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...ergent, uniformly in ε, only for a short time interval. To complete the proof it was enough to exploit the term by term convergence holding by virtue of geometric and measure–zero arguments (see also =-=[23, 28, 27, 9, 29]-=-). The original argument of Lanford was qualitative, in the sense that (1.3) was shown without an explicit rate of convergence. Recently, explicit estimates on the rate of convergence have been obtain...

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...ergent, uniformly in ε, only for a short time interval. To complete the proof it was enough to exploit the term by term convergence holding by virtue of geometric and measure–zero arguments (see also =-=[23, 28, 27, 9, 29]-=-). The original argument of Lanford was qualitative, in the sense that (1.3) was shown without an explicit rate of convergence. Recently, explicit estimates on the rate of convergence have been obtain...

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...ions and velocities. If pn = (1/n!) ∫ W ε0,n, then∑ n pn = 1 and the average number of particles is 〈N〉 = ∑ n n pn. We are interested in analyzing a low–density limit, namely the Boltzmann–Grad limit =-=[11, 12]-=-, defined by 〈N〉 → ∞, ε→ 0 and 〈N〉ε2 → λ−1 > 0 , (1.1) where λ is a fixed constant proportional to the mean free path. Since 〈N〉 and ε are related in the Boltzmann–Grad limit, let us use a single para...

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.... . . 68 References 69 1 Introduction In 1975 O. E. Lanford III presented his celebrated proof of the mathematical validity of the Boltzmann equation for hard spheres, in a time interval small enough =-=[17]-=-. To remind his result, let us consider a system of identical hard spheres of diameter ε moving in the whole space R3, with collisions governed by the usual laws of elastic reflection. A state of the ...

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...ions and velocities. If pn = (1/n!) ∫ W ε0,n, then∑ n pn = 1 and the average number of particles is 〈N〉 = ∑ n n pn. We are interested in analyzing a low–density limit, namely the Boltzmann–Grad limit =-=[11, 12]-=-, defined by 〈N〉 → ∞, ε→ 0 and 〈N〉ε2 → λ−1 > 0 , (1.1) where λ is a fixed constant proportional to the mean free path. Since 〈N〉 and ε are related in the Boltzmann–Grad limit, let us use a single para...

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...context will be discussed later on. 1 It may be worth to mention that the only validity result holding globally in time refers to the special situation of a rare cloud of gas expanding in the vacuum, =-=[13, 14]-=-. We remind here the Boltzmann equation for the unknown f = f(x, v, t), with hard sphere kernel and mean free path λ [4], (∂t+v·∇x)f(x, v, t) = λ−1 ∫ R3×S2+ dv1dω (v−v1)·ω { f(x, v′1, t)f(x, v ′, t)−f...

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...context will be discussed later on. 1 It may be worth to mention that the only validity result holding globally in time refers to the special situation of a rare cloud of gas expanding in the vacuum, =-=[13, 14]-=-. We remind here the Boltzmann equation for the unknown f = f(x, v, t), with hard sphere kernel and mean free path λ [4], (∂t+v·∇x)f(x, v, t) = λ−1 ∫ R3×S2+ dv1dω (v−v1)·ω { f(x, v′1, t)f(x, v ′, t)−f...

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... original argument of Lanford was qualitative, in the sense that (1.3) was shown without an explicit rate of convergence. Recently, explicit estimates on the rate of convergence have been obtained in =-=[10]-=- (see also [19] for a different class of potentials). Furthermore the explicit control of the error has been used to reach hydrodynamic regimes: see [3], where the heat equation is derived from the ha...

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... the rate of convergence have been obtained in [10] (see also [19] for a different class of potentials). Furthermore the explicit control of the error has been used to reach hydrodynamic regimes: see =-=[3]-=-, where the heat equation is derived from the hard sphere dynamics in a low–density regime, by studying one tagged particle in a gas close to equilibrium. In the present paper we give a different and ...

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...hat Equation (1.15) is a reformulation of Lanford’s result together with an explicit representation of the error. 6 The quantities EBJ (t), under the name “v–functions”, were previously introduced in =-=[5, 6, 7]-=- in dealing with kinetic limits of stochastic particle systems. We note finally that the result (1.15) provides a further information. Suppose to have a nice solution to the Boltzmann equation satisfy...

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...ergent, uniformly in ε, only for a short time interval. To complete the proof it was enough to exploit the term by term convergence holding by virtue of geometric and measure–zero arguments (see also =-=[23, 28, 27, 9, 29]-=-). The original argument of Lanford was qualitative, in the sense that (1.3) was shown without an explicit rate of convergence. Recently, explicit estimates on the rate of convergence have been obtain...

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...ent of Lanford was qualitative, in the sense that (1.3) was shown without an explicit rate of convergence. Recently, explicit estimates on the rate of convergence have been obtained in [10] (see also =-=[19]-=- for a different class of potentials). Furthermore the explicit control of the error has been used to reach hydrodynamic regimes: see [3], where the heat equation is derived from the hard sphere dynam...

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...(t)→ f(t)⊗j for t < t̄ (1.3) almost everywhere in (R3 ×R3)j, where f(t) is a solution of the Boltzmann equation with initial datum f0. Note that we found convenient to recall the theorem as stated in =-=[16]-=- (or also in [2, 25]), namely without fixing the total number of particles. The advantage of this formulation in our context will be discussed later on. 1 It may be worth to mention that the only vali...

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...ergent, uniformly in ε, only for a short time interval. To complete the proof it was enough to exploit the term by term convergence holding by virtue of geometric and measure–zero arguments (see also =-=[23, 28, 27, 9, 29]-=-). The original argument of Lanford was qualitative, in the sense that (1.3) was shown without an explicit rate of convergence. Recently, explicit estimates on the rate of convergence have been obtain...

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...ting the analysis of the low–density regime, let us describe the time evolution for any fixed ε > 0. The evolution equations for the considered quantities were first derived formally by Cercignani in =-=[8]-=-. Assuming some explicit bound and sufficient smoothness, he deduced the hard sphere version of the well known BBGKY hierarchy of equations, which for the rescaled correlation functions takes the form...

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...3 dωdvj+1B ε(ω; vj+1 − vk)f εj+1(zj, xk + ωε, vj+1, ·) . 11 Rigorous derivations of the hard sphere hierarchy, under rather weak assumptions on the initial measure, have been discussed later on, e.g. =-=[24, 15, 22]-=-. The latter references focus mainly on the validity of the series expansion (2.1.15). Let us formulate the result in a form useful for our analysis. We shall assume that there exist constants z, β > ...

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... < t̄ (1.3) almost everywhere in (R3 ×R3)j, where f(t) is a solution of the Boltzmann equation with initial datum f0. Note that we found convenient to recall the theorem as stated in [16] (or also in =-=[2, 25]-=-), namely without fixing the total number of particles. The advantage of this formulation in our context will be discussed later on. 1 It may be worth to mention that the only validity result holding ...

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...s to the special situation of a rare cloud of gas expanding in the vacuum, [13, 14]. We remind here the Boltzmann equation for the unknown f = f(x, v, t), with hard sphere kernel and mean free path λ =-=[4]-=-, (∂t+v·∇x)f(x, v, t) = λ−1 ∫ R3×S2+ dv1dω (v−v1)·ω { f(x, v′1, t)f(x, v ′, t)−f(x, v1, t)f(x, v, t) } (1.4) where S2+ = {ω ∈ S2| (v − v1) · ω ≥ 0}, S2 is the unit sphere in R3 (with surface measure d...

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...defines the flow of the n–particle dynamics, t 7→ Tεn(t)zn. Observe that these rules do not cover all possible situations, e.g. triple collisions are excluded. Nevertheless, as proved by Alexander in =-=[1]-=-, there exists a full–measure subset ofMn, over which Tεn(t) is uniquely defined for all t (see also [18, 9]). Thus T ε n(t) can be defined as a one–parameter group of Borel maps on Mn, leaving invari...

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...hat Equation (1.15) is a reformulation of Lanford’s result together with an explicit representation of the error. 6 The quantities EBJ (t), under the name “v–functions”, were previously introduced in =-=[5, 6, 7]-=- in dealing with kinetic limits of stochastic particle systems. We note finally that the result (1.15) provides a further information. Suppose to have a nice solution to the Boltzmann equation satisfy...

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...hat Equation (1.15) is a reformulation of Lanford’s result together with an explicit representation of the error. 6 The quantities EBJ (t), under the name “v–functions”, were previously introduced in =-=[5, 6, 7]-=- in dealing with kinetic limits of stochastic particle systems. We note finally that the result (1.15) provides a further information. Suppose to have a nice solution to the Boltzmann equation satisfy...

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...3 dωdvj+1B ε(ω; vj+1 − vk)f εj+1(zj, xk + ωε, vj+1, ·) . 11 Rigorous derivations of the hard sphere hierarchy, under rather weak assumptions on the initial measure, have been discussed later on, e.g. =-=[24, 15, 22]-=-. The latter references focus mainly on the validity of the series expansion (2.1.15). Let us formulate the result in a form useful for our analysis. We shall assume that there exist constants z, β > ...

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...3 dωdvj+1B ε(ω; vj+1 − vk)f εj+1(zj, xk + ωε, vj+1, ·) . 11 Rigorous derivations of the hard sphere hierarchy, under rather weak assumptions on the initial measure, have been discussed later on, e.g. =-=[24, 15, 22]-=-. The latter references focus mainly on the validity of the series expansion (2.1.15). Let us formulate the result in a form useful for our analysis. We shall assume that there exist constants z, β > ...

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...ergent, uniformly in ε, only for a short time interval. To complete the proof it was enough to exploit the term by term convergence holding by virtue of geometric and measure–zero arguments (see also =-=[23, 28, 27, 9, 29]-=-). The original argument of Lanford was qualitative, in the sense that (1.3) was shown without an explicit rate of convergence. Recently, explicit estimates on the rate of convergence have been obtain...

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...specifies the quantitative information given by the estimate of Ej(t), on how the statistical independence is achieved. For previous results on the fluctuation field in the Boltzmann– Grad limit, see =-=[2, 25, 26]-=-. Note that no quantity associated to the Boltzmann equation appears yet in the previous formulas. Indeed the functions Ej describe a part, but not all, of the total dynamical correlation between part...

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... < t̄ (1.3) almost everywhere in (R3 ×R3)j, where f(t) is a solution of the Boltzmann equation with initial datum f0. Note that we found convenient to recall the theorem as stated in [16] (or also in =-=[2, 25]-=-), namely without fixing the total number of particles. The advantage of this formulation in our context will be discussed later on. 1 It may be worth to mention that the only validity result holding ...