langle\\pi^{\\varepsilon}_{t},h\\rangle])\\\\\n&\\quad=\\frac{1}{\\delta}\\mathbb{E}_{\\varepsilon}\\biggl[\\frac{1}{\\mu_{\\varepsilon}}\\sum_{\\begin{subarray}{c}j\\\\\n\\text{no collision}\\end{subarray}}\\bigl(h(z^{\\varepsilon}_{j}(t+\\delta))-h(z^{\\varepsilon}_{j}(t))\\bigr)\\biggr]\\\\\n&\\quad\\qquad+\\frac{1}{\\delta}\\mathbb{E}_{\\varepsilon}\\biggl[\\frac{1}{2\\mu_{\\varepsilon}}\\sum_{\\begin{subarray}{c}(i,j)\\\\\n\\text{one collision}\\end{subarray}}\\bigl(h(z^{\\varepsilon}_{i}(t+\\delta))+h(z^{\\varepsilon}_{j}(t+\\delta))\\\\[-17.22217pt]\n&\\hskip 140.00021pt-h(z^{\\varepsilon}_{i}(t))-h(z^{\\varepsilon}_{j}(t))\\bigr)\\biggr]\\\\[-8.61108pt]\n&\\quad\\qquad+\\cdots.\\end{split}"
}
},
{
"type": "Paragraph",
"content": [
"To simplify, ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ziε(t)",
"meta": {
"altText": "z^{\\varepsilon}_{i}(t)"
}
},
" denotes the coordinates ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "(xiε(t),viε(t))",
"meta": {
"altText": "(x^{\\varepsilon}_{i}(t),v^{\\varepsilon}_{i}(t))"
}
},
" of the ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "i",
"meta": {
"altText": "i"
}
},
"-th particle at time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "t",
"meta": {
"altText": "t"
}
},
".\nSince the left-hand side of (",
{
"type": "Cite",
"target": "S2-E4",
"content": [
"2.4"
]
},
") formally converges when ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "δ→0",
"meta": {
"altText": "\\delta\\to 0"
}
},
" to the time derivative of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝔼ε[⟨πtε,h⟩]",
"meta": {
"altText": "\\mathbb{E}_{\\varepsilon}[\\langle\\pi^{\\varepsilon}_{t},h\\rangle]"
}
},
", we will analyze the limit ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "δ→0",
"meta": {
"altText": "\\delta\\to 0"
}
},
" of the first two terms in the right-hand side of (",
{
"type": "Cite",
"target": "S2-E4",
"content": [
"2.4"
]
},
"), which should lead to a transport term and a collision term as in (",
{
"type": "Cite",
"target": "S2-E1",
"content": [
"2.1"
]
},
").\nWe will also explain why the remainder terms, involving two or more collisions in the short time interval ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "δ",
"meta": {
"altText": "\\delta"
}
},
", tend to 0 with ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "δ",
"meta": {
"altText": "\\delta"
}
},
" (showing that they are of order ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "δ",
"meta": {
"altText": "\\delta"
}
},
").\nSince the particles move in a straight line and at constant speed in the absence of collisions, if the distribution ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "F1ε",
"meta": {
"altText": "F_{1}^{\\varepsilon}"
}
},
" is sufficiently regular, the definition (",
{
"type": "Cite",
"target": "S2-E3",
"content": [
"2.3"
]
},
") of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "F1ε",
"meta": {
"altText": "F_{1}^{\\varepsilon}"
}
},
" formally implies that, when ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "δ",
"meta": {
"altText": "\\delta"
}
},
" tends to 0, the first term in the right-hand side of (",
{
"type": "Cite",
"target": "S2-E4",
"content": [
"2.4"
]
},
") is asymptotically equal to"
]
},
{
"type": "MathBlock",
"id": "S2.Ex3",
"mathLanguage": "mathml",
"text": "∫F1ε(t,z)v⋅∇xh(z)𝑑z=−∫(v⋅∇xF1ε(t,z))h(z)𝑑z.",
"meta": {
"altText": "\\int F_{1}^{\\varepsilon}(t,z)v\\cdot\\nabla_{x}h(z)\\,dz=-\\int\\bigl(v\\cdot\\nabla_{x}F_{1}^{\\varepsilon}(t,z)\\bigr)h(z)\\,dz."
}
},
{
"type": "Paragraph",
"content": [
"The transport term in (",
{
"type": "Cite",
"target": "S2-E1",
"content": [
"2.1"
]
},
") is thus well obtained in the limit.\nLet us now consider the second term in the right-hand side of (",
{
"type": "Cite",
"target": "S2-E4",
"content": [
"2.4"
]
},
").\nTwo particles of configurations ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "(x1,v1)",
"meta": {
"altText": "(x_{1},v_{1})"
}
},
" and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "(x2,v2)",
"meta": {
"altText": "(x_{2},v_{2})"
}
},
" at time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "t",
"meta": {
"altText": "t"
}
},
" collide at a later time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "τ≤t+δ",
"meta": {
"altText": "\\tau\\leq t+\\delta"
}
},
" if there exists ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ω∈𝕊d−1",
"meta": {
"altText": "\\omega\\in{\\mathbb{S}}^{d-1}"
}
},
" such that"
]
},
{
"type": "MathBlock",
"id": "S2.E5",
"mathLanguage": "mathml",
"text": "x1−x2+(τ−t)(v1−v2)=−εω.",
"meta": {
"altText": "x_{1}-x_{2}+(\\tau-t)(v_{1}-v_{2})=-\\varepsilon\\omega."
}
},
{
"type": "Paragraph",
"content": [
"This implies that their relative position must belong to a tube of length ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "δ|v1−v2|",
"meta": {
"altText": "\\delta\\lvert v_{1}-v_{2}\\rvert"
}
},
" and width ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ε",
"meta": {
"altText": "\\varepsilon"
}
},
" oriented in the ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "v1−v2",
"meta": {
"altText": "v_{1}-v_{2}"
}
},
" direction.\nThe Lebesgue measure of this set is of the order ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "δεd−1|v2−v1|=O(δεd−1)",
"meta": {
"altText": "\\delta\\varepsilon^{d-1}\\lvert v_{2}-v_{1}\\rvert=O(\\delta\\varepsilon^{d-1})"
}
},
" (neglecting large velocities).\nMore generally, a sequence of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k−1",
"meta": {
"altText": "k-1"
}
},
" collisions between ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k",
"meta": {
"altText": "k"
}
},
" particles imposes ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k−1",
"meta": {
"altText": "k-1"
}
},
" constraints of the previous form, and this event can be shown to have probability less than ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "(δεd−1)k−1=(δμε−1)k−1",
"meta": {
"altText": "(\\delta\\varepsilon^{d-1})^{k-1}=(\\delta\\mu_{\\varepsilon}^{-1})^{k-1}"
}
},
" (again neglecting large velocities).\nSince there are, on average, ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "μεk",
"meta": {
"altText": "\\mu_{\\varepsilon}^{k}"
}
},
" ways to choose these ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k",
"meta": {
"altText": "k"
}
},
" colliding particles, we deduce that the occurrence of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k−1",
"meta": {
"altText": "k-1"
}
},
" collisions in (",
{
"type": "Cite",
"target": "S2-E4",
"content": [
"2.4"
]
},
") has a probability of order ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "δk−1με",
"meta": {
"altText": "\\delta^{k-1}\\mu_{\\varepsilon}"
}
},
".\nThis explains why the probability of having ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k≥3",
"meta": {
"altText": "k\\geq 3"
}
},
" colliding particles can be estimated by ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "O(δ2)",
"meta": {
"altText": "O(\\delta^{2})"
}
},
" and thus can be neglected in (",
{
"type": "Cite",
"target": "S2-E4",
"content": [
"2.4"
]
},
").\nIt remains to examine more closely the collision term involving two particles in (",
{
"type": "Cite",
"target": "S2-E4",
"content": [
"2.4"
]
},
"), in order to obtain the collision operator ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "C(f,f)",
"meta": {
"altText": "C(f,f)"
}
},
" of the Boltzmann equation (",
{
"type": "Cite",
"target": "S2-E1",
"content": [
"2.1"
]
},
").\nThis term involves the two-particle correlation function ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "F2ε",
"meta": {
"altText": "F_{2}^{\\varepsilon}"
}
},
".\nFor any ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k≥1",
"meta": {
"altText": "k\\geq 1"
}
},
", we define"
]
},
{
"type": "MathBlock",
"id": "S2.E6",
"mathLanguage": "mathml",
"text": "∫Fkε(t,Zk)hk(Zk)dZk=𝔼ε(1μεk∑(i1,…,ik)hk(zi1ε(t),…,zikε(t))),",
"meta": {
"altText": "\\begin{split}&\\mathop{\\smash[b]{\\int}}F_{k}^{\\varepsilon}(t,Z_{k})h_{k}(Z_{k})\\,dZ_{k}\\\\\n&\\qquad=\\mathbb{E}_{\\varepsilon}\\biggl(\\frac{1}{\\mu_{\\varepsilon}^{k}}\\sum_{(i_{1},\\dots,i_{k})}h_{k}\\bigl(z^{\\varepsilon}_{i_{1}}(t),\\dots,z^{\\varepsilon}_{i_{k}}(t)\\bigr)\\biggr),\\end{split}"
}
},
{
"type": "Paragraph",
"content": [
"where ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "i1,…,ik",
"meta": {
"altText": "i_{1},\\dots,i_{k}"
}
},
" are all distinct and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "Zk=(xi,vi)1≤i≤k",
"meta": {
"altText": "Z_{k}=(x_{i},v_{i})_{1\\leq i\\leq k}"
}
},
".\nWe can then show that, in the limit ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "δ→0",
"meta": {
"altText": "\\delta\\to 0"
}
},
","
]
},
{
"type": "MathBlock",
"id": "S2.E7",
"mathLanguage": "mathml",
"text": "∂tF1ε+v⋅∇xF1ε⏟transport=Cε(F2ε)⏟collisionat distanceε,",
"meta": {
"altText": "\\partial_{t}F^{\\varepsilon}_{1}+\\underbrace{v\\cdot\\nabla_{x}F^{\\varepsilon}_{1}}_{\\mathstrut\\text{transport}}=\\smash{\\underbrace{C^{\\varepsilon}(F^{\\varepsilon}_{2})}_{\\begin{subarray}{c}\\mathstrut\\text{collision}\\\\\n\\mathstrut\\text{at distance}\\ \\varepsilon\\end{subarray}}},"
}
},
{
"type": "Paragraph",
"content": [
"where"
]
},
{
"type": "MathBlock",
"id": "S2.Ex4",
"mathLanguage": "mathml",
"text": "Cε(F2ε)(t,x,v)=∬[F2ε(t,x,v′,x+εω,v1′)⏟gain term−F2ε(t,x,v,x−εω,v1)⏟loss term]×((v−v1)⋅ω)+⏟cross sectiondv1dω.",
"meta": {
"altText": "\\begin{split}&C^{\\varepsilon}(F^{\\varepsilon}_{2})(t,x,v)\\\\\n&\\quad=\\iint[\\underbrace{F^{\\varepsilon}_{2}(t,x,v^{\\prime},x+\\varepsilon\\omega,v^{\\prime}_{1})}_{\\mathstrut\\text{gain term}}-\\underbrace{F^{\\varepsilon}_{2}(t,x,v,x-\\varepsilon\\omega,v_{1})}_{\\mathstrut\\text{loss term}}]\\\\\n&\\hskip 40.00006pt\\times\\underbrace{\\bigl((v-v_{1})\\cdot\\omega\\bigr)_{+}}_{\\text{cross section}}\\,dv_{1}\\,d\\omega.\\end{split}"
}
},
{
"type": "Paragraph",
"content": [
"The key step in closing the equation is the ",
{
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" postulated by Boltzmann, which states that the pre-collisional particles remain independently distributed at all times so that, with the convention (",
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"content": [
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") fixing the sign of ",
{
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},
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"text": "0.\" display=\"block\">F2ε(t,z1,z2)≃F1ε(t,z1)F1ε(t,z2)if(v1−v2)⋅ω>0.",
"meta": {
"altText": "F_{2}^{\\varepsilon}(t,z_{1},z_{2})\\simeq F_{1}^{\\varepsilon}(t,z_{1})F_{1}^{\\varepsilon}(t,z_{2})\\quad\\text{if}\\ (v_{1}-v_{2})\\cdot\\omega>0."
}
},
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", ",
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"13"
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", ",
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", ",
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"content": [
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"), which is valid only for specific configurations."
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"content": [
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"."
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{
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" and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "2⋆",
"meta": {
"altText": "2^{\\star}"
}
},
" trees may interact.\nWe can thus decompose the trees constituting ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "F2ε",
"meta": {
"altText": "F_{2}^{\\varepsilon}"
}
},
" into two categories: those such that there is at least one collision involving a particle from each tree (such a recollision will be called ",
{
"type": "Emphasis",
"content": [
"external"
]
},
"), and the others (Figure ",
{
"type": "Cite",
"target": "S3-F5",
"content": [
"5"
]
},
")."
]
},
{
"type": "Figure",
"id": "S3-F6",
"caption": [
{
"type": "Paragraph",
"content": [
"Decomposition of the dynamical exclusion condition."
]
}
],
"content": [
{
"type": "ImageObject",
"contentUrl": "mag-124-fig-6.png",
"mediaType": "image/png",
"meta": {
"inline": false
}
}
]
},
{
"type": "Paragraph",
"id": "S3.SS1.p4",
"content": [
"Note, however, that two collision-free trees do not correspond to independent trees, precisely because of the dynamical exclusion condition.\nThis exclusion condition can itself be decomposed as ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝟏1⋆≁2⋆=1−𝟏1⋆∼2⋆",
"meta": {
"altText": "\\mathbf{1}_{1^{\\star}\\not\\sim 2^{\\star}}=1-\\mathbf{1}_{1^{\\star}\\sim 2^{\\star}}"
}
},
" (Figure ",
{
"type": "Cite",
"target": "S3-F6",
"content": [
"6"
]
},
"), where ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝟏1⋆∼2⋆",
"meta": {
"altText": "\\mathbf{1}_{1^{\\star}\\sim 2^{\\star}}"
}
},
" means that there is an overlap at some point between a particle from the ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "1⋆",
"meta": {
"altText": "1^{\\star}"
}
},
" tree and a particle from the ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "2⋆",
"meta": {
"altText": "2^{\\star}"
}
},
" tree.\nThis decomposition is a pure mathematical artifact, and the ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "1⋆∼2⋆",
"meta": {
"altText": "1^{\\star}\\sim 2^{\\star}"
}
},
" overlap condition does not affect the dynamics (the overlapping particles are not deflected)."
]
},
{
"type": "Figure",
"id": "S3-F7",
"caption": [
{
"type": "Paragraph",
"content": [
"The second-order cumulant corresponds to the occurrence of at least one external recollision or an overlap."
]
}
],
"content": [
{
"type": "ImageObject",
"contentUrl": "mag-124-fig-7.png",
"mediaType": "image/png",
"meta": {
"inline": false
}
}
]
},
{
"type": "Paragraph",
"content": [
"Let us now define the ",
{
"type": "Emphasis",
"content": [
"second-order rescaled cumulant"
]
}
]
},
{
"type": "MathBlock",
"id": "S3.E1",
"mathLanguage": "mathml",
"text": "f2ε≔με(F2ε−F1ε⊗F1ε).",
"meta": {
"altText": "f_{2}^{\\varepsilon}≔\\mu_{\\varepsilon}(F_{2}^{\\varepsilon}-F_{1}^{\\varepsilon}\\otimes F_{1}^{\\varepsilon})."
}
},
{
"type": "Paragraph",
"content": [
"The previous discussion indicates that this cumulant is represented by trees that are coupled by external collisions or overlaps (Figure ",
{
"type": "Cite",
"target": "S3-F7",
"content": [
"7"
]
},
").\nIn view of definition (",
{
"type": "Cite",
"target": "S3-E1",
"content": [
"3.1"
]
},
") and the discussion in Section ",
{
"type": "Cite",
"target": "S2-SS3",
"content": [
"2.3"
]
},
" giving an ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "O(t/με)",
"meta": {
"altText": "O(t/\\mu_{\\varepsilon})"
}
},
" estimate of the Lebesgue measure of configurations giving rise to a collision, one can expect ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f2ε",
"meta": {
"altText": "f_{2}^{\\varepsilon}"
}
},
" to be uniformly bounded in ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "L1",
"meta": {
"altText": "L^{1}"
}
},
" and therefore to have a limit ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f2",
"meta": {
"altText": "f_{2}"
}
},
" in the sense of the measures.\nOne can prove in addition that ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f2",
"meta": {
"altText": "f_{2}"
}
},
" corresponds to trees with ",
{
"type": "Emphasis",
"content": [
"exactly"
]
},
" one external recollision or overlap on ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "[0,t]",
"meta": {
"altText": "[0,t]"
}
},
": any other interaction between the trees gives rise to additional smallness and is therefore negligible."
]
},
{
"type": "Claim",
"id": "S3.Thm124Thm1",
"label": "Remark 3.1.",
"title": [
{
"type": "Strong",
"content": [
"Remark 3.1"
]
},
{
"type": "Strong",
"content": [
"."
]
}
],
"content": [
{
"type": "Paragraph",
"id": "S3.Thm124Thm1.p1",
"content": [
"The initial measure does not factorize exactly ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "(F2ε,0≠F1ε,0⊗F1ε,0)",
"meta": {
"altText": "(F_{2}^{\\varepsilon,0}\\neq F_{1}^{\\varepsilon,0}\\otimes F_{1}^{\\varepsilon,0})"
}
},
" because of the static exclusion condition.\nThus, the initial data also induce a small contribution to ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f2ε",
"meta": {
"altText": "f_{2}^{\\varepsilon}"
}
},
", but this contribution is significantly smaller than the dynamical correlations (by a factor ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ε",
"meta": {
"altText": "\\varepsilon"
}
},
")."
]
}
]
},
{
"type": "Heading",
"id": "S3.SS2",
"depth": 2,
"content": [
"3.2 The cumulant generating function"
]
},
{
"type": "Figure",
"id": "S3-F8",
"caption": [
{
"type": "Paragraph",
"content": [
"The cumulant of order ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k",
"meta": {
"altText": "k"
}
},
" corresponds to trees with roots in ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "1⋆,…,k⋆",
"meta": {
"altText": "{1^{\\star}},\\dots,{k^{\\star}}"
}
},
" that are completely connected by external collisions or overlaps."
]
}
],
"content": [
{
"type": "ImageObject",
"contentUrl": "mag-124-cumulantk-gazette.svg",
"mediaType": "image/svg+xml",
"meta": {
"inline": false
}
}
]
},
{
"type": "Paragraph",
"content": [
"For a Gaussian process, the first two correlation functions ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "F1ε",
"meta": {
"altText": "F_{1}^{\\varepsilon}"
}
},
" and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "F2ε",
"meta": {
"altText": "F_{2}^{\\varepsilon}"
}
},
" determine completely all other ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k",
"meta": {
"altText": "k"
}
},
"-particle correlation functions ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "Fkε",
"meta": {
"altText": "F_{k}^{\\varepsilon}"
}
},
", but in general, part of the information is encoded in the cumulants of higher order (",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k≥3",
"meta": {
"altText": "k\\geq 3"
}
},
")"
]
},
{
"type": "MathBlock",
"id": "S3.Ex1",
"mathLanguage": "mathml",
"text": "fkε(t,Zk)≔μεk−1∑ℓ=1k∑σ∈𝒫kℓ(−1)ℓ−1(ℓ−1)!∏i=1ℓF|σi|ε(t,Zσi),",
"meta": {
"altText": "f_{k}^{\\varepsilon}(t,Z_{k})≔\\mu_{\\varepsilon}^{k-1}\\sum_{\\ell=1}^{k}\\sum_{\\sigma\\in\\smash{\\mathcal{P}^{\\ell}_{k}}\\vphantom{\\ell}}(-1)^{\\ell-1}(\\ell-1)!\\prod_{i=1}^{\\ell}F^{\\varepsilon}_{\\lvert\\sigma_{i}\\rvert}(t,Z_{\\sigma_{i}}),"
}
},
{
"type": "Paragraph",
"content": [
"where ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝒫kℓ",
"meta": {
"altText": "\\mathcal{P}^{\\ell}_{k}"
}
},
" is the set of partitions of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "{1,…,k}",
"meta": {
"altText": "\\{1,\\dots,k\\}"
}
},
" into ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℓ",
"meta": {
"altText": "\\ell"
}
},
" parts with ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "σ={σ1,…,σℓ}",
"meta": {
"altText": "\\sigma=\\{\\sigma_{1},\\dots,\\sigma_{\\ell}\\}"
}
},
", ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "|σi|",
"meta": {
"altText": "\\lvert\\sigma_{i}\\rvert"
}
},
" being the cardinality of the set ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "σi",
"meta": {
"altText": "\\sigma_{i}"
}
},
" and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "Zσi=(zj)j∈σi",
"meta": {
"altText": "Z_{\\sigma_{i}}=(z_{j})_{j\\in\\sigma_{i}}"
}
},
".\nEach cumulant encodes finer and finer correlations.\nContrary to the correlation functions ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "(Fkε)",
"meta": {
"altText": "(F_{k}^{\\varepsilon})"
}
},
", the cumulants ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "(fkε)",
"meta": {
"altText": "(f_{k}^{\\varepsilon})"
}
},
" do not duplicate the information which is already encoded at lower orders.\nFrom a geometric point of view, we can extend the analysis of the previous section and show that the cumulant ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "fkε",
"meta": {
"altText": "f_{k}^{\\varepsilon}"
}
},
" of order ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k",
"meta": {
"altText": "k"
}
},
" can be represented by ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k",
"meta": {
"altText": "k"
}
},
" trees that are completely connected either by external collisions, or by overlaps (Figure ",
{
"type": "Cite",
"target": "S3-F8",
"content": [
"8"
]
},
").\nThese dynamical correlations can be classified by a signed graph with ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k",
"meta": {
"altText": "k"
}
},
" vertices representing the different trees, coding tree collisions (the corresponding edges take a + sign) and overlaps (the corresponding edges take a ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "−",
"meta": {
"altText": "-"
}
},
" sign).\nWe can then systematically extract a minimally connected graph ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "T",
"meta": {
"altText": "T"
}
},
" by identifying ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k−1",
"meta": {
"altText": "k-1"
}
},
" “aggregations” of tree collisions or overlaps.\nWe then expect ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "fkε",
"meta": {
"altText": "f_{k}^{\\varepsilon}"
}
},
" to decompose into a sum of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "2k−1kk−2",
"meta": {
"altText": "2^{k-1}k^{k-2}"
}
},
" terms, where the factor ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "kk−2",
"meta": {
"altText": "k^{k-2}"
}
},
" is the number of trees with ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k",
"meta": {
"altText": "k"
}
},
" numbered vertices (from Cayley’s formula).\nFor each given signed minimally connected graph, the collision/overlap conditions correspond to ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k−1",
"meta": {
"altText": "k-1"
}
},
" independent constraints on the configuration ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "z1⋆,…,zk⋆",
"meta": {
"altText": "z_{1^{\\star}},\\dots,z_{k^{\\star}}"
}
},
" at time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "t",
"meta": {
"altText": "t"
}
},
".\nTherefore, neglecting the issue of large velocities, this contribution to the cumulant ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "fkε",
"meta": {
"altText": "f_{k}^{\\varepsilon}"
}
},
" has a Lebesgue measure of size ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "O((t/με)k−1)",
"meta": {
"altText": "O((t/\\mu_{\\varepsilon})^{k-1})"
}
},
", and we derive the estimate"
]
},
{
"type": "MathBlock",
"id": "S3.E2",
"mathLanguage": "mathml",
"text": "∥fkε∥L1≤μεk−1Ck×2k−1kk−2×(t/με)k−1≤k!C(Ct)k−1.",
"meta": {
"altText": "\\lVert f_{k}^{\\varepsilon}\\rVert_{L^{1}}\\leq\\mu_{\\varepsilon}^{k-1}C^{k}\\times\n2^{k-1}k^{k-2}\\times(t/\\mu_{\\varepsilon})^{k-1}\\leq k!\\,C(Ct)^{k-1}."
}
},
{
"type": "Paragraph",
"content": [
"A geometric argument similar to the one developed in Lanford’s proof and recalled in the analysis of the second-order cumulant above shows that ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "fkε",
"meta": {
"altText": "f_{k}^{\\varepsilon}"
}
},
" converges to a limiting cumulant ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "fk",
"meta": {
"altText": "f_{k}"
}
},
" and that only graphs with exactly ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "k−1",
"meta": {
"altText": "k-1"
}
},
" external collisions or overlaps (and no cycles) contribute in the limit.\nNote further that a classical and rather simple calculation (based on the series expansions of the exponential and logarithm) shows that the cumulants are nothing but the coefficients of the series expansion of the exponential moment:"
]
},
{
"type": "MathBlock",
"id": "S3.E3",
"mathLanguage": "mathml",
"text": "ℐtε(h)≔1μεlog𝔼ε[exp(με⟨πtε,h⟩)]=∑k=1∞1k!∫fkε(t,Zk)∏i=1k(eh(zi)−1)dZk.",
"meta": {
"altText": "\\begin{split}\\mathcal{I}^{\\varepsilon}_{t}(h)&≔\\frac{1}{\\mu_{\\varepsilon}}\\log\\mathbb{E}_{\\varepsilon}[\\exp(\\mu_{\\varepsilon}\\langle\\pi^{\\varepsilon}_{t},h\\rangle)]\\\\\n&=\\sum_{k=1}^{\\infty}\\frac{1}{k!}\\int f_{k}^{\\varepsilon}(t,Z_{k})\\prod_{i=1}^{k}(e^{h(z_{i})}-1)\\,dZ_{k}.\\end{split}"
}
},
{
"type": "Paragraph",
"content": [
"The quantity ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℐtε(h)",
"meta": {
"altText": "\\mathcal{I}^{\\varepsilon}_{t}(h)"
}
},
" is called the ",
{
"type": "Emphasis",
"content": [
"cumulant generating function"
]
},
".\nEstimate (",
{
"type": "Cite",
"target": "S3-E2",
"content": [
"3.2"
]
},
") provides the analyticity of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℐtε(h)",
"meta": {
"altText": "\\mathcal{I}^{\\varepsilon}_{t}(h)"
}
},
" in short time as a function of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "eh",
"meta": {
"altText": "e^{h}"
}
},
", and this uniformly with respect to ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ε",
"meta": {
"altText": "\\varepsilon"
}
},
" (sufficiently small).\nThe limit ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℐt",
"meta": {
"altText": "\\mathcal{I}_{t}"
}
},
" of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℐtε",
"meta": {
"altText": "\\mathcal{I}^{\\varepsilon}_{t}"
}
},
" can then be determined as a series in terms of the limiting cumulants ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "fk",
"meta": {
"altText": "f_{k}"
}
},
","
]
},
{
"type": "MathBlock",
"id": "S3.Ex2",
"mathLanguage": "mathml",
"text": "ℐt(h)=∑k=1∞1k!∫fk(t,Zk)∏i=1k(eh(zi)−1)dZk.",
"meta": {
"altText": "\\mathcal{I}_{t}(h)=\\sum_{k=1}^{\\infty}\\frac{1}{k!}\\int f_{k}(t,Z_{k})\\prod_{i=1}^{k}(e^{h(z_{i})}-1)\\,dZ_{k}."
}
},
{
"type": "Paragraph",
"content": [
"In a suitable functional setting [",
{
"type": "Cite",
"target": "bib-bib5",
"content": [
"5"
]
},
"], it can be shown that this functional satisfies a Hamilton–Jacobi equation"
]
},
{
"type": "MathBlock",
"id": "S3.Ex3",
"mathLanguage": "mathml",
"text": "∂tℐt(h)=∫𝑑z∂ℐt(h)∂hv⋅∇xh+ℋ(∂ℐt(h)∂h,h)",
"meta": {
"altText": "\\partial_{t}\\mathcal{I}_{t}(h)=\\int dz\\,\\frac{\\partial\\mathcal{I}_{t}(h)}{\\partial h}v\\cdot\\nabla_{x}h+\\mathcal{H}\\Bigl(\\frac{\\partial\\mathcal{I}_{t}(h)}{\\partial h},h\\Bigr)"
}
},
{
"type": "Paragraph",
"content": [
"with initial condition ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℐ(0,h)=∫𝑑zf0(eh−1)",
"meta": {
"altText": "\\mathcal{I}(0,h)=\\int dz\\,f^{0}(e^{h}-1)"
}
},
" and Hamiltonian ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℋ",
"meta": {
"altText": "\\mathcal{H}"
}
},
" given by"
]
},
{
"type": "MathBlock",
"id": "S3.E4",
"mathLanguage": "mathml",
"text": "ℋ(φ,h)≔12∫φ(z1)φ(z2)(eΔh−1)𝑑μ(z1,z2,ω),",
"meta": {
"altText": "\\mathcal{H}(\\varphi,h)≔\\frac{1}{2}\\int\\varphi(z_{1})\\varphi(z_{2})(e^{\\Delta h}-1)\\,d\\mu(z_{1},z_{2},\\omega),"
}
},
{
"type": "Paragraph",
"content": [
"where ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "Δh(z1,z2,ω)=h(z1′)+h(z2′)−h(z1)−h(z2)",
"meta": {
"altText": "\\Delta h(z_{1},z_{2},\\omega)=h(z^{\\prime}_{1})+h(z^{\\prime}_{2})-h(z_{1})-h(z_{2})"
}
},
".\nWe use here notation (",
{
"type": "Cite",
"target": "S2-E2",
"content": [
"2.2"
]
},
") for the pre-collisional velocities and the definition"
]
},
{
"type": "MathBlock",
"id": "S3.Ex4",
"mathLanguage": "mathml",
"text": "dμ(z1,z2,ω)≔δx1−x2((v1−v2)⋅ω)+dωdv1dv2dx1.",
"meta": {
"altText": "d\\mu(z_{1},z_{2},\\omega)≔\\delta_{x_{1}-x_{2}}\\bigl((v_{1}-v_{2})\\cdot\\omega\\bigr)_{+}\\,d\\omega\\,dv_{1}\\,dv_{2}\\,dx_{1}."
}
},
{
"type": "Paragraph",
"content": [
"The successive derivatives of this functional being precisely the limit cumulants ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "fk",
"meta": {
"altText": "f_{k}"
}
},
", the successive derivatives of the Hamilton–Jacobi equation provide the evolution equations of these cumulants: for example, differentiating this equation once produces the Boltzmann equation, differentiating it twice produces the equation of the covariance described in the next paragraph."
]
},
{
"type": "Heading",
"id": "S3.SS3",
"depth": 2,
"content": [
"3.3 Fluctuations"
]
},
{
"type": "Paragraph",
"id": "S3.SS3.p1",
"content": [
"The control of the cumulant generating function allows in particular to obtain the convergence of the fluctuation field defined in (",
{
"type": "Cite",
"target": "S1-E4",
"content": [
"1.4"
]
},
") and thus to analyze the dynamical fluctuations over a time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "T⋆",
"meta": {
"altText": "T^{\\star}"
}
},
" of the same order of magnitude as the convergence time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "TL",
"meta": {
"altText": "T_{L}"
}
},
" of Theorem ",
{
"type": "Cite",
"target": "S2-Thm124Thm1",
"content": [
"2.1"
]
},
"."
]
},
{
"type": "Claim",
"id": "S3.Thm124Thm2",
"claimType": "Theorem",
"label": "Theorem 3.2(Bodineau, Gallagher, Saint-Raymond, Simonella [5]).",
"title": [
{
"type": "Strong",
"content": [
"Theorem 3.2"
]
},
{
"type": "Strong",
"content": []
},
"(Bodineau, Gallagher, Saint-Raymond, Simonella [",
{
"type": "Cite",
"target": "bib-bib5",
"content": [
"5"
]
},
"])",
{
"type": "Strong",
"content": [
"."
]
}
],
"content": [
{
"type": "Paragraph",
"content": [
{
"type": "Emphasis",
"content": [
"The fluctuation field ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ζtε",
"meta": {
"altText": "\\zeta^{\\varepsilon}_{t}"
}
},
" converges, in the low density limit and on a time interval ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "[0,T⋆]",
"meta": {
"altText": "[0,T^{\\star}]"
}
},
", towards a process ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ζt",
"meta": {
"altText": "\\zeta_{t}"
}
},
", solution to the fluctuating Boltzmann equation"
]
}
]
},
{
"type": "MathBlock",
"id": "S3.E5",
"mathLanguage": "mathml",
"text": "{dζt=ℒtζtdt⏟linearized Boltzmann operator+dηt⏟Gaussian noise,ℒth=−v⋅∇xh⏟𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡+C(ft,h)+C(h,ft)⏟linearized collision operator,",
"meta": {
"altText": "\\left\\{\\begin{aligned} d\\zeta_{t}&=\\underbrace{\\mathcal{L}_{t}\\zeta_{t}\\,dt}_{\\text{linearized Boltzmann operator}}+\\underbrace{\\;d\\eta_{t}\\;}_{\\text{Gaussian noise}},\\\\\n\\mathcal{L}_{t}h&=\\underbrace{-v\\cdot\\nabla_{x}h}_{\\text{transport}}+\\underbrace{C(f_{t},h)+C(h,f_{t})}_{\\text{linearized collision operator}},\\end{aligned}\\right."
}
},
{
"type": "Paragraph",
"content": [
{
"type": "Emphasis",
"content": [
"where ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ft",
"meta": {
"altText": "f_{t}"
}
},
" is the solution at time ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "t",
"meta": {
"altText": "t"
}
},
" to the Boltzmann equation (",
{
"type": "Cite",
"target": "S2-E1",
"content": [
"2.1"
]
},
") with initial data ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "f0",
"meta": {
"altText": "f^{0}"
}
},
", and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "dηt",
"meta": {
"altText": "d\\eta_{t}"
}
},
" is a centered Gaussian noise delta-correlated in ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "t,x",
"meta": {
"altText": "t,x"
}
},
" with covariance"
]
}
]
},
{
"type": "MathBlock",
"id": "S3.Ex5",
"mathLanguage": "mathml",
"text": "Covt(h1,h2)=12∫dz1dz2dω((v2−v1)⋅ω)+δx2−x1f(t,z1)f(t,z2)Δh1Δh2(z1,z2,ω),",
"meta": {
"altText": "\\begin{split}\\operatorname{Cov}_{t}(h_{1},h_{2})&=\\smash[b]{\\frac{1}{2}}\\mathop{\\smash[b]{\\int}}dz_{1}\\,dz_{2}\\,d\\omega((v_{2}-v_{1})\\cdot\\omega)_{+}\\delta_{x_{2}-x_{1}}\\\\\n&\\hskip 40.00006ptf(t,z_{1})f(t,z_{2})\\Delta h_{1}\\Delta h_{2}(z_{1},z_{2},\\omega),\\end{split}"
}
},
{
"type": "Paragraph",
"content": [
{
"type": "Emphasis",
"content": [
"where ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "Δh(z1,z2,ω)=h(z1′)+h(z2′)−h(z1)−h(z2)",
"meta": {
"altText": "\\Delta h(z_{1},z_{2},\\omega)=h(z^{\\prime}_{1})+h(z^{\\prime}_{2})-h(z_{1})-h(z_{2})"
}
},
"."
]
}
]
}
]
},
{
"type": "Paragraph",
"id": "S3.SS3.p2",
"content": [
"The limiting process (",
{
"type": "Cite",
"target": "S3-E5",
"content": [
"3.5"
]
},
") was conjectured by Spohn in [",
{
"type": "Cite",
"target": "bib-bib25",
"content": [
"25"
]
},
"], and this reference also presents a large panorama on the theory of fluctuations in physics.\nIn the context of dynamics with random collisions, a similar result is shown by Rezakhanlou in [",
{
"type": "Cite",
"target": "bib-bib24",
"content": [
"24"
]
},
"].\nIn the deterministic setting, the noise obtained in the limit is a consequence of the asymptotically unstable structure of the microscopic dynamics (Figure ",
{
"type": "Cite",
"target": "S1-F2",
"content": [
"2"
]
},
") combined with the randomness of the initial data at small scales."
]
},
{
"type": "Heading",
"id": "S3.SS4",
"depth": 2,
"content": [
"3.4 Large deviations"
]
},
{
"type": "Paragraph",
"content": [
"The strength of the cumulant generating function becomes really apparent at the level of large deviations, i.e., for very improbable trajectories that are at a “distance” ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "O(1)",
"meta": {
"altText": "O(1)"
}
},
" from the averaged dynamics: roughly speaking, we can show that the probability of observing an empirical distribution close to the density ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "φ(t,x,v)",
"meta": {
"altText": "\\varphi(t,x,v)"
}
},
" during the time interval ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "[0,T]",
"meta": {
"altText": "[0,T]"
}
},
" decays exponentially fast with a rate quantified by a functional ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℱ[0,T]",
"meta": {
"altText": "\\mathcal{F}_{[0,T]}"
}
},
" which evaluates the cost of this deviation in the low density asymptotics"
]
},
{
"type": "MathBlock",
"id": "S3.E6",
"mathLanguage": "mathml",
"text": "ℙε(πtε≃φt,∀t≤T)∼exp(−μεℱ[0,T](φ)).",
"meta": {
"altText": "\\mathbb{P}_{\\varepsilon}(\\pi^{\\varepsilon}_{t}\\simeq\\varphi_{t},\\forall t\\leq T)\\sim\\exp\\bigl(-\\mu_{\\varepsilon}\\mathcal{F}_{[0,T]}(\\varphi)\\bigr)."
}
},
{
"type": "Paragraph",
"content": [
"The proximity between ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "πε",
"meta": {
"altText": "\\pi^{\\varepsilon}"
}
},
" and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "φ",
"meta": {
"altText": "\\varphi"
}
},
" is measured in the weak topology on the Skorokhod space of measure-valued functions.\nA precise formulation of (",
{
"type": "Cite",
"target": "S3-E6",
"content": [
"3.6"
]
},
") and a proof can be found in [",
{
"type": "Cite",
"target": "bib-bib6",
"content": [
"6"
]
},
"].\nThe result of [",
{
"type": "Cite",
"target": "bib-bib6",
"content": [
"6"
]
},
"] can be summarized as follows: for a class of functions ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "φ",
"meta": {
"altText": "\\varphi"
}
},
" in a neighborhood of the solution to the Boltzmann equation, there exists a time interval ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "[0,T⋆]",
"meta": {
"altText": "[0,T^{\\star}]"
}
},
" where the asymptotic (",
{
"type": "Cite",
"target": "S3-E6",
"content": [
"3.6"
]
},
") is characterized by a functional ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℱ[0,T⋆]",
"meta": {
"altText": "\\mathcal{F}_{[0,T^{\\star}]}"
}
},
" obtained by a certain Legendre transform of the Hamiltonian ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℋ",
"meta": {
"altText": "\\mathcal{H}"
}
},
" defined by (",
{
"type": "Cite",
"target": "S3-E4",
"content": [
"3.4"
]
},
").\nThis functional is identical to the one conjectured in [",
{
"type": "Cite",
"target": "bib-bib24",
"content": [
"24"
]
},
", ",
{
"type": "Cite",
"target": "bib-bib9",
"content": [
"9"
]
},
"], by analogy with stochastic collision models of Kac type [",
{
"type": "Cite",
"target": "bib-bib23",
"content": [
"23"
]
},
", ",
{
"type": "Cite",
"target": "bib-bib18",
"content": [
"18"
]
},
", ",
{
"type": "Cite",
"target": "bib-bib15",
"content": [
"15"
]
},
", ",
{
"type": "Cite",
"target": "bib-bib2",
"content": [
"2"
]
},
"].\nLet us also note that the limiting SPDE (",
{
"type": "Cite",
"target": "S3-E5",
"content": [
"3.5"
]
},
") could be predicted by the same analogy with Kac’s model for which collisions are modeled by a Markov process [",
{
"type": "Cite",
"target": "bib-bib19",
"content": [
"19"
]
},
", ",
{
"type": "Cite",
"target": "bib-bib20",
"content": [
"20"
]
},
"].\nThus, the statistical analysis of the fluctuations and large deviations of the empirical measure confirms the robustness of Boltzmann’s intuition (cf. Section ",
{
"type": "Cite",
"target": "S2-SS1",
"content": [
"2.1"
]
},
"): even on exponentially small scales, the behavior of the empirical measure of a hard sphere gas is identical to that of a model of particles with random collisions depending only on the local density.\nThis does not contradict the Hamiltonian structure of the microscopic dynamics.\nMemory effects persist, but they are encoded in ways that are “transverse” to the empirical measure (or at different spatial scales)."
]
},
{
"type": "Heading",
"id": "S4",
"depth": 1,
"content": [
"4 Conclusion"
]
},
{
"type": "Paragraph",
"content": [
"Over a short time, Lanford’s theorem states the convergence of the empirical measure of a hard sphere gas to the solution to the Boltzmann equation (Theorem ",
{
"type": "Cite",
"target": "S2-Thm124Thm1",
"content": [
"2.1"
]
},
").\nThis result is completed by the analysis of fluctuations (Theorem ",
{
"type": "Cite",
"target": "S3-Thm124Thm2",
"content": [
"3.2"
]
},
") and large deviations (Section ",
{
"type": "Cite",
"target": "S3-SS4",
"content": [
"3.4"
]
},
") of the empirical measure.\nThese stochastic corrections are proved on times of the same order of magnitude as Lanford’s theorem.\nThe strategy of the proof consists in tracking how the randomness of the initial measure is transported by the dynamics of hard spheres and how the instability of this dynamics transfers, in the low density asymptotics, the initial randomness into a dynamical white noise (space/time).\nThe convergence time is limited because the current proof gives only rough estimates of the dynamical correlations, obtained by considering that collisions only destroy the initial chaos by forming larger and larger aggregates of correlated particles.\nAn important step to progress in the mathematical understanding of these models would be to show that the disorder is not simply the result of the initial data, but that it can be regenerated by the mixing properties of the dynamics.\nA more favorable framework for controlling long time evolution is to consider an initial measure obtained as a perturbation of an equilibrium measure.\nThe stationarity of the equilibrium measure then becomes a key tool to control dynamical correlations.\nThe simplest case consists in perturbing only one particle, which shall be called the tagged particle, and to study its evolution over time.\nIn [",
{
"type": "Cite",
"target": "bib-bib3",
"content": [
"3"
]
},
"], it is established that this particle follows a Brownian motion for large times.\nAnother case where we know how to use the invariant measure is the study of the fluctuation field at equilibrium.\nIn a series of recent works [",
{
"type": "Cite",
"target": "bib-bib7",
"content": [
"7"
]
},
", ",
{
"type": "Cite",
"target": "bib-bib5",
"content": [
"5"
]
},
"],\nTheorem ",
{
"type": "Cite",
"target": "S3-Thm124Thm2",
"content": [
"3.2"
]
},
" has been generalized to arbitrarily large, and even slightly divergent, kinetic times.\nThis allows in particular to derive the fluctuating hydrodynamic Stokes–Fourier equations."
]
},
{
"type": "Paragraph",
"content": [
{
"type": "Emphasis",
"content": [
"Acknowledgements. "
]
},
"The ",
{
"type": "Emphasis",
"content": [
"EMS Magazine"
]
},
" thanks ",
{
"type": "Emphasis",
"content": [
"La Gazette des Mathématiciens"
]
},
" for authorization to republish this text, which is an English translation of the paper entitled “Sur la dynamique des gaz dilués” and published in [",
{
"type": "Emphasis",
"content": [
"La Gazette des Mathématiciens"
]
},
", Number G174, October 2022].\nThe main part of the text is extracted from an article published in the ICM 2022 proceedings.\nThe authors warmly thank Stéphane Baseilhac for his attentive proofreading and his numerous suggestions.\nThey are also grateful to J.-B. Bru and M. Gellrich Pedra for the English translation of the original paper."
]
},
{
"type": "Paragraph",
"id": "authorinfo",
"content": [
"\nThierry Bodineau is CNRS researcher working at Laboratoire A. Grothendieck, IHÉS.\nHis research focuses on the probabilistic study of interacting particle systems.\n",
{
"type": "Link",
"target": "mailto:thierry.bodineau@ihes.fr",
"content": [
"thierry.bodineau@ihes.fr"
]
},
"\nIsabelle Gallagher is professor in mathematics at Université Paris Cité and École Normale Supérieure de Paris.\nHer research focuses on the analysis of partial differential equations.\nShe is currently director of the Fondation Sciences Mathématiques de Paris.\n",
{
"type": "Link",
"target": "mailto:gallagher@math.ens.fr",
"content": [
"gallagher@math.ens.fr"
]
},
"\nLaure Saint-Raymond is professor at IHÉS.\nShe is working in the field of partial differential equations, at the interface between mathematics and fluid mechanics, with a special focus on multiscale problems.\n",
{
"type": "Link",
"target": "mailto:laure@ihes.fr",
"content": [
"laure@ihes.fr"
]
},
"\nSergio Simonella is professor at the Mathematics Institute of Sapienza University of Rome.\nHe is interested in problems of kinetic theory of gases at the boundary between PDEs and statistical mechanics.\n",
{
"type": "Link",
"target": "mailto:sergio.simonella@uniroma1.it",
"content": [
"sergio.simonella@uniroma1.it"
]
}
]
}
]
}