Thermostat MD

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Thermostats in Molecular Dynamics Simulations

Alden Johnson, Teresa Johnson, Aimee KhanUniversity of Massachusetts Amherst

December 6th, 2012

Alden Johnson, Teresa Johnson, Aimee Khan (UMass Amherst)

Thermostats in MD Simulations December 6th, 2012 1 / 29


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An Interest in Thermostats

Thermostat: A modification of the Newtonian MD scheme with thepurpose of generating a statistical ensemble at a constant temperature.

Match experimental conditions

Manipulate temperatures in algorithms such as simulated annealing

Avoid energy drifts caused by accumulation of numerical errors.




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Statistical Ensembles

Ensemble: a large collection of microscopically defined states of asystem, with certain constant macroscopic properties

Microcanonical (NVE)

Arises in Newtonian MD simulation Conserves total energy

Canonical (NVT)

Implement thermostats to sample from here Relevant to real behavior in experiment




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Microcanonical Ensemble

Consider an isolated system

Microstate: complete description of a state of the system,microscopically

The probabilities of being in a certain microstate for themicrocanonical ensemble are uniform over all possible states

Alden Johnson, Teresa Johnson, Aimee Khan (UMass Amherst)Thermostats in MD Simulations December 6th, 2012 4 / 29


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Canonical Ensemble

Follows a Gibbs distribution for probability pj of being in a givenmicrostate j with energy Ej

pj =eEj

Z , Z =

j

e

Ej

=1

kBT

Derivation follows from maximizing entropy



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Ergodic Hypothesis

The ergodic hypothesis says the long time average of an observable fcoincides with an ensemble average of the observable f.

f = limt

1

tt

0 f(x(s)) ds

f =

f(x) d(x)

where is the ensemble measure and is the phase space of theobservable.



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Molecular Dynamics Simulation

Phase space is collection of positions q and momenta p of particles insystem

The Hamiltonian Formdqt = pH(qt, pt) dtdpt = qH(qt, pt) dt

H(q, p) = Ekin

(p) + V(q), Ekin

(p) =1

2pTM1p



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Sampling from Ensembles

A canonical ensemble (constant average energy) is a distribution ofmicrocanonical ensembles (constant energy)

To sample from the canonical ensemble, the following thermostatsmodulate the energy entering and leaving the boundaries of thesystem



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Velocity Rescaling

Velocities are described by Maxwell-Boltzmann distribution

P(vi,) = m

2kBT12

e

mv2i,2kBT

Adjust instantaneous temperature by scaling all velocities

Average Ekin per degree of freedom related to T via the equipartitiontheorem

mv2i,2

=1

2 kBT



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Velocity Rescaling

Ensemble average average over velocities of all particles: defineinstantaneous temperature Tc for a finite system

kBTc =1

Nfi,

mv2

i,

Tc = T until rescaling

v

i, =T

Tcvi,



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Velocity Rescaling

Disadvantages

Results do not correspond to any ensemble

Does not allow the proper temperature fluctuations

Localized correlation not removed

Not time reversible

Advantages

Straightforward to implement

Good for use in warmup / initialization phase



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Nose-Hoover

Based on extended Lagrangian formalism

Deterministic trajectory

Simulated system contains virtual variables related to real variables

Coordinates qi = qi

Momenta pi = pi/s

Time t =t

0dts

s: additional degree of freedom, acts as external system



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Nose-Hoover

Hamiltonian given by

H =Ni=1

p2i

2mis2+ V(q) +

Qs2

2+ (3N + 1)kBTln s

Logarithmic term required for proper time scaling: canonical ensemble

Effective mass Q associated with s Determines thermostat strength Q too small: system not canonical

Q too large: temperature control inefficientMicrocanonical dynamics on extended system give canonicalproperties



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Nose-Hoover

Disadvantages

Extended system not guaranteed to be ergodic

Advantages

Easy to implement and use

Implement as a chain

Each link: apply thermostatting to the previous thermostat variable

Increasing Q lengthens decay time of response to instantaneous

temperature jump

Deterministic and time reversible



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Langevin

Consider the motion of large particles through a continuum of smallerparticles

dqidt

=pi

mi

dpidt

= V(q)qi

pi + Gi

Viscous drag force proportional to velocity pi Smaller particles give random pushes to large particle

Fluctuation-dissipation relation

2 = 2mikBT



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Langevin

Disadvantages

Difficult to implement drag for non-spherical particles: related toparticle radius

Momentum transfer lost: cannot compute diffusion coefficients

Advantages

Damping + random force correct canonical ensemble

Ergodic Can use larger time step


A d


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Andersen

Couple a system to a heat bath to impose desired temperature

Equations of motion are Hamiltonian with stochastic collision term

Strength of coupling specified by , the stochastic collision frequency

When particle has collision, new velocity is sampled from N(0,T)


A d


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Andersen

Disadvantages

Newtonian dynamics + stochastic collisions Markov chain

Algorithm randomly decorrelates velocity: dynamics are not physical

Advantages

Allows sampling from canonical ensemble


Th S


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Thermostats Summary

Thermostat Description Canonical? Stochastic?

Velocity Rescaling KE fixed to match T no no

Nose-Hoover extra degree of freedomacts as thermal reser-voir

yes no

Langevin noise and drag balanceto give correct T

yes yes

Andersen momenta occasionally

re-randomized

yes yes


A li d M th M t P j t


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Applied Math Masters Project

Molecular Dynamics Simulation

Implemented in MATLAB and C++

Simulations in 2 and 3 dimensions

Periodic and walled boundary conditions

External fields such as gravity

Optimization

OpenMP

CUDA


L d J P t ti l


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Lennard-Jones Potential


Numeric Integration


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Numeric Integration

Verlet Algorithm

pn+1/2 = pn t2V(qn)

qn+1 = qn + tM1pn+1/2

p

n+1

= p

n+1/2

t

2 V(qn+1

)

Preserves modified Hamiltonian


Thermostat Implementation


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Thermostat Implementation

Velocity Rescaling

pi

T

Tcpi

Anderson = 1%, 0.1%, 0.05% Velocities are sampled from N(0,T)

Langevin

BBK algorithmpn+1/2 = pn t

2V(qn) t

2(qn)M1pn +

t2(qn)Gn

qn+1 = qn + tM1pn+1/2

pn+1

= pn+1/2

t

2 V(qn+1

)t

2 (qn+1

)M1

pn+1

+t

2 (qn+1

)Gn+1

chosen to be constant =

2MkBT

G sampled from a standard normal distribution


Velocity Rescaling Instantaneous Temperature


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Velocity Rescaling - Instantaneous Temperature

0 2000 4000 6000 8000 100000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Temp

time step


Anderson Instantaneous Temperature


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Anderson - Instantaneous Temperature

0 2000 4000 6000 8000 100000

1

2

3

4

5

6

Temp

time step

= 1%

= 0.1%

= 0.05%


Anderson - Histogram of Velocity Distribution


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Anderson - Histogram of Velocity Distribution

10 5 0 5 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

velocity

p(v)


Langevin Instantaneous Temperature


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Langevin Instantaneous Temperature

0 2 4 6 8 10

x 104

1

2

3

4

5

6

7

Temp

time step

= 0.1

= 0.01

= 0.001


Langevin - Histogram of Velocity Distribution


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Langevin Histogram of Velocity Distribution

10 5 0 5 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

p(v)

velocity



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Documents

Thermostat MD