DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2020135 DYNAMICAL SYSTEMS SERIES B Volume 25, Number 10, October 2020 pp. 4001–4038

CLASSICAL LANGEVIN DYNAMICS DERIVED FROM QUANTUM MECHANICS

H˚akon Hoel Chair of Mathematics for Uncertainty Quantification RWTH Aachen University Germany Anders Szepessy∗ Institutionen f¨orMatematik Kungl. Tekniska H¨ogskolan 100 44 Stockholm, Sweden

Abstract. The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig’s model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction ma- trix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.

1. Langevin molecular dynamics. Langevin dynamics for (unit mass) particle N systems with position coordinates XL : [0, ∞) → R and momentum coordinates N PL : [0, ∞) → R , defined by

dXL(t) = PL(t)dt (1)
1/2 dPL(t) = −∇λ XL(t) dt − κP¯ L(t)dt + (2¯κT ) dW (t) , is used for instance to simulate molecular dynamics in the canonical ensemble of constant temperature T , volume and number of particles, where W denotes the standard Wiener process with N independent components. The purpose of this work is to precisely determine both the potential λ : RN → R and the friction matrixκ ¯ ∈ RN×N in this equation, from a quantum mechanical model of a molecular system including weak coupling to a heat bath. Molecular systems are described by the Schr¨odinger equation with a potential based on Coulomb interaction of all nuclei and electrons in the system. This quan- tum mechanical model is complete in the sense that no unknown parameters enter - the observables in the canonical ensemble are determined from the Hamiltonian and

2020 Mathematics Subject Classification. Primary: 82C31, 82C10, 60H10; Secondary: 65C30. Key words and phrases. Quantum mechanics, heat bath dynamics, generalized Langevin dy- namics, classical Langevin dynamics, weak convergence. The second author is supported by the Swedish Research Council grants 2014-04776 and 2019- 03725. ∗ Corresponding author: Anders Szepessy ([email protected]).

4001 4002 HAKON˚ HOEL AND ANDERS SZEPESSY the temperature. The classical limit of the quantum formulation, yields an accurate approximation of the observables based on the nuclei only, for large nuclei–electron mass ratio M. Ab initio molecular dynamics based on the electron ground state eigenvalue can be used when the temperature is low compared to the first elec- tron eigenvalue gap. A certain weighted average of different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for any temperature, see [10], also in the case of observables including time correlation and many particles. The elimination of the electrons provides a substantial com- putational reduction, making it possible to simulate large molecular systems, cf. [18]. In molecular dynamics simulations one often wants to determine properties of a large macroscopic system with many particles, say N ∼ 1023. Such large particle systems cannot yet be simulated on a computer and one may then ask for a setting where a smaller system has similar properties as the large. Therefore, we seek an equilibrium density that has the property that the marginal distribution for a subsystem has the same density as the whole system. In [10] it is motivated how this assumption leads to the Gibbs measure, i.e. the canonical ensemble; this is also the motivation to use the canonical ensemble for the composite system in this work, although some studies on heat bath models use the microcanonical ensemble for the composite whole system. Langevin dynamics is often introduced to sample initial configurations from the Gibbs distribution and to avoid to simulate the dynamics of all heat bath particles. The friction/damping parameter in the Langevin equation is then typically set small enough to not perturb the dynamics too much and large enough to avoid long sampling times. The purpose of this work is to show that Langevin molecular dynamics, for the non heat bath nuclei, with a certain friction/damping parameter determined from the Hamiltonian, approximates the quantum system, in the canonical ensemble for any temperature, more accurately than any Hamiltonian dynamics (for the non heat bath particles) in the case the system is weakly coupled to a heat bath of many fast particles. Our heat bath model is based on the assumption of weak coupling - in the sense that the perturbation in the system from the heat bath is small and vice versa - which we show leads to Zwanzig’s model for nonlinear generalized Langevin equations in [28], with a harmonic oscillator heat bath. Zwanzig also derives a pure Langevin equation: he assumes first a continuous Debye distribution of the eigenvalues of the heat bath potential energy quadratic form; in the next step he lets the coupling to the heat bath have a special form, so that the integral kernel for the friction term in the generalized Langevin equation becomes a Dirac-delta measure. Our derivation uses a heat bath based on nearest neighbour interaction on an infinite cubic lattice, so that the continuum distribution of eigenvalues is rigorously obtained by considering a difference operator with an infinite number of nodes. Our convergence towards a pure Langevin equation is not based on a Dirac- delta measure for the integral kernel but obtained from the time scale separation of the fast heat bath particles and the slower system particles, which allows a general coupling and a bounded covariance matrix for the fluctuations. The fast heat bath dynamics is provided either from light heat bath particles or a stiff heat bath potential. By a stiff heat bath we mean that the smallest eigenvalue of the Hessian of the heat bath potential energy is of the order χ−1, where χ  1. We show LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4003 that the friction/damping coefficient in the pure Langevin equation is determined by the derivative of the forces on the heat bath particles with respect to the system particle positions. We also prove that the observables of the system coordinates in the system-heat bath quantum model can be approximated using this Langevin dynamics with accuracy O(m log m−1 + (mM)−1), where M  1 is the system nuclei–electron mass ratio and m  1 is the heat bath nuclei – system nuclei mass ratio; the approximation by a Hamiltonian system yields the corresponding larger error estimate O(m1/2 + (mM)−1), in the case of light heat bath particles. In this sense, our Langevin equation is a better approximation. The case with a stiff heat bath system has the analogous error estimate O(χ2δ log χ−1 + M −1) with the correct friction/damping parameter, while a Hamiltonian system gives the larger error O(χ2δ−1/2 + M −1), where χδ  χ1/4 measures the coupling between the heat bath and the system. Our main assumptions are: • the coupling between the system and the heat bath is weak and localized, • either the heat bath particles are much lighter than the system particles or the heat bath is stiff, • the harmonic oscillator heat bath is constructed from nearest neighbour in- teraction on an infinite cubic lattice in dimension three, • the heat bath particles are initially randomly Gibbs distributed (conditioned by the system particle coordinates), and • the system potential energy and the observables are sufficiently regular. The system and the heat bath is modelled by a Hamiltonian where the system potential energy is perturbed by V¯ (x, X) with system particle positions X ∈ RN and heat bath particle positions x ∈ Rn. Our assumption of weak coupling is formulated as the requirement that the potential V¯ satisfies min V¯ (x, X) = V¯ a(X),X
= 0 . n x∈R Theorem 3.4 proves that the obtained minimizer a(X) ∈ Rn determines the friction matrixκ ¯ in (1) by the rank one N × N matrix 1/2 00 00 0 ¯ ¯ κ¯`` =cm ¯ hV (a, X)∂X` a, V (a, X)∂X`0 ai , (2) where V¯ and a have limits as n → ∞; here V¯ 00 denotes the Hessian of V¯ with respect to x, the brackets h·, ·i denote the scalar product in Rn andc ¯ is a constant (related to the density of states). In a setting when the temperature is small compared to the difference of the two smallest eigenvalues of the Hamiltonian symbol, with given nuclei coordinates, it is well known that ab initio molecular dynamics is based on the ground state electron eigenvalue as the potential λ in (1), cf. [18]. When the temperature is larger, excited electron states influence the nuclei dynamics. The work [10] derives a molecular dynamics approximation of quantum observables, including time correlations, as a certain weighted average of ab initio observables in the excited states, and these ideas are put into the context of this work in Section5. The analysis of particles in a heat bath has a long history, starting with the work by Einstein and Smoluchowski. The Langevin equation was introduced in [14] to study Brownian motion mathematically, before the Wiener process was available. Early results on the elimination of the heat bath degrees of freedom to obtain a Langevin equation are [7,6] and [28],[27, Section 9.3]. The work [7,6] include in addition a derivation of a quantum Langevin equation, which is based on an operator version of the classical Langevin equation. Our study of the classical 4004 HAKON˚ HOEL AND ANDERS SZEPESSY

Langevin equation from quantum mechanics is not related to this quantum Langevin equation. We start with an ab initio quantum model of the system coupled to a heat bath in the canonical ensemble and use its classical limit to eliminate the electron degrees of freedom. Then the classical system coupled to the heat bath is analyzed by separation of time scales. The separation of time scales of light bath particles and heavier system particles was first used in [15], see also [20, 23] to determine a Fokker-Planck equation for the system particle, from the Liouville equation of the coupled system using a formal expansion in the small mass ratio. Section 8 in [22] presents a proof, and several references to related work, where the Langevin equation is derived from the generalized Langevin equation, using exponential decay of the kernel in the memory term of the generalized Langevin equation. Our contribution employs the separation of time scales approach previously used in [15, 20, 23], but a novelty is that we here present mathematical proofs for the weak convergence rate of Langevin dynamics towards quantum mechanics. Our work also differs from [22, Section 8], wherein the kernel of the generalized Langevin equation is assumed to be such that by adding a finite dimensional variable the system becomes Markovian and the kernel tends to a point mass. For instance, our kernel vanishes as m → 0+ and we use the density of heat bath states to determine the kernel. Using the precise information from the density of states in the case of heat bath nearest neighbor interaction in a cubic lattice we obtain a positive definite friction matrixκ ¯, while if the nearest neighbor interaction would be related to a lattice in dimension four the friction matrix would vanish, see Remark4. The main new ideas in our work are the first principles formulation from quantum mechanics, the weak coupling condition as a minimization, the precise use of the density of heat bath states, that the error estimate uses stability of the Kolmogorov backward equation for the Langevin equation evaluated along the dynamics of the coupled system, and formulation of numerical schemes and numerical results related to Langevin dynamics approximation of particles systems. Although our heat bath formulation starts from a quantum mechanical descrip- tion, our analysis impose restrictions on the generality. In a first step the classical limit of a quantum system and heat bath interaction is derived by semi-classical analysis in Section5 and the classical interaction potential is approximated by pair interactions in Section 3.1.1. The finally result is proved based on an example of pair interactions. Therefore the main limitation in our work is the analysis restriction to a harmonic oscillator heat bath model based on nearest neighbour interaction on periodic cubic lattices, with n points. The use of the restrictive heat bath model here is due to the mathematical techniques in the convergence analysis. The anal- ysis impose in particular four additional conditions for the interaction potential V¯ (x, X), namely Assumption 1.1. The Hessian matrix V¯ 00 in (2) is constant, i.e. we use a har- monic oscillator approximation as motivated in (4). Assumption 1.2. The classical force on each heat bath particle j has a constant 0 ¯ 00 derivative F`j = (V ∂`a(X))j with respect to each system particle position X`, see (11) and (13). 0 Assumption 1.3. The force derivative F`j is localized, in the sense that for a fixed system particle `, the force derivative decays with a certain rate with respect j as measured by the distance between the heat bath particle j and the system particle `, see (13). LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4005

Assumption 1.4. The potentials for different number of lattice points n are related 0 so that the force derivative F`j has a limit as the number of lattice points n tends to infinity, see (13). General heat bath models could also satisfy assumptions 1.3 and 1.4 while as- sumptions 1.1 and 1.2 would typically not hold exactly but may be local approxi- mations, as motivated in Section 3.1.1. Section2 formulates a classical model for the system and the heat bath, including the weak coupling, and derives the corresponding generalized Langevin equation, based on a memory term with a specific integral kernel, in a classical molecular dynamics setting. The generalized Langevin equation is analyzed in Section3, with subsections related to the dissipation, fluctuations and approximation by pure Langevin dynamics in the case of light heat bath particles. The main result of Sec- tion3 is Theorem 3.4, where a certain Langevin dynamics is shown to approximate a classical system weakly coupled to a heat bath. Section4 extends the analysis to the case of stiff heat baths and derives a corresponding approximation result in Theorem 4.1. Section5 relates the classical model to a quantum formulation and provides background and error estimates on quantum observables in the canonical ensemble approximated by classical molecular dynamics. The main result of the work is Theorem 5.2, which proves that for any temperature canonical quantum observables based on the system coordinates can be accurately approximates by the Langevin dynamics obtained in Theorems 3.4 and 4.1. Section6 includes numerical results of the system particle density and autocorrelation of the coupled system and heat bath approximated by the particle density and autocorrelation for the Langevin dynamics. The aim of the numerical examples is to show for a simple model problem that the classical system coupled to the heat bath model, described by a generalized Langevin equation, is well approximated by the derived Langevin equation as the mass ratio of heat bath and system particles vanish. The important comparison of the derived Langevin equation to experiments based on real matter remains for future work.

2. The model of the system and the heat bath. We consider in this section a classical model of a molecular system, with position coordinates X ∈ RN and mo- mentum coordinates P ∈ RN , coupled to a heat bath, with position and momentum coordinates x ∈ Rn and p ∈ Rn, respectively, represented by the Hamiltonian |P |2 |p|2 + + λ(X) + V¯ (x, X) , 2 2m where λ : RN → R is the potential energy for the system and V¯ : Rn × RN → R is the potential energy for the heat bath including the coupling to the system. The parameter m is the mass ratio between heat bath nuclei and system nuclei. We have set the time scale so that the system nuclei mass is one. In Section5 we show that this model is the classical limit of a quantum model and study the accuracy of the classical approximation, also in the case with system nuclei that have different masses. We study small perturbations of the equilibrium bath state x = a(X) ∈ Rn where min V¯ (x, X) = V¯ a(X),X
. n x∈R Taylor expansion around the equilibrium yields 4006 HAKON˚ HOEL AND ANDERS SZEPESSY 1 V¯ (x, X) = V¯ a(X),X
+ x − a(X), V¯ 00 a(X) + ξ(x − a(X)),X
x − a(X)
2 xx ¯ n×n for some ξ ∈ [0, 1] depending on (x, X), where Vxx is the Hessian matrix in R , with respect to the x coordinate, and the notation h·, ·i is the standard scalar product in Cn. We assume that the coupling between the system and the heat bath is weak, which means that the perturbation in the system from the heat bath must be small. As the perturbation |x − a(X)| → 0 we therefore require that V¯ (a(X),X) = 0 . (3) Weak coupling also means that the perturbation in the heat bath from the system is ¯ 00 small, so that the influence on the Hessian Vxx from X is negligible, and we assume following Assumption 1.1 therefore that ¯ 00
Vxx a(X) + ξ(x − a),X = C (4) where C is constant, symmetric and positive definite. Hence the vibration frequen- cies for x are assumed to be constant for all x and X. We use the notation C = V¯ 00 below. The assumptions (3) and (4) of weak coupling lead to the Hamiltonian H : RN × RN × Rn × Rn → R |P |2 |p|2 1 H(X, P, x, p) = + + λ(X) + hV¯ 00x − a(X)
, x − a(X)i . (5) 2 2m 2 This model (5) is of the same form as the model of interaction with a heat bath introduced and analysed by Zwanzig in the seminal work [28], although here the motivation with weak coupling and several system particles is different. The Hamiltonian (5) yields the dynamics

X˙ t = Pt 00
P˙t = −∇λ(Xt) + hV¯ xt − a(Xt) , ∇a(Xt)i

x˙ t = pt/m 00
p˙t = −V¯ xt − a(Xt)

00 1/2 and the change of variables (mV¯ ) rt := pt implies −1/2 001/2 x˙ t = m V¯ rt −1/2 001/2
r˙t = −m V¯ xt − a(Xt) .

Define ϕt := xt − a(Xt) + irt to obtain

X˙ t = Pt 00 P˙t = −∇λ(Xt) + RehV¯ ϕt, ∇a(Xt)i (6) −1/2 001/2 iϕ ˙ t = m V¯ ϕt − iX˙ t · ∇a(Xt) , where the third equation uses the notation v · w for the standard scalar product in RN . Assume that x0 and p0 are Gaussian with the distributions provided by the marginals of the Gibbs density e−H(X,P,x,p)/T , R −H(X,P,x,p)/T 2N+2n e dXdP dxdp R LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4007 that is, the momentum p is multivariate normal distributed with mean zero and covariance matrix mT I and independent of x, which is multivariate normal dis- tributed with mean a(X) and covariance matrix T (V¯ 00)−1. Consequently the initial data can be written n X 0 ϕ0 = γkνk (7) k=1 0 0 0 using the orthogonal eigenvectors νm, normalized as hνk, νki = 1, and eigenvalues 00 µm of V¯ ¯ 00 0 0 0 n V νm = µmνm , νm ∈ R and µm ∈ R+ (8) and the definition r i γm := γm + iγm (9) r i with γm and γm, m = 1, . . . , n, independent and normal distributed real scalar random numbers with mean zero and variance T/µm. Duhamel’s principle shows that Z t −iV¯ 001/2m−1/2t −iV¯ 001/2m−1/2(t−s) ϕt = e ϕ0 − e a˙(Xs)ds 0 which implies a form of Zwanzig’s generalized Langevin equation Z t (t − s)V¯ 001/2 X¨ = −∇λ(X ) − hV¯ 00 cos
a˙(X ), ∇a(X )ids t t 1/2 s t 0 m (10) 00 −iV¯ 001/2m−1/2t + RehV¯ e ϕ0, ∇a(Xt)i , with non Markovian friction term given by the integral and a noise term including the stochastic initial data ϕ0. We study two different cases: • either the mass ratio m  1 is small, or • the smallest eigenvalue of V¯ 00 is large of the order χ−1 while the coupling derivative k∇ak is small of size χδ with δ > 1/4. In these cases, both the friction and the noise terms are based on highly oscillatory functions, which will make these contributions small, as explained in the next sec- tion. To simplify the analysis we assume that Assumptions 1.1 and 1.2 hold, which imply that ∇a(·) is a constant matrix, (11) since the matrices V¯ 00 and F 0 are constant, V¯ 00 is non singular and F 0 = V¯ 00∇a(X).

3. Analysis of the generalized Langevin equation for m  1. In this sec- 001/2 R t ¯ 00 (t−s)V¯
tion we first study the dissipation term 0 hV cos m1/2 a˙(Xs), ∇aids and the 00 −iV¯ 001/2m1/2t fluctuation term RehV¯ e ϕ0, ∇ai in (10), as the number of bath par- ticles tend to infinity and the mass ratio, m, between light heat bath nuclei and heavier system nuclei is small. Then in Section 3.3 we prove an a priori estimate 2 ˙ 2 ¨ 2 of sups

dXL(t) = PL(t)dt (12)
1/2 1/2 1/2 dPL(t) = −∇λ XL(t) dt − m κPL(t)dt + (2m κT ) dW (t) , with a certain symmetric friction matrix κ ∈ RN×N and a Wiener process W : [0, ∞) × Ω → RN , with N independent components; here Ω is the set of out- comes for the process (X,P ) : [0, ∞) × Ω → R2N . Finally, we use the solution 4008 HAKON˚ HOEL AND ANDERS SZEPESSY of the Kolmogorov backward equation for the Langevin dynamics along a solution path (Xt,Pt) of (10) to derive an error estimate of the approximation, namely −1 E[g(Xt,Pt)] − E[g(XL(t),PL(t)] = O(m log m ), for any given smooth bounded 2N observable g : R → R and equal initial data (X(0),P (0)) = (XL(0),PL(0)).

3.1. The dissipation term and a precise heat bath. The change of variables t − s = τ m1/2 yields Z t 001/2 00 (t − s)V
hV¯ cos a˙(X ), ∂ ` aids 1/2 s X 0 m √ Z t/ m 1/2 ¯ 00 ¯ 001/2 = m hV cos(τV )˙a(Xt−m1/2τ ), ∂X` aidτ 0 √ t/ m Z 0 X 1/2 ¯ 00 ¯ 001/2 ˙ ` √ = m hV cos(τV )∂X`0 a, ∂X` aiXt− mτ dτ . `0 0 We require this dissipation term to be small, so that the coupling to the heat bath yields a small perturbation of the dynamics for X and P . If V¯ 00 and ∇a are of order one, the mass m needs to be small, or if ∇a is small we can have V¯ 00 large. The case with small m is studied in this section and the case with large V¯ 00 is in Section 4. A small mass also requires the integrand to decay as τ → ∞. We will use Fourier analysis to study the decay of the kernel ``0 ¯ 00 ¯ 001/2 Kn := hV cos(τV )∂X`0 a, ∂X` ai , by writing limn→∞ Kn as an integral, which is the next step in the analysis. The kernel Kn is based on the equilibrium heat bath position derivative ∂X` a and this derivative is determined by the derivative of the force on bath particle xj, with respect to X`, by (5) as 0 ¯ ¯ 00 F`,j := −∂X` ∂xj V (x, X) = (V ∂X` a)j , which, for fixed ` and j, by assumptions (4) and (11) is independent of x and X. We assume that Assumptions 1.3 and 1.4 hold, and more precisely that V¯ is constructed so that this force derivative is localized in the sense X 0 ˆ 2 ˆ |F`,j|(1 + |j − `| ) = O(1) , as n → ∞, for ` := argminj|X` − xj| j (13) 0 limn→∞ F`,j exists. 0 0 ¯ 00 Let νk(j) := νk(j)/ maxj |νk(j)| be the set of orthogonal eigenvectors of V that are normalized to one in the maximum norm.The eigenvalue representation of V¯ 00 in (8) and the definition 2 ωk = µk yields

X 0 ∗ 0 β`,k := F`,jνk (j) = hνk,F`·i , j LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4009

0 ¯ 00 X β`,k F`,j = (V ∂X` a)j = 2 νk(j) , kνkk2 k (14) X β`,k ∂X` aj = 2 2 νk(j) , ω kνkk k k 2 P 2 1/2 where kνkk2 := ( j |νk(j)| ) . Consequently we obtain ``0 ¯ 00 ¯ 001/2 Kn (τ) = hV cos(τV )∂X` a, ∂X`0 ai ∗ X β`,kβ`0,k (15) = cos(τωk) 2 2 . ω kνkk k k 2 00 If n is finite and we make a tiny perturbation of V¯ so that all ωk become rational, the function Kn will be periodic in τ and consequently it will not decay for large τ. To obtain a decaying kernel we will therefore consider a heat bath with infinite number of particles, n = ∞, which requires a more precise formulation of the heat bath. The next section presents heat bath models based on pair interactions and motivates the related Assumptions 1.1-1.4 on V¯ . Then in Section 3.1.2 a particular simple pair interaction model is formulated to study the limit of Kn, as n → ∞. 3.1.1. Motivation for Assumptions 1.1-1.4. To provide a motivation for Assump- tions 1.1-1.4 of the heat bath potential V¯ , consider a classical heat bath model based 3 0 on pair interactions with one system particle X = x1 ∈ R and n − 1 = n/3 heat 3(n0−1) bath particles x = (x2, . . . , xn0 ) ∈ R n0 n0 1 X X V¯ (x, X) = φ(|x − x |) . 2 i j i=1 j=1 σ 12 σ 6
Let for instance φ be a Lennard-Jones potential φ(r) = 4 ( r ) − ( r ) , which for given positive constants  and σ model noble gases, see [26]. The force on particle j becomes ¯ X −∇xj V (x, X) = − ∇xj φ(|xj − xi|) i6=j and the force gradient with respect to X = x1 is 0 ¯ f1j = −∂X ∂xj V (x, X) = −∂x1 ∂xj φ(|xj − x1|) . (16) We have min V¯ (x, X) = V¯ (a(X),X) x 0 where aj(X), j = 1, . . . , n form a face-centered-cubic lattice including also the point a1(X) = X = x1, see [26]. By considering small values of |xj − aj| compared to |aj − ai|, for j 6= i, we obtain the harmonic oscillator approximation 1 V¯ (x, X) ' V¯ (a(X),X) + hx − a(X), V¯ 00x − a(X)
i , (17) 2 where ¯ 00 ¯ 00 Vij := (Vxx(a(X),X))ij =  0  P 00 aj −ak aj −ak φ (|aj −ak|) aj −ak aj −ak
φ (|aj − ak|) ⊗ − ⊗ − , i = j,  |aj −ak| |aj −ak| |aj −ak| |aj −ak| |aj −ak| I k6=j 0 00 aj −ai aj −ai φ (|aj −ai|) aj −ai aj −ai
−φ (|aj − ai|) ⊗ + ⊗ − , i 6= j. |aj −ai| |aj −ai| |aj −ai| |aj −ai| |aj −ai| I

Here the distances between lattice points, |aj −ai|, are at least of the order one while (6) and Lemma 3.3 show that in our heat bath model with constant V¯ 00 we have 4010 HAKON˚ HOEL AND ANDERS SZEPESSY

|xj − aj| small compared to one, if the temperature T and mass m both are small compared to one. Therefore the assumption that |xj −aj| is small compared to |aj − ai| is consistent with the study here. The constant force derivative approximation is then by (16) related to the harmonic oscillator approximation and becomes 0 ¯ 00 0 0 f1j ' −V1j =: F1j, j = 2, . . . , n . which is localized due to the decay of φ. Relating to the constraints (13), we note ¯ 00 0 for this particular model problem that since V1j is constant and independent of n , 0 it follows that limn→∞ F1j exists and that Z ∞ X 0 ˆ 2 00 2 2
|F1,j|(1 + |j − `| ) = O |φ (r)|(1 + r )r dr = O(1) . j 1 In conclusion a heat bath model based on pair interactions with a sufficiently lo- calized potential φ provides a motivation for Assumptions 1.1-1.4. The model can directly be extended to several system particles and different pair interactions for heat bath and system particles, so that 0 ¯ 00 F`j = −(V ∂X` a)j 0 00 aj − a` aj − a` φ`(|aj − a`|) aj − a` aj − a`
= φ` (|aj − a`|) ⊗ − ⊗ − I , |aj − a`| |aj − a`| |aj − a`| |aj − a`| |aj − a`| which determines the constant matrix ∇a. Here φ` is the interaction potential between system particle ` and heat bath particle j.

3.1.2. A precise heat bath. To study limn→∞ Kn we choose an example of a heat bath model (17) where the equilibrium points aj form a cubic lattice and the inter- actions are limited to the nearest neighbor lattice points, namely we use the periodic lattice 3 3 En¯ := {−n/¯ 2, −n/¯ 2 + 1,..., n/¯ 2 − 1} ⊂ R in dimension three to form the equilibrium positions for the heat bath. The position deviation from the equilibrium, namelyx ¯j := xj − aj(X) ∈ R for each particle j ∈ En¯ , is then determined by the potential by nearest neighbour interaction in the lattice 3 3 ¯ 00 2 X X 2 2 X X 2 hV x,¯ x¯i = c |x¯j+ei − x¯j| + ηn |x¯j+ei | i=1 j∈E i=1 j∈E n¯ n¯ (18) 3 3 2 X X 2 X X 2 = c (−x¯j+ei + 2¯xj − x¯j−ei )¯xj + ηn |x¯j+ei |

i=1 j∈En¯ i=1 j∈En¯ with periodic boundary conditionsx ¯j+¯nei =x ¯j, where

j = (j1, j2, j3), e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1) , 3 n¯ = n and c is a positive constant independent of n. The sequence {ηn}n ⊂ (0, ∞) is introduced to make the potential strictly convex for any fixed n, while we also assume that for some 0 < δ1 < δ2 < 1/2 and constants c2 > c1 > 0

−1/2+δ1 −1/2+δ2 c1n ≤ ηn ≤ c2n . (19) We will see in (32) that the upper bound in (19) is needed in our model to obtain non zero friction matrices κ, and by (14), the lower bound in (19) implies that k∂X` ak`∞ ¯ 00 remains bounded, provided kβ`,·k`1 is bounded. We note that V consists of the standard finite difference matrix with mesh size one related to the Laplacian in R3 LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4011

2 and a small positive definite perturbation ηn I, where I is the corresponding identity matrix. The minimum of the potential is obtained for the position deviationsx ¯j = 0, j ∈ En¯ , which we may view as the n heat bath particles located on the n different 3 lattice points in En¯ . This heat bath model can be extended to positionsx ¯j ∈ R , see Remark1, and we note that the first double sum in (18) forms a matrix which for each row has a positive diagonal element which is equal to the negative sum of all off diagonal elements elements as for V¯ 00 in (17). In each coordinate direction the discrete Laplacian is a circulant matrix so that the eigenvectors and eigenvalues can be written

2πij·k/n¯ νk(j) = e , ji = −n/¯ 2,..., n/¯ 2 − 1, and ki = −n/¯ 2,..., n/¯ 2 − 1 , 3 X 2πki (20) ω2 = c2 21 − cos( )
+ η2 , k n¯ n i=1

2 P3 Pn/¯ 2−1 2 3 k1 k2 k3 and kνkk = |νk(j)| =n ¯ . Let r := ( , , ) and r := |r|. We 2 i=1 ji=−n/¯ 2 n¯ n¯ n¯ have for k =n ¯r by (19)

2πij·r νk(j) = e =: ν(r, j) , 3 2 2 X

2 lim ωk = c 2 1 − cos(2πri) =: ω(r) , n→∞ i=1 and we have assumed in (13) that the derivative of the force has a limit

0 ¯ lim F`,j = F`,j , (21) n→∞ which implies

X ∗ lim β`,k = F¯`,jν (r, j) =: β`(r) , n → ∞ k = r j∈E∞ n¯ (22) X β`(0) = F¯`,j ,

j∈E∞

3 ¯ that is, the function β` :[−1/2, 1/2] → R has the Fourier coefficients F`,j. The value β`(0) will be used in (32) to determine the friction matrix κ.

Remark 1. The heat bath model (18) can be extended to havex ¯j = xj − aj(X) ∈ 3 1 2 3 R , withx ¯j = (¯xj , x¯j , x¯j ). In the case

3 3 3 X X X X X hV¯ 00x,¯ x¯i = c2 (−x¯k + 2¯xk − x¯k )¯xk + η2 |x¯ |2 k j+ei j j−ei j n j+ei k=1 i=1 j∈En¯ i=1 j∈En¯

2 2 we still have the same eigenvalues if ck = c , so the model does not change in princi- ple. If on the other hand the constants ck are different for the different components ofx ¯j the spectrum may change and we obtain a different heat bath model. 4012 HAKON˚ HOEL AND ANDERS SZEPESSY

3.1.3. The limit friction matrix. We can take the limit asn ¯ → ∞ in (15), while r = k/n¯ is constant, to obtain an integral ∗ 0 ``0 X β`,kβ` ,k lim Kn (τ) = lim cos(τωk) 2 2 n→∞ n→∞ ω kνkk k k 2 Z ∗
β` (r)β`0 (r) (23) = cos τω(r) 2 dr1dr2dr3 1 1 3
[− 2 , 2 ] ω(r) ``0 =: K∞ (τ) . 1/2 p The change of variables ωi = 2 c 1 − cos(2πri) sgn(ri) yields, with the spherical coordinate ω = (cos α cos θ, sin α cos θ, sin θ)ω, ∗ X β`,kβ`0,k lim cos(τωk) 2 2 n→∞ ω kνkk k k 2 q 3
Z 1 − cos 2πri(ωi) ∗ Y   dω = cos(τω)β (r(ω))β 0 (r(ω)) ` ` 1/2
2 1 1 3 2 πc sin 2πri(ωi) |ω| ω([− 2 , 2 ] ) i=1 (24) Z 0 dω = cos(τω)f(ω, `, ` ) 2 1 1 3 |ω| ω([− 2 , 2 ] ) Z π Z 2π Z ∞ = cos(τω)f(ω, `, `0) sin θdωdαdθ 0 0 0 where ( ∗ Q3 −1 2 2
−1/2 β 0 (ω)β (ω) π 4c − ω , −2c < ω < 2c , f(ω, `, `0) := ` ` i=1 i i (25) 0 , otherwise .

2 2 −1/2 with β(ω) := β(r(ω)). The term (4c − ωi ) is unbounded (but integrable) at the boundary where ωi = ±2c. For the purpose of simplifying later proofs we will assume that β`(ω) is two times differentiable and that it vanishes at the boundary 1 1 3 3 of ω([− 2 , 2 ] ) = [−2c, 2c] as follows: β(ω) = 0 for ω ∈ [−2c, 2c]3 satisfying ω = |ω| > c . (26) Remark 2 (Density of states relates to c). We also note that the Jacobian deter- minant in (24) 3 3 dr(ω) Y dri Y −1/2 | | = = π−14c2 − ω2
(27) dω dω i i=1 i i=1 is related to the density |∇ω(r)|−1 = π−1(12c2 − ω2)−1/2 of the density of states Z Z dS δω¯ − ω(r)
dr = [−1/2,1/2]3 ω(r)=¯ω |∇ω(r)| evaluated by the co-area formula, where dS is the Lebesque surface measure on the surface {r : ω(r) =ω ¯} for a given frequencyω ¯ ∈ [0, ∞). We are now ready to formulate the limit as n → ∞ in the friction term based on the constant N × N friction matrix κ. LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4013

Lemma 3.1. Let 1 X
X
κ 0 := F¯ F¯ 0 `` 4πc3 `j ` j j∈E∞ j∈E∞ and assume that (3),(4), (7), (11), (13), (18), (26) hold and for each t > 0 and (X0,P0) there is a constant C such that ˙ 2 1/2 ˙ 2 1/2 sup (E[|Ps| ]) + sup (E[|Xs| ]) ≤ C, 0≤s≤t 0≤s≤t then for any function h ∈ L∞(R2N ) √ Z t ¯ 00 ¯ 00 (t − s) V
lim E[h(Xt,Pt) hV cos √ a˙(Xs), ∂X` aids] n→∞ 0 m (28) 1/2 ˙ −1 = E[m h(Xt,Pt)κXt] + O(m log m ) . Remark 3 (Dirchlet boundary condition). If we replace the periodic boundary conditions in the heat bath model (20) with homogenous Dirichlet conditions, we have instead 3 Y πki πjiki ν (j) = sin( + ) , j = −n/¯ 2,..., n/¯ 2 − 1, and k = 1,..., n¯ , k 2 n¯ + 1 i i i=1 3 X πki ω2 = c2 21 − cos( )
+ η2 , k n¯ + 1 n i=1 n¯3 lim = 8 . n¯→∞ 2 kνkk2

We can write the eigenvectors as functions of ki/(¯n + 1) since  sin( πjiki ) if mod (k , 4) = 0 ,  n¯+1 i πk πj k  cos( πjiki ) if mod (k , 4) = 1 , sin( i + i i ) = n¯+1 i 2 n¯ + 1 − sin( πjiki ) if mod (k , 4) = 2 ,  n¯+1 i  πjiki − cos( n¯+1 ) if mod (ki, 4) = 3 , and split the sum over ki into one sum over odd ki, where the eigenvector is based on cosine functions, and one sum over even ki, where the eigenvector is based on sine functions. With these changes, the derivation of κ follows as in the case with periodic boundary conditions.

Proof of Lemma 3.1. To study the decay as τ → ∞ of the kernel K∞, we use (26) and the shorthand f(ω) := f(ω, ·, ·) and integrate by parts Z dω K∞(τ) = cos(τω)f(ω) 2 3 ω R Z π Z 2π Z ∞ = cos(τω)f(ω) sin θdωdαdθ 0 0 0 Z π Z 2π Z ∞ sin(τω) = − ∂ωf(ω) sin θdωdαdθ 0 0 0 τ Z π Z 2π 1 = − 2 ∂ωf(ω) ω=0 sin θdαdθ 0 0 τ Z π Z 2π Z ∞ cos(τω) 2 − 2 ∂ωf(ω) sin θdωdαdθ . 0 0 0 τ 4014 HAKON˚ HOEL AND ANDERS SZEPESSY

Since fω
has its support in ω = |ω| ≤ c, cf. (26), and the second derivative 2 ∂ωf(ω) is bounded, we obtain Z dω −2
kK∞(τ)k = k cos(τω)f(ω) 2 k = O (1 + τ) . (29) 3 ω R For a given t > 0 and bounded function h : RN × RN → R we next study the expectation of √ Z t ¯ 00 ¯ 00 (t − s) V
lim h(Xt,Pt) hV cos √ a˙(Xs), ∂X` aids n→∞ 0 m √ √ Z t/ m ¯ 00 ¯ 001/2 √ = m lim h(Xt,Pt) hV cos(τV )˙a(Xt− mτ ), ∂X` aidτ. n→∞ 0 −1/2 Let τ∗ = m min(t, 1) and split the integral √ Z t/ m ¯ 00 ¯ 001/2 √ h(Xt,Pt) hV cos(τV )˙a(Xt− mτ ), ∂X` aidτ 0 Z τ∗ ¯ 00 ¯ 001/2 √ = h(Xt,Pt) hV cos(τV )˙a(Xt− mτ ), ∂X` aidτ (30) 0 √ Z t/ m ¯ 00 ¯ 001/2 √ + h(Xt,Pt) hV cos(τV )˙a(Xt− mτ ), ∂X` aidτ. τ∗ −1/2 The magnitude of the second integral is zero if t ≤ 1, and otherwise τ∗ = m < tm−1/2 and the integral is bounded in expectation using (15), (23) and (29): √ h Z t/ m i ¯ 00 ¯ 001/2 √ E h(Xt,Pt) √ hV cos(τV )˙a(Xt− mτ ), ∂X` aidτ 1/ m √ Z t/ m ˙ CE[|h(Xt,Pt)||Xt−m1/2τ |] ≤ √ 2 dτ 1/ m τ √ ˙ 2 1/2 2 1/2 ≤ C sup (E[|Xs| ]) (E[|h(Xt,Pt)| ]) m . 0≤s≤t The expectation and the limit n → ∞ of the first integral in the right hand side of (30) can be written as

Z τ∗ Z π Z 2π Z ∞ ˙ E[h(Xt,Pt) cos(τω)f(ω) sin θdωdαdθ Xt dτ] 0 0 0 0 Z τ∗ Z π Z 2π Z ∞ ˙ ˙ √ − E[h(Xt,Pt) cos(τω)f(ω) sin θdωdαdθ (Xt − Xt−τ m) dτ], 0 0 0 0 where, using (29), h Z τ∗ Z π Z 2π Z ∞ i ˙ ˙ √ E h(Xt,Pt) cos(τω)f(ω) sin θdωdαdθ (Xt − Xt−τ m) dτ 0 0 0 0 h Z τ∗ Z π Z 2π Z ∞ Z 0 i ≤ h(X ,P ) cos(τω)f(ω) sin θdωdαdθ X¨ ds dτ E t t √ t+s 0 0 0 0 −τ m √ Z τ∗ τ ¨ 2 1/2 2 1/2 ≤ C m 2 dτ sup (E[|Xs| ]) (E[|h(Xt,Pt)| ]) 0 1 + τ 0≤s≤t √ −1 ¨ 2 1/2 2 1/2 ≤ C m min(1, log m ) sup (E[|Xs| ]) (E[|h(Xt,Pt)| ]) . 0≤s≤t LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4015

˙ 2 ¨ 2 We prove in Section 3.3 that the expected value sup0≤s≤t(E[|Xs| + |Xs| ]) is 2 bounded, and these properties imply that E[|h(Xt,Pt)| ] also is bounded. Con- sequently, √ Z t ¯ 00 ¯ 00 (t − s) V
lim E[h(Xt,Pt) hV cos √ a˙(Xs), ∂X` aids] n→∞ 0 m Z τ∗ Z π Z 2π Z ∞ 1/2 X 0 ˙ `0 − E[m h(Xt,Pt) cos(τω)f(ω, `, ` ) sin θdωdαdθdτXt ] `0 0 0 0 0 = O(m log m−1) . (31) The Fourier transform of f(ω) with respect to ω is integrable and since f(ω) is continuous, the Fourier inversion property and (22) yield that

Z τ∗ Z ∞ Z 2π Z π X 0 ˙ `0 lim cos(τω)f(ω, `, ` ) sin θdθdαdωdτXt m→0+ `0 0 0 0 0 Z ∞ Z ∞ Z 2π Z π X 0 ˙ `0 = cos(τω)f(ω, `, ` ) sin θdθdαdωdτXt `0 0 0 0 0 Z 2π Z π π X 0 ˙ `0 = lim f(ω, `, ` ) sin θdθdαXt 2 ω→0 `0 0 0 (32) 2 X 0 ˙ `0 = 2π f(0, `, ` )Xt `0 1 X X
X
`0 ¯ ¯ 0 ˙ = 3 F`j F` j Xt 4πc 0 ` j∈E∞ j∈E∞ X ˙ `0 = κ``0 Xt . `0 That is, the friction matrix, 1 X
X
κ 0 := F¯ F¯ 0 `` 4πc3 `j ` j j∈E∞ j∈E∞ in the Langevin equation is determined by the ∂X` -derivative of the sum of forces on all bath particles. Equation (28) follows from (31) and (32).

Remark 4 (Vanishing friction). If the heat bath model is modified to have nearest neighbor interactions in a lattice in dimension d, we obtain as in (32)

0 d−1 2 ∗ d−3 κ``0 = Sd lim f(ω, `, ` )ω /ω = Sd lim β`(ω)β`0 (ω)ω ω→0 ω→0 where Sd is a positive constant related to the dimension d. We see that under the + + assumption β`(0 )β`0 (0 ) > 0, it is only in dimension d = 3 that this heat bath generates a positive definite friction matrix κ. For d < 3 we obtain κ = ∞ and for d > 3 we have κ = 0. If we change the heat bath potential energy to be based on any circulant matrix in each dimension, we have the requirement r2(ω) dr lim dω = constant (33) ω→0 ω2 4016 HAKON˚ HOEL AND ANDERS SZEPESSY to obtain a positive definite friction matrix κ, in dimension d = 3. This limit for the eigenvalues ω2 implies that V¯ 00 becomes a difference quotient approximation of the Laplacian with mesh size one. We conclude that (33) leads to a choice of V¯ 00 in (18) that is another discretization of the Laplacian or discretizations that tends to the Laplacian as n → ∞.

P r i 0 3.2. The fluctuation term. The initial distribution of ϕ0 = k(γk + iγk)νk, determined by the Gibbs distribution of the Hamiltonian system in (7) and (9), r i shows that all γk and γk are independent and normal distributed with mean zero 2 and variance T/ωk. This initial data provides the fluctuation term in (10), namely √ √ ` −it V¯ 00/ m ¯ 00 ζt = Rehe ϕ0, V ∂X` ai √ γ∗ X itωk/ m k ¯ 00 = Re(e hνk, V ∂ ` ai) n¯3/2 X (34) k X √ γ∗ = Re(eitωk/ m k β ) , n1/2 k,` k which has the special property that its covariance satisfies √ √ √ √ ` `0 −it V¯ 00/ m ¯ 00 −is V¯ 00/ m ¯ 00 E[ζt ζs ] = E[Rehe ϕ0, V ∂X` ai Rehe ϕ0, V ∂X`0 ai] p √ (35) ¯ 00 ¯ 00 = T hcos (t − s) V / m)V ∂X` a, ∂X`0 ai that is, the covariance is T times the friction integral kernel, as observed in [28]. We repeat a proof for our setting here, where we also show that the limit of ζt as n → ∞ is well defined. √ √ √ √ −it V¯ 00/ m −it V¯ 00/ m Proof of (35). Let e =: St0 and write e ϕ0 = StsSs0ϕ0. Since P 0 0 0 the operator Ss0 is unitary, we have Ss0ϕ0 = k γkνk where {νk} is the set of normalized real valued eigenvectors of V¯ 00, defined in (7). The random coefficients 0 0 0 0 0 γk = γk,r + iγk,i are based on γk,r and γk,i which are independent normal distri- 2 butioned with mean zero and variance T/ωk, for k = (k1, k2, k3) and ki = 1,..., n¯. 0 Use the orthonormal real-valued basis {νk} to obtain ` `0 E[ζt ζs ] h X r i 0
00 r i 0
00 i ¯ 0 0 0 ¯ = E Re St0(γk + iγk)νk , V ∂X` a Re Ss0(γk + iγk )νk , V ∂X`0 a k,k0 h X 0 0 0
00 0 0 0
00 i ¯ 0 0 0 ¯ = E Re Sts(γk,r + iγk,i)νk , V ∂X` a Re (γk ,r + iγk ,i)νk , V ∂X`0 a k,k0 h X 2 −1/2
0 −1/2
0  0 = E ωk cos (t − s)ωkm γk,r − sin (t − s)ωkm γk,i νk, ∂X` a k,k0 0 0 00 i 0 0 ¯ × γk ,rνk , V ∂X`0 a

X  2 −1/2
0 0 = ωk cos (t − s)ωkm E[γk,rγk0,r] k,k0 −1/2
0 0  0 0 00  0 0 ¯ − sin (t − s)ωkm E[γk,iγk ,r] νk, ∂X` a νk , V ∂X`0 a p √ X ¯ 00
0 0 ¯ 00 = T cos (t − s) V / m νk, ∂X` a νk, V ∂X`0 a k LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4017 p √ ¯ 00
¯ 00 = T hcos (t − s) V / m ∂X` a, V ∂X`0 ai . We conclude that ζ is a Gaussian process with mean zero and covariance (35). The fluctuation-dissipation property (35), the limit (23) and the bound (29) show that the covariance matrix has a limit as n → ∞: t − s
lim E[ζtζs] = TK∞ √ . (36) n→∞ m

2 R τ 2 1/2 Therefore ζ has a well defined limit, in the L -space with norm ( 0 E[|ζt| ]dt) , for any finite time τ as n → ∞.

3.3. The system dynamics. We can write the system dynamics (6) and (10) as Z t Xt = X0 + Psds , 0 Z t Z t 00 Pt = P0 − ∇λ(Xs)ds + Rehϕs, V¯ ∇aids 0 0 Z t Z t Z s −i(s−σ)V¯ 00m−1/2 00 = P0 − ∇λ(Xs)ds − Rehe (Pσ · ∇)a, V¯ ∇aidσds 0 0 0 Z t + ζsds . 0 (37) We assume that λ is three times continuously differentiable and 2 3 (38) supX kD λ(X)k + supX kD λ(X)k is bounded and apply Cauchy’s inequality on (37) to obtain d Z t |X |2 + |P |2ds = |X |2 + |P |2 dt s s t t 0 (39) Z t Z t  3 2 2 2  ≤ C 1 + (t + t ) (|Xs| + |Ps| )ds + t |ζs| ds , 0 0 which implies there is a constant C, depending on (X0,P0), such that Z t Z t Z t d 2 2 3 2 2 2
E[|Xs| + |Ps| ]ds ≤ C 1 + (t + t ) E[|Xs| + |Ps| ]ds + t E[|ζs| ]ds . dt 0 0 0 | {z } 2 ≤t T trace(K∞(0)) Here Dkλ denotes the set of all partial derivatives of order k and if k = 1 or k = 2 we identify it with the gradient and the Hessian, respectivly. Integration yields the Gronwall inequality Z t 2 2 Ct4 E[|Xs| + |Ps| ]ds ≤ Ce , 0 which by (39) and the mean square of Z t −i(t−s)V¯ 00m−1/2 00 P˙t = −∇λ(Xt) + Rehe (Ps · ∇)a, V¯ ∇aids + ζt 0 establishes 4018 HAKON˚ HOEL AND ANDERS SZEPESSY

Lemma 3.2. Suppose that the assumptions in Lemma 3.1 and (38) hold, then for each t > 0 and (X0,P0) there is a constant C such that 2 2 ˙ 2 Ct4 sup E[|Xs| + |Ps| + |Ps| ] ≤ C(1 + e ) . 0≤s≤t 3.4. Approximation by Langevin dynamics. In this section we approximate the Hamiltonian dynamics (6) by the Langevin dynamics (12)

dXL(t) = PL(t)dt ,
1/2 1/2 1/2 dPL(t) = −∇λ XL(t) dt − m κPL(t)dt + (2m κT ) dW (t) ,

XL(0) = X(0) ,

PL(0) = P (0) . To analyse the approximation we use that, for any infinitely differentiable function g : R2N → R with compact support, (40) the expected value

2N u(z, s) = E[g XL(t∗),PL(t∗) | XL(s),PL(s) = z] , s < t∗ and z ∈ R ,
is well defined, since the Langevin equation (12), for Zt := XL(t),PL(t) , has Lipschitz continuous drift and constant diffusion coefficient. The assumption (38) 2 ∂Zt ∂ Zt implies that the stochastic flows and 2 also are well defined, which implies ∂Zs ∂Zs that the function u has bounded and continuous derivatives up to order two in z and to order one in t, see [12] and [8], and solves the corresponding Kolmogorov equation √
∂su(X, P, s) + P · ∇X u(X, P, s) − ∇λ(X) + mκP · ∇P u(X, P, s) N X √ + mT κjk∂Pj Pk u(x, P, s) = 0 , s < t∗ , j,k=1

u(X, P, t∗) = g(X,P ) . We have


E[g X(t∗),P (t∗) | X0,P0] − E[g XL(t∗),PL(t∗) | XL(0),PL(0) = (X0,P0)]

= E[u X(t∗),P (t∗), t∗ − u X(0),P (0), 0 | X0,P0] Z t∗
= E[du X(t),P (t), t | X0,P0] 0 Z t∗ ˙ = E[∂tu(Xt,Pt, t) + Pt · ∇X u(Xt,Pt, t) + Pt · ∇P u(Xt,Pt, t) | X0,P0]dt 0 Z t∗ √  ˙ = E (Pt + ∇λ(Xt) + mκPt) · ∇P u(Xt,Pt, t) 0 √ 2  − mT κ :DPP u(Xt,Pt, t) | X0,P0 dt Z t∗ √  ¯ 00
= E Rehϕt, V ∇ai + mκPt · ∇P u(Xt,Pt, t) 0 √ 2  − mT κ :DPP u(Xt,Pt, t) | X0,P0 dt , (41) 2 P 0 where κ :DPP u := `,`0 κ`,` ∂P` ∂P`0 u. LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4019

Lemma 3.3. Suppose that the assumptions in Lemma 3.2 and (40) hold, then √ Z t ¯ 00 (t − s) V
¯ 00 lim E[ hcos √ a˙(Xs), V ∇a · ∇P u(Xt,Pt)ids | X0,P0] n→∞ 0 m (42) 1/2 ˙ −1 = m E[κXt · ∇P u(Xt,Pt) | X0,P0] + O(m log m ) . and √ −itV 001/2/ m ¯ 00 lim E[ Re(e ϕ0), V ∇a · ∇P u(Xt,Pt, t) | X0,P0] n→∞ (43) 1/2 −1 = m T E[κ :DPP u(Xt,Pt, t) | X0,P0] + O(m log m ) . The limit (42) is verified in (31) and we prove (43) below. The lemma and the error estimates (31) and (31) inserted in (41) imply Theorem 3.4. Provided the assumtions in Lemma 3.3 hold, then the Langevin dynamics (12) approximates the Hamiltonian dynamics (6) and (10), with the error estimate

[g(X ,P ) | X ,P ] − [gX (t ,P (t )
| X (0),P (0)
= (X ,P )] E t∗ t∗ 0 0 E L ∗ L ∗ L L 0 0 1 = O(m log ) , m where the rank one friction matrix is determined by the force F¯ from (14) and (21) 1 X
X
κ 0 = F¯ F¯ 0 0 `` 4πc3 `j ` j j∈E j0∈E ∞ ∞ (44) 1 X 00
X 00
= V¯ ∂ ` a(X) V¯ ∂ `0 a(X) . 4πc3 X j X j0 0 0 (j,j )∈E∞ j ∈E∞ We see also that the alternative dynamics, with any friction coefficientκ ¯ ≥ 0,

dX¯t = P¯tdt 1/2 dP¯t = −∇λ(X¯t)dt − κ¯P¯tdt + (2¯κT ) dWt approximates (6) and (10) with the larger error ¯ ¯ ¯ ¯ 1/2
E[g(Xt∗ ,Pt∗ ) | X0,P0]−E[g(Xt∗ , Pt∗ ) | X0 = X0, P0 = P0] = O max(m , kκ¯k) , (45) unlessκ ¯ = m1/2κ. Proof of (43). We have

00 −itV¯ 00/m1/2 00 Rehϕt, V¯ ∇ai = hRe(e ϕ0), V¯ ∇ai | {z } =:ϕ1(t) (46) Z t t − s 001/2
00 − hcos √ V¯ a˙(Xs), V¯ ∇aids 0 m √ and (42) shows that the second term cancels mκPt · ∇P u(Xt,Pt, t) to leading order in (41). It remains to show that the scalar product of ∇P u(Xt,Pt, t) and the first term in the right hand side (46) cancels the last term in (41). −itV¯ 00/m1/2 00 To analyze the dependence between ∂P u(Xt,Pt, t) and Rehe ϕ0, V¯ ∇ai from the initial data ϕ0 we will study how a small perturbation of ϕ0 influences ∂P u(Xt,Pt, t). The proof has three steps: 4020 HAKON˚ HOEL AND ANDERS SZEPESSY

(1) to derive a representation of Zt = (Xt,Pt) in terms of perturbations of the P 0 initial data ϕ0 = k γkνk by removing one term, −itV¯ 001/2m−1/2 (2) to determine expected values E[he ϕ0, h(Zt)i], for a given function h, using Step (1), and (3) to evaluate (43), using the conclusion from Step (2) and the covariance result (35).

Step 1. Claim. Consider two different functions: ϕ1(t) and ϕ2(t) = ϕ1(t) + v(t), where   1, then the corresponding solution paths Z1 = (X1,P1) and Z2 = (X2,P2) of the dynamics (6) (which can be written as (10)) based on the functions ϕ1 and ϕ2, respectively, satisfy R t ¯ 00  0 GX1P` (t, s)Rehv(s), V ∂X` aids  .   .  N  t  X R G (t, s)Rehv(s), V¯ 00∂ aids Z (t) = Z (t) +  0 XN P` X`  2 1  R t ¯ 00   0 GP1P` (t, s)Rehv(s), V ∂X` aids  `=1    .  (47)  .  R t ¯ 00 0 GPN P` (t, s)Rehv(s), V ∂X` aids N Z t X ¯ 00 = Z1(t) + G·P` (t, s)Rehv(s), V ∂X` aids `=1 0 where   GXX (t, s) GXP (t, s) 2N×2N G(t, s) = ∈ R GPX (t, s) GPP (t, s) solves the linear equation

∂tGXy(t, s) = GP y(t, s) , Z 1 2
∂tGP y(t, s) = − D λ τX1(t) + (1 − τ)X2(t) dτ GXy(t, s) 0 Z t (t − r)V¯ 001/2 − hV¯ 00 cos
∇a · G (r, s), ∇aidr , t > s, y = X,P, 1/2 P y s m  I 0  G(s, s) = . 0 I (48) Proof of the claim. The linearized problem corresponding to (10) becomes d X¯ = P¯ , dt t t Z 1 d 2
P¯t = − D λ τX1(t) + (1 − τ)X2(t) dτ X¯t dt 0 (49) Z t (t − s)V¯ 001/2 − hV¯ 00 cos
∇a · P¯ , ∇aids 1/2 s 0 m −itV¯ 00/m1/2 00 + Rehe ϕ¯0, V¯ ∇ai , t > 0.

−itV¯ 00/m1/2 00 00 Consider R(t) := Rehe ϕ¯0, V¯ ∇ai = hv(t), V¯ ∇ai as a perturbation to the linearized equation (49) with Z¯(t) := (X¯t, P¯t) = Z2(t) − Z1(t) and R¯(t) := LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4021

2N 0,R(t) ∈ R and write (47) as Z t Z¯(t) = G(t, s)R¯(s)ds (50) 0 and (48) in abstract form as Z t ∂tG(t, s) = A(t)G(t, s) − L(t, v)G(v, s)dv , t > s , s (51) G(s, s) = I . Differentiation and change of the order of integration imply by (50) and (51) d Z t Z¯(t) = G(t, t)R¯(t) + ∂tG(t, s)R¯(s)ds dt 0 Z t Z t Z t = R¯(t) + A(t) G(t, s)R¯(s)ds − L(t, v)G(v, s)R¯(s)dvds 0 0 s Z t Z v = R¯(t) + A(t)Z¯t − L(t, v) G(v, s)R¯(s)dsdv 0 0 Z t = R¯(t) + A(t)Z¯(t) − L(t, v)Z¯(v)dv , 0 which shows that (47) satisfies the linearized equation (49), and the claim is proved. The main reason we use assumption (11), namely that ∇a(X) is constant, is to obtain this linearized equation for G, which otherwise would include the term Rehϕ, V¯ 00D2ai that would introduce the fast time scale of ϕ in G, which we now avoid. Step 2. We will use

X −itV¯ 00m−1/2 r i 0 ϕ1(t) := e (γj + iγj)νj , j

0 based on the orthonormal eigenvectors {νk} in (7), and for a given k define ϕ2(t) := 00 −1/2 −1/2 P −itV¯ m r i 0 −itωkm r i 0 j6=k e (γj + iγj)νj, that is v(t) = −e (γk + iγk)νk, so that Z1 = (X1,P1) is the path corresponding to ϕ1 and Z2 = (X2,P2) corresponds to 0 ϕ2 where γkνk is removed from the sum in ϕ1. For any bounded and differentiable 2N n P 0 function h : R → C the perturbation property (47), with ϕ0 = k γkνk, implies

−itV¯ 001/2m−1/2
E[he ϕ0, h Z1(t) i] X ∗ 0 −itV¯ 001/2m−1/2
= E[γk hνk, e h Z1(t) i] k | {z } =:hk(Z1(t)) Z 1 X h ∗
= E γk hk(Z2) + ∇hk σZ2 + (1 − σ)Z1 dσ k 0 (52) t Z −1/2 i X −isωkm 0 ¯ 00 · G·P` (t, s)Rehe γkνk, V ∂X` aids ` 0 Z 1 X ∗ X ∗
= E[γkhk(Z2)] + E[γk ∇hk σZ2 + (1 − σ)Z1 dσ 0 k | ∗ {z } k =E[γk ]E[(hk(Z2)]=0 4022 HAKON˚ HOEL AND ANDERS SZEPESSY

t Z −1/2 X −isωkm 0 ¯ 00 · G·P` (t, s)Rehe γkνk, V ∂X` aids] ` 0 t Z −1/2 X ∗ 0 X −isωkm 0 ¯ 00 = E[γk∇hk(Z ) · G·P` (t, s)Rehe γkνk, V ∂X` aids] , k ` 0 0 where Z is between Z1(t) and Z2(t) and satisfies Z 1 0

 ∇h Z (t) = ∇h Z2(t) + σ Z1(t) − Z2(t) dσ . 0

We will use that the difference between Z1 and Z2 is small, namely

t Z −1/2 X −isωkm 0 ¯ 00 ∆Z(t) := Z2(t) − Z1(t) = G·P` (t, s)Rehe γkνk, V ∂X` aids 0 ` (53) Z t 1/2 X T −1/2 = G (t, s) Re(e−isωkm ξ β )ds , ·P` ω n1/2 k k,` ` 0 k

−1/2 where ξk = ωkT γk = ξk,r + iξk,i, with ξk,r and ξk,i independent and standard normal distributed with√ mean zero and variance one. The eigenvalue representation (20) and (19) show that n ωk → ∞ as n → ∞. Consequently we have |Z2−Z1| → 0 as n → ∞, uniformly for all realizations. −itV¯ 00/m1/2 00 Step 3. The dependence between ∂P u(Xt,Pt, t) and Rehe ϕ0, V¯ ∇ai 0 0 0 from the initial data ϕ0 with Z = (X ,P ) can by (3.4) and (34) be written √ √ −it V¯ 00/ m ¯ 00 E[hRe(e ϕ0), V ∇a · ∇P u(Xt,Pt, t)i] √ √ X h −it V¯ 00/ m 0 ¯ 00 = E hRe(e γkνk), V ∂X` ai 0 00 | {z } k,`,` ,` =St0 Z t   × D2 u(X0,P 0, t)G (t, s) + D2 u(X0,P 0, t)G (t, s) P`0 P` t t P`0 P`00 X`0 P` t t X`0 P`00 0 | {z } =:U`,`0,`00 (Xt,Pt,G,t,s) √ √ −is V¯ 00/ m 0 ¯ 00 i × hRe(e γkνk), V ∂X`00 aids so that by (14) √ √ −it V¯ 00/ m ¯ 00 E[hRe(e ϕ0), V ∇a · ∇P u(Xt,Pt, t)i] 1/2 t −1/2 T β Z X itωkm k,` 0 0 r 2
= Re(e ) [U 0 00 (X ,P , G, t, s)|ξ | ]Re  00 (s) ds 1/2 E `,` ,` t t k k,` 0 00 ωkn 0 k,`,` ,` | {z } =:k,`(t) Z t X
0 0 i 2
+ Im k,`(t) E[U`,`0,`00 (Xt,Pt , G, t, s)|ξk| ]Im k,`00 (s) ds k,`,`0,`00 0 Z t X
0 0 r i
− Im k,`(t) E[U`,`0,`00 (Xt,Pt , G, t, s)ξkξk]Re k,`00 (s) ds k,`,`0,`00 0 Z t X
0 0 r i
− Re k,`(t) E[U`,`0,`00 (Xt,Pt , G, t, s)ξkξk]Im k,`00 (s) ds . k,`,`0,`00 0 LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4023

0 0 r i To determine the expected value E[U`,`0,`00 (Xt,Pt , G, t, s)ξkξk] we use Z2 = (X2,P2) in (47) based on ϕ2 and the Green’s function   G2,XX (t, s) G2,XP (t, s) 2N×2N G2(t, s) = ∈ R , G2,P X (t, s) G2,P P (t, s) based on X2, that is for t ≥ s defined by

∂tG2,Xy(t, s) = G2,P y(t, s) , 2 ∂tG2,P y(t, s) = −D λ X2(t))G2,Xy Z t (t − r)V¯ 001/2 − hV¯ 00 cos
∇a · G (r, s), ∇aidr , y = X,P, 1/2 2,P y s m  I 0  G (s, s) = . 2 0 I

The difference of the two Green’s functions satisfy by (47) the perturbation repre- sentation

G(t, s) − G2(t, s) Z t  0 0  = G(t, r) 2
R 1 2
G2(r, s)dr s D λ X2(r) − 0 D λ τX1(r) + (1 − τ)X2(r) dτ 0 Z t  0 0  = G(t, r) R 1 R 1 3
G2(r, s)dr . s 0 0 D λ τX2(r) + τσ∆X(r) τ∆X(r)dτdσ 0

The processes Z2 and G2 do not depend on ξk. The expected value can therefore be evaluated using the Jacobian U 0 = DU, with respect to Z and G, and the small perturbation ∆Z in the path from (53) as

0 0 r i E[U`,`0,`00 (Xt,Pt , G, t, s)ξkξk] r i = E[U`,`0,`00 (Z2(t),G2, t, s)ξkξk] Z 1 Z 1 0
+ E[ U`,`0,`00 Z2(t) + στ(Z1(t) − Z2(t)),G2 + τ(G − G2), t, s dτ 0 0 
 r i σ Z1(t) − Z2(t) × ξkξk dσ] G − G2

r i 1 = [U 0 00 (Z (t), t, s)] [ξ ξ ] +O( ) E `,` ,` 2 E k k 1/2 | {z } n ηn =0 with analogous splittings for

0 0 i i 0 0 r r U`,`0,`00 (Xt,Pt , G, t, s)ξkξk and U`,`0,`00 (Xt,Pt , G, t, s)ξkξk,

r r i i based on E[ξkξk] = E[ξkξk] = 1. Dominated convergence, using the assumption 1/2 n ηn → ∞ in (19) as n → ∞, implies therefore

√ √ −it V¯ 00/ m 00 lim [hRe(e ϕ0), V¯ ∇a · ∇P u(Xt,Pt, t)i] n¯→∞ E 4024 HAKON˚ HOEL AND ANDERS SZEPESSY Z t X

= lim Re k,`(t) [U`,`0,`00 (Xt,Pt, G, t, s)]Re k,`00 (s) ds n¯→∞ E k,`,`0,`00 0 Z t X

+ lim Im k,`(t) [U`,`0,`00 (Xt,Pt, G, t, s)]Im k,`00 (s) ds n¯→∞ E k,`,`0,`00 0 Z t X ∗
= lim Re k,`(t)k,`00 (s) [U`,`0,`00 (Xt,Pt, G, t, s)]ds , n¯→∞ E k,`,`0,`00 0 which shows that (Xt,Pt) in the limit becomes independent of the small perturba- 0 ¯ 00 tion caused by hγkνk, V ∇ai. We have √ X ∗
X 0 i(t−s)V¯ 00/ m 0 00 00 ¯ Re k,`(t)k,` (s) = T Re( hνk, e ∂X` aihνk, V ∂X`00 ai) k k √ ¯ 00
¯ 00 = T hcos (t − s)V / m ∂X` a, V ∂X`00 ai , and we obtain then as in the proof of (35)

X h X r i 0 ¯ 00 lim RehSt0(γk + iγk)νk, V ∂X` ai n¯→∞ E `,`0,`00 k Z t  2 r i 0 00 × D u G (t, s)RehS (γ + iγ )ν , V¯ ∂ 00 aids P`0 P` P`0 P`00 s0 k k k ` 0 Z t 2 r i 0 00 i + D u G (t, s)RehS (γ + iγ )ν , V¯ ∂ 00 aids X`0 P` X`0 P`00 s0 k k k ` 0 Z t X 2 2
= T lim [ DP 0 P uGP 0 P 00 (t, s) + DX 0 P uGX 0 P 00 (t, s) n¯→∞ E ` ` ` ` ` ` ` ` `,`0,`00 0 p √ ¯ 00
¯ 00 × hcos (t − s) V / m ∂X` a, V ∂X`00 aids] √ X Z t/ m  √ = m1/2T [ D2 u(X ,P , t)G (t, t − mτ) E P`0 P` t t P`0 P`00 `,`0,`00 0 √  p + D2 u(X ,P , t)G (t, t − mτ) hcos(τ V¯ 00)∂ a, V¯ 00∂ aidτ] . X`0 P` t t X`0 P`00 X` X`00

√ √ √ t √ R √ Lemma 3.1 with G(t, t− mτ) = G(t− mτ, t− mτ)+ t− mτ ∂rG(r, t− mτ)dr replacing √ √ Z t

˙ ˙
X(t − mτ),P (t − mτ) = (X(t)P (t) − √ X(s), P (s) ds , t− mτ √ √ √ √ using also (48), GPP (t − mτ, t − mτ) = I and GXP (t − mτ, t − mτ) = 0, verifies (43): √ X Z t/ m  √ m1/2T [ D2 u(X ,P , t)G (t, t − mτ) E P`0 P` t t P`0 P`00 `,`0,`00 0 √  p + D2 u(X ,P , t)G (t, t − mτ) hcos(τ V¯ 00)∂ a, V¯ 00∂ aidτ] X`0 P` t t X`0 P`00 X` X`00 1/2 2 −1 = m T E[DPP u(Xt,Pt, t): κ] + O(m log m ) . LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4025

4. Analysis of the generalized Langevin equation for χ  1. In this section we study a time scale separation for system particles and fast heat bath particles due to a stiff heat bath obtained by changing the scaling to V¯ 00 = χ−1V˜ 00 , a(X) = χδa˜(X) , X 0 1/2 X 0 1/2 ϕ0 = γkνk = χ γ˜kνk = χ ϕ˜0 , (54) k k 2 γ˜k ∼ N(0,T/ω˜k) , 2 ˜ 00 ω˜k are the eigenvalues of V , and assume that m and the elements of V˜ 00 anda ˜ are of size one while χ  1. (55) The nonlinear generalized Langevin equation then becomes Z t (t − s)V˜ 001/2 X¨ = −∇λ(X ) − χ2δ−1 hV˜ 00 cos
a˜˙(X ), ∇a˜(X )ids t t 1/2 s t 0 (χm) δ−1/2 00 −iV˜ 001/2(χm)−1/2t + χ Re V˜ e ϕ˜0, ∇a˜(Xt) .

The analysis in Section3 can be applied by replacing m by χm; V¯ 00 by V˜ 00; and a bya ˜ and we obtain

2δ−1/2 1/2 Theorem 4.1. Let m0 := χ m with δ > 1/4 and assume that (54)-(55) hold together with the assumptions in Theorem 3.4, then the Langevin dynamics

dXL(t) = PL(t)dt
1/2 dPL(t) = −∇λ XL(t) dt − m0κPL(t)dt + (2m0κT ) dW (t) ,

XL(0) = X(0) ,

PL(0) = P (0) approximates the Hamiltonian dynamics (6) and (10), with the error estimate

[g(X ,P ) | X ,P ] − [gX (t ,P (t )
| X (0),P (0)
= (X ,P )] E t∗ t∗ 0 0 E L ∗ L ∗ L L 0 0 1/2 −1 2δ −1 = O(m0χ log χ ) = O(χ log χ ) ,

˜ ˜ 00 where the friction matrix is determined by the force F`j = limn→∞(V ∂X` a˜)(j) as 1 X
X
κ 0 = F˜ F˜ 0 . (56) `` 4πc3 `j ` j j∈E∞ j∈E∞

5. Molecular dynamics approximation of a quantum system. The purpose of this section is to present a molecular dynamics approximation for observables of a quantum particle system consisting of nuclei and electrons coupled to a heat bath. The observables may include correlations in time. The first subsection provides background to quantum observables approximated by molecular dynamics in the canonical ensemble. The next subsection combines these quantum approximation results of with the classical Langevin approximation in Theorems 3.4 and 4.1. 4026 HAKON˚ HOEL AND ANDERS SZEPESSY

5.1. Canonical quantum observables approximated by molecular dynam- ics. The quantum formulation is based on wave functions Φ : RN × Rn → Cd and the Hamiltonian M −1 M −1 − s I∆ − b I∆ + V (X) + V (x, X) , 2 X 2 x b where X ∈ RN and x ∈ Rn are the nuclei coordinates of the system and heat bath positions, respectively, and Ms and Mb are the diagonal matrices of the mass of the system nuclei and heat bath nuclei, respectively, measured in units of the N d2 n N d2 electron mass. The functions V : R → C and Vb : R × R → C are finite difference approximations of the electron kinetic energy and nuclei–nuclei, nuclei– electron and electron–electron interactions, related to the system and the heat bath. The matrix I is the identity on Cd. This simplification to replace the Laplacians for the electron kinetic energy by difference approximations makes it easier to derive the 1/2 1/2 classical limit. Another simplification is to change coordinates M X¯ = Ms X 1/2 1/2 ¯ N+n and M x¯ = Mb x and letx ˜ = (X, x¯) ∈ R , where M  1 is a reference nuclei–electron mass ratio. In these coordinates the Hamiltonian takes the form 1 Hˆ = − I∆ + v(˜x) , 2M x˜ −1/2 1/2 ¯ −1/2 1/2 −1/2 1/2 ¯ where v(˜x) := V (Ms M X) + Vb(Mb M x,¯ Ms M X). We will use the eigenvalues λk(˜x) ∈ R and eigenvectors ψk(˜x) of the Hermitian matrix v(˜x) defined by

v(˜x)ψk(˜x) = λk(˜x)ψk(˜x) . We assume that the eigenvalues satisfy

λ1(˜x) < λ2(˜x) < . . . < λd(˜x) , (57) λ1(˜x) → ∞ as |x˜| → ∞ . The first assumption is in order to have differentiable eigenvectors and the second condition implies that the spectrum of Hˆ is discrete, see [5]. The aim here is to study canonical quantum observables, including correlations in time, namely ∞ X trace(Aˆτ Cˆ0) = (Φn, Aˆτ Cˆ0Φn) , n=1 ∞ 2 where {Φn}n=1 is a normalized basis of L (d˜x), e.g. the set of normalized eigenfunc- ˆ R ∗ ˆ tions to H and (f, g) := N+n f (˜x)g(˜x)d˜x. An operator B is the Weyl quantization R that maps L2(RN+n) to [L2(RN+n)]d and is defined, from a d × d matrix valued 2 symbol B : RN+n → Cd in the Schwartz class, by 1/2 M Z 1/2 x˜ + y BˆΦ(˜x) = ( )N+n eiM (˜x−y)·p˜B( , p˜)Φ(y)d˜pdy . 2π 2(N+n) 2 R |p˜|2\ ˆ ˆ For instance, we have 2 + v(˜x) = H. The time dependent operator Bτ is defined by ˆ iτM 1/2Hˆ ˆ −iτM 1/2Hˆ Bτ := e Be , τ ∈ R, which implies the von Neumann-Heisenberg equation d Bˆ = iM 1/2[H,ˆ Bˆ ] (58) dτ τ τ LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4027 where [H,ˆ Bˆτ ] := Hˆ Bˆτ − Bˆτ Hˆ is the commutator. The example of the observable for the diffusion constant (N+n)/3 1 3 X 1 |x˜ (τ) − x˜ (0)|2 = |x˜(τ)|2 + |x˜(0)|2 − 2˜x(τ) · x˜(0)
6τ N + n k k 2(N + n)τ k=1 uses the time-correlation x˜ˆ(τ) · x˜ˆ(0) where Aˆτ = x˜ˆτ I and Cˆ0 = x˜ˆ0I and (N+n)/3 3 ˆ ˆ X X iτM 1/2Hˆ ˆ −iτM 1/2Hˆ ˆ x˜τ · x˜0 = e x˜kj e x˜kj . k=1 j=1 A main tool to determine the classical limit is to diagonalize (58), which is based on the following composition of Weyl quantizations: the symbol C for the product of two Weyl operators AˆBˆ = Cˆ is determined by i 1/2 (∇x˜0 ·∇p˜−∇x˜·∇p˜0 ) 0 0 C(˜x, p˜) = e 2M A(˜x, p˜)B(˜x , p˜ ) 0 =: (A#B)(˜x, p˜) , (59) x˜ =x ˜ p˜ =p ˜0

2 see [29]. Assume that Ψ : RN+n → Cd and Ψ(˜x) is any unitary matrix with the 2 Hermitian transpose Ψ∗(˜x) and define A¯ : RN+n × [0, ∞) → Cd by ˆ ∗ Aˆτ = Ψ(˜ˆ x)A¯τ Ψˆ (˜x) so that ˆ ∗ A¯τ = Ψˆ (˜x)Aˆτ Ψ(˜ˆ x) . Then ∗ ˆ ∗ [H,ˆ Aˆτ ] = Ψ[ˆ Ψˆ Hˆ Ψˆ , A¯τ ]Ψˆ and consequently ˆ 1/2 ∗ ˆ ∂τ A¯τ = iM [Ψˆ Hˆ Ψˆ , A¯τ ] . ∗ ∗ ∗ The composition rule (59) implies Ψˆ Hˆ Ψˆ = (Ψ #H#Ψ)b and A¯τ = Ψ #Aτ #Ψ. The next step is to determine Ψ so that H¯ := Ψ∗#H#Ψ is almost diagonal. Having H¯ diagonal implies that H¯ˆ is diagonal and then A¯ˆ remains diagonal if it initially were diagonal, since then d A¯ˆ (τ) = iM 1/2H¯ˆ A¯ˆ (τ) − A¯ˆ (τ)H¯ˆ
= 0 , for j 6= k. dτ jk jj jk jk kk The composition rule (59) with |p˜|2 H(˜x, p˜) = I + v(˜x) 2 implies that H¯ = Ψ∗#H#Ψ |p˜|2 1 = I + Ψ∗vΨ + ∇Ψ∗ · ∇Ψ 2 4M |p˜|2 1 = Ψ∗( I + v + Ψ∇Ψ∗ · ∇ΨΨ∗)Ψ , 2 4M as verified in [10, Lemma 3.1]. Therefore, the aim is to choose the unitary matrix Ψ so that it becomes an approximate solution to the nonlinear eigenvalue problem 1 v + Ψ∇Ψ∗ · ∇ΨΨ∗
Ψ = ΨΛ¯ 4M 4028 HAKON˚ HOEL AND ANDERS SZEPESSY in the sense that 1 v + Ψ∇Ψ∗ · ∇ΨΨ∗
Ψ = ΨΛ¯ + O(M −2) (60) 4M where Λ:¯ RN+n → Cd×d is diagonal, that is  ¯ 0 j 6= k Λjk(˜x) = ¯ . λj(˜x) j = k Such a solution Ψ then implies

|p˜|2 H¯ (˜x, p˜) = I + Λ(˜¯ x) + r (˜x) , 2 0 −2 where the remainder satisfies kr0kL∞(RN+n) = O(M ). A solution, Ψ, to this nonlinear eigenvalue problem is an O(M −1) perturbation of the eigenvectors to v(˜x) provided the eigenvalues do not cross and M is sufficiently large. The work [10, (3.18)] shows that (60) has a solution Ψ, if v is twice differentiable, the eigenvalues of v are distinct and M is sufficiently large. ˆ ˆ The canonical ensemble is typically based on trace(Ceˆ −H/T¯ )/trace(e−H/T¯ ). We will instead use the related trace(Cˆe\−H/T¯ )/trace(e\−H/T¯ ). If the density operators ˆ e−H/T¯ and e\−H/T¯ would differ only little it would not matter which one we use as a reference. Since we do not know if this difference is small in the case of a large number of particles, we may ask which density operator to use. The density −H/T¯ˆ operatorρ ˆq = e is a time-independent solution to the quantum Liouville-von Neumann equation 1/2 ˆ ∂tρˆt = iM [ˆρt, H¯ ] while the classical Gibbs density e−H/T¯ is a time-independent solution to the clas- sical Liouville equation

∂tρ¯t = −{ρ¯t, H¯ } , with the Poisson bracket in the right hand side. The corresponding density matrix symbol ρq is not a time-independent solution to the classical Liouville equation, since 1/2 0 = iM (ρq#H¯ − H¯ #ρq) 6= {ρq, H¯ } , and the classical Gibbs density is not a time-independent solution to the quantum Liouville-von Neumann equation, since

iM 1/2(e−H/T¯ #H¯ − H¯ #e−H/T¯ ) 6= {e−H/T¯ , H¯ } = 0 .

However, it is shown in [10] that a solution to the quantum Liouville equationρ ˆt −H/T¯ with initial data ρ0 = e generates only a small time dependent perturbation on observables up to time t < M, which motivates our use of e\−H/T¯ . The following result for approximating non equilibrium quantum observables by classical molecular dynamics observables is proved in [10].

Theorem 5.1. Assume that v satisfies (57), the d × d matrices A¯ ≡ A¯0 and B¯ are diagonal, the d × d matrix valued Hamiltonian H has distinct eigenvalues, and that LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4029 there is a constant C such that X α k∂ ψ k ∞ N ≤ C , k = 1, . . . , d , x˜ k L (R ) |α|≤2 X α ¯ ∞ N+n max k∂x˜ ∂x˜i λjkL ( ) ≤ C, i R |α|≤3 X α k∂ A¯ k 2 2(N+n) ≤ C, z˜ jj L (R ) |α|≤3 −H¯ (˜z)/T kB¯(˜z)e k 2 2(N+n) ≤ C, L (R ) then there is a constant C0, depending on C, such that the canonical ensemble average satisfies ¯ d ¯ ˆ ˆ ¯ −H/T ˆ ∗
Z ¯ j ¯ −Hjj (˜z0)/T 0 trace Aτ Ψ(Be )bΨ X Ajj(˜z (˜z0))Bjj(˜z0)e C − τ d˜z ≤ , d ¯ 0 ¯ 2(N+n) P R −Hkk(˜z)/T M ˆ \−H/T ˆ ∗ 2(N+n) e d˜z trace(Ψe Ψ ) j=1 R k=1 R j where z˜τ = (˜xτ , p˜τ ) is the solution to the Hamiltonian system

x˜˙ τ =p ˜τ ¯ p˜˙τ = −∇λj(˜xτ ), τ > 0 , 2 ¯ based on the Hamiltonian H¯jj(˜x, p˜) = |p˜| /2 + λj(˜x), with initial data (˜x0, p˜0) = 2(N+n) z˜0 ∈ R . 5.2. Langevin dynamics derived from quantum mechanics. Assume that the potential V (X) has the eigenvalues λj(X) and eigenvectors ψj(X), j = 1, . . . , d. The eigenvalues of the potential v(˜x(X, x)) = V (X) + Vb(x, X) will to leading order ∗ in kVb(·,X)k be given by λj(X) + ψj Vb(x, X)ψj. We assume now that all coupling ∗ potentials ψj Vb(x, X)ψj satisfy the weak coupling assumptions (3) and (4), namely ∗ ∗ min ψj (X)Vb(x, X)ψj(X) = ψj (X)Vb(aj(X),X)ψj(X) = 0 x so that ¯
¯ 00
λj x˜(x, X) = λj(X) + hx − aj(X), Vj x − aj(X) i ¯ 00 ∗ 00 ∗ where Vj = ψj Vbxxψj is a constant matrix as in (4) and each coupling ψj Vbψj ¯ 00 for j = 1, . . . , d yields one equilibrium position aj(X) and one coupling matrix Vj . The eigenvalues of the Hamiltonian symbol M M ( P · M −1P + p · M −1p) I + V (X) + V (x, X) 2 s 2 b b are then M M H¯ (X, P,x, p) = P · M −1P + p · M −1p + λ¯ x˜(x, X)
jj 2 s 2 b j M M = P · M −1P + p · M −1p + λ (X) + hx − a (X), V¯ 00x − a (X)
i . 2 s 2 b j j j j Let m be the mass ratio for a reference heat bath nuclei to a reference system nuclei. Theorems 3.4 and 4.1 show the classical dynamics provided by the Hamiltonians H¯jj are accurately approximated by the Langevin dynamics −1 dXt = MMs Ptdt ,

1/2 j −1 p 1/2 j dPt = −∇λj(Xt)dt − m κ Ms Ptdt + 2m κ T dWt , 4030 HAKON˚ HOEL AND ANDERS SZEPESSY

j 1 00 00 ¯ ¯ 0 where κ``0 = 3 hVj ∂X` aj, Vj ∂X aji, for m  1. The next step is to relate this 4πcj ` approximation by Langevin dynamics also to quantum observables in the canonical ensemble. In particular we need to obtain the Gibbs distribution of the heat bath particles from the quantum observables in the canonical ensemble. Define the probability, qj, to be in electron state j as

R −H¯jj (˜z)/T 2(N+n) e d˜z q := R , j = 1, . . . , d , (61) j d 0 P R −H¯kk(˜z )/T 0 2(N+n) e d˜z k=1 R then the molecular dynamics observable becomes a sum of observables in the dif- ferent electron states with initial Gibbs distribution, namely d ¯ Z ¯ j ¯ −Hjj (˜z0)/T X Ajj(˜z (˜z0))Bjj(˜z0)e τ d˜z d ¯ 0 2(N+n) P R −Hkk(˜z)/T 2(N+n) e d˜z j=1 R k=1 R d ¯ Z ¯ j ¯ −Hjj (˜z0)/T X Ajj(˜zτ (˜z0))Bjj(˜z0)e = qj d˜z0 . R −H¯jj (˜z)/T 2(N+n) 2(N+n) e d˜z j=1 R R

For instance if only the ground state matters we have q1 = 1 and qk = 0, k = 2, 3, 4, . . . , d. We simplify by letting the system nuclei have the same mass M and the heat bath nuclei the same mass mM. Then the diagonalized Hamiltonian can by (5) be written as |P |2 |p|2 H¯ (˜z) = + λ (X) + + h(x − a (X), V¯ 00(x − a (X)i , jj 2 j 2m j j j N N n n ¯ wherez ˜ = (X, P, x, p) ∈ R × R × R × R . Assume that the observables Ajj and B¯jj only depend on the system coordinates X and P , then the classical molec- ular dynamics approximation of the canonical quantum observables in Theorem 5.1 satisfies ¯ Z −Hjj (˜z0)/T ¯ j j
¯ e qjAjj Xτ (˜z0),Pτ (˜z0) Bjj(X0,P0) d˜z0 R −H¯jj (˜z)/T 2(N+n) 2(N+n) e d˜z R R 2 −( |P0| +λ (X ))/T Z Z e 2 j 0 = q A¯ Xj(˜z ),P j(˜z )
B¯ (X ,P ) j jj τ 0 τ 0 jj 0 0 |P |2 2n 2N R −( 2 +λj (X))/T R R 2N e dXdP R 2 −( |p| +hx−a (X ),V¯ 00(x−a (X ))/T e 2m j 0 j j 0 × 2 dX0dP0dxdp |p| ¯ 00 R −( 2m +hx−aj (X0),Vj (x−ak(X0))/T 2n e dxdp R 2 " −( |P0| +λ (X ))/T # Z e 2 j 0 = q A¯ Xj(˜z ),P j(˜z )
B¯ (X ,P ) dX dP E j jj τ 0 τ 0 jj 0 0 |P |2 0 0 2N R −( 2 +λj (X))/T R 2N e dXdP R where the expected value is with respect to the Gibbs measure

2 −( |p| +hx−a (X ),V¯ 00(x−a (X ))/T e 2m j 0 j j 0 2 (62) |p| ¯ 00 R −( 2m +hx−aj (X0),Vj (x−ak(X0))/T 2n e dxdp R of the heat bath coordinates conditioned on the initial system coordinates. We note that the Gibbs measure (62) is the invariant measure used to sample the initial heat bath configurations in theorems 3.4 and 4.1. Therefore the combintion of theorems 3.4, 4.1 and 5.1 show that canonical observables for a quantum system coupled LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4031 to a heat bath can be accurately approximated by Langevin dynamics, with the friction coefficient determined by (44) and (56), including several electron surfaces λj, j = 1, . . . , d. Theorem 5.2. Suppose that the assumptions in Theorems 5.1 and (3.4 or 4.1) hold and the observables A¯jj(·) and B¯jj(·) depend only on the system coordinates (X0,P0), then   ¯
b ¯ −H/T¯
b traceAˆ Ψˆ (Be¯\−H/T¯ )Ψˆ ∗
trace Aτ (X,P ) B(X,P )e τ = trace(Ψˆ e\−H/T¯ Ψˆ ∗) trace(e\−H/T¯ ) d Z X ¯ j j
¯ = E[qj Ajj Xτ (X0,P0),Pτ (X0,P0) Bjj(X0,P0) 2N j=1 R 2 −( |P0| +λ (X ))/T e 2 j 0 × dX dP ] |P |2 0 0 R −( 2 +λj (X))/T 2N e dXdP R + O(M¯ −1 +χ ¯) , where (¯χ, m,¯ M¯ ) := (m log m−1, m1/2, mM) for m → 0+ in Theorem 3.4 or (¯χ, m,¯ M¯ ) := (χ2δ log χ−1, χ2δ−1/2m1/2,M)
for χ → 0+ in Theorem 4.1,

j j and (Xt ,Pt ), for t > 0, is the solution to the Langevin equation j j dXt = Pt dt j j
j j j 1/2 dPt = −∇λj Xt dt − mκ¯ Pt dt + (2mκ ¯ T ) dWt , j j with initial data (X0 ,P0 ) = (X0,P0), 1 κj = hV¯ 00∂ a , V¯ 00∂ a i ``0 3 j X` j j X`0 j 4πcj and cj set by the Jacobian determinant of heat bath states at zero frequency in (27). The probability qj to be in electron state j is determined by (61) and the expected value is with respect to the Wiener process W , with N independent components.

6. Numerical example. We consider a single heavy particle in R3, the nearest neighbour lattice interaction V¯ 00, cf. (18), λ(X) = |X|2/2, c = 1 and

3 Y 1/2 2 1/4 β`(ω) = 1|ω|≤1 π (4 − ωi ) for all ` ∈ {1, 2, 3} , i=1 implying by (25) that 0 0 f(ω, `, ` ) = 1|ω|≤1 for all `, ` ∈ {1, 2, 3} . Since this contradicts that β is twice differentiable, an assumption that was used in the proof of Lemma 3.1, let us demonstrate that said lemma also applies in the current setting with jump-discontinuous β: √ Z t ¯ 00 −1/2 ¯ 00 (t − s) V
lim lim m hV cos √ a˙(Xs), ∂X` aids m→0+ n→∞ 0 m 4032 HAKON˚ HOEL AND ANDERS SZEPESSY Z t Z   X (t − s)ω dω 0 −1/2 √ 0 ˙ ` = lim m cos f(ω, `, ` ) 2 dτXs ds m→0+ 3 m ω `0 0 R Z t Z 1 Z 2π Z π   X −1/2 (t − s)ω ˙ `0 = lim m cos √ sin(θ)dθdαdωXs ds m→0+ m `0 0 0 0 0 Z t √ X sin ((t − s)/ m) ˙ `0 = 4π lim Xs ds m→0+ (t − s) `0 0 Z tm−1/2 X sin (τ) ˙ `0 = 4π lim Xt−m1/2τ dτ m→0+ τ `0 0 2 X ˙ `0 = 2π Xt , `0 R ∞ sin τ ˙ where the last equality follows from 0 τ dτ = π/2 and Xt being continuous with respect to t. We conclude that 1 2   κ = 2π 1 1 1 1 . (63) 1 To prove the convergence (28), observe first that Z ``0 dω sin(τ) K∞ (τ) = cos(τω)f(ω) 2 = 4π . 3 ω τ R

Introducing the mesh τk = 2kπ and √ bt/( m2π)c X Y˙ ` √ := 1 X˙ ` √ , t−τ m τ∈[τk,τk+1) t−τk m k=0 it follows that for any τ ∈ [τk, τk+1), there exists a path-dependent s ∈ [τk, τk+1) such that √ ˙ √ ˙ √ ¨ √ |Xt−τ m − Yt−τ m| = |Xt−s m|2π m. −1/2 By the splitting (30) with τ∗ = m min(t, 1), we have

Z τ∗ X 0 0 0 [h(X ,P ) K`` (τ)(X˙ ` √ − X˙ ` ) dτ] E t t ∞ t− mτ t 0 `0

Z τ∗ X 0 0 0 ≤ [h(X ,P ) K`` (τ)(X˙ ` √ − Y˙ ` √ ) dτ] E t t ∞ t−τ m t−τ m 0 `0

Z τ∗ X 0 0 0 + [h(X ,P ) K`` (τ)(Y˙ ` √ − X˙ ` ) dτ] E t t ∞ t−τ m t 0 `0 √ b1/( mπ)c √ X Z τk+1 | sin(τ)| ≤ C m dτ τ k=0 τk √ d1/ mπe √ X ≤ C m k−1 k=1 √ ≤ C m max 1, log(m−1)
, LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4033 and, in the case t > 1, √ Z t/ m ¯ 00 ¯ 001/2 √ |E[h(Xt,Pt) hV cos(τV )˙a(Xt− mτ ), ∂X` aidτ]| τ∗ √ t/ m Z 0 0 0 X `` ˙ ` √ ˙ ` √ ≤ |E[h(Xt,Pt) √ K∞ (τ)(Xt−τ m − Yt−τ m)dτ]| 1/ m `0 √ t/ m Z 0 0 X `` ˙ ` √ + |E[h(Xt,Pt) √ K∞ (τ)Yt−τ mdτ]| 1/ m `0 √ bt/( mπ)c  Z τk+1 Z τk+1  X √ | sin(τ)| sin(τ) ≤ C m dτ + dτ √ τ τ τ τ k=b1/( mπ)c k k √ ≤ C m max 1, log(m−1)
,

R τk+1 sin(τ) R 2(k+1)π cos(τ) where dτ = − 2 dτ to bound the last summand. For the τk τ 2kπ τ current setting, equation (28) follows by the same reasoning as in the proof of Lemma 3.1.

6.1. Dynamical systems. For a given mass ratio m, the generalized Langevin equation of the heat bath dynamics takes the form X˙ (t) = P (t) Z t t − s P˙ (t) = −∇λ(X(t)) − K∞ √ P (s) ds + ζ(t) , 0 m where ζ(t) denotes a mean-zero Gaussian process with ζ1 = ζ2 = ζ3 and 1 1 11 √ E[ζ (s)ζ (t)] = TK∞ ((t − s)/ m) , cf. (35) and (36). The associated Langevin dynamics is

X˙ L(t) = PL(t) 1/2 1/2 1/2 P˙L(t) = −∇λ(XL(t)) − m κPL(t) + (2m κT ) W˙ (t) with κ given by (63). We will compare the dynamical systems numerically for the initial data XL(0) = X(0) = ξX (1, 1, 1) and PL(0) = P (0) = ξP (1, 1, 1), where ξX and ξP are independent identically distributed standard Gaussians that are sampled pathwise. Due to the initial data and ∇λ(X) = X, it holds that X(t),P (t) ∈ Span((1, 1, 1)) for all t ≥ 0. For this particular example, it therefore suffices to 1 1 1 1 study the reduced dynamics (X ,P ) and (XL,PL) rather than the respective 6 dimensional full systems. The respective reduced dynamics are equal in distribution to X˙ 1(t) = P 1(t) √ √ Z t sin((t − s)/ m) P˙ 1(t) = −X1(t) − 12π m P 1(s) ds + ζ1(t), 0 t − s and ˙ 1 1 XL(t) = PL(t) √ (64) ˙ 1 1 2 1 1/4 1/2 ˙ 1 PL(t) = −XL(t) − 6π mPL(t) + 2πm T W (t) . 4034 HAKON˚ HOEL AND ANDERS SZEPESSY

6.2. Numerical integration schemes. Langevin dynamics (based on St¨ormer– Verlet/Ornstein–Uhlenbeck [21]): r √ √ 1 − exp(−12π2 m∆t/2) P ∗ = exp(−6π2 m∆t/2)P 1 + T 1/2 ξ L,n+1/2 L,n 3 2n−1 ∆t P 1 = P ∗ − X1 L,n+1/2 L,n+1/2 L,n 2 1 1 1 XL,n+1 = XL,n + PL,n+1/2∆t ∆t P ∗ = P 1 − X1 L,n+1 L,n+1/2 L,n+1 2 r √ √ 1 − exp(−12π2 m∆t/2) P 1 = exp(−6π2 m∆t/2)P ∗ + T 1/2 ξ , L,n+1 L,n+1 3 2n where ξn is a sequence of independent and identically distributed standard normals. The scheme is motivated from the splitting method with symplectic integration of the Hamiltonian system ˙ 1 1 ˙ 1 1 XL(t) = PL(t), PL(t) = −XL(t) and exact solution of the Ornstein–Uhlenbeck equation √ ˙ 1 2 1 1/4 1/2 ˙ 1 PL(t) = −6π mPL(t) + 2πm T W (t), cf. [21]. For the heat bath dynamics we construct a splitting scheme which for a uniform 1 1 mesh tk = k∆t computes the position at every timestep (X (t0),X (t1),...) and 1 1 1 the momentum at every half-timestep (P (t0),P (t1/2),P (t1),...). The damping term’s integral is approximated as follows:

√ n √ Z tn+1 Z tk+1/2 sin((tn+1 − s)/ m) 1 X 1 sin((tn+1 − s)/ m) P (s)ds ≈ P (tk) ds tn+1 − s tn+1 − s 0 k=0 tk n √ Z tk+1 X 1 sin((tn+1 − s)/ m) + P (tk+1/2) ds tn+1 − s k=0 tk+1/2 n X h 1 √ √ = P (tk)(Si(tn+1−k/ m) − Si(tn+1/2−k/ m)) k=0 1 √ √ i + P (tk+1/2)(Si(tn+1/2−k/ m) − Si(tn−k/ m)) , (65) where the last equality follows from √ √ Z b Z (tn+1−a)/ m sin((tn+1 − s)/ m) sin(s) ds = √ ds a tn+1 − s (tn+1−b)/ m s and Z t sin(s) Si(t) := ds . 0 s Introducing the notation √ √ ∆Si` := Si(t`+1/2/ m) − Si(t`/ m) , LANGEVIN DYNAMICS FROM QUANTUM MECHANICS 4035 the approximation (65) takes the compact form √ n Z tn+1 sin((tn+1 − s)/ m) 1 X 1 1 P (s)ds ≈ P (tk)∆Sin+1/2−k + P (tk+1/2)∆Sin−k , tn+1 − s 0 k=0 and, similarly, √ n Z tn+1/2 sin((tn+1/2 − s)/ m) X P 1(s)ds ≈ P 1(t )∆Si t − s k n−k 0 n+1/2 k=0 n−1 X 1 + P (tk+1/2)∆Sin+1/2−k . k=0 We make use of the above approximations of the damping term integral in the following splitting scheme for the heat bath dynamics: " ∆t P 1 = P 1 − X1 − ζ1(t ) n+1/2 n 2 n n

n n−1 !# √ X 1 X 1 + 12π m Pk ∆Sin−k + Pk+1/2∆Sin+1/2−k k=0 k=0 1 1 1 Xn+1 = Xn + Pn+1/2∆t " ∆t P 1 = P 1 − X1 − ζ1(t ) n+1 n+1/2 2 n+1 n+1

n # √ X 1 1 + 12π m Pk ∆Sin+1/2−k + Pk+1/2∆Sin−k . k=0 1 1 1 The mean-zero Gaussian vector ζ = (ζ (t0), ζ (t1), . . . , ζ (tN )) is sampled by com- puting the square root of the Toeplitz matrix with first row vector + √ √ √ (TK∞(0 / m),TK∞(t1/ m),...,TK∞(tN / m)) √ and multiplying the square root matrix,√ say K, with an (N + 1)−vector ξ of iid standard normals components: ζ = Kξ. See [11] for further details on sampling of Gaussian processes, [4, 24, 16, 19,3,1, 17] for numerical methods for Langevin dynamics and [2,9, 13] and [16, Chapter 8.7] for an alternative numerical method for generalized Langevin equations based on a truncated Prony series approximation of the kernel K∞. 6.3. Observations. By setting the temperature to T = 3, the stationary distribu- tion of the exact Langevin dynamics (64) becomes N(0,I) for any m > 0, with I denoting the identity matrix in R2. In order to reduce the computational challenges of long time numerical integration, we sample the initial data from this stationary, i.e., (ξX , ξP ) ∼ N(0,I), as we assume this yields initial data for the numerical 1 1 1 1 dynamics (XL,0,PL,0) and (X0 ,P0 ) that are very close to their respective station- ary distributions. The numerical computations are performed using ∆t = 0.005, 5 N = 4000 integration steps relating to the final time tN = 20, and M = 2 × 10 solution realizations of the respective dynamics. Figure1 shows good correspon- 1 1 dence between the final time joint distribution of the heat bath dynamics (XN ,PN ) 1 1 and the Langevin dynamics (XL,N ,PL,N ) over a range of m-values (all being ap- proximately N(0,I)-distributed). Figures2 and3 show that the autocorrelation 4036 HAKON˚ HOEL AND ANDERS SZEPESSY

Figure 1. (Left column) unit-volume scaled histogram for the fi- nal time position and momentum of the heat bath dynamics for a series of m-values and (right column) corresponding histograms for the Langevin dynamics. for both the position and the momentum of the heat bath dynamics converges to the corresponding ones for the Langevin dynamics as m → 0+. The computa- tions of the autocorrelation functions are made under the assumption that both kinds of dynamics are wide-sense stationary. For any m > 0 this property holds for the Langevin dynamics, and in the limit m → 0+ it also holds for the heat bath dynamics. In addition to our above observations, we believe it would be of great interest to obtain numerical verification for the heat bath dynamics weak convergence rate O(m1/2) in (45). But, most likely due to computational constraints, we are unable to achieve this currently since even at the quite computationally demanding level of generating M = 200000 heat bath dynamics sample paths, it seems that the sample error dominates errors pertaining to the parameter m.

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1 1

0.5 0.5

-10 -5 0 5 10 -10 -5 0 5 10

1 1 0.5 0.5 0 0 -10 -5 0 5 10 -10 -5 0 5 10

1 1 0.5 0.5 0 0 -10 -5 0 5 10 -10 -5 0 5 10

1 1 0.5 0.5 0 0 -10 -5 0 5 10 -10 -5 0 5 10

Figure 2. (Left column) heat bath dynamics autocorrelation function E[X1(t)X1(10)] for a series of m-values and (right column) corresponding Langevin dynamics autocorrelation functions 1 1 E[XL(t)XL(10)].

1 1 0 0.5 -1 0 -4 -2 0 2 4 -4 -2 0 2 4

1 1 0.5 0.5 0 0 -4 -2 0 2 4 -4 -2 0 2 4

1 1 0.5 0.5 0 0 -4 -2 0 2 4 -4 -2 0 2 4

1 1 0.5 0.5 0 0 -4 -2 0 2 4 -4 -2 0 2 4

Figure 3. (Left column) heat bath dynamics autocorrelation function E[P 1(t)P 1(10)] for a series of m-values and (right column) corresponding Langevin dynamics autocorrelation functions 1 1 E[PL(t)PL(10)]. 4038 HAKON˚ HOEL AND ANDERS SZEPESSY

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