##
**Diffusivity in multiple scattering systems.**
*(English)*
Zbl 1332.60134

The paper under review aims to motivate the main question regarding partical-surface systems and to discuss the relation between the surface scattering properties encoded in the transition probability operator \(P\) and the constant diffusivity of a Brownian motion from an appropriate limit of the random flight in the channel.

An ideal experiment is to inject a small amount of gas composed of point-like non-interacting masses into a 2-dimensional channel and to record the amount of outflowing gas per unit time. Possible gas transport characteristics from this experiment are the mean value and higher moments of the molecular time of escape. The main question is: what can these time characteristics of the gas outflow tell about the microscopic interaction (scattering properties) between gas molecules and the inner surface of the channel walls? Let \(\tau = \tau(L, r, s)\) be the time of escape (expected exit time) of the flight, where \(s\) is the molecular root-mean square velocity, \(L\) is the position relative to the pulse of the gas, and the random flight is in the region \(\mathbb{R}^2\times [-r, r]\) between two parallel plates. By the central limit theorem for reversible Markov chains, the asymptotic expression of \(\tau(L, r, s)\) is given by \[ \tau(L, r, s) \sim \frac{L^2}{D(r, s) \ln (L/r)}, \] where the diffusivity \(D(r, s)\) of a limit Brownian motion is \[ D(r, s) = \frac{4rs}{\pi}\eta, \] by a simple dimension argument, where \(\eta\) depends only on the scattering characteristics at the microscopic scale determined by the transition probability operator \(P\). The main problem amounts to finding the functional dependence of \(\eta\) on geometric parameters of the surface microstructure. These parameters are scale invariant and are typically length ratios and angles. Definition 1 in Section 1 states the natural collision operator \(P\) with respect to the unique stationary distribution \(\mu_{\beta}\). The process defined by \(P\) does not change the particle speed and \(\mu\) is a stationary probability measure for \(P\) for all \(s\). Various natural collision operators are discussed in Section 1 and for a number of cases \(P\) is quasi-compact (the spectral radius of \(P\) restricted to the orthogonal complement of the constant function is strictly less than 1). Section 2 first discusses between-collision displacements, times, diffusivity, spectrum and mean exit time. Under an additional assumption in order to apply a result of [C. Kipnis and S. R. S. Varadhan, Commun. Math. Phys. 104, 1–19 (1986; Zbl 0588.60058)], Theorem 1 provides conditions under which a piecewise linear path \(X_{a, t}\) in the channel region with \(a\)-scaled random flight converges to the Brownian motion for large \(a\). The asymptotic expression of \(\tau(L, r, s)\) follows in Proposition 3. For natural collision operators, a more dispersing one has slower diffusion (Proposition 4). Interesting examples are illustrated in Section 2.4 on the diffusivity. Section 3 gives the main central limit theorem for a quasi-compact natural collision operator \(P\) in Theorem 2, and the weak invariance principle for \(X_{a, t}\) (converging weakly to \(B_t\)) in Theorem 3. The rest of this section is devoted to the proofs of Theorem 2 by using Bernstein’s big-small block technique and Theorem 3 by using the techniques of truncation and Bernstein’s method. Section 4 gives examples to show how the diffusivity encodes surface microscopic structure when the natural collision operator \(P\) represents a random reflection. It outlines a computation of the diffusivity \(D(r, s)\) independent of any given surface microscopic structure, and examples in the case that the surface of the walls are given by a periodic arrangement of focusing semicircles and in the case that certain parametric families of the surfaces are derived from the semicircle. Section 5 collects more technical propositions and lemmas with proofs. The paper is well-written and contains more techniques and ideas for random flights of point particles inside \(n\)-dimensional channels.

An ideal experiment is to inject a small amount of gas composed of point-like non-interacting masses into a 2-dimensional channel and to record the amount of outflowing gas per unit time. Possible gas transport characteristics from this experiment are the mean value and higher moments of the molecular time of escape. The main question is: what can these time characteristics of the gas outflow tell about the microscopic interaction (scattering properties) between gas molecules and the inner surface of the channel walls? Let \(\tau = \tau(L, r, s)\) be the time of escape (expected exit time) of the flight, where \(s\) is the molecular root-mean square velocity, \(L\) is the position relative to the pulse of the gas, and the random flight is in the region \(\mathbb{R}^2\times [-r, r]\) between two parallel plates. By the central limit theorem for reversible Markov chains, the asymptotic expression of \(\tau(L, r, s)\) is given by \[ \tau(L, r, s) \sim \frac{L^2}{D(r, s) \ln (L/r)}, \] where the diffusivity \(D(r, s)\) of a limit Brownian motion is \[ D(r, s) = \frac{4rs}{\pi}\eta, \] by a simple dimension argument, where \(\eta\) depends only on the scattering characteristics at the microscopic scale determined by the transition probability operator \(P\). The main problem amounts to finding the functional dependence of \(\eta\) on geometric parameters of the surface microstructure. These parameters are scale invariant and are typically length ratios and angles. Definition 1 in Section 1 states the natural collision operator \(P\) with respect to the unique stationary distribution \(\mu_{\beta}\). The process defined by \(P\) does not change the particle speed and \(\mu\) is a stationary probability measure for \(P\) for all \(s\). Various natural collision operators are discussed in Section 1 and for a number of cases \(P\) is quasi-compact (the spectral radius of \(P\) restricted to the orthogonal complement of the constant function is strictly less than 1). Section 2 first discusses between-collision displacements, times, diffusivity, spectrum and mean exit time. Under an additional assumption in order to apply a result of [C. Kipnis and S. R. S. Varadhan, Commun. Math. Phys. 104, 1–19 (1986; Zbl 0588.60058)], Theorem 1 provides conditions under which a piecewise linear path \(X_{a, t}\) in the channel region with \(a\)-scaled random flight converges to the Brownian motion for large \(a\). The asymptotic expression of \(\tau(L, r, s)\) follows in Proposition 3. For natural collision operators, a more dispersing one has slower diffusion (Proposition 4). Interesting examples are illustrated in Section 2.4 on the diffusivity. Section 3 gives the main central limit theorem for a quasi-compact natural collision operator \(P\) in Theorem 2, and the weak invariance principle for \(X_{a, t}\) (converging weakly to \(B_t\)) in Theorem 3. The rest of this section is devoted to the proofs of Theorem 2 by using Bernstein’s big-small block technique and Theorem 3 by using the techniques of truncation and Bernstein’s method. Section 4 gives examples to show how the diffusivity encodes surface microscopic structure when the natural collision operator \(P\) represents a random reflection. It outlines a computation of the diffusivity \(D(r, s)\) independent of any given surface microscopic structure, and examples in the case that the surface of the walls are given by a periodic arrangement of focusing semicircles and in the case that certain parametric families of the surfaces are derived from the semicircle. Section 5 collects more technical propositions and lemmas with proofs. The paper is well-written and contains more techniques and ideas for random flights of point particles inside \(n\)-dimensional channels.

Reviewer: Weiping Li (Stillwater)

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60F05 | Central limit and other weak theorems |

60F17 | Functional limit theorems; invariance principles |

60J65 | Brownian motion |

60J60 | Diffusion processes |

82B40 | Kinetic theory of gases in equilibrium statistical mechanics |

### Keywords:

particle-surface system; multiple scattering system; natural collision operator; expected exit time; random flight; diffusivity; central limit theorem; reversible Markov chain; weak invariance principle; surface microstructure; quasi-compactness; generalized Maxwell-Smoluchowski model### Citations:

Zbl 0588.60058
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\textit{T. Chumley} et al., Trans. Am. Math. Soc. 368, No. 1, 109--148 (2016; Zbl 1332.60134)

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