New Insights from One-Dimensional Spin Glasses

a r

X i

v

:0

8

3

.

3

4

1

7v

1

[

c o

n

d

-

m

a

t

.

d

i

s

-

n

n

]

2

4

M

a

r

2

08

New Insights from One-Dimensional Spin Glasses Helmut G.Katzgraber 1,Alexander K.Hartmann 2,and A.P.Young 31Theoretische Physik,ETH Z¨u rich,CH-8093Z¨u rich,Switzerland 2Institute for Physics,University of Oldenburg,Germany 3Department of Physics,University of California Santa Cruz,CA 95064,USA Abstract.The concept of replica symmetry breaking found in the solution of the mean-?eld Sherrington-Kirkpatrick spin-glass model has been applied to a variety of problems in science ranging from biological to computational and even ?nancial analysis.Thus it is of paramount importance to understand which predictions of the mean-?eld solution apply to non-mean-?eld systems,such as realistic short-range spin-glass models.The one-dimensional spin glass with random power-law interactions promises to be an ideal test-bed to answer this question:Not only can large system sizes—which are usually a shortcoming in simulations of high-dimensional short-range system—be studied,by tuning the power-law exponent of the interactions the universality class of the model can be continuously tuned from the mean-?eld to the short-range universality class.We present details of the model,as well as recent applications to some questions of the physics of spin glasses.First,we study the existence of a spin-glass state in an external ?eld.In addition,we discuss the existence of ultrametricity in short-range spin glasses.Finally,because the range of interactions can be changed,the model is a formidable test-bed for optimization algorithms.1Introduction Spin glasses [1–3]are paradigmatic models which can be applied to a wide va-riety of problems and ?elds ranging from economical to biological,as well as sociological problems,to name a few.Most prominent is the replica symme-try breaking solution of Parisi [4]of the mean-?eld Sherrington-Kirkpatrick (SK)spin glass.Unfortunately,an analytical solution for short-range realistic spin-glass models,such as the Edwards-Anderson Ising spin glass [5],remain to be found and generally phenomenological descriptions,such as the droplet picture [6]or numerical simulations are used to understand these systems.Given the lack of rigorous results for short-range spin glasses,it is of impor-tance to understand the applicability of di?erent predictions made by the

mean-?eld solution of the SK model,as well as other theoretical pictures.Unfortunately,numerical studies of spin glasses are di?cult to accom-plish and in general only small to moderate system sizes can be accessed.Despite huge technological advances in the last decade which have enabled the construction of large computer clusters out of commodity components,brute force computation alone will not su?ce to probe considerably larger

2Helmut G.Katzgraber et al.

system sizes.The source of this problem lies in the diverging equilibration times of Monte Carlo simulations of spin glasses;the systems are generally NP hard.Furthermore,to obtain thermodynamically sound results,calcula-tions need to be disorder averaged,thus adding considerable overheard to any simulation.To properly probe the thermodynamic limiting behavior it is thus important to use e?cient algorithms,improved models,as well as large computer clusters.

We discuss in detail a one-dimensional spin-glass model with power-law interactions[6–8]where,by tuning the exponent of the power-law interactions di?erent universality classes from in?nite-range SK to short-range can be probed.Furthermore,because the model is one-dimensional,a wide range of system sizes can be probed.In the past we have applied the model to di?erent problems in the physics of spin glasses[8–13].In this work we study two questions which lie at the core of the applicability of the mean-?eld solution to short-range spin glasses:Do short-range spin glasses order in an externally-applied magnetic?eld?Are short-range spin glasses ultrametric? Our results suggest that new theoretical descriptions are needed:While there are indications of an ultrametric structure of phase space,spin-glass order is destroyed in a?eld for short-range systems.

Finally,we also discuss extensions as well modi?cations of the model to study di?erent related problems in the physics of spin glasses and present applications to algorithm development and testing.

2Model&Numerical Method

We?rst introduce the one-dimensional Ising chain in detail and explain its rich phase diagram.Furthermore,we describe exchange(parallel tempering) Monte Carlo,a numerical method which is very e?cient to study spin-glass systems at low temperatures.

2.1The one-dimensional Ising chain

The Hamiltonian of the one-dimensional Ising chain with power-law interac-tions[7,14,8]is given by

H1D=? i

New Insights from One-Dimensional Spin Glasses3

Fig.1.Left panel:Schematic phase diagram of the one-dimensional Ising chain with power-law interactions[14].The white horizontal arrow corresponds to d=1. Forσ≤1/2we expect in?nite-range(IR)behaviour reminiscent of the SK model. For1/2<σ≤2/3we have mean-?eld(MF)behaviour corresponding to an e?ective space dimension d e?≥6,whereas for2/3<σ 1we have a long-range(LR+) spin glass with a?nite ordering temperature T c.In these regimes d e?≈2/(2σ?1) [1].For1≤σ 2we have a long-range spin glass with T c=0(LR0)and forσ 2 the model displays short-range(SR)behaviour with T c=0.Figure adapted from Ref.[8].Right panel:Graphical representation of the one-dimensional Ising chain with L=16spins.

phase is T MF

c =1;see Ref.[8]for details.The model has a very rich phase

diagram in the d–σplane,see Fig.1(left panel).In this work we are in-terested in d=1which corresponds to the thick horizontal white arrow in the phase diagram.The universality class and range of the interactions of the model can be continuously tuned by changing the power-law exponentσ. Furthermore,there are theoretical predictions for the critical exponents[7]:ν=1/(2σ?1)forσ≤2/3,andη=3?2σ.Therefore,predictions made for the mean-?eld spin-glass can be probed when the e?ective space dimension (range of the interactions)is reduced.Furthermore,because the e?ciency of di?erent algorithms often depends strongly on the range of the interactions, the one-dimensional chain is an ideal test bed to benchmark the e?ciency of optimization algorithms.

In one space dimension,forσ≤1/2the model is in the Sherrington-Kirkpatrick[15]in?nite-range universality class where the energy of the sys-tem needs to be rescaled with the system size to avoid divergencies.In par-ticular,forσ=0the SK model is recovered exactly.For1/2<σ<1the model has a?nite-temperature spin-glass ordering transition.Furthermore, for1/2<σ≤2/3the system is in the mean-?eld universality class cor-responding to a high-dimensional short-range spin-glass system above the

4Helmut G.Katzgraber et al.

upper critical dimension d u =6.For 2/3<σ<1the system is non-mean ?eld,whereas for σ≥1the spin-glass phase only exists at T =0,i.e.,the lower critical dimension of short-range spin glasses corresponds to σ=1.

2.2Numerical method

Because of a rough energy landscape and diverging relaxation times,spin glasses are extremely di?cult to study numerically.Any numerical method used must have the potential to e?ciently cross energy barriers and thus sample the phase space evenly.Probably one of the simplest,yet most e?-cient methods to study problems with rough energy landscapes (beyond spin glasses)is the exchange (parallel tempering)Monte Carlo method [16].

The idea behind the method is to allow for a Markov process in temper-ature space.M copies of the system are simulated at di?erent temperatures,where the largest temperature is generally chosen to be of the order of 2T MF c .Besides the simple Monte Carlo updates [17]on each spin of the system,after a certain number of lattice sweeps the energies of neighboring temperatures are compared and a Monte Carlo move which swaps the temperatures of neighboring con?gurations is proposed.With this approach,a con?guration stuck in a metastable state has the possibility to heat up and then cool back down to the true equilibrium state thus e?ectively speeding up equilibration by orders of magnitude.The position of the temperatures has to be chosen with care:If neighboring temperatures are chosen too far apart,a bottleneck in the temperature-space Markov process emerges thus reducing the e?ciency of the method.If the temperatures are too close extra unnecessary overhead is introduced.To select the position of the temperatures,it is convenient to study the acceptance probabilities of the global Monte Carlo moves.Because in spin glasses the susceptibility does not diverge,a generally good thumb-rule is to select the position of the temperatures such that the probabilities are between 0.2and 0.9and roughly independent of temperature.This is not necessarily the case for other systems.We also refer the reader to Ref.[18]where an iterative feedback method is introduced which ensures that the random walk of each con?guration in temperature space is optimal.

When using Gaussian-distributed disorder we test equilibration of the Monte Carlo simulations by equating the link overlap q l to the energy U =?(1/N ) i,j [J ij S i S j ]av of the system [19],i.e.,

U (q l )=(T MF c )

2N i,j

[J 2ij ]av

New Insights from One-Dimensional Spin Glasses5

Fig.2.Equilibration test using

Eq.(2).Once the energy U com-

puted directly and from the link

overlap[U(q l)]agree,the system

is in thermal equilibrium.Data for

L=512,T=0.20andσ=0.75.

equilibrium.Note that the method can be easily extended to system with (Gaussian distributed)external?elds[11].

3Selected results

We have applied the one-dimensional Ising chain to several problems in the physics of spin glasses.Below we present in more detail two questions which lie at the core of the applicability of the mean-?eld solution to short-range spin glasses.In the following we compare the mean-?eld SK model(σ=0) to the one-dimensional Ising chain forσ=0.75where the model is in the non-mean-?eld universality class.

3.1Do spin glasses order in a magnetic?eld?

The applicability of spin-glass models to other?elds of science relies heavily on the existence of a spin-glass phase in a?eld.Many mappings onto spin-glass models produce external?eld terms.While the mean-?eld model has been shown to have a spin-glass phase in a?eld,it has been unclear until recently if short-range spin glasses order in a?eld as well[20–29].Simulations of three-dimensional spin-glass models[28,30]suggest that the de Almeida-Thouless line[31],which separates the spin-glass from the paramagnetic state in the H–T phase diagram does not exist for realistic short-range Ising spin glasses.Although the aforementioned studies in three space dimensions using the?nite-size two-point correlation length[32]provide clear evidence that short-range spin glasses do not order in a?eld,they do not shed any light on the behavior of short-range spin glasses with space dimensions above the upper critical dimension.

6Helmut G.Katzgraber et al.

In Ref.[11]the one-dimensional Ising chain has been studied in an exter-nally applied Gaussian-distributed random?eld—which has a similar behav-ior than a uniform?eld—for di?erent exponentsσof the power-law inter-actions.For exponents which correspond to e?ective space dimensions above the upper critical dimension,a spin-glass state in a?eld is found,whereas for exponentsσ>2/3which correspond to e?ective space dimensions less than six,no de Almeida-Thouless line could be found for simulations down to very low temperatures.Technical details about the simulation,and in particular the parameters of the simulation can be found in Ref.[11].

In order to probe the existence of a spin-glass state we add an external (random)?eld to the Hamiltonian,i.e.,H1D→H1D? i h i S i.The reasons for using random?elds are the ability to thoroughly test for equilibration of the Monte Carlo method(for detail see Refs.[28]and[11]).Furthermore, exchange Monte Carlo performs better.

To test for the existence of the transition forσ>1/2,we compute the ?nite-size correlation length from the Fourier transform of the spin-glass sus-ceptibility[33,32]:

1

χSG(k)=

New Insights from One-Dimensional Spin Glasses7

Fig.4.Left panel:Finite-size correlation length divided by the system size as a function of temperature for the one-dimensional Ising spin chain withσ=0.75at zero?eld.In this regime the system is not in the mean-?eld universality class.The data cross cleanly at T c(H=0)≈0.69.Right panel:Same observable and model as the left panel,except for H=0.10.Note that for temperatures as low as T=0.1 there is no crossing visible,suggesting that there is no spin-glass state in a?eld. Figure adapted from Ref.[11].

After performing an Ornstein-Zernicke approximation we obtain for the two-point?nite-size correlation length

ξL=1

χSG(k min)

?1 1/(2σ?1),(4)

whereχSG(0)is the standard spin-glass susceptibility and k min=2π/L.The ?nite-size correlation length divided by the system size is a dimensionless quantity which scales asξL/L=?X[L1/ν(T?T c)].Because in the in?nite-range universality class no correlation length can be computed,we exploit the fact that the critical exponentη=1/3is exactly known for the SK model [34,30].Therefore,we locate the transition in the SK model by studying χSG/N1/3=?C[L1/ν(T?T c)],whereχSG=χSG(k=0).Once the respective observables for di?erent system sizes cross we have a spin-glass state for T≤T c,where T c is given by the crossing point.

In Fig.3we showχSG/N1/3for the SK model(σ=0)as a function of temperature for zero,as well as an external?eld of strength H=0.10.In both cases the data cross,indicative of a transition in zero as well as?nite ?elds.This is not the case for the one-dimensional model withσ=0.75. While the data of the?nite-size correlation length at zero?eld clearly show a transition at T c≈0.69(see Fig.4,left panel),this is not the case for

8Helmut G.Katzgraber et al.

H=0.10where the data do not cross even for temperatures considerably

lower than the critical temperature(see Fig.4,right panel).

The presented results clearly show the numerical existence of an AT line

for the mean-?eld SK model,whereas for the model atσ=0.75(outside the

mean-?eld universality class there is no sign of a transition in a small but ?nite?eld).Together with results presented in Ref.[11]we thus conclude

that short-range spin glasses below the upper critical dimension do not order in an externally-applied magnetic?eld.

3.2Are spin glasses ultrametric?

Ultrametricity is an intrinsic property of the Parisi solution of the mean-?eld model[35]and it can be described in the following way:Consider an

equilibrium ensemble of states at T

1,i.e.,the states lie on an isosceles triangle.

To date,the existence of ultrametricity for short-range spin glasses—which would validate the applicability of the mean-?eld solution to short-range systems—is highly controversial.Recent results[36]suggest that short-range systems are not ultrametric,whereas other opinions exist[37–39].Be-cause the one-dimensional Ising chain allows for tuning the system away from the mean-?eld universality class,it presents itself as the ideal test-bed for this problem.Below we present results forσ=0.0(SK)as well as0.75 (non-mean-?eld regime)using an approach closely related to the one used by Hed et al.[36].

We generate1000equilibrium states(spin con?gurations)for1000–4000

disorder instances of the model using exchange Monte Carlo at T=0.4T c

(i.e.,T=0.4for the SK model and T=0.27for the one-dimensional chain withσ=0.75).The temperature used is chosen such that we probe deep in the spin-glass phase,but not too low to avoid trivial state triangles.The generated states are in turn sorted using Ward’s hierarchical clustering ap-proach[40](see Fig.5).The clustering procedure starts with L clusters which contain one state and the two closest lying clusters are merged.Distances are measured in terms of the hamming distance dαβ=(1?qαβ)/2.This pro-cedure is repeated until one large cluster is obtained.Once the states are clustered,we select three states from di?erent branches of the left sub-tree (see Ref.[36]for details)and sort the distances:d max≥d med≥d min.We compute the correlator

d max?d med

K=

New Insights from One-Dimensional Spin Glasses9

Fig.5.Dendrograms and distance matrices.Darker colors correspond to closer distances in phase space.Left panel:SK model at T=0.4(L=1024).The distance matrix shows clear structure below T c.Middle panel:One-dimensional Ising chain forσ=0.75and T=0.40

Fig.6.Left panel:Distribution P(K)for the mean-?eld SK model at T=0.4T c. The data peak for K→0with increasing system size showing clearly that phase space is ultrametric.Right panel:Same observable as in the left panel for the one-dimensional Ising chain withσ=0.75(non-mean-?eld universality class)at T=0.27≈0.4T c.While the divergence at K=0is less pronounced,the data show a similar behavior than in the left panel.

where?(d)is the width of the distribution of distances.If the space is ultra-metric,we expect d max=d med for L→∞.This means for the distribution P(K)→δ(K=0)for L→∞.

In Fig.6(left panel)we show data for the distribution P(K)for the SK model at T=0.4T c.For increasing system size the data seem to converge to

10Helmut G.Katzgraber et al.

a limiting delta function.This is not the case for T=T c(not shown)where the data are independent of system size and show no divergence for K→0.This suggests that the used observable correctly captures the underlying ultrametric behavior.Furthermore,studies of cophonetic distances show that the structures found in the dendrograms are not arbitrary.Figure6(right panel)shows P(K)for the one-dimensional Ising chain withσ=0.75at T=0.27≈0.4T c for a range of system sizes.The data show a similar behavior than for the SK model,although the e?ect is not as pronounced. Further simulations atσvalues larger than0.75as well as a quantitative study of the number of clusters and RSB layers shall clarify with certainty if short-range spin glasses have an ultrametric phase structure or not.

4Future directions

In the past,we have studied several properties of spin glasses using the one-dimensional Ising chain,such as the nature of the spin-glass state[8,9,41], ground-state energy distributions of spin glasses[10],the existence of a spin-glass state in a?eld[11](see above),?eld chaos in spin glasses[13],local-?eld distributions in spin glasses[12],as well as ultrametricity in spin glasses[13] (see above).Furthermore,other groups have also studied other open ques-tions in the physics of spin glasses with this model,such as nonequilibrium problems[42]or di?erent cumulants of the order parameter distribution[43]. All previous studies had been done on the model presented in Eq.(1)using Ising spins.In this section we mention some extensions,as well as modi?ca-tions of the model which can be used to study di?erent problems.

4.1Variations on the model

Recently,a one-dimensional spin-glass chain with Heisenberg spins has been studied in Ref.[44]to test the controversial spin-chirality decoupling scenario [45–48]proposed by Kawamura.It is unclear to date what the nature of the spin-glass state in Heisenberg spin glasses is.In particular,it is unclear if spin and chirality degrees of freedom decouple.To test this scenario,simulations of the one-dimensional Heisenberg chain[44]atσ=1.1have been performed. Forσ=1.1the spin degrees of freedom only order at T=0,whereas results suggest that the chirality degrees of freedom order at?nite nonzero temper-atures.Similar studies could be performed for models with planar XY spin degrees of freedom,as well as Potts spins(work in progress).

Finally,the Hamiltonian can also be modi?ed to include,for example, p-spin interactions to study structural glasses[49](work in progress).Pre-liminary results suggest that the model has a?nite ordering temperature in the mean-?eld regime.

New Insights from One-Dimensional Spin Glasses11 4.2Studying larger systems with dilution

While the linear system sizes L studied with the one-dimensional Ising chain are considerably larger than the system sizes accessible in short-range sys-tems,the fact that the model is fully-connected makes it di?cult to access large numbers of spins because any algorithm would have to do O(L2)up-dates at every Monte Carlo sweep.This is because the system has O(L2) interactions between the spins.Recently,Leuzzi and collaborators suggested a variation of the model which is diluted,thus drastically reducing the num-ber of neighbors for each spin[50].In their version,a random bimodally-distributed bond between two spins is placed with a power-law dependent probability adjusted such that the mean connectivity z is always6for allσ. This has the e?ect that forσ→0we recover the Viana-Bray model with ?xed connectivity[51].Because of the dilution,systems of104spins can be studied to temperatures as low as～0.4T c.

In Fig.7we present data for a diluted system with Gaussian-distributed random interactions andσ=0.75.In this case,the probability to place a bond between two spins is P(J ij=0)=r?2σ,where r is the distance between the spins.The mean connectivity z of the model is then given by z=2ζ(2σ)in the thermodynamic limit,whereζis the Riemann zeta function.Forσ=0 we recover the SK model,whereas,for example,forσ=0.75the mean connectivity is only z≈5.22thus allowing the study of large systems(note that the interactions are rescaled such that T MF

=1).In the left panel of

c

Fig.7.Left panel:Finite-size correlation length for the power-law diluted one-dimensional Ising spin glass with variable connectivity.The data cross at T c≈0.54illustrating the existence of a transition.Right panel:Distribution of the spin overlap function P(q)at T=0.4for di?erent system sizes.The width of the lines corresponds to the error bars.

12Helmut G.Katzgraber et al.

Fig.7the?nite-size correlation length as a function of temperature is shown. The data cross at T c≈0.54signaling the existence of a spin-glass transition.

In the right panel of Fig.7we show the distribution of the spin overlap q= L?1 i Sαi Sβi.While the data show corrections due to critical?uctuations, they converge to a seemingly system-size independent value around|q|≈

0.This would agree with the replica symmetry breaking scenario by Parisi [4,52,53,2]although lower temperatures are needed to properly address this question.Current work focuses on revisiting the existence of a spin-glass state in a?eld using the model with dilution.

5Benchmarking of algorithms

Benchmarking optimization algorithms[55,56]plays a crucial role in the?eld of disordered and complex systems,as well as many other interdisciplinary applications.Knowing the range of applicability of a given algorithms can be of great importance when trying to solve a given problem.For example, whereas the branch,cut&price algorithm[57–59]works best for short-range systems,it is least e?cient when the interactions are long range[10].

Recently,the hysteretic optimization heuristic[60]has been introduced to estimate ground states of spin-glass systems.The method is known to work well for the mean-?eld SK model,as well as the traveling salesman problem[56].The idea behind hysteretic optimization is successive demag-netization at zero temperature.With additional shake-ups(?eld increases to further randomize the system)close-to-ground-state con?gurations can be obtained.Recently,Gon?c alves and Bottcher[54]have studied the e?ciency of the method on the one-dimensional Ising chain.Data adapted from their work shown in Fig.8clearly show that the method works best for in?nite-

Fig.8.Percentage error in the

ground-state energies obtained

with hysteretic optimization in

comparison to exact ground states

as a function of the exponent

σfor di?erent system sizes L.

Clearly,the algorithm works best

forσ 1/2(vertical dashed line),

i.e.,in the in?nite-range regime.

Outside the in?nite-range regime,

avalanches do not percolate the

system and thus the algorithm is

less e?cient.Figure adapted from

reference[54].

New Insights from One-Dimensional Spin Glasses13 range models(σ≤1/2)where avalanches in the hysteresis loops proliferate easily.While the error in?nding the ground states increases slightly with system size forσ 1/2the increase is considerably stronger for larger values ofσ.As soon as the system is not in?nite-ranged,avalanches are small and the method is not e?cient.

6Concluding remarks

By using a one-dimensional spin-glass model with random power-law interac-tions we have been able to shed some light on some of the open questions in the physics of spin glasses.The one-dimensional spin-glass chain has the ad-vantage over conventional models that large linear system sizes can be stud-ied.Furthermore,by changing the power-law exponent of the interactions, di?erent universality classes ranging from the mean-?eld to the short-range universality class can be probed.The latter feature of the model allows also for e?cient benchmarking of optimization algorithms. Acknowledgments

We would like to thank Stefan B¨o ttcher,Ian Campbell,Thomas J¨o rg,Flo-rent Krz?a ka l a,Wolfgang Radenbach,David Sherrington and Gergely Zimanyi for discussions.In particular,we would like to thank B.Gon?c alves and S.B¨o ttcher for sharing their data from Ref.[54].The simulations have been performed on the Asgard,Brutus,Gonzales and Hreidar clusters at ETH Z¨u rich.H.G.K acknowledges support from the Swiss National Science Foun-dation under Grant No.PP002-114713.A.P.Y.acknowledges support from the National Science Foundation under Grant No.DMR0337049. References

1.K.Binder,A.P.Young,Rev.Mod.Phys.58,801(1986)

2.M.M′e zard,G.Parisi,M.A.Virasoro,Spin Glass Theory and Beyond(World

Scienti?c,Singapore,1987)

3. A.P.Young(ed.),Spin Glasses and Random Fields(World Scienti?c,Singa-

pore,1998)

4.G.Parisi,Phys.Rev.Lett.43,1754(1979)

5.S.F.Edwards,P.W.Anderson,J.Phys.F:Met.Phys.5,965(1975)

6. D.S.Fisher,D.A.Huse,Phys.Rev.Lett.56,1601(1986)

7.G.Kotliar,P.W.Anderson,D.L.Stein,Phys.Rev.B27,R602(1983)

8.H.G.Katzgraber,A.P.Young,Phys.Rev.B67,134410(2003)

9.H.G.Katzgraber,A.P.Young,Phys.Rev.B68,224408(2003)

10.H.G.Katzgraber,M.K¨o rner,F.Liers,M.J¨u nger,A.K.Hartmann,Phys.Rev.

B72,094421(2005)

11.H.G.Katzgraber,A.P.Young,Phys.Rev.B72,184416(2005)

14Helmut G.Katzgraber et al.

12.S.Boettcher,H.G.Katzgraber,D.S.Sherrington,Local?eld distributions in

spin glasses(2007).(arXiv:cond-mat/0711.3934)

13.H.G.Katzgraber,J.Phys.:Conf.Ser.95,012004(2008)

14. D.S.Fisher,D.A.Huse,Phys.Rev.B38,386(1988)

15. D.Sherrington,S.Kirkpatrick,Phys.Rev.Lett.35,1792(1975)

16.K.Hukushima,K.Nemoto,J.Phys.Soc.Jpn.65,1604(1996)

17.N.Metropolis,S.Ulam,J.Am.Stat.Assoc.44,335(1949)

18.H.G.Katzgraber,S.Trebst,D.A.Huse,M.Troyer,J.Stat.Mech.P03018

(2006)

19.H.G.Katzgraber,M.Palassini,A.P.Young,Phys.Rev.B63,184422(2001)

20.R.N.Bhatt,A.P.Young,Phys.Rev.Lett.54,924(1985)

21.J.C.Ciria,G.Parisi,F.Ritort,J.J.Ruiz-Lorenzo,J.Phys.I France3,2207

(1993)

22.N.Kawashima,A.P.Young,Phys.Rev.B53,R484(1996)

23. A.Billoire,B.Coluzzi,Phys.Rev.E68,026131(2003)

24. E.Marinari,C.Naitza,F.Zuliani,J.Phys.A31,6355(1998)

25.J.Houdayer,O.C.Martin,Phys.Rev.Lett.82,4934(1999)

26. F.Krzakala,J.Houdayer,E.Marinari,O.C.Martin,G.Parisi,Phys.Rev.Lett.

87,197204(2001)

27.H.Takayama,K.Hukushima,Field-shift aging protocol on the3D Ising spin-

glass model:dynamical crossover between the spin-glass and paramagnetic states(2004).(cond-mat/0307641)

28. A.P.Young,H.G.Katzgraber,Phys.Rev.Lett.93,207203(2004)

29.P.E.J¨o nsson,H.Takayama,J.Phys.Soc.Jap.74,1131(2005)

30.T.J¨o rg,H.G.Katzgraber,F.Krzakala,Do spin glasses order in a?eld?(2007).

(arXiv:cond-mat/0712.2009)

31.J.R.L.de Almeida,D.J.Thouless,J.Phys.A11,983(1978)

32.H.G.Ballesteros,A.Cruz,L.A.Fernandez,V.Martin-Mayor,J.Pech,J.J.

Ruiz-Lorenzo,A.Tarancon,P.Tellez,C.L.Ullod,C.Ungil,Phys.Rev.B62, 14237(2000)

33.M.Palassini,S.Caracciolo,Phys.Rev.Lett.82,5128(1999)

34. A.Billoire,Some aspects of in?nite range models of spin glasses:theory and

numerical simulations(2007).(arXiv:cond-mat/0709.1552)

35.M.M′e zard,G.Parisi,N.Sourlas,G.Toulouse,M.Virasoro,Phys.Rev.Lett.

52,1156(1984)

36.G.Hed,A.P.Young,E.Domany,Phys.Rev.Lett.92,157201(2004)

37.S.Franz,F.Ricci-Tersenghi,Phys.Rev.E61,1121(2000)

38.P.Contucci,C.Giardin`a,C.Giberti,G.Parisi,C.Vernia,Phys.Rev.Lett.99,

057206(2007)

39.T.J¨o rg,F.Krzakala,Comment on”Ultrametricity in the Edwards-Anderson

Model”(2007).(arXiv:cond-mat/0709.0894)

40.J.Ward,J.of the Am.Stat.Association58,236(1963)

41.H.G.Katzgraber,M.K¨o rner,F.Liers,A.K.Hartmann,Prog.Theor.Phys.

Supp.157,59(2005)

42.M.A.Montemurro,F.A.Tamarit,Int.J.Mod.Phys.C14,1(2003)

43.L.Leuzzi,J.Phys.A32,1417(1999)

44. A.Matsuda,M.Nakamura,H.Kawamura,J.Phys.C19,5220(2007)

45.H.Kawamura,Phys.Rev.Lett.68,3785(1992)

46.L.W.Lee,A.P.Young,Phys.Rev.Lett.90,227203(2003)

New Insights from One-Dimensional Spin Glasses15

47.K.Hukushima,H.Kawamura,Phys.Rev.B72,144416(2005)

48.I.Campos,M.Cotallo-Aban,V.Martin-Mayor,S.Perez-Gaviro,A.Tarancon,

Phys.Rev.Lett.97,217204(2006)

49.M.A.Moore,Phys.Rev.Lett.96,137202(2006)

50.L.Leuzzi,G.Parisi,F.Ricci-Tersenghi,J.J.Ruiz-Lorenzo,(2008).(arxiv:cond-

mat/0801.4855)

51.L.Viana,A.J.Bray,J.Phys.C18,3037(1985)

52.G.Parisi,J.Phys.A13,1101(1980)

53.G.Parisi,Phys.Rev.Lett.50,1946(1983)

54. B.Gon?c alves,S.Boettcher,J.Stat.Mech.P01003(2008)

55. A.K.Hartmann,H.Rieger,Optimization Algorithms in Physics(Wiley-VCH,

Berlin,2001)

56. A.K.Hartmann,H.Rieger,New Optimization Algorithms in Physics(Wiley-

VCH,Berlin,2004)

57. F.Barahona,M.Gr¨o tschel,M.J¨u nger,G.Reinelt,Oper.Res.36,493(1988)

58. F.Liers,M.J¨u nger,G.Reinelt,G.Rinaldi,in New Optimization Algorithms

in Physics,ed.by A.K.Hartmann,H.Rieger(Wiley-VCH,Berlin,2004) 59.M.J¨u nger,G.Reinelt,S.Thienel,in DIMACS Series in Discrete Mathemat-

ics and Theoretical Computer Science,vol.20,ed.by W.Cook,L.Lovasz, P.Seymour(American Mathematical Society,1995)

60.K.F.Pal,Physica A367,261(2006)

清华大学《控制工程基础》课件-4

则系统闭环传递函数为 假设得到的闭环传递函数三阶特征多项式可分解为 令对应项系数相等，有 二、高阶系统累试法 对于固有传递函数是高于二阶的高阶系统，PID校正不可能作到全部闭环极点的任意配置。但可以控制部分极点，以达到系统预期的性能指标。 根据相位裕量的定义，有 则有 则 由式可独立地解出比例增益，而后一式包含两个未知参数和，不是唯一解。通常由稳态误差要求，通过开环放大倍数，先确定积分增益，然后计算出微分增益。同时通过数字仿真，反复试探，最后确定、和三个参数。 设单位反馈的受控对象的传递函数为 试设计PID控制器，实现系统剪切频率 ，相角裕量。 解： 由式,得 由式,得 输入引起的系统误差象函数表达式为

令单位加速度输入的稳态误差，利用上式，可得 试探法 采用试探法，首先仅选择比例校正，使系统闭环后满足稳定性指标。然后，在此基础上根据稳态误差要求加入适当参数的积分校正。积分校正的加入往往使系统稳定裕量和快速性下降，此时再加入适当参数的微分校正，保证系统的稳定性和快速性。以上过程通常需要循环试探几次，方能使系统闭环后达到理想的性能指标。 齐格勒－尼柯尔斯法 （Ziegler and Nichols ） 对于受控对象比较复杂、数学模型难以建立的情况，在系统的设计和调试过程中，可以考虑借助实验方法，采用齐格勒－尼柯尔斯法对PID调节器进行设计。用该方法系统实现所谓“四分之一衰减”响应（”quarter-decay”），即设计的调节器使系统闭环阶跃响应相临后一个周期的超调衰减为前一个周期的25%左右。 当开环受控对象阶跃响应没有超调，其响应曲线有如下图的S形状时，采用齐格勒－尼柯尔斯第一法设定PID参数。对单位阶跃响应曲线上斜率最大的拐点作切线，得参数L 和T，则齐格勒－尼柯尔斯法参数设定如下： (a) 比例控制器： (b) 比例－积分控制器： ， (c) 比例－积分－微分控制器： ， 对于低增益时稳定而高增益时不稳定会产生振荡发散的系统，采用齐格勒－尼柯尔斯第二法(即连续振荡法)设定参数。开始只加比例校正，系统先以低增益值工作，然后慢慢增加增益，直到闭环系统输出等幅度振荡为止。这表明受控对象加该增益的比例控制已达稳定性极限，为临界稳定状态，此时测量并记录振荡周期Tu和比例增益值Ku。然后，齐格勒－尼柯尔斯法做参数设定如下： (a) 比例控制器：

新八年级下册英语课文语法填空和短文改错

一、 HanselandGretel______(live)nearaforestwithhisfatherandstepmother.Oneyear,theweather______(be)sodry thatnofood_____(grow).Thewifetoldherhusbandthatunlesshe________(leave)hischildren______(die)inthe forest,thewholefamilywoulddie.Gretel_________(hear)thattheirstepmotherplanned________(kill)herandh erbrother.ButHanselhadaplan________(save)himselfandhissister.Hewenttogetsomewhitestonesbeforehew enttobedthatnight.Thenextday,thewife_________(send)thechildrentotheforest.Hansel___________(drop)t https://www.sodocs.net/doc/392642570.html,terthatnight,they________(see)thestonesbecauseoftheshiningmoon.Thestones__ ace)forthisistheHimalayas.TheHimalayasrunalongthe______________(southwest)partofChina.Ofallthem ountains,Qomolangma________(rise)thehighestandis____________(famous).Itis8,844.43metershighands oisverydangerous________(climb).Thickcloudscoverthetopandsnow__________(fall)veryhard.Evenmore serious_______(difficulty)includefreezingweatherconditionsandheavystorms.Itisalsoveryhard_____(take) inairasyougetnearthetop. Thefirstpeople_____(reach)thetopwereTenzingNorgayandEdmundHillaryonMay29,1953.ThefirstChi neseteam__(do)soinI960,whilethefirstwoman_____(succeed)wasJunkoTabeifromJapanin1975. Whydosomanyclimbersrisktheir_____(life)?Oneofthemain_______(reason)isbecausepeoplewant______(c hallenge)themselvesinthefaceofdifficulties. Thespiritoftheseclimbers_____(show)usthatweshouldnevergiveup____(try)toachieveourdreams.Itals

中国文化常识-中国传统艺术I 中国民乐与戏剧(英文讲授中国文化)Chinese Folk Music and Local Operas

Folk Music 中国民乐 Beijing Opera 京剧 Local Operas 地方戏 Outline ?China, a nation of long history with profound culture, ?inherited and developed a great variety of traditional art forms which suit both refined and popular tastes 雅俗共赏. ?From the melodious and pleasant folk music to the elaborate and touching local dramas, ?from the simple but elegant inkwash painting(水墨画) to the flexible and powerful calligraphy, ?one can always discern the light of sparkling wisdom. ?Traditional Chinese arts have tremendously impressed the world. ?Chinese folk music, with strong nationalistic features, is a treasure of Chinese culture. ?As early as the primitive times, Chinese people began to use musical instruments, ?which evolved today into four main types categorized by the way they are played 吹拉弹打.materials they are made of （丝竹管弦）. stringed instrument 丝：弦乐器 wind instrument 竹：管乐器 ?吹：The first type is wind instrument 管乐器, as show in xiao （a vertical bamboo flute ）箫，flute 笛子, suona horn 唢呐, etc. ?拉：The second type is string instrument 弦乐器, represented by urheen 二胡, jinghu 京胡（a two-string musical instrument similar to urheen ）, banhu fiddle 板胡, etc.

《交通规划》课程教学大纲

《交通规划》课程教学大纲 课程编号：E13D3330 课程中文名称：交通规划 课程英文名称：Transportation Planning 开课学期：秋季 学分/学时：2学分/32学时 先修课程：管理运筹学，概率与数理统计，交通工程学 建议后续课程：城市规划，交通管理与控制 适用专业/开课对象：交通运输类专业/3年级本科生 团队负责人：唐铁桥责任教授：执笔人：唐铁桥核准院长： 一、课程的性质、目的和任务 本课程授课对象为交通工程专业本科生，是该专业学生的必修专业课。通过本课程的学习，应该掌握交通规划的基础知识、常用方法与模型。课程具体内容包括：交通规划问题分析的一般方法，建模理论，交通规划过程与发展历史，交通调查、出行产生、分布、方式划分与交通分配的理论与技术实践，交通网络平衡与网络设计理论等，从而在交通规划与政策方面掌握宽广的知识和实际的操作技能。 本课程是一间理论和实践意义均很强的课程，课堂讲授要尽量做到理论联系实际，模型及其求解尽量结合实例，深入浅出，使学生掌握将交通规划模型应用于实际的基本方法。此外，考虑到西方在该领域内的研究水平，讲授时要多参考国外相关研究成果，多介绍专业术语的英文表达方法以及相关外文刊物。课程主要培养学生交通规划的基本知识、能力和技能。 二、课程内容、基本要求及学时分配 各章内容、要点、学时分配。适当详细，每章有一段描述。 第一章绪论（2学时） 1. 交通规划的基本概念、分类、内容、过程、发展历史、及研究展望。 2. 交通规划的基本概念、重要性、内容、过程、发展历史以及交通规划中存在的问题等。

第二章交通调查与数据分析（4学时） 1. 交通调查的概要、目的、作用和内容等；流量、密度和速度调查；交通延误和OD调查；交通调查抽样；交通调查新技术。 2. 交通中的基本概念，交通流量、速度和密度的调查方法，调查问卷设计与实施，调查抽样，调查结果的统计处理等。 第三章交通需求预测（4学时） 1. 交通发生与吸引的概念；出行率调查；发生与吸引交通量的预测；生成交通量预测、发生与吸引交通量预测。 2. 掌握交通分布的概念；分布交通量预测；分布交通量的概念，增长系数法及其算法。 3. 交通方式划分的概念；交通方式划分过程；交通方式划分模型。 第四章道路交通网络分析（4学时） 1. 交通网络计算机表示方法、邻接矩阵等 2. 交通阻抗函数、交叉口延误等。 第五章城市综合交通规划（2学时） 1. 综合交通规划的任务、内容；城市发展战略规划的基本内容和步骤 2. 城市中长期交通体系规划的内容、目标以及城市近期治理规划的目标与内容 第六章城市道路网规划（2学时） 城市路网、交叉口、横断面规划及评价方法。 第七章城市公共交通规划（2学时） 城市公共交通规划目标任务、规划方法、原则及技术指标。 第八章停车设施规划（2学时） 停车差设施规划目标、流程、方法和原则。 第九章城市交通管理规划（2学时） 城市交通管理规划目标、管理模式和管理策略。 第十章公路网规划（2学时） 公路网交通调查与需求预测、方案设计与优化。 第十一章交通规划的综合评价方法（2学时） 1. 交通综合评价的地位、作用及评价流程和指标。 2. 几种常见的评价方法。 第十二章案例教学（2学时）