A note on Fisher information hypocoercive decay for the linear Boltzmann equation
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A note on Fisher information hypocoercive decay for the linear Boltzmann equation Pierre Monmarché1 Received: 18 June 2020 / Revised: 27 October 2020 / Accepted: 4 November 2020 © Springer Nature Switzerland AG 2020
Abstract This note deals with the linear Boltzmann equation in the non-compact setting with a confining potential which is close to quadratic. We prove that in this situation, starting from a smooth initial datum, the Fisher Information (and hence, the relative entropy) with respect to the stationary state converges exponentially fast to zero. Keywords Hypocoercivity · Linear Boltzmann equation · Fisher information · Randomized HMC Mathematics Subject Classification 35K99 · 60J25
1 Introduction We are interested in the long-time convergence to equilibrium of the solution f of the so-called linear Boltzmann (or BGK) equation ∂t f t (x, y) = −v · ∇x f t (x, y) + ∇U (x) · ∇ y f t (x, y) + λQ f t (x, y)
(1)
where (x, y) ∈ R2d , d ∈ N∗ , λ > 0 is constant, U ∈ C 2 Rd , R and Q is either Q 1 or Q 2 with
Data sharing not applicable to this article as no datasets were generated or analysed during the current study. Pierre Monmarché acknowledges support from the French ANR-17-CE40-0030 - EFI - Entropy, flows, inequalities.
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Pierre Monmarché [email protected] Sorbonne Université, LJLL and LCT, 4 place Jussieu, 75 005 Paris, France 0123456789().: V,-vol
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P. Monmarché
Q 1 f (x, y) = γd,σ (y) Q 2 f (x, y) =
Rd
f (x, v)dv − f (x, y)
d f (x, y1 , . . . , yk−1 , w, yk+1 , . . . , yd )dw − f (x, y) γ1,σ (yk ) R
k=1
with, for some σ > 0, −
1
|y|2
e 2σ 2 γ p,σ (y) = p/2 2π σ 2 a Gaussian measure on R p . We assume that f 0 is a probability density so that, mass and positivity being conserved through time, f t is a probability density for all t ≥ 0. Denoting H (x, y) = U (x) + |y|2 /2, we suppose that exp(−H /σ 2 ) is integrable and we denote by μ the probability law with density proportional to it (we also write μ this density). Then μ is a fixed point of (1). Our goal is to give a quantitative estimate for the convergence of a solution of (1) toward μ. In fact we will rather work with the relative density h t = f t /μ, which solves ∂t h t = Lh t
(2)
with Lh(x, y) = −v · ∇x h(x, y) + ∇U (x) · ∇ y h(x, y) + η (Ph − h) , where (P, η) is either (P1 , η1 ) or (P2 , η2 ) with η1 = λ, η2 = λd and P1 h(x, y) = P2 h(x, y) =
Rd
h(x, v)γd,σ (v)dv
d 1 h(x, y1 , . . . , yk−1 , w, yk+1 , . . . , yd )γ1,σ (w)dw . d R k=1
Remark that P1 and P2 are Markov operators. Equation (1) is a classical model in statistical physics, modelling the motion of a particle influenced by an external potential U and by random collisions with other particles with Gaussian velocities. We refer the interested reader to [12] and references within for details. Moreover, it intervenes in Markov Chain Monte Carlo methods. More precisely, denote L ∗ the dual of L in L 2 (μ). Integrating by parts, we see that L ∗ ϕ(x, y) = v · ∇x ϕ(x, y) − ∇U (x) · ∇ y ϕ(x, y) + η (Pϕ − ϕ) .
(
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