The Langevin Limit of the Nosé-Hoover- Langevin Thermostat

Transcription

1 Journal of Statistical Physics manuscrit No. (will be inserted by the editor) Jason Frank Georg Gottwald The Langevin Limit of the Nosé-Hoover- Langevin Thermostat the date of receit and accetance should be inserted later Abstract In this note we study the asymtotic limit of large variance in a stochastically erturbed thermostat model, the Nosé-Hoover-Langevin device. We show that in this limit, the model reduces to a Langevin euation with one-dimensional Wiener rocess, and that the erturbation is in the direction of the conjugate momentum vector. Numerical exeriments with a double well otential corroborate the asymtotic analysis. Keywords thermostat methods Nosé-Hoover dynamics Langevin dynamics homogenization methods canonical samling Introduction Deterministic thermostats for canonical samling were introduced by Nosé [6, 7] and Hoover [6] and are commonly used in molecular dynamics to simulate systems at constant temerature. These methods extend the hase sace of a Hamiltonian system by one or more degrees of freedom such that the extended dynamics when rojected back onto the original hase sace reserve the Gibbs distribution, i.e. the canonical distribution is a steady state of the Liouville flow associated with the rojected dynamics. Bulgac and Kusnezov [] generalized the Hoover thermostat to noncanonical Hamiltonian systems. Proving ergodicity of deterministic thermostats has resented a challenge (see for examle [3, 4]), and this has led recently to the construction of stochastically forced thermostat euations, referred to here as Nosé-Hoover-Langevin (NHL) thermostats, in which the augmented degrees of freedom satisfy a Langevin euation [2,2]. J. E. Frank Centrum Wiskunde & Informatica, P.O. Box 9479, 9 GB Amsterdam, The Netherlands jason@cwi.nl G. A. Gottwald School of Mathematics and Statistics, University of Sydney, NSW 26, Australia georg.gottwald@sydney.edu.au

2 2 Consider a Hamiltonian system dx dt = J H(x), () where x R n, J T = J and H : R n R. Assuming that the system is in thermal euilibrium with an external heat bath of constant temerature T, the NHL thermostated dynamics is given by dx = [J H(x)+yg(x)] dt [ ] ασ 2 dy = ( g(x) β H g(x)) α 2 y dt + σ dw, (2) with y R and initial conditions x() = x and y() = y. Here W is a onedimensional indeendent Brownian motion, and β = /(k B T) is roortional to the inverse temerature with k B being the Boltzman constant. We denote by the gradient oerator with resect to the x-variables only. The NHL system (2) is constructed to reserve the extended measure ρ(x,y) = ex( βh(x) αy 2 /2) and differs from a classical Langevin euation in a number of ways. For one the Wiener rocess W does not directly erturb the dynamics in x, but instead does so indirectly through y after integration. In essence this can be seen as roviding a memory effect; that is, the erturbed dynamics in x is non-markovian. Furthermore, (2) makes use of only a scalar stochastic variable, and it is a uestion how this scalar erturbation ervades the rest of the dynamics. To assure ergodicity, the (otherwise arbitrary) vector field g(x) can be chosen to satisfy a so-called Hörmander controllability condition [5]. The Hörmander condition for the couled system (2) can be formulated by writing it in the form of a stochastic differential euation with degenerate noise term dz = r(z)dt + s(z)dw, where r and s are vector fields in R n+ and W(t) is a scalar Wiener rocess. The Hörmander condition states that the Lie algebra generated by the vector fields r and s sans the whole R n+, and therefore ensures that the noise eventually ervades all dimensions of hase sace. In the resent case, this reduces to the reuirement that the Lie algebra generated by vector fields J H and g sans R n, see [2]. In this aer, we are interested in the limiting case when the thermostat variable y can have large values due to an increased variance /α. In a heuristic way, it can be understood that the limiting dynamics is a Langevin euation (see below) as the thermostat variable y will tend to Brownian motion in the case when its relaxation time /(ασ 2 ) tends to zero. We will in the following make this heuristic more recise. In articular we will see that additional drift terms arise. A general Langevin thermostat for () is [ dx = J H(x)+ 2 (Σ(x)Σ T (x)) β ] 2 Σ(x)Σ T (x) H(x) dt + Σ(x)dw, (3) where w(t) is a Wiener rocess in R n and Σ(x) R n n induces multilicative noise. The form of this euation is dictated by the demand that the Gibbs measure ρ = ex( β H(x)) be stationary under the flow of the associated Fokker-Planck euation, for arbitrary choice of Σ(x). We show that the limiting dynamics of the NHL model (2) is a Langevin model (3) with rank-one multilicative noise Σ(x). For mechanical systems with

3 3 x = (, ) and searable Hamiltonian H = 2 T M +V(), we will see that in the large noise limit the NHL model (2) is euivalent to the momentum directed Langevin thermostat roosed in [], of which the stochastic velocity rescaling thermostat of Bussi et al. [2,3] is a articular case. We introduce the scaling α ε 2 α, y ε y, σ ε 2 σ (4) with ε. The NHL euation (2) then becomes dx = [J H(x)+ ε ] yg(x) dt ] 2 y [ ασ 2 dy = ( g(x) β H g(x)) ε α ε 2 dt + σ dw. (5) ε This articular scaling in ε allows for the effective reduced dynamics to cature diffusive effects rather than classical averaging. 2 Homogenization We will analyse the NHL system (5) in the framework of the backward Kolmogorov euation for the conditional exectation value of some sufficiently smooth observable φ(x, y) defined as v(x,y,t) = E[φ(x(t),y(t)) x() = x,y() = y ]. Here the exectation value is taken with resect to Brownian motion driving aths. Droing the subscrits, we study the following Cauchy roblem for t [, ) with the generator where L = ασ 2 v (x,y,t) = L v(x,y,t) t v(x,y,) = φ(x,y), (6) L = ε 2 L + ε L +L 2, 2 y y + 2 σ 2 yy (7) L = yg(x) + α ( g(x) β H g(x)) y (8) L 2 = J H. (9) Pioneered by Khasminsky [7], Kurtz [8,9,] and Paanicolaou [8] singular erturbation theory can be formulated for a erturbation exansion according to v(x,y,t) = v + εv + ε 2 v 2 +. ()

4 4 A comrehensive exosition of the theory of stochastic model reductions and their imlementation is given for examle in [2, 9]. Substituting the series () into the backward Kolmogorov euation (6) we obtain at lowest order, O(/ε 2 ), L v =. () The fast dynamics associated with the generator L is given by an Ornstein- Uhlenbeck rocess and is therefore ergodic. Ergodicity of the fast rocess imlies that exectation values do not deend on initial conditions y. Hence the constant solution v = v (x,t) is the only solution of (). Ergodicity is euivalent to the existence of a uniue invariant density, i.e. that the euation L ρ =, has a uniue solution ρ (y). Here L is the formal L 2-adjoint of the generator L. For later reference we resent the uniue invariant density of the fast Ornstein- Uhlenbeck rocess associated with L α ρ (y) = 2π ex( α 2 y2 ). (2) At the next order, O(/ε), we obtain L v = L v. (3) To assure boundedness of v (and thereby of the asymtotic exansion ()) the solvability condition rescribed by the Fredholm alternative has to be satisfied. Euation (3) is solvable only rovided the right-hand-side is in the sace orthogonal to the (one-dimensional) null sace of the adjoint L, i.e. if L v ρ = yg(x) ρ v (x,t) =, where h ρ := ρ (y)h(x,y)dy denotes the average of an observable h(x,y) over the invariant density. Since y ρ =, there exists a solution of (3), which is readily calculated as v (x,y,t) = 2 ασ 2 yg v + R(x), (4) where R(x) lies in the kernel of L. Note that the vanishing of the average of the thermostat erturbation with resect to the invariant measure induced by the fast thermostat variable (i.e. yg ρ = ) imlies that classical averaging would roduce a trivial reduced dynamics with dx = J Hdt without any trace of the fast stochastic thermostat variable. At the next order, O(), we obtain the desired evolution euation for v, L v 2 = t v L v L 2 v. (5) As discussed above, the Ornstein-Uhlenbeck rocess was introduced recisely into the framework of thermostat euations to render the full system ergodic.

5 5 Again a solvability condition has to be imosed which reads t v = L 2 v ρ + L v (x,t) ρ. (6) Using y 2 ρ = /α we obtain the full reduced slow backward-kolmogorov euation ( t v = J H + 2 [ ( gg T ) α 2 σ 2 βgg T H ] ) v + 2 α 2 σ 2 ggt : v. (7) Here we define that the divergence oerator acting on matrix valued functions h acts by contraction as { h} i = j (h i j ), and the inner roduct of matrices is defined as A : B = a i j b i j = Tr(AB T ), which induces the Frobenius norm. Note that R(x) does not contribute to the dynamics. We can therefore choose R(x) = in order to assure that v ρ = v +O(ε 2 ) The slow reduced Langevin euation associated with the reduced backward Kolmogorov euation (7) is then dx = F(X)dt + S(X)dW with X() = x, (8) with one-dimensional Wiener rocess W and where the drift coefficient vector F(X) and the diffusion coefficient vector S(X) are given by F(X) = J H + 2 α 2 σ 2 [ ( gg T ) βgg T H ] S(X) = 2 ασ g. (9) We conclude that (8) (9) is in the form of the Langevin euation (3) for the case Σ(x) = ασ 2 g(x) and scalar Wiener rocess W. 3 Reduced model for molecular dynamics The original thermostat devices of Nosé and Hoover [6, 7, 6] were develoed for molecular dynamics roblems in canonical Hamiltonian form. (For an alication of the NHL thermostat to oint vortices, see [4].) Since Nosé-Hoover thermostats are rimarily used in the molecular dynamics context, in this section we secify the reduced Langevin model derived in the revious section to the articular case of mechanical systems. To this end we take n = 2d, and consider a mechanical system with ositive diagonal mass matrix M and hase sace coordinates x = (, ),, R d, and with Hamiltonian H(x) = H(, ) = 2 T M +V(),

6 6 and canonical structure matrix J = ( I I ). The deterministic euations of motion () are then given by d dt = M, d dt = V(). (2) For the NHL system (2) we make the secific choice g(x) = g(, ) = (, ). Note that the Hörmander condition is immediate here, since the original vector field M, V/ and the erturbation vector field g(, ) = (, ) are trivially linearly indeendent. In molecular dynamics, this choice is hysically motivated, since the thermostat acts to slow or accelerate the motion. One would then need to check the Hörmander condition for the given otential V. Our numerical examle in the next section is in R 2, so the Hörmander condition is satisfied if J H and g are linearly indeendent. The reduced Langevin euation (8) (9) then takes the form d = M dt [ d = V()+ 2 ( d + β T α 2 σ 2 M ) ] dt + 2 dw, (2) ασ from which it is aarent that the noise and dissiation act in the direction of the generalized momentum vector. In articular, we note that the reduced Langevin euation (2) and the NHL system (2) reserve the set of euilibria of the full deterministic dynamics (). We also oint out that the reduced Langevin euation (2) is euivalent to the momentum directed Langevin thermostat described in [] as a generalization of the stochastic velocity rescaling thermostat of Bussi et al. [2,3]. 4 Numerical verification In this Section we will numerically illustrate that the reduced Langevin euation (8) is a good athwise model of the small-ε limit of the NHL model (5). We show this for a simle system with one degree of freedom (d = ). Parameters for the NHL model (2) and its reduced Langevin euation (2) are chosen to be α = ε 2, σ = /ε 2. Both the NHL model (2) and the reduced model (2) were imlemented using a slitting method, euivalent to the Störmer-Verlet method for the Hamiltonian vector field J H, and solving the thermostat dynamics using a comosition of a half-ste each of the slit-ste backward Euler method [5] and the Euler- Maruyama method, which effects the midoint rule in the absence of the noise

7 7 term. For the reduced model (2) this becomes n+/2 = n + t 2 M n n+/2 = n t 2 V( n+/2 ) n+/2 = n+/2 t 2 Γ( n+/2 ) n+/2 = n+/2 + 2 ασ n+/2 (W n+/2 W n ) n+ = n+/2 t 2 Γ( n+/2)+ 2 ασ n+/2(w n+ W n+/2 ) n+ = n+ t 2 V( n+/2 ) n+ = n+/2 + t 2 M n+, (22a) (22b) (22c) where Γ() = 2(ασ) 2 (β T M (d+)). The variables W n, W n+/2, etc. in (22b) (22c) denote successive values of a Wiener rocess, i.e., the increments W n+/2 W n are drawn from a normal distribution with variance t/2. The slitste backward Euler method (22a) (22b) is imlicit in the momenta, which for the simle examle we consider is comutationally feasible. Numerical exeriments were done for the following double well otential V() = , with M =. In all exeriments we took as initial conditions (, ) = (,/4) and set β = as the inverse temerature. In this case, the canonical euilibrium density of the system (2) is given by ] ρ(, ) = ex [ β( ). (23) Figure demonstrates that the NHL model (2) and the reduced Langevin model (2) both samle the canonical euilibrium measure, as designed. The canonical euilibrium measure (23) is shown on the left, along with nearly-converged emirical measures obtained from a single, long simulation each of the NHL model (2) and the reduced Langevin model (2), resectively, in the middle and on the right. Each simulation was run on the interval t = [, 5 ] with t =.. If the time interval is increased by a factor, the samled emirical densities are indistinguishable from the theoretical density (23). The agreement is exected, by construction of the models, if the simulation is ergodic. We next resent results of simulations of the NHL model (2) for varying values of ε. Recall that the variance of the thermostat variable y is ε 2. We thus exect that for large values of ε the effect of the NHL thermostat will be weak on the chosen finite time interval t [, 3 ], and the trajectories will behave nearly deterministically. The reduced Langevin model (8) is derived in the limit ε, so

8 Fig. Contour lot of the canonical euilibrium measure for a article in a double well otential. Left: theoretical measure (23), middle: emirical measure calculated from a simulation of the NHL model (2) with ε =, right: emirical measure calculated from a simulation of the reduced Langevin model (2). we exect that the NHL dynamics will be more erratic in this limit. This behavior is confirmed by Figures 2 and 3. For ε = the trajectories are smooth and nearly eriodic. The left otential well is only samled once on this time interval. Note, however, that the NHL model satisfies the Hörmander condition [5] and is exected to be ergodic for this roblem. Indeed, for a simulation on the much longer interval t [, 7 ] the samled measure is indistinguishable from the theoretical measure shown in Figure (left) for all values of ε. For ε = and ε =, the solution is still uite smooth but both wells are freuently visited on the given time interval. For ε =. the hase trajectories are much more erratic, with large fluctuations in the momentum. In this regime the trajectories look very similar to those of the Langevin model (8) (not shown). On the other hand, Figure 3 shows that the samling behavior of the double well otential is ualitatively similar for ε = and ε =., suggesting that the time series for is aroximately converged even for ε =. We can uantify the convergence by numerically estimating the mean residence time τ, i.e. the average time the trajectory sends in the resective otential wells. We find that the mean residence times for ε = and ε =., which are τ = and τ = 2 resectively, are both close to the mean residence time of the reduced Langevin model with τ = 3. Whereas the dynamics of the osition variable of the NHL model (2) has converged in the sense that its statistics converges to the statistics of the reduced model (8) for ε =, this is not the case for the momentum variable. In Figure 2 it is clearly seen that the variance of the momentum variable strongly differs for ε = and ε =.. We conclude that the small-ε limit corresonds to large stochastic forcing and large influence of the thermostat variable y, consistent with the scaling we roosed in the derivation of the reduced Langevin euation (8) (9) from the NHL model (2). We now show that trajectories of the reduced Langevin euation (8) (9) converge ath-wise to solutions of the full NHL model (2) in the limit of ε. To study convergence we introduce the suremum error ε = su NHL (t) red (t), (24) t [,t ] between solutions of the full NHL model (2), denoted by NHL, and solutions of the reduced Langevin model (2), denoted by red (t), on a fixed time interval

9 9 ε =.5.5 ε = ε =. ε=.5 Fig. 2 Phase sace trajectories for diminishing ε, comuted with the NHL model (2) using an integration stesize of t = 3. [,t ]. To investigate ath-wise convergence we use identical Wiener increments for both the NHL model (2) and the reduced Langevin model (2). How the error ε scales with ε is illustrated in Figure 4 where a clear uadratic scaling is seen with ε ε 2. Although we do exect a scaling with ε, we are not aware of any rigorous results that exlain the uadratic behavior. It is remarkable that the limiting Langevin rocess that we derive here by means of a stochastic singular erturbation analysis [9] has been heuristically roosed before e.g. by Leimkuhler et al. [] and Bussi et al. [2,3] to construct a thermostat with a mild effect on the dynamics, only along the direction of motion. Acknowledgements We are grateful to the Isaac Newton Institute where arts of this research were erformed during the rogramme Mathematical and Statistical Aroaches to Climate Modelling and Prediction (CLP). GAG acknowledges suort from the Australian Research Council. JEF acknowledges suort from the Netherlands Organization for Scientific Research (NWO). References. A. B ULGAC AND D. K USNEZOV, Canonical ensemble averages from seudomicrocanonical dynamics, Phys. Rev. A 42 (99), G. B USSI, D. D ONADIO AND M. PARINELLO, Canonical samling through velocity rescaling, J. Chem. Phys 26 (27), 4,.

10 2 ε = 2 ε = 2 5 t 2 ε = 2 5 t 2 ε =. 2 5 t 2 5 t Fig. 3 Time series of osition for diminishing ε with the NHL model (2). The integration stesize is t = 3. 2 ε ε Fig. 4 Log-log lot of the suremum error (24) on t [, ] between the numerical solution comuted with the NHL dynamics (2) and that comuted with the reduced Langevin model (2). A time ste of t = e 6 was used and identical Wiener increments have been emloyed for both models. The sloe of the dashed line indicates uadratic convergence in ε.

11 3. G. BUSSI AND M. PARINELLO, Stochastic thermostats: comarison of local and global schemes, Com. Phys. Comm. 79 (28), S. DUBINKINA, J. FRANK, AND B. LEIMKUHLER, Simlified Modelling of Energetic Interactions with a Thermal Bath, with Alication to a Fluid Vortex System, Multiscale Model. Simul. (2), to aear. 5. D. J. HIGHAM, X. MAO, AND A. M. STUART, Strong convergence of Euler-tye methods for nonlinear stochastic differential euations, SIAM J. Numer. Anal. 4 (22), W. HOOVER, Canonical dynamics: euilibrium hase sace distributions, Phys. Rev. A 3 (985), R. Z. KHASMINSKY, On stochastic rocesses defined by differential euations with a small arameter. Theory Prob. Alications (966), T. G. KURTZ, A limit theorem for erturbed oerator semigrous with alications to random evolutions. J. Functional Analysis 2 (973), T. G. KURTZ, Limit theorems and diffusion aroximations for density deendent Markov chains. Math. Prog. Stud. 5 (976), T. G. KURTZ, Strong aroximation theorems for density deendent Markov chains. Stochast. Proc. Al. 6 (978), B. LEIMKUHLER, E. NOORIZADEH, AND O. PENROSE, Comaring the efficiencies of stochastic isothermal molecular dynamics methods, J. Stat. Phys. (2), to aear. 2. B. LEIMKUHLER, E. NOORIZADEH, AND F. THEIL, A gentle stochastic thermostat for molecular dynamics, J. Stat. Phys. 35 (29), F. LEGOLL, M. LUSKIN AND R. MOECKEL, Non-ergodicity of Nosé-Hoover thermostatted harmonic oscillator, Arch. Rational Mech. Anal. 84 (27), F. LEGOLL, M. LUSKIN AND R. MOECKEL, Non-ergodicity of Nosé-Hoover dynamics, Nonlinearity 22 (29), J.C. MATTINGLY, A.M. STUART AND D.J. HIGHAM, Ergodicity for SDEs and aroximations: locally Lischitz vector fields and degenerate noise, Stoch. Proc. Al. (22), S. NOSÉ, A molecular dynamics methods for simulations in the canonical ensemble, Mol. Phys. 52 (984), S. NOSÉ, A unified formulation of the constant temerature molecular dynamics method, J. Chem. Phys. 8 (984), G. C. PAPANICOLAOU, Introduction to the asymtotic analysis of stochastic euations, Modern Modeling of Continuum Phenomena ed R C DiPrima (Providence, RI: AMS), (974). 9. G. A. PAVLIOTIS AND A. M. STUART, Multiscale Methods Averaging and Homogenization. Texts in Alied Mathematics 53, Sringer, New York, (28). 2. A. SAMOLETOV, M. A. J. CHAPLAIN, AND C. P. DETTMANN, Thermostats for slow configurational modes, J. Stat. Phys. 28 (27), D. GIVON, R. KUPFERMAN, AND A. STUART, Extracting macroscoic dynamics: model roblems and algorithms, Nonlinearity 7 (24), R55 R27.