Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from...
Transcript of Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from...
![Page 1: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/1.jpg)
Results from combining ensembles
at several values of chemical potential
Xiao-Yong Jin
RIKEN AICS
In collaboration with
Y. Kuramashi, Y. Nakamura, S. Takeda and A. Ukawa
Mainz, Germany, Lattice 2013
![Page 2: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/2.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Introduction
• Simulations with 4 flavors [arXiv:1307.7205] described in S. Takeda's talk
• Phase quenched simulations of grand canonical ensembles
Z||(µ) =∫
[dU ] e−Sg (U )+Nf ln|detD(µ;U )|
• (β, κ )= (1.58,0.1385) and (1.60,0.1371).
• Nt = 4 with spatial volumes from 63 to 103.
• 50,000 up to ∼ 300,000 trajectories with 1/10 measured.
• µ-reweighting using Taylor expansion of ln detD works well.
Suscep>bility�
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
!P
"=1.58
63
668688
83
103
"=1.60
0
10
20
30
40
50
60
0.1 0.12 0.14 0.16 0.18 0.2
!q
aµ
0.16 0.18 0.2 0.22 0.24
aµ
Plaqueee�
Quark&Number�
First order at β = 1.58 Weak at β = 1.60
![Page 3: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/3.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Contents of this talk
Multi-ensemble reweighting and probability density.
Pitfalls and how to avoid them.
Zeros of the partition function with a complex µ .
![Page 4: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/4.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Reweighting
• Probability density
Prob(U ) = 1
Z exp[−S(U )]MC−→ 1
N
N∑l=1
δ(U −Ul)
• Reweight from simulation Ss to target St using weight wt(U )
Probt(U ) = Probs(U )exp[Ss(U )− St(U )]Zs
Zt
• From multiple simulations of S1, S2, . . .
Probt(U ) =∑sps (U ) Probs (U )exp[Ss (U )− St(U )]
ZsZt
• F.–W. (1989) ⇒ Choose ps (U ) to minimize σ 2[Probt(U )] (Z1 chosen arbitrary)
Probt(U ) =(∑s ′Ns ′ Probs ′ (U )
)(∑sNs exp[St(U )− Ss (U )]
Z1
Zs
)−1Z1
Zt
• In general (ratio of Z determined by∫D[U ] Prob(U ) = 1)
Probt(U ) = 1
Zt
exp[−St(U )]MC-RW−→ wt(U )
∑conf
δ(U −Uconf)
![Page 5: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/5.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Reweighted probability distribution
• Any quantity f (U ) with real valued 〈f 〉t for real actions
〈f 〉t =∫D[U ]f (U ) Probt(U ) =
∫D[U ]f (U )wt(U )
∑conf
δ(U −Uconf)
• Its probability density is
Probft (X) =
∫D[U ]δ
(X − f (U )
)Probt(U )
=∫D[U ]δ
(X − f (U )
)wt(U )
∑conf
δ(U −Uconf)
• With complex actions, for real partition functions, keep the real probability
〈f 〉t =∫D[U ]f (U ) Probt(U )
=∫D[U ]
(Re f (U )− Im f (U )
Imwt(U )
Rewt(U )
)Rewt(U )
∑conf
δ(U −Uconf)
• Define the probability density with real probability
Probf̃t (X) =
∫D[U ]δ
(X − f̃ (U )
)Re Probt(U )
![Page 6: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/6.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Quark number using different ensembles
-5 0 5 10 15 20 25 30 350
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
traje
cto
ry
Nq = ∂ ln Z∂(µ/T )
µ = 0.13
µ = 0.14
µ = 0.15
µ = 0.16
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
PD
F
-5 0 5 10 15 20 25 30 350
500
1000
1500
2000
2500
3000
3500
4000
traje
cto
ry
Nq = ∂ ln Z∂(µ/T )
µ = 0.02
µ = 0.30
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
PD
F
• Reweighted to µ = 0.139
• Quite visible defect if reweighted from too far
![Page 7: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/7.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Quark number—Compare
-5 0 5 10 15 20 25 30 350
5000
10000
15000
20000
25000
traje
cto
ry
Nq = ∂ ln Z∂(µ/T )
µ = 0.13
µ = 0.14
µ = 0.15
µ = 0.16
µ = 0.02
µ = 0.30
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
PD
F
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
-5 0 5 10 15 20 25 30 35
PD
F
Nq = ∂ ln Z∂(µ/T )
µ = 0.13, 0.14, 0.15, 0.16µ = 0.02, 0.30µ = 0.02, 0.13, 0.14, 0.15, 0.16 0.30
• Very slight change from remote ensembles
• More noise than signal
• We only use ensembles near the transition
![Page 8: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/8.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Partition function zeros with complex µ
• Ratios of partition functions can be determined
ZaZb
=∑
confwa(Uconf)∑confwb(Uconf)
• Define normalized partition function at µ + iµ I
Znorm =Z(µ + iµ I )Z(µ)
• Use the symmetry to reduce the noise
Znorm =Z(µ + iµ I )+Z∗(µ − iµ I )
2 ReZ(µ)
• Additionally remove a factor exp〈lnw〉, which includes a large real factor
and an oscillating complex phase
![Page 9: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/9.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Locations of the first zero, β = 1.58, κ = 0.1385
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.125 0.13 0.135 0.14 0.145 0.15 0.155 0.16
𝜇u�
𝜇
𝜇 = 0.13, 𝑉 = 63
𝜇 = 0.14, 𝑉 = 63
𝜇 = 0.15, 𝑉 = 63
𝜇 = 0.16, 𝑉 = 63
multi 𝜇, 𝑉 = 63
𝜇 = 0.13, 𝑉 = 62 × 8𝜇 = 0.14, 𝑉 = 62 × 8𝜇 = 0.15, 𝑉 = 62 × 8𝜇 = 0.16, 𝑉 = 62 × 8multi 𝜇, 𝑉 = 62 × 8𝜇 = 0.13, 𝑉 = 6 × 82
𝜇 = 0.14, 𝑉 = 6 × 82
𝜇 = 0.15, 𝑉 = 6 × 82
𝜇 = 0.16, 𝑉 = 6 × 82
multi 𝜇, 𝑉 = 6 × 82
𝜇 = 0.13, 𝑉 = 83
𝜇 = 0.14, 𝑉 = 83
𝜇 = 0.15, 𝑉 = 83
multi 𝜇, 𝑉 = 83
𝜇 = 0.14, 𝑉 = 103
𝜇 = 0.15, 𝑉 = 103
𝜇 = 0.16, 𝑉 = 103
multi 𝜇, 𝑉 = 103
Better statistics after combining multiple ensembles.
![Page 10: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/10.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Volume scaling of µ I of first zeros
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
1⁄103 1⁄83 1⁄826 1⁄628 1⁄63
𝜇𝐼
1⁄𝑉
0
0.01
0.02
0.03
0.04
0.05
0.06
1⁄103 1⁄83 1⁄826 1⁄628 1⁄63
𝜇𝐼
1⁄𝑉
β = 1.58, κ = 0.1385 β = 1.60, κ = 0.1371
Consistent with first order PT Inconsistent with first order
![Page 11: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/11.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Estimate the confidence region in µ-reweighting
• How far can we reweight in µ? Zeros of Z(µ + iµ I ) with larger µ I?
• Complex β studied by Alves, Berg, and Sanielevici (1992)
• Introduce imaginary parameter by Fourier transform probability density
Z̃(µ,µ I ) =∫dX exp[iX
µ I
T] Probf (X) = 1
Z(µ)
∫D[U ] exp[if (U )
µ I
T] exp[−S(µ;U )]
• Taking f (U ) to be the quark number, Z̃(µ,µ I ) approximates Znorm up to the
first derivative of µ/T
• ApproximateNq distribution with Gaussian peaks, k stdev away from zero
with L i.i.d. samples, and consider one Gaussian (note: variance is additive)
|||||||||⟨
exp[Nq(∆µ + i∆µ I )/T ]⟩||||||||| ≥ k
√σ 2(| exp[Nq(∆µ + i∆µ I )/T ]|)
L
√∆µ2 + (∆µ I )2 ≤
√√√√ ln(Lk2 + 1
)σ 2(Nq)/T 2
k=1,Nl=63
−−−−−→ 0.2
• Circle around the simulation, volume factor hidden in the width of Nq• Effects of cancellations between two peaks not included
![Page 12: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/12.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
|Znorm| vs. µ , fixing µ I at the first zero
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
|Z|
µ
![Page 13: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/13.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Z with complex µ , 63 (β = 1.58, κ = 0.1385)
|Znorm|0.50.20.1
0.050.020.01
0.0050.0020.001
0.00050.00020.0001
0.15 0.2 0.25 0.3
µ
0
0.05
0.1
0.15
0.2
µI
−π
−π2
0
π2
π
![Page 14: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/14.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Spurious zeros within the confidence region
-1
0
1
2
3
4
5
0 0.05 0.1 0.15 0.2
|Z|
µ I
-1
0
1
2
3
4
5
0 0.05 0.1 0.15 0.2
|Z|
µI
• Plot |Znorm| vs. µ I , at the two second lowest zeros
• Statistically they are the same zero
![Page 15: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/15.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Z with complex µ , 83, (β = 1.58, κ = 0.1385)
|Znorm|0.50.20.1
0.050.020.01
0.0050.0020.001
0.00050.00020.0001
0.15 0.2 0.25 0.3
µ
0
0.05
0.1
0.15
0.2
µI
−π
−π2
0
π2
π
![Page 16: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/16.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Zeros of the partition function with complex µ
0
0.05
0.1
0.15
0.2
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
µI
µ
V = 63
V = 62 × 8V = 6× 82
V = 83
V = 103
![Page 17: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/17.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Volume scaling of µ I for the first two zeros
0
0.02
0.04
0.06
0.08
0.1
0.12
1/103 1/83 1/826 1/628 1/63
µI
1/V
![Page 18: Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin Contents of this](https://reader030.fdocuments.net/reader030/viewer/2022040701/5d61ad5d88c993d6258be413/html5/thumbnails/18.jpg)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Summary
• Taylor expansions of the logarithm of the fermion determinant enables us
to do µ-reweighting accurately and efficiently.
• µ-reweighting is excellent in the range of parameters we are interested in.
• Carefully combining ensembles at multiple µ gives better statistics.
• Locations of partition function zeros with complex chemical potential
form a curve starting from a finite real µ and extending to larger values
of both real and imaginary parts of the chemical potential.
• Volume dependence of the locations of zeros is interesting.