eport no. OxPDE-/8 Kramers formula for chemical reactions in the context of Wasserstein gradient flows by Michael Herrmann Mathematical Institute, University of Oxford & Barbara Niethammer Mathematical Institute, University of Oxford Oxford Centre for Nonlinear PDE Mathematical Institute, University of Oxford adcliffe Observatory Quarter, Gibson Building Annexe Woodstock oad Oxford, England OX2 6HA Email: herrmann@maths.ox.ac.uk August 2
Kramers formula for chemical reactions in the context of Wasserstein gradient flows Michael Herrmann and Barbara Niethammer August 2, 2 Abstract We derive Kramers formula as singular limit of the Fokker-Planck equation with double-well potential. The convergence proof is based on the ayleigh principle of the underlying Wasserstein gradient structure and complements a recent result by Peletier, Savaré and Veneroni. Keywords: Kramers formula, Fokker-Planck equation, Wasserstein gradient flow, ayleigh principle MSC 2: 35Q84, 49S5, 8A3 Introduction In 94 Kramers derived chemical reaction rates from certain limits in a Fokker-Planck equation that describes the probability density of a Brownian particle in an energy landscape [3]. The limit of high activation energy has been revisited in a recent paper by Peletier et al [6], where a spatially inhomogeneous extension of Kramers formula is rigorously derived for unimolecular reactions between two chemical states A and B. Their derivation relies on passing to the limit in the L 2 -gradient flow structure of the Fokker-Planck equation. It is well-known by now that the Fokker- Planck equation has also an interpretation as a Wasserstein gradient flow [2] and the question was raised in [6] whether Kramers formula can also be derived and interpreted within this Wasserstein gradient flow structure. This concept has also been investigated on a formal level for more complicated reaction-diffusion systems in [4]. A further motivation for studying the Fokker-Planck equation within the Wasserstein framework comes from applications that additionally prescribe the time evolution of a moment []. In this note we present a rigorous derivation which is based on passing to the limit within the Wasserstein gradient flow structure. To keep things simple we restrict ourselves to the spatially homogeneous case and consider the simplest case of a unimolecular reaction between two chemical states A and B, which are represented as two wells of an enthalpy function H. To avoid unimportant technicalities we assume that the enthalpy function H is a typical double-well potential, meaning smooth, nonnegative, even, and such that = H± Hx H = H±2 = for x 2, xh x c > for x > 2 with H < < H ± and H ±2. Oxford Centre for Nonlinear PDE OxPDE, michael.herrmann@maths.ox.ac.uk, niethammer@maths.ox.ac.uk
The probability density of a molecule with chemical state x is in the following denoted by ρ. In Kramers approach the molecule performs Brownian motion in an energy landscape given by H. Then the evolution of ρ is governed by the Kramers-Smoluchowski equation t ρ = x x ρ + H x ρ, where is the so-called viscosity coefficient. We consider the high activation energy limit 2. The leading order dynamics of can be derived by formal asymptotics and governs the evolution of a + t = Using WKB methods, for instance, one finds da ± dt 2 ρt, xdx, a t = ρt, xdx. = τ k a a ±, τ = 2 e /2, k = π H H. 2 2 We emphasize that the constant k depends on the details of the function H near its two local minima and its local maximum, and that the time scale is exponentially slow in the height barrier H/ 2 = / 2 between the two wells. Our goal is to derive 2 rigorously by passing to the limit in. In order to derive a non-trivial limit we have to rescale time accordingly. Thus we consider in the following the probability distribution = t, x that is a solution of t = τ x x + H x 2, x, t >. 3 An important role in the analysis will be played by the unique invariant measure γ x = Z 8π e Hx/2, Z = e Hy/2 dy = H + o, which converges in the weak topology of measures to 2 δ + + δ : φxγ xdx 2 φ + φ φ C. As in [6] it is often convenient to switch to the density of with respect to γ, that is u = /γ. Heuristically we expect that u is to leading order piecewise constant for x < and x >, where the respective values correspond to 2a + and 2a as introduced above. In the following we denote the value for x > by u = ut. The normalization of and γ then implies that the value for x < is given by 2 u. For the derivation of the limit equation we assume that our data are well-prepared. Theorem. Let τ and k be as in 2, and assume that the initial data ρ M satisfy γ u 2 dx + γ x u τ 2 dx C, u x = ρ x/γ x c > x. Then for all t we have that t, 2 utδ + + ut δ weakly in the sense of measures, and ut satisfies u = ku. 4 2
This result has already been derived in [6] in the more general setting with spatial diffusion and under slightly weaker assumptions on the initial data. Our main contribution here is therefore not the result as such, but the method of proof. We answer the question posed in [6], how the passage to the limit can be performed within a Wasserstein gradient flow structure and we identify the corresponding structure for the limit. We present the formal gradient flow structures of 3 as well as 4 in section 2. In order to derive the limit equation we pass to the limit in the ayleigh principle that is associated to any gradient flow. This strategy is inspired by the notion of Γ-convergence and has already been successfully employed in other singular limits of gradient flows e.g. in [5]. In section 3 we first obtain some basic a priori estimates as well as an approximation for u that will be essential in the identification of the limit gradient flow structure. It is somewhat unsatisfactory that we cannot derive suitable estimates solely from the energy estimates associated to the Wasserstein gradient flow. Instead we use the estimates that correspond to the L 2 -gradient flow structure that is satisfied by u. It is not obvious to us how this can be avoided. Section 4 finally contains our main result, that is the novel proof of Theorem. We show that the limit of t satisfies the ayleigh principle that one obtains as a limit of the ayleigh principle associated to the Wasserstein gradient flow structure of 3. As a consequence the limit is a solution of 4. 2 Gradient flows and ayleigh principle Given an energy functional E defined on a manifold M, whose tangent space T x M is endowed with a metric tensor g x, the gradient flow xt is defined such that g xt ẋt, v + DExtv = 5 for all v T xt M and for all t >. Here DExtv denotes the directional derivative of E in direction v. Our convergence result relies on the ayleigh principle, that simply amounts to the observation that a curve xt on M solves 5 if and only if ẋt minimizes 2 g xt v, v + DExtv 6 among all v T xt M. Since we lack sufficient regularity in time, we use 6 in the following time integrated version: For each T > the function ẋ minimizes T βt 2 g xt vt, vt + DExtvt dt for all nonnegative β C [, T] among all vt such that vt T xt M for all t [, T]. Gradient flow structure of the problem To describe the Wasserstein gradient structure of 3 we consider the formal manifold { } { } M := : [,, dx =, T ρ M = v : with metric tensor g v, v = τ j 2 dx, where v = x j. 3
The energy is given by E = ln + Hx 2 dx, and has the directional derivative DE v = v lnρ + Hx dx = 2 j x lnρ + Hx 2. The direction of steepest descent v is hence characterized by the requirement that This means g v, ṽ = DE ṽ ṽ T ρ M. j = x ln + Hx τ 2 and implies that 3 is in fact the g -gradient flow of E on M. Gradient flow structure of the limit problem The limit structure is extremely simple. The corresponding manifold { } { } M = u, 2, T u M = v is one-dimensional and the metric tensor reduces to g u v, v = v 2 lnu/2 u 2ku with k > as in 2. The limit energy is Eu = 2 u lnu + 2 uln2 u and satisfies DEuv = 2 v ln u/2 u. It is now easily checked that 4 is the g-gradient flow of E on M. 3 A priori estimates and weak compactness The density of with respect to γ, that is u = /γ, satisfies the equation and hence we readily justify that γ u 2 dx + 2 t τ 2τ γ x u 2 dx + τ γ t u = x γ x u, 7 t γ x u 2 dx ds = t 2 dx ds = γ 2τ γ u 2 dx, 8 γ x u 2 dx. 9 These a priori estimates are direct consequences of the H -gradient and the L 2 -gradient flow structures of 7, see [6] for details. The Wasserstein structure, however, corresponds to ln + Hx t 2 dx + τ x ln + Hx 2 2 dx ds = ρ lnρ + Hx 2 dx. 4
and dx =. As mentioned before, our analysis employs 8 and 9 but does not really make use of. An important implication of the above estimates is compactness of. In fact, the properties of γ combined with 9 imply that for any φ C we have φ t, x t 2, x t2 t2 t 2 /2 /2 ds = φ t dx dt dx φγ dx dt γ t C sup φx t 2 t 2. x Therefore, by the Arzelà-Ascoli theorem and the estimates and 8 there exist a function u :, and a subsequence again denoted by such that φ t, dx 2 utφ + 2 ut φ 2 holds locally uniformly in [, for any φ C. Furthermore, 9 implies t t t dx ds t t 2 /2 t dx ds γ γ dx ds C, so t is a finite measure on +. Thus, for possibly another subsequence, we have t u 2 δ + 2 δ in the sense that T βt φx t dx dt βt u 2 φ 2 φ dt 3 for all φ C and β C [, T]. By the iesz theorem we also deduce that t u L 2 [, T], and 8 and 9 ensure that the convergence in both 2 and 3 can be extended to any φ C with lim x φxγ x =. As already pointed out in [6], for δ > with 2 = oδ we obtain sup t [,T] x u 2 dx C 2 e hδ 2 ω δ, where ω δ = 2 + δ, δ δ, 2 δ and h δ = sup x ωδ Hx <. Consequently, and by embedding, u { u 2 u } {, 2 locally uniformly on [, T] 2, The assumption from Theorem combined with the maximum principle implies that }. 4 inf u t, x c >. 5 t,x We therefore have minu, 2 u >, where u describes the limit of both ρ and u. Without loss of generality we assume u, 2 and remark that the limit equation gives ut, 2 for all t >. In particular, we can assume in the following that both lnu/2 u and u are always positive. 5
Lemma 2. For δ > we have where T sup x +δ, δ u t, x C x γ C = 2ut / satisfies sup t> C /τ 2 kut. Proof. Integrating 7 twice, we find u = C x γ dy + C 2 + τ x dy 2 dt C e for some constants C, C 2, and a direct computation reveals γ ydy = Z = Z e /2 Moreover, we estimate y δ t dz t 2 γ dx y γ y t dz dy, 2 H δ 2 / 2, e Hy/2 dy = Z e /2 e H y 2 /2 2 dy + o 2π 4π 2 e /2 4 + o = + o = + o. H H H τ k γ /2 δ dz γ dz /2 e H δ/22 C t 2 γ /2 dz. The constants C, C 2 are determined by 4, and combining the previous estimates with 9 we find C2 = + o and that C has the desired properties. Lemma 3. Let δ = α for some α,. Under the assumptions of Theorem we find T δ φt, x τ dx 2 ln u/2 u φt, 2 dt ku +δ for any φ L 2 [, T], C L. Proof. With = + δ, δ and ũ = + C φ τ dx = φ τ γ ũ dx x γ φ and 4 combined with Lemma 2 provides that φ τ dx = τ τ γ ũ C x lnũ φdx = C 2 ln u/2 u k φt, u dy we write τ γ u ũ u ũ dx, lnũ x φdx + τ C [ lnũ φ ] δ +δ holds locally uniformly in time. Moreover, thanks to 5 we estimate T τ 2 T u ũ dx γ u ũ dt C sup φt, x 2 sup u t, x ũ t, x 2 dt x x φ and Lemma 2 completes the proof. φ L 2 L u ũ L 2 L, 6
4 Passing to the limit in the ayleigh principle We are now able to prove our main result, which then implies Theorem. Theorem 4. The limit of t that is characterized by u satisfies T βt 2 g T u u, u + DEu u dt βt 2 g u v, v + DEuv dt for all nonnegative β C [, T] and for all functions v L 2 [, T]. Theorem 4 is implied by the following three Lemmas. These ensure that the limit u of the minimizers t of the ayleigh principle associated to the -problems is indeed a minimizer of the ayleigh principle associated to the limit problem. Notice that the assertions of Lemmas 5-7 are closely related, but not equivalent to the Γ-convergence of the ayleigh principles. In fact, in Lemma 5 we prove lower semi-continuity of the metric tensor only for the minimizers t. However, this is sufficient to conclude that the limit u is again a minimizer of the limit problem. Lemma 5 lim-inf estimate for metric tensor. For all nonnegative β C [, T] we have lim inf T βt T 2 g t, t dt βt 2 g u u, u dt. Lemma 6 lim-inf estimate for energy. For all nonnegative β C [, T] we have T βt DE t dt βt DEu u dt. Lemma 7 Existence of recovery sequence. For all v L 2 [, T] there exists a sequence v such that v t T ρ M for all t [, T] such that T lim sup βt 2 g v, v + DE v dt for all nonnegative β C [, T]. βt 2 g u v, v + DEuv dt Proof of Lemma 5. For the following considerations we introduce the functions h t, x = x +δ τ dy, f t, x = βt x t dy, and for small δ > we define = + δ, δ as well as τ µ t = h t, δ, ϕ t = f t, δ, b t = f dx. By Lemma 3 we then find µ µ = 2 ln u/2 u ku strongly in L 2 [, T], whereas 3 yields ϕ t 2 βt u weakly in L 2 [, T]. 7
Moreover, we have sup x Iδ γ and thus T T t x f h dx dt C Iδ 2 γ Since b satisfies we further conclude that /2 T /2 dx dt γ h 2 dx dt. b = f x h dx = x f h dx + ϕ µ b b = lnu/2 u β u ku We now estimate T βtg T t, t dt = T f 2 τ dx dt b 2 t µ t dt, weakly in L [, T], T τ dx τ 2 f dx dt and apply Egorov s Theorem as follows. Since /µ /µ almost everywhere on, T, for each η > there exists E η, T with, T\E η η such that /µ /µ uniformly on E η. In particular, b is bounded in L 2 E η for sufficiently small, and this implies b b in L 2 E η due to the uniqueness of weak limits. Hence b lim inf Eη 2 b dt lim inf µ Eη 2 dt lim sup b 2 b dt µ µ µ Eη 2 dt µ and, consequently, lim inf T b 2 b 2 dt dt = µ E η µ 2 βt u 2 lnu/2 u dt. E η ku Finally, the desired result follows from taking the limit η and using the Monotone Convergence Theorem. Proof of Lemma 6. ecall that DE t = The convergences in 3 and 4 imply that T βt t lnu dx dt ω δ 2 E η t ln + Hx 2 dx = t lnu dx. T βt u lnu/2 udt with ω δ = 2+δ, δ δ, 2 δ. Furthermore, since u c > and thus ln u C u /2 +, we find T δ T δ t 2 /2 T δ βt t lnu dx dt C dx dt β 2 t γ u + /2 dx dt δ δ γ δ δ C sup u 2 + /4 γ dx δ /4 γ dx t T δ δ uniformly in due to the properties of γ and the estimate 8. Finally, the contributions on, 2 + δ and on 2 δ, can be estimated by a similar argument. 8 δ
Proof of Lemma 7. By approximation it is sufficient to prove the assertion for v C [, T]. Given such a vt we choose δ = α for some α, and define v = 2 vt ψ δ x ψ δ x+, where ψ δ is a standard Dirac sequence with support in δ, δ. Then j = x v t, ydy is smooth, zero for both x δ and x + δ, and equal to 2vt on + δ, δ. We now split T βtτ j 2 dy dt = 4 T δ βtv 2 τ T j 2 t dy dt + βtτ dy dt, +δ ω δ where ω δ = δ, + δ δ, + δ. Due to 5 and the properties of γ we find T βtτ j 2 ω δ dy dt Cδτ T βtv 2 tdt, 4 inf x ωδ γ x so Lemma 3 yields T j 2 T βtτ dy dt βtg u v, vdt. Finally, using the properties of v and the convergence properties of u we show T exactly as in the proof of Lemma 6. T βt v lnu dx dt 2 βtvtln ut/2 ut dt Acknowledgments The authors gratefully acknowledge discussions with Alexander Mielke. This work was supported by the EPSC Science and Innovation award to the Oxford Centre for Nonlinear PDE EP/E3527/. eferences [] W. Dreyer, C. Guhlke and M. Herrmann, Hysteresis and phase transition in many-particle storage systems, WIAS-Preprint No. 48 2 [2]. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29, -7 998 [3] H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7,4, 284-34 94 [4] A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion processes, Nonlinearity 2, to appear [5] B. Niethammer and F. Otto. Ostwald ipening: The screening length revisited. Calc. Var. and PDE, 3-, 33 68, 2. [6] M. Peletier, G. Savaré and M. Veneroni, From diffusion to reaction via Γ-convergence, SIAM J. Math. Anal. 2, to appear 9