Nosé-Hoover Thermostats

Transcription

1 Nosé- Nosé- Texas A & M, UT Austin, St. Olaf College October 29, 2013

2 Physical Model Nosé- q Figure: Simple Oscillator p

3 Physical Model Continued Heat Bath T p Nosé- q Figure: Simple Oscillator in a heat bath

4 and Notations Nosé- for simple harmonic oscillator q = p, ṗ = q Nosé- T denotes temperature ξ denotes feedback term ε denotes feedback constant Extended energy of system E = 1 2 (q2 + p 2 + ξ 2 ), de dt = εξt (2)

5 and Notations Nosé- for simple harmonic oscillator q = p, ṗ = q εξp, ξ = ε(p 2 T ) (1) Nosé- T denotes temperature ξ denotes feedback term ε denotes feedback constant Extended energy of system E = 1 2 (q2 + p 2 + ξ 2 ), de dt = εξt (2)

6 History Nosé- History S. Nosé found a Hamiltonian system in 1984 that included a time-scaled variable s. q = p ṗ = s V (3) ms ṡ = sp s ṗ s = [ p 2 ] ms 2 kt William added to Nosé s work in 1985, scaling out extra time variable. q = p m ṗ = V ξp (4) ξ = 1 m p 2 T

7 Canonical Density Nosé- Solving for density The canonical density, denoted Ω = f(q, p, ξ; T )dv, (5) satisfies dω dt 0 (6) refore we get following PDE f v + f div(v) = 0 (7) We solved this equation using separation of variables: f(q, p, ξ; T ) = (q 2 +p 2 +ξ 2 ) e 2T Z(T ) = 1 Z(T ) e 1 T E (8)

8 Special Cases Nosé- Special Solutions Consider initial conditions (0, 0, ξ 0 ) where ξ 0 R, n re is an exact solution of form q(t) = 0, p(t) = 0, ξ(t) = ξ 0 εt t (9) q ξ ξ 0 p

9 Special Cases cont. Zero Temperature For T = 0, E(t) is constant and all solutions lie on a sphere of radius equal to magnitude of initial conditions. Nosé- Figure: [q 0, p 0, ξ 0 ] = [0.1, 0.1, 1], ɛ =.5, T = 0

10 Special Cases Nosé- Free Particle [ 1985] Introducing spring constant ω 2, we get system q = p, ṗ = ω 2 q εξp, ξ = ε(p 2 T ). (10) Setting τ = 1 2 p2 and ω 2 = 0 leads to system τ = 2εξτ, ξ = ε (2τ T ) (11) which is integrable with trajectories lying on curves given by level sets of H(τ, ξ) = 1 2 ξ2 + τ 1 T ln τ = K (12) 2

11 Periodicity Nosé- Motivation Prove nonexistence of a function h : R 3 R such that h is deformable to a union of standard tori. Fourier Analysis The original differential equation is γ(t) = f(γ(t)). Applying Fourier transform we find G(t) = a n/ k 1 a k cos( 2πkt τ ) + b k sin( 2πkt ). (13) τ We can compare G(t) γ(t) and 1 n n 1 G n(t) γ n (t) 2 to check for periodicity.

12 Pictures Nosé- [q 0, p 0, ξ 0 ] = [0, 1.55, 0], β = 1, ε = 1, τ = 5.58, Error= [ 1985] [q 0, p 0, ξ 0 ] = [0.01, 0.01, 2], β = , ε = 0.4, τ = 136, Error=

13 More Pictures Nosé- [q 0, p 0, ξ 0 ] = [0.48, 0.620, 0], β = 0.1, ε = 0.1, τ = 125 [q 0, p 0, ξ 0 ] = [0.48, 0.685, 0], β = 0.1, ε = 0.1, τ = 842

14 Orbit Averages Nosé- Theorem (Birkhoff Ergodic Theorem) If a system has a stationary density and solutions exist for all time, n for any continuous function f(x), 1 f(x 0 ) = lim τ τ τ 0 f(x(s))ds exists with probability 1. Here, f(x 0 ) is average of f along orbit with initial value x 0.

15 Orbit Averages- Global Existence Nosé- Gronwall s Inequality Let A, B : [a, b] R be constants and u differentiable in (a, b) satisfying u(t) A u(t) + B, t [a, b] (14) n we have following inequality u(t) Ce A(t a) (15) for some constant C > 0 depending on A, B and u(a).

16 Orbit Averages- Global Existence Continued Proposition Since Ė(t) ε ξ ε 2E 2ε(1 + E), (16) Nosé- n by Gronwall s inequality we know E(t) Ae 2εt. (17) Plugging initial condition x 0 = (q 0, p 0, ξ 0 ), n E(t) (εt )2 t 2 εt ξ 0 t + E(0). (18) 2 Theorem (Global Existence of Solutions) For any initial condition, x 0 = (q 0, p 0, ξ 0 ), re exists a solution for all time t.

17 Orbit Averages- Recurrence Nosé- Definition Consider system of ODE ẋ = F (x), x(0) = y. (19) A point y is a recurrent point for above system if re exists a sequence (τ n ) such that x(τ n ) y Theorem (Poincaré Recurrence Theorem) If a system has a stationary density and solutions exists for all time, n set of recurrent points has probability one.

18 Orbit Averages- Recurrence Continued Nosé- Definition (Recurrence Time) Let τ n be a Poincaré recurrence time, that is time elapsed until a recurrence. Given any C > 0 τ n such that x(τ n ) x 0 < C. (20) Note, x(τ n ) = (q(τ n ), p(τ n ), ξ(τ n )).

19 Average Values Nosé- Theorem (Gross, Shi, Weiss) For almost all initial conditions, p = ξ = 0, p 2 = T, q = ε cov(ξ, p), q 2εĒ.

20 Average Values Proof Nosé- Average Values of p and ξ By Birkhoff Ergodic Theorem, we know q, p, and ξ exist. Using this and Poincaré Recurrence Theorem we find: p n = 1 τ n τn 0 p(s)ds = 1 τ n τn 0 dq(s) ds (21) ds = 1 (q(τ n ) q(0)) 1 0, τ n τ n ξ n = 1 τn ξ(s)ds = 1 τn de ds (22) τ n 0 τ nεt 0 ds = 1 τ n εt (E(τ n) E(0)) C τ n εt 0,

21 Average Values Proof continued Nosé- Average Value of q q n = 1 τ n τn 0 q(s)ds = 1 τn τ n 0 τn ( dp ) ds εξp ds (23) = 1 (p(0) p(τ n )) ε ξp ds ε cov(ξ, p). τ n τ n 0 Alternatively, since E = 1 2 (q2 + p 2 + ξ 2 ) p 2E and ξ 2E, n q n = ε τ n τn 0 ξp ds 2ε τ n τn By same reasoning, we find p 2 = T. 0 E ds 2εĒn. (24)

22 Integrators Partially Linearized System Nosé- ϕ n + εξ n ϕ n + ϕ n = 0, ϕ n (0) = q n, ϕ n (0) = p n (25) q n+1 = ϕ n (h) p n+1 = ϕ n (h) ξ n+1 = ξ n + h 0 ( ϕ n (t) 2 T )dt

23 Integrators cont. Nosé- Variant of Partially Linearized System - Leap Frog q n+1/2 = ϕ n (h/2), p n+1/2 = ϕ n (h/2) (26) ξ n+1 = ξ n + h 0 ( ϕ n (t) 2 T )dt q n+1 = ϕ n+1/2 (h/2), p n+1 = ϕ n+1/2 (h/2)

24 Runge-Kutta Method Nosé- 4-th Order Runge-Kutta Method y n+1 = y n (k 1 + 2k 2 + 2k 3 + k 4 ) where f(y n ) = ẏ n, k 1 = hf(y n ), k 2 = hf(y n + h 2 k 1), k 3 = hf(y n + h 2 k 2), k 4 = hf(y n + hk 3 ).

25 Comparison of Analysis Methods Nosé-

26 Accuracy of Analysis Nosé- Stationarity Defect Measure of exactness of probability of flow of (y i+1 ) region where n (y in ) is change in volume and E i is n change in energy. SD(h) = (y i+1 ) n ln (y i ) β E i (27) n n SD(h, k) is Maclaurin polynomial of SD(h) to degree k.

27 Stationarity Defect Runge-Kutta Method Nosé-

28 Motivating Question Nosé- Tori For any solutions that lie on some invariant surface, we know that surface should be deformable to standard torus. Do se tori exist as level sets of functions H : R 3 R?

29 Nosé- Coordinate Transfomation Action-Angle [Legoll et. al.] Consider Nosé- with T = 1. We first make action-angle coordinate transformations: q = 2τ cos(θ), p = 2τ sin(θ) (28) Under this transformation, Nosé- become: θ = 1 εξ sin(θ) cos(θ) τ = 2ετξ sin 2 (θ) (29) ξ = ε(2τ sin 2 (θ) 1)

30 , cont d Nosé- Coordinate Transformation Averaging [Legoll et. al.] Next we make averaging transformation τ = ˆτ + εˆτ ˆξ sin(θ) cos(θ) (30) ξ = ˆξ εˆτ sin(θ) cos(θ) which has result of averaging out sin 2 (θ) terms from equation in action-angle coordinates: θ = 1 εˆξ sin(θ) cos(θ) + O(ε 2 ) ˆτ = εˆτ ˆξ + O(ε 2 ) ˆξ = ε(ˆτ 1) + O(ɛ 2 ) (31)

31 , cont d Nosé- Hamiltonian, No [Legoll et. al.] By (31), Theorem (Gross, Shi, Weiss) H 1 2 ˆξ 2 + ˆτ ln(ˆτ) Ḣ = O(ε 2 ) With perturbation, Hamiltonian becomes, H 2 = 1 2 ξ2 + τ ln(τ) + 1 ɛξ sin(2θ) 2 + ɛ 2 [ τ 2 sin4 (θ) cos(2θ) ξ cos(4θ)], Ḣ 2 =O(ε 3 )

32 Level Set H 2 = 0.4, ε = 0.1 Nosé-

33 Order 3 Nosé- Theorem (Gross, Shi, Weiss) Repeating our perturbation process again gives us [ ] τξ H 3 =H 2 ɛ 3 (28θ 15 sin(4θ) + sin(6θ)) 16 [ ξ ɛ 3 ξ3 ( 4θ + 2 sin(2θ) + sin(4θ) sin3 (2θ)) but this contains terms multiplied by θ, and since θ is not uniquely defined, n H 3 is undefined. Conjecture ], There is no function H : R 3 R such that solutions to Nosé- equations lie on level sets of H.

34 Acknowledgements Nosé- Thank you Dr. Leo Butler, Dr. Sivaram Narayan and Central Michigan University, NSF-REU grant DMS

35 References Nosé- Frédéric Legoll, Mitchel Luskin, Richard Moeckel (2009) Non-ergodicity of Nosé- dynamics Nonlinearity. Frédéric Legoll, Mitchel Luskin, Richard Moeckel (2007) Non-ergodicity of Nosé- rmostatted harmonic oscillator Archive for Rational Mechanics and Analysis 3:184 pps Benedict Leimkuhler, Sebastian Reich (2009) A Hamiltonian formulation for recursive multiple rmostats in a common timescale SIAM Journal on Applied Dynamical Systems 1:4 pps Harald A. Posch, William G., Franz J. Vesely (1986) Canonical dynamics of Nosé oscillator: stability, order, and chaos Physical Review. A. Third Series 6:33 pps Shuichi Nosé (1984) A unified formulation of constant temperature molecular dynamics method J. Chem. Phys. 81 pps