Thermostats Lora Weiss Nos e-Hoover Thermostats · Thermostats Karl Gross, Brian Shi, Lora Weiss...
Transcript of 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
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
Physical Model
q
p
Figure: Simple Harmonic Oscillator
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
Physical Model Continued
q
pHeat Bath
T
Figure: Simple Harmonic Oscillator in a heat bath
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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)
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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)
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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)
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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)
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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.
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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.
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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).
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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.
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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.
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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)).
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
Average Values
Theorem (Gross, Shi, Weiss)
For almost all initial conditions,
p = ξ = 0, p2 = T,
q = −ε cov(ξ, p), |q| ≤ 2εE.
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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,
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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 .
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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)
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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).
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
Comparison of Numerical Analysis Methods
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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.
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
Stationarity Defect Runge-Kutta Method
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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?
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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)
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
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)
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
Perturbation Results, cont’d
Hamiltonian, No Perturbation [Legoll et. al.]
H ≡ 1
2ξ2 + τ − ln(τ)
By (31),H = O(ε2)
Theorem (Gross, Shi, Weiss)
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)
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
Level Set H2 = 0.4, ε = 0.1
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
Perturbation Order 3
Theorem (Gross, Shi, Weiss)
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.
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
Acknowledgements
Thank you
Dr. Leo Butler,Dr. Sivaram Narayan and Central Michigan University,NSF-REU grant DMS 11-56890
Thermostats
Karl Gross,Brian Shi,Lora Weiss
BackgroundInformation
Analysis ofthe Nose-HooverEquations(SimpleHarmonicOscillator)
NumericalAnalysis oftheEquations
PerturbationResults
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.