self-avoiding trails with nearest neighbour interactions on the … · 2012-12-18 · self-avoiding...

Self-avoiding trails with nearest neighbourinteractions on the square lattice

Andrea Bedini

MASCOS and Department of Mathematics and Statistics, The University of Melbourne

Inaugural ANZAMP Annual Meeting 2012, Lorne

arXiv:1210.7092Work in collaboration with A. L. Owczarek and T. Prellberg

Collapse ISAW ISAT Tri-KSAT Canonical model eISAT Conclusion

Model dependency in polymer critical phenomena

†Aleks Owczarek, †Andrea Bedini, †Jason Doukas,and ‡Thomas Prellberg

†MASCOS and Department of Mathematics and Statistics, The University of Melbourne

‡School of Mathematical Sciences, Queen Mary, University of London

December, 2010

Australian Statistical Mechanics Annual Meeting, 2010

Model dependency in polymer critical phenomena Owczarek

Collapse ISAW ISAT Tri-KSAT Canonical model eISAT Conclusion

Model dependency in polymer critical phenomena

†Aleks Owczarek, †Andrea Bedini, †Jason Doukas,and ‡Thomas Prellberg

†MASCOS and Department of Mathematics and Statistics, The University of Melbourne

‡School of Mathematical Sciences, Queen Mary, University of London

December, 2010

Australian Statistical Mechanics Annual Meeting, 2010

Model dependency in polymer critical phenomena Owczarek

Andrea Bedini (MASCOS) NNISAT ANZAMP 1 / 22

Outline

1 Lattice polymers

2 Collapsing polymers

3 Nearest-Neighbour Interacting Self-Avoiding Trails

4 Conclusions


Outline

1 Lattice polymers



4 Conclusions


Self-Avoiding Walk (SAW)

Consider a walk on a some regular lattice

φn = {x0 ≡ 0, x1, . . . , xn}

where xi and xi+1 are lattice neighbours.Require xi to be all distinct (e.g. xi 6= xj if i 6= j).


Fundamental quantities

We are interested in their number

Zn ' µnnγ−1,

and in their size (ad. es. end-to-end distance)

R2n = 〈|xn|2〉 ' n2ν

γ and ν are universal exponent.These exponents can be understood as those of a magneticsystem with O(N) symmetry in the limit N → 0.Exact values can be obtained using Coulomb Gas arguments

ν = 3/4 and γ = 43/32

“Dilute polymers” phase


Self-Avoiding Trail (SAT)

A model for polymers with loops or polymers in thin layers.

φn = {x0 ≡ 0, x1, . . . , xn}where we now require xixi+1 6= xjxj+1 if i 6= j (bond avoidance)CG predicts crossings to be an irrelevant perturbation of the diluteuniversality class.Indeed, there is numerical evidence that the SAT exponents arethe same as SAW


Outline

1 Lattice polymers



4 Conclusions


Interacting Self-Avoiding Walks (ISAW)

We introduce an attractive self interaction (contacts mc).

and define the partition function:

Zn(ω) =∑

φ∈SAWn

ωmc(φ)

Energy: un = 〈mc〉/n, Specific heat: cn = (〈m2c〉 − 〈mc〉2)/n


Collapse transition

As the interaction increases we reach a critical point.

dilute denseω

θ-point

ωc

The collapse transition corresponds to a tri-critical point of theO(N → 0) magnetic system.Finite-size quantities are expected to obey a scaling form

cn(ω) ∼ nαφ C((ω − ωc)nφ

)where C(x) is a scaling function and 0 < φ ≤ 1.Exponents α and φ satisfy the tri-critical relation

2− α =1φ


Exact θ-point exponents

The presence of vacancies induceshort-range interactions on SAWs.θ-point is obtained at the point wherethe vacancies percolate

Full set of exponents can be obtained

φ = 3/7, α = −1/3 and ν = 4/7.

Specific heat does not diverge (exponent αφ = −1/7)Third derivative does diverge (exponent (α+ 1)φ = 2/7)


Interacting Self-Avoiding Trails (ISAT)

Introduce a same-site interaction on trails

Let mt be the number of doubly visited sites, we define

Z ISATn (ω) =

∑ψ∈SATn

ωmt (ψ).

Energy: un = 〈mt〉/n, Specific heat: cn = (〈m2t 〉 − 〈mt〉2)/n


ISAT Collapse

As shown by Owczarek and Prellberg on the square lattice thereis a collapse transition with estimated exponents

φIT = 0.84(3) and αIT = 0.81(3)

Additionally, the scaling of end-to-end distance was found to beconsistent with

R2n ' n (log n)2

Clearly different from the θ-pointNo predictions for these exponentsPhase diagram

dilute ??ω

ωc


ISAT collapsed stateIf we consider that proportion pn of sites which are not doublyoccupied

pn =n − 2〈mt〉

n.

it is found1 that in the low temperature region

pn ∼ n−1/2 → 0 as n→∞.

1AB, A.L. Owczarek, T. Prellberg, arXiv:1210.7196Andrea Bedini (MASCOS) NNISAT ANZAMP 13 / 22

We have seen two models of the polymer collapse.that implement the same ideas(excluded volume + short range attraction)

whose collapse transitions lie in different universality classes.

There is no clear understanding of why this is the case.

Geometry

self-avoiding walksvs

self-avoiding trails

Interaction

nearest-neighboursvs

multiply visited sites


Outline

1 Lattice polymers



4 Conclusions


Definition

Let mc(ψn) be the number of contacts, we define

Z NTn (ω) =

∑ψn∈SATn

ωmc(ψn).

Energy: un = 〈mc〉/n, Specific heat: cn = (〈m2c〉 − 〈mc〉2)/n


Specific-heat behaviour

Let cpn be specific-heat at peak

at length n.We plotted the quantity

log2

[cp

n − cpn/2

cpn/2 − cp

n/4

]n→∞−−−→ αφ

We find

αNTφNT = −0.16(3),

vs a −1/7 ≈ −0.14 (θ-point)and ≈ +0.68 (ISAT).

0.00 0.02 0.04 0.06 0.08 0.10 0.12

n−3/7

−0.20

−0.15

−0.10

−0.05

0.00

log

2[(c n−c n/2)/

(cn/2−c n/4)]


Third-derivative behaviour

Let tpn be the peak of the third

derivative.The quantity

log2

[tpn

tpn/2

]n→∞−−−→ (1 + α)φ

We find

(αNT + 1)φNT = 0.23(5)

vs a θ-point value of2/7 ≈ 0.28

0.00 0.02 0.04 0.06 0.08 0.10 0.12

n−3/7

0.2

0.3

0.4

0.5

0.6

0.7

0.8

log

2(tc,n/tc,n/2)

positive peaknegative peak


Radius scaling

Assuming ISAW crossoverexponent φ = 3/7, we can

determine precisely thecritical point go to greater

lengths.

ν ' 0.575(5)

vs a θ-point value of

ν = 4/7 ' 0.571..2 3 4 5 6 7 8 9 10

log n

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

1/2

log〈R

2〉 n

slope 0.575


Characterisation of the low-temperature region

Plot of the proportion ofsteps visiting the same siteonce, at differenttemperatures above andbelow the critical point.

The scale n−1/2 chosen isthe natural low temperaturescale.

In all cases: limn→∞ pn > 0.0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

n−1/2

0.70

0.75

0.80

0.85

0.90

0.95

1.00

p nω = 1.2

ω = 1.8

ω = 2.4


Outline

1 Lattice polymers



4 Conclusions


Summary

We simulated a model of self-avoiding trails withnearest-neighbour interactionWe presented evidence that its collapse transition is in the sameuniversality class as the θ-point.The θ-point seems to be robust when allowing crossings.While crossings are expected to be relevant in the dense phase,the dense phase seems also unaffected.CG predictions might not hold in presence of crossings.

Thanks.


self-avoiding trails with nearest neighbour interactions on the … · 2012-12-18 · self-avoiding...

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