Statistical mechanics of random billiard systems Renato Feres Washington University, St. Louis U. Houston, Summer Course, 2014 1 / 31
Diffusion in (straight) channels Idealized diffusion experiment. Channel inner surface has micro-structure. L r exit flow pulse of gas time How does the micro-structure influence diffusivity? 2 / 31
Surface scattering operators (definition of P ) velocity space (upper-half space) random scattered velocity surface Scattering characteristics of gas-surface interaction encoded in operator P . ( Pf )( v ) = E [ f ( V ) | v ] where f is any test function on H n and E [ ·| v ] is conditional expectation given � 1 if V ∈ U initial velocity v . If f ( V ) = ∈ U then 0 if V / ( Pf )( v ) = probability that V lies in U given initial v . 3 / 31
Standard model I - Knudsen cosine law � n + 1 � 1 d µ ∞ ( V ) = C n , s cos θ d Vol s ( V ) C n , s = 2 Γ n − 1 2 s n π probability density of scattered directions � ( Pf )( v ) = H n f ( V ) d µ ∞ ( V ) independent of v 4 / 31
Standard model II - Maxwellian at temperature T � n + 1 � β M � − β M � 2 2 | V | 2 d µ β ( V ) = 2 π cos θ exp d Vol ( V ) 2 π where β = 1 /κ T . probability density of scattered directions � ( Pf )( v ) = H n f ( V ) d µ β ( V ) independent of v 5 / 31
Natural requirements on a general P We say that P is natural if µ β is a stationary probability distribution for the velocity Markov chain. ◮ ◮ The stationary process defined by P and µ β is time reversible. I.e., in the stationary regime, all V j have the surface Maxwellian distribution µ β and the process satisfies P ( dV 2 | V 1 ) d µ β ( V 1 ) = P ( dV 1 | V 2 ) d µ β ( V 2 ) Time reversibility a.k.a. reciprocity a.k.a. detailed balance. 6 / 31
Deriving P from microstructure: general idea scattering process molecule system molecule system before collision after collision wall system with fixed Gibbs state, ◮ Sample pre-collision condition of wall system from fixed Gibbs state ◮ Compute trajectory of deterministic Hamiltonian system ◮ Obtain post-collision state of molecule system. Theorem (Cook-F, Nonlinearity 2012) Resulting P is natural. The stationary distribution is given by Gibbs state of molecule system with same parameter β as the wall system. 7 / 31
Purely geometric microstructures V is billiard scattering of initial v with random initial point q over period cell. cell of periodic microstructure uniformly distributed random point Theorem (F-Yablonsky, CES 2004) The resulting scattering operator is natural with stationary distribution µ ∞ . ◮ The stationary probability distribution is Knudsen cosine law � n + 1 � C n = Γ 1 2 d µ ∞ ( V ) = C n cos θ dV sphere ( V ) , n − 1 | V | n π 2 ◮ No energy exchange: | v | = | V | 8 / 31
Microstructures with moving parts random initial velocity of wall random entrance point range of free motion of wall random initial height Assumptions: Hidden variables are initialized prior to each scattering event so that: ◮ ◮ random displacements are uniformly distributed in their range; ◮ initial hidden velocities are Gaussian satisfying energy equipartition. ◮ Statistical state of wall is kept constant . Theorem (Thermal equilibrium. Cook-F, Nonlinearity 2012) Resulting operator P is natural . The stationary probability distribution is µ β , where β = 1 /κ T and T is the mean kinetic energy of each moving part. 9 / 31
Cylindrical channels Define for the random flight of a particle starting in the middle of cylinder: ◮ s rms root-mean square velocity of gas molecules ◮ τ = τ ( L , r , s rms ) expected exit time of random flight in channel 10 / 31
CLT and Diffusion in channels (anomalous diffusion) Theorem (Chumley, F., Zhang, Transactions of AMS, 2014) Let P be quasi-compact (has spectral gap) natural operator. Then � L 2 1 if n − k ≥ 2 D k τ ( L , r , s rms ) ∼ L 2 1 if n − k = 1 D k ln ( L / r ) where D = C ( P ) rs rms . Values of C ( P ) are described next. Useful for comparison to obtain values of diffusion constant D for the i.i.d. velocity process before looking at specific micro-structures. We call these reference values D 0 . 11 / 31
Values of D 0 for reference (Trans. AMS, 2014) For any direction u in R k diffusivities for the i.i.d. processes are: 4 √ n − k when n − k ≥ 2 and ν = µ β ( n − k ) 2 − 1 rs β 2 π ( n + 1 ) Γ ( n 2 ) 2 n − k when n − k ≥ 2 and ν = µ ∞ ( n − k ) 2 − 1 rs √ π Γ ( n + 1 2 ) D 0 = √ 4 2 π ( n + 1 ) rs β when n − k = 1 and ν = µ β Γ ( n 2 ) 2 2 ) rs when n − k = 1 and ν = µ ∞ √ π Γ ( n + 1 where s β = ( n + 1 ) /β M and M is particle mass. We are, therefore, interested in η u ( P ) := D u P / D 0 (coefficient of diffusivity in direction u ) a signature of the surface’s scattering properties. ( u is a unit vector in R k .) 12 / 31
Numerical example (F-Yablonsky, CES 2006) A = 0.7 A = 0.4 A = 0.1 R = 1 A = molecular radius Diffusivity increases with the radius of probing molecule. 13 / 31
Examples in 2-D (C. F. Z., Trans. AMS, 2014) Top: D is smaller then in i.i.d. (perfectly diffusive) case. Middle and bottom: D increases by adding flat top. 14 / 31
Examples in 2-D (C. F. Z., Trans. AMS, 2014) Diffusivity can be discontinuous on geometric parameters: Peculiar effects when 15 / 31
Diffusivity and the spectrum of P Consider the Hilbert space L 2 ( H n , µ β ) of square-integrable functions on velocity space with respect to the stationary measure µ β (0 < β ≤ ∞ ). Proposition (F-Zhang, Comm. Math. Physics, 2012) The natural operator P is a self-adjoint operator on L 2 ( H n , µ β ) with norm 1. In particular, it has real spectrum in the interval [ − 1 , 1 ] . In many special cases we have computed, P has discrete spectrum (eigenvalues) or at least a spectral gap. Z ( d λ ) := � Z u � − 2 � Z u , Π( d λ ) Z u � , Π the spectral measure of P . Then Let Π u � 1 1 + λ η u ( P ) = 1 − λ Π u ( d λ ) . − 1 Example: Maxwell-Smolukowski model: η = 1 + λ 1 − λ , λ = prob. of specular reflec. 16 / 31
Remarks about diffusivity and spectrum ◮ From random flight determined by P ⇒ Brownian motion limit via C.L.T. Random flight in channel Random walk in velocity space ◮ D determined by rate of decay of time correlations (Green-Kubo relation) ◮ All the information needed for D is contained in the spectrum of P ◮ It is difficult to obtain detailed information about the spectrum of P ; would like to find approximation more amenable to analysis. 17 / 31
Weak scattering and diffusion in velocity space Assume weakly scattering microstructure: P is close to specular reflection. In example below small h ⇒ small ratio m / M and small surface curvature. In such cases, the sequence V 0 , V 1 , V 2 , . . . of post-collision velocities can be approximated by a diffusion process in velocity space. If ρ ( v , t ) is the probability density of velocity distribution ∂ρ ∂ t = Div MB Grad MB ρ where MB stands for “Maxwell-Boltzmann.” 18 / 31
Weak scattering and diffusion in velocity space ◮ Λ square matrix of (first derivatives in perturbation parameter of) mass-ratios and curvatures. ◮ C is a covariance matrix of velocity distributions of wall-system. Definition (MB-grad, MB-div, MB-Laplacian ) ◮ On Φ ∈ C ∞ 0 ( H m ) ∩ L 2 ( H m , µ β ) (smooth, comp. supported) define √ � Λ 1 / 2 ( v m grad v Φ − Φ m ( v ) v ) + Tr ( C Λ) 1 / 2 Φ m e m � ( Grad MB Φ)( v ) := 2 where e m = ( 0 , . . . , 0 , 1 ) and Φ m is derivative in direction e m . ◮ On the pre-Hilbert space of smooth, compactly supported square-integrable vector fields on H m with inner product � ξ 1 , ξ 2 � := � H m ξ 1 · ξ 2 d µ β , define Div MB as the negative of the formal adjoint of Grad MB . ◮ Maxwell-Boltzmann Laplacian: L MB Φ = Div MB Grad MB Φ . 19 / 31
Weak scattering limit Theorem (F.-Ng-Zhang, Comm. Math. Phys. 2013) Let µ be a probability measure on R k with mean 0, covariant matrix C and finite 2nd and 3rd moments. Let P h be the collision operator of a family of microstructures parametrized by flatness parameter h . Then ◮ L MB is second order, essent. self-adjoint, elliptic on C 0 ( H m ) ∩ L 2 ( H m , µ β ) . P h Φ − Φ ◮ The limit L MB Φ = lim h → 0 0 ( H m ) . holds uniformly for each Φ ∈ C ∞ h ◮ The Markov chain defined by ( P h , µ β ) converges to an Itô diffusion with diffusion PDE ∂ρ ∂ t = L MB ρ 20 / 31
Example: 1-D billiard thermostat Define γ = m 2 / m 1 . P γ is operator on L 2 (( 0 , ∞ ) , µ ) . Theorem (Speed of convergence to thermal equilibrium) The following assertions hold for γ < 1 / 3: 1. P γ is a Hilbert-Schmidt; µ is the unique stationary distribution. Its density relative to Lebesgue measure on ( 0 , ∞ ) is − v 2 � � ρ ( v ) = σ − 1 v exp . 2 σ 2 2. For arbitrary initial µ 0 and small γ 1 − 4 γ 2 � n → 0 . � µ 0 P n � γ − µ � TV ≤ C 21 / 31
Approach to thermal equilibrium initial velocity distribution 1 limit Maxwellian probability density 0.5 0 1 2 3 4 5 speed 22 / 31
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