Thermostats Lora Weiss Nos e-Hoover Thermostats · Thermostats Karl Gross, Brian Shi, Lora Weiss...

Thermostats

Karl Gross,Brian Shi,Lora Weiss

BackgroundInformation

Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)

NumericalAnalysis oftheEquations

PerturbationResults

Nose-Hoover Thermostats

Karl Gross, Brian Shi, Lora Weiss

Texas A & M, UT Austin, St. Olaf College

October 29, 2013

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PerturbationResults

Physical Model

q

p

Figure: Simple Harmonic Oscillator

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Physical Model Continued

q

pHeat Bath

T

Figure: Simple Harmonic Oscillator in a heat bath

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Equations and Notations

Nose-Hoover Equations for the simple harmonic oscillator

q = p, p = −q

− εξp, ξ = ε(p2 − T ) (1)

• T denotes temperature

• ξ denotes the feedback term

• ε denotes feedback constant

Extended energy of the system

E =1

2(q2 + p2 + ξ2),

dE

dt= −εξT (2)

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Equations and Notations

Nose-Hoover Equations for the simple harmonic oscillator

q = p, p = −q − εξp, ξ = ε(p2 − T ) (1)

• T denotes temperature

• ξ denotes the feedback term

• ε denotes feedback constant

Extended energy of the system

E =1

2(q2 + p2 + ξ2),

dE

dt= −εξT (2)

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History

History

S. Nose found a Hamiltonian system in 1984 that included atime-scaled variable s.

q =p

msp = −s∇V (3)

s = sps ps =∑[

p2

ms2− kT

]William Hoover added to Nose’s work in 1985, scaling outthe extra time variable.

q =p

mp = −∇V − ξp (4)

ξ =1

m||p||2 − T

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Canonical Density

Solving for the density

The canonical density, denoted

Ω = f(q, p, ξ;T )dV, (5)

satisfiesdΩ

dt≡ 0 (6)

therefore we get the following PDE

∇f · v + f div(v) = 0 (7)

We solved this equation using separation of variables:

f(q, p, ξ;T ) =e−

(q2+p2+ξ2)2T

Z(T )=

1

Z(T )e−

1TE (8)

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Special Cases

Special Solutions

Consider the initial conditions (0, 0, ξ0) where ξ0 ∈ R, thenthere is an exact solution of the form

q(t) = 0, p(t) = 0, ξ(t) = ξ0 − εT t (9)

ξ0

ξ

p

q

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Special Cases cont.

Zero Temperature

For T = 0, E(t) is constant and all solutions lie on a sphereof radius equal to the magnitude of the initial conditions.

Figure: [q0, p0, ξ0] = [0.1, 0.1,−1], ε = .5, T = 0

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Special Cases

Free Particle [Hoover 1985]

Introducing the spring constant ω2, we get the system

q = p, p = −ω2q − εξp, ξ = ε(p2 − T ). (10)

Setting τ = 12p

2 and ω2 = 0 leads to the system

τ = −2εξτ, ξ = ε (2τ − T ) (11)

which is integrable with trajectories lying on the curves given bythe level sets of

H(τ, ξ) =1

2ξ2 + τ − 1

2T ln τ = K (12)

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Periodicity

Motivation

Prove nonexistence of a function h : R3 → R such that h isdeformable to a union of standard tori.

Fourier Analysis

The original differential equation is γ(t) = f(γ(t)). ApplyingFourier transform we find

G(t) =a0

2+

n/2∑k≥1

ak cos(2πkt

τ) + bk sin(

2πkt

τ). (13)

We can compare G(t)− γ(t) and 1n

∑n1 |Gn(t)− γn(t)|2 to

check for periodicity.

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Pictures

[q0, p0, ξ0] = [0, 1.55, 0],β = 1, ε = 1, τ =5.58, Error= 0.00012268[Hoover 1985]

[q0, p0, ξ0] = [0.01, 0.01, 2],β = 2.3920, ε = 0.4,τ = 136, Error=0.0092897

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More Pictures

[q0, p0, ξ0] = [0.48, 0.620, 0],β = 0.1, ε = 0.1, τ = 125

[q0, p0, ξ0] = [0.48, 0.685, 0],β = 0.1, ε = 0.1, τ = 842

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Orbit Averages

Theorem (Birkhoff Ergodic Theorem)

If a system has a stationary density and solutions exist forall time, then for any continuous function f(x),

f(x0) = limτ→∞

1

τ

∫ τ

0f(x(s))ds

exists with probability 1.Here, f(x0) is the average of f along the orbit with initialvalue x0.

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Orbit Averages- Global Existence

Gronwall’s Inequality

Let A,B : [a, b]→ R be constants and u differentiable in(a, b) satisfying

u(t) ≤ Au(t) +B, t ∈ [a, b] (14)

then we have the following inequality

u(t) ≤ CeA(t−a) (15)

for some constant C > 0 depending on A,B and u(a).

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Orbit Averages- Global Existence Continued

Proposition

Since|E(t)| ≤ ε|ξ| ≤ ε

√2E ≤ 2ε(1 + E), (16)

then by Gronwall’s inequality we know

E(t) ≤ Ae2εt. (17)

Plugging initial condition x0 = (q0, p0, ξ0), then

E(t) ≤ (εT )2

2t2 − εTξ0t+ E(0). (18)

Theorem (Global Existence of Solutions)

For any initial condition, x0 = (q0, p0, ξ0), there exists asolution for all time t.

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Orbit Averages- Recurrence

Definition

Consider the system of ODE

x = F (x), x(0) = y. (19)

A point y is a recurrent point for the above system if thereexists a sequence (τn)→∞ such that x(τn)→ y

Theorem (Poincare Recurrence Theorem)

If a system has a stationary density and solutions exists forall time, then the set of recurrent points has probability one.

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Orbit Averages- Recurrence Continued

Definition (Recurrence Time)

Let τn be a Poincare recurrence time, that is the timeelapsed until a recurrence. Given any C > 0

∃ τn →∞ such that |x(τn)− x0| < C. (20)

Note, x(τn) = (q(τn), p(τn), ξ(τn)).

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Average Values

Theorem (Gross, Shi, Weiss)

For almost all initial conditions,

p = ξ = 0, p2 = T,

q = −ε cov(ξ, p), |q| ≤ 2εE.

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Average Values Proof

Average Values of p and ξ

By the Birkhoff Ergodic Theorem, we know q, p, and ξ exist.Using this and the Poincare Recurrence Theorem we find:

pn =1

τn

∫ τn

0

p(s)ds =1

τn

∫ τn

0

dq(s)

dsds (21)

=1

τn(q(τn)− q(0)) ≤ 1

τn→ 0,

ξn =1

τn

∫ τn

0

ξ(s)ds = − 1

τnεT

∫ τn

0

dE

dsds (22)

=1

τnεT(E(τn)− E(0)) ≤ C

τnεT→ 0,

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Average Values Proof continued

Average Value of q

qn =1

τn

∫ τn

0q(s)ds =

1

τn

∫ τn

0

(−dpds− εξp

)ds (23)

=1

τn(p(0)− p(τn))− ε

τn

∫ τn

0ξp ds→ −ε cov(ξ, p).

Alternatively, sinceE = 1

2(q2 + p2 + ξ2)⇒ |p| ≤√

2E and |ξ| ≤√

2E, then

|qn| =∣∣∣∣ ετn

∫ τn

0ξp ds

∣∣∣∣ ≤ 2ε

τn

∫ τn

0E ds ≤ 2εEn. (24)

By the same reasoning, we find p2 = T .

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Numerical Integrators

Partially Linearized System

ϕn + εξnϕn + ϕn = 0, ϕn(0) = qn, ϕn(0) = pn (25)

qn+1 = ϕn(h)

pn+1 = ϕn(h)

ξn+1 = ξn +

∫ h

0(ϕn(t)2 − T )dt

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Numerical Integrators cont.

Variant of Partially Linearized System - Leap Frog

qn+1/2 = ϕn(h/2), pn+1/2 = ϕn(h/2) (26)

ξn+1 = ξn +

∫ h

0(ϕn(t)2 − T )dt

qn+1 = ϕn+1/2(h/2), pn+1 = ϕn+1/2(h/2)

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Runge-Kutta Method

4-th Order Runge-Kutta Method

yn+1 = yn +1

6(k1 + 2k2 + 2k3 + k4)

wheref(yn) = yn,

k1 = hf(yn),

k2 = hf(yn +h

2k1),

k3 = hf(yn +h

2k2),

k4 = hf(yn + hk3).

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Comparison of Numerical Analysis Methods

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Accuracy of Numerical Analysis Equations

Stationarity Defect

Measure of the exactness of the probability of the flow of the

region where

∣∣∣∣∂(y i+1n

)

∂(y in

)

∣∣∣∣ is the change in volume and ∆E in

is

the change in energy.

SD(h) =∑

ln

∣∣∣∣∣∂(y i+1n

)

∂(y in

)

∣∣∣∣∣− β∑∆E in

(27)

SD(h, k) is the Maclaurin polynomial of SD(h) to degree k.

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Stationarity Defect Runge-Kutta Method

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Motivating Question

Tori

For any solutions that lie on some invariant surface, weknow that the surface should be deformable to the standardtorus.

• Do these tori exist as level sets of functionsH : R3 → R?

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Perturbation Results

Coordinate Transfomation–Action-Angle [Legoll et. al.]

Consider the Nose-Hoover Equations with T = 1. We firstmake the action-angle coordinate transformations:

q =√

2τ cos(θ), p = −√

2τ sin(θ) (28)

Under this transformation, the Nose-Hoover Equationsbecome:

θ = 1− εξ sin(θ) cos(θ)

τ = −2ετξ sin2(θ) (29)

ξ = ε(2τ sin2(θ)− 1)

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PerturbationResults

Perturbation Results, cont’d

Coordinate Transformation–Averaging [Legoll et. al.]

Next we make the the averaging transformation

τ = τ + ετ ξ sin(θ) cos(θ) (30)

ξ = ξ − ετ sin(θ) cos(θ)

which has the result of averaging out the sin2(θ) terms fromequation in the action-angle coordinates:

θ = 1− εξ sin(θ) cos(θ) +O(ε2)

˙τ = −ετ ξ +O(ε2)

˙ξ = ε(τ − 1) +O(ε2) (31)

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Perturbation Results, cont’d

Hamiltonian, No Perturbation [Legoll et. al.]

H ≡ 1

2ξ2 + τ − ln(τ)

By (31),H = O(ε2)


With perturbation, the Hamiltonian becomes,

H2 =1

2ξ2 + τ − ln(τ) +

1

2εξ sin(2θ)

+ ε2[−τ2

sin4(θ)− cos(2θ)

4− ξ2

16cos(4θ)],

H2 =O(ε3)

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Level Set H2 = 0.4, ε = 0.1

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Perturbation Order 3


Repeating our perturbation process again gives us

H3 =H2 − ε3[τξ

16(28θ − 15 sin(4θ) + sin(6θ))

]− ε3

[ξ

32(−4θ + 2 sin(2θ) + sin(4θ) +

ξ3

24sin3(2θ))

],

but this contains terms multiplied by θ, and since θ is notuniquely defined, then H3 is undefined.

Conjecture

There is no function H : R3 → R such that the solutions tothe Nose-Hoover equations lie on the level sets of H.

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Acknowledgements

Thank you

Dr. Leo Butler,Dr. Sivaram Narayan and Central Michigan University,NSF-REU grant DMS 11-56890

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References

Frederic Legoll, Mitchel Luskin, Richard Moeckel (2009)Non-ergodicity of Nose-Hoover dynamicsNonlinearity.

Frederic Legoll, Mitchel Luskin, Richard Moeckel (2007)Non-ergodicity of Nose-Hoover thermostatted harmonic oscillatorArchive for Rational Mechanics and Analysis 3:184 pps. 449-463.

Benedict Leimkuhler, Sebastian Reich (2009)A Hamiltonian formulation for recursive multiple thermostats in acommon timescaleSIAM Journal on Applied Dynamical Systems 1:4 pps. 187-216.

Harald A. Posch, William G. Hoover, Franz J. Vesely (1986)Canonical dynamics of the Nose oscillator: stability, order, and chaosPhysical Review. A. Third Series 6:33 pps. 4253-4265.

Shuichi Nose (1984)A unified formulation of the constant temperature molecular dynamicsmethodJ. Chem. Phys. 81 pps. 511-519.

Thermostats Lora Weiss Nos e-Hoover Thermostats · Thermostats Karl Gross, Brian Shi, Lora Weiss...

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