Overcoming strong metastabilities with the LLR method · LLR alhgorithm for metastable systems...

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod

Replicaexchangemethod

Numericalresults forPotts modelsin D=2,3

Conclusionsand outlook

Overcoming strong metastabilitieswith the LLR method

Biagio Lucini

(Work in collaboration with W. Fall and K. Langfeld)

Lattice 2016, University of Southampton, 26th July 2016

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Outline

1 The LLR method

2 Replica exchange method

3 Numerical results for Potts models in D=2,3

4 Conclusions and outlook

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Outline

1 The LLR method




LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




LLR cheat sheet

Divide the (continuum) energy interval in N sub-intervals of amplitude δE/2For each interval, given its centre En, define

log ρ(E) = an (E − En − δE/2) + cn for En − δE/2 ≤ E ≤ En + δE/2

Obtain an as the root of the stochastic equation

〈〈∆E〉〉an = 0⇒∫ En+

δE2

En−δE2

(E − En − δE/2) ρ(E)e−anEdE = 0

using the Robbins-Monro iterative method

limm→∞

a(m)n = an , a(m+1)

n = a(m)n −

1m〈〈∆E〉〉

a(m)n

Define

cn =δ

2a1 + δ

n−1∑k=2

ak +δ

2an (piecewise continuity of log ρ(E))

[Langfeld, Lucini and Rago, Phys. Rev. Lett. 109 (2012) 111601; Langfeld, Lucini,

Pellegrini and Rago, Eur. Phys. J. C76 (2016) no.6, 306]

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




LLR method – rigorous results

One can prove that:1 For small δE, ρ(E) converges to the density of states ρ(E), i.e.

limδE→0

ρ(E) = ρ(E)

“almost everywhere”2 With βµ(E) the microcanonical temperature at fixed E

limδE→0

an =d log ρ(E)

dE

∣∣∣∣E=En

= βµ(En)

3 For ensemble averages of observables of the form O(E)

〈O〉β =

∫O(E)ρ(E)e−βEdE∫ρ(E)e−βEdE

= 〈O〉β +O(δ2

E

)4 ρ(E) is measured with constant relative error (exponential error reduction)

∆ρ(E)

ρ(E)' constant

[Langfeld, Lucini, Pellegrini and Rago, Eur. Phys. J. C76 (2016) no.6, 306]

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




LLR method – hints and directions

Available numerical results [see Kurt Langfeld’s plenary] suggest that1 The convergence is precocious in δE

2 Potential ergodicity issues can be resolved with the replica exchangemethod

3 The cost of the algorithm is quadratic in V even at first order phasetransitions (where importance sampling has an exponentially long tunnellingtime between the degenerate equilibrium states)

4 The method allows to compute partition functions (and hence interfaces)5 A simple modification of the method can be used to simulate efficiently

systems afflicted by a sign problem6 The method can be extended to generic observables, for which one still get

quadratic convergence in δE to the correct result7 At finite δE, δ2

E errors can be corrected with a multicanonical algorithm

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




LLR method – hints and directions

Available numerical results [see Kurt Langfeld’s plenary] suggest that1 The convergence is precocious in δE

2 Potential ergodicity issues can be resolved with the replica exchangemethod

3 The cost of the algorithm is quadratic in V even at first order phasetransitions (where importance sampling has an exponentially long tunnellingtime between the degenerate equilibrium states)

4 The method allows to compute partition functions (and hence interfaces)5 A simple modification of the method can be used to simulate efficiently

systems afflicted by a sign problem6 The method can be extended to generic observables, for which one still get

quadratic convergence in δE to the correct result7 At finite δE, δ2

E errors can be corrected with a multicanonical algorithm

This talk will focus on the ergodicity properties and the efficiency of the LLRalgorithm with replica exchange at first order phase transitions, by applying themethod to Potts models in D=2 and D=3 [See also Guagnelli, arXiv:1209.4443 ]

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Outline

1 The LLR method




LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Trapping and ergodicity

Trapping occurs when, due to the small value of δE, a disconnection inconfiguration space is created between regions with the same energy

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Trapping and ergodicity

Trapping occurs when, due to the small value of δE, a disconnection inconfiguration space is created between regions with the same energy

Ergodicity can be recovered by having suitably overlapping energy intervals andallowing exchange of configurations if both energies are compatible with therestrictions of both intervals

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Replica exchange

We use a second set of simulations, with centres of intervals shifted by δE/2

i2n-2 i2n i2n+2

i2n-3

i2n-1

i2n+1

i2n+3

E2n-2E2n E2n+2 E2n+4

E2n-3 E2n-1 E2n+1 E2n+3 E2n+5

Ei2n

Ei2n-1

After a certain number m of Robbins-Monro steps, we check if both energies in twooverlapping intervals are in the common region and if this happens we swapconfigurations with probability

Pswap = min(

1, e(

a(m)2n −a(m)

2n−1

)(Ei2n−Ei2n−1

))Subsequent exchanges allow any of the configuration sequences to travel throughall energies, hence overcoming trapping

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Outline

1 The LLR method




LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




The q-state Potts model

On a D-dimensional lattice, the Hamiltonian of the q-state Potts model is given by

H = 2J∑〈ij〉

(1q− δσi,σj

), J > 0

with the spin variables σi = 0, . . . , q− 1

In D=2,3 the system undergoes an order-disorder phase transition driven by theZq symmetry

For D=2, the transition is first order for q > 4, second order for q ≤ 4

For D=3, the transition is second order only for q = 2 (Ising case), first order forq > 2

The strength of the transition increases with q⇒ strong metastabilities pose achallenge for simulations at large q

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




The phase transition in D=2

〈E〉 vs β, lattice size L = 64

0 0.2 0.4 0.6 0.8 1

β

-4

-3

-2

-1

0

<E

>/V

q=2

q=4

q=6

q=10

Exact βc = 12 log

(1 +√

q)

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




The phase transition in D=3

〈E〉 vs β, lattice size L = 16

0 0.2 0.4 0.6 0.8 1

β

-6

-5

-4

-3

-2

-1

0

<E

>/V

q=2

q=4

q=6

q=10

βc from Bazavov, Berg and Dubey, Nucl. Phys. B802 (2008) 421-434

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Efficiency of replica swapping (D=2 q=20)

0 500 1000 1500 2000

MC time

2000

3000

4000

5000

6000

7000

acti

on

The hopping of configurations across intervals is reminiscent of a random walk

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Replica swap and diffusive dynamics

0 500 1000 1500 2000MC time

0

200

400

600

800

1000

dif

fusi

on

len

tgh

fit: 19.67 (time)1/2

q=20 Potts model, 642 lattice, dE = 29

Mean path in energy space: 〈(Ef − Eı)2〉1/2 = Dt1/2

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Diffusion coefficient vs. Eı

0 500 1000 1500 2000MC time

3000

4000

5000

6000

7000

8000

dif

fusi

on

len

tgh

fit: 19.67 (time)1/2

fit::19.80 (time)1/2

fit: 18.73 (time)1/2

q=20 Potts model, 642 lattice, dE = 29

D seems independent from Eı

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Probability density at criticality

-4 -3 -2 -1 0

E/V

-50

-40

-30

-20

-10

0

log(P

(E/V

))

L= 32

L= 64

L= 96

L=128

The value of β for which P(E/V) has two equal-height maxima is a possibledefinition of βc(V−1)

The minimal depth of the valley between the peaks is related to theorder-disorder interface

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Finite Size Scaling – βc

0 0.0002 0.0004 0.0006 0.0008 0.001

V-1

0.8484

0.8486

0.8488

0.849

0.8492

0.8494

0.8496

0.8498

βc(

V-1)

Fit

Fitted βc(0)

Analytic βc(0)

For first order phase transitions

βc(V−1) = βfitc +

aβV

+ . . .

With a linear fit, we find

βfitc = 0.8498350(21) ,

βfitc − βexact

c

βexactc

= 1.7(2.5)× 10−6

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Finite Size Scaling – order-disorder interface

-4 -3 -2 -1 0

E/V

-50

-40

-30

-20

-10

0

log(P

(E/V

))

L= 32

L= 64

L= 96

L=128

At finite L

2σod(L) = −1L

log Pmin,valley

Ansatz

2σod(L)−log L2L

= 2σod +cσL

⇒ 2σod = 0.36853(88)

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Finite Size Scaling – order-disorder interface

0 0.01 0.02 0.03 0.04

1/L

0.3

0.31

0.32

0.33

0.34

0.35

0.36

0.37

0.38

2σ

od(L

) -

log

(L)/

(2L

)

Fit

Fitted σod

At finite L

2σod(L) = −1L

log Pmin,valley

Ansatz

2σod(L)−log L2L

= 2σod +cσL

⇒ 2σod = 0.36853(88)

LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Outline

1 The LLR method




LLRalhgorithm for

metastablesystems

Biagio Lucini

The LLRmethod




Conclusions and outlook

LLR algorithm (supplemented with replica exchange) works for Potts modelsin D=2 and D=3⇒ perhaps because E/V→ continuous when V →∞?

The replica exchange method behaves as expected, providing a randomwalk in energy space even when the system is metastable⇒ cost of thealgorithm scaling as V2?

To my knowledge, first determination of critical properties for q = 20, D=2(including σod)

Future directions:I study the dependence V of the diffusion coefficient DI determine σoo and check if the perfect wetting condition is satisfiedI perform a similar precision study in D=3

Date post:	27-Jul-2018
Category:	Documents
Upload:	phungcong
View:	219 times
Download:	0 times

Overcoming strong metastabilities with the LLR method · LLR alhgorithm for metastable systems...

Documents