Renormalized phonons in nonlinear lattices: A variational approach

0.44 MB·English

by  Junjie Liu

Renormalizedphonons innonlinear lattices: A variationalapproach
Junjie Liu,1 Sha Liu,2 Nianbei Li,3 Baowen Li,2,3,4,5,∗ and Changqin Wu1,6,†
1StateKeyLaboratoryofSurfacePhysicsandDepartmentofPhysics,FudanUniversity,Shanghai 200433,China
2DepartmentofPhysicsandCentreforComputationalScienceandEngineering,NationalUniversityofSingapore,117546Singapore
3CenterforPhononicsandThermalEnergyScience,SchoolofPhysicsScienceandEngineering,TongjiUniversity,Shanghai200092,China
4NUS Graduate School for Integrative Sciences and Engineering, 117456 Singapore
5GrapheneResearchCentre, FacultyofScience, NationalUniversityofSingapore, 117542 Singapore
5 6CollaborativeInnovationCenterofAdvancedMicrostructures, NanjingUniversity,Nanjing210093, China
1 (Dated:)
0
2 Weproposeavariationalapproachtostudyrenormalizedphononsinmomentumconservingnonlinearlattices
witheithersymmetricorasymmetricpotentials.Toinvestigatetheinfluenceofpressuretophononproperties,we
n
deriveaninequalitywhichprovidesboththelowerandupperboundoftheGibbsfreeenergyastheassociated
a
variational principle. This inequality is a direct extension to the Gibbs-Bogoliubov inequality. Taking the
J
symmetryeffectintoaccount,thereferencesystemforthevariationalapproachischosentobeharmonicwith
7
an asymmetric quadratic potential which contains variational parameters. We demonstrate the power of this
1
approach by applying it to one dimensional nonlinear lattices with a symmetric or asymmetric Fermi-Pasta-

Ulamtypepotential. Forasystemwithasymmetricpotentialandzeropressure, werecoverexistingresults.
]
h Forothersystemswhichbeyondthescopeofexistingtheories,includingthosehavingthesymmetricpotential
c andpressure,andthosehavingtheasymmetricpotentialwithorwithoutpressure,wealsoobtainaccuratesound
e velocity.
m
- PACSnumbers:63.20.-e,63.20.Ry,45.10.Db,44.05.+e
t
a
t
s I. INTRODUCTION usedtoobtainapproximateinformationofanonlinearsystem
.
t viaareferencesystemandanassociatedvariationalprinciple
a
[16]. Thereferencesystemwhosepropertiescanbeeasilyob-
m Heat conduction in low dimensional anharmonic systems
tainedcontainsseveralvariationalparameters.Inavariational
has attracted considerable interest in recent years [1–3].
-
scheme,anoptimalreferencesystemcanbeobtainedbyvary-
d Phonon,asthepredominantheatcarrierininsulatingmateri-
n ingthoseparameterssuchthatboundsofthevariationalprin-
als,undoubtedlyliesintheheartofheatconduction.However,
o ciplegotoarelativeminimumormaximum.
phonon bears a solid basis only in harmonic systems. The
c In this paper, we develop a variational approach to study
[ intrinsic nonlinearity in anharmonic systems will inevitably
affect the behavior of phonon. Therefore, understandingthe phononsinnonlinearlattices. Wechooseageneralharmonic
1 system as the reference system for this purpose and regard
phonon properties in anharmonic systems represents an im-
v theso-obtainedoptimalharmonicsystemasanapproximation
portantquestionintheheatconductionfield.
9 to the originalsystem fromwhich we identify the properties
6 Arenormalizedphonon(r-ph)picturewasthenputforward
of r-phs. We consider applications to one dimensional (1D)
1 independentlybyseveralgroupsusingvaryingtechniques[4–
nonlinearlatticeswitheithersymmetricorasymmetricinter-
4 11]. Within the scope of this picture, one can successfully
particleFermi-Pasta-Ulam(FPU)potentials[17]anddemon-
0
interpret and understand a wide range of physical phenom-
. stratethepowerofourapproachbycomparingwithmolecule
1 ena, includinga theoreticaldescriptionofthe soundvelocity
dynamics(MD)results.
0 [12]aswellasscalinglawsofthermalconductivityκ(T)with
Thispaperisorganizedasfollows. InSec.II,weintroduce
5 temperatureT [10,13].
1 ourvariationalapproach.InSec.III,wepresentnumericalde-
However, the generality of existing, state-of-the-artquasi-
: tails. In Sec.IV and Sec.V, we study nonlinear lattices with
v harmonic theories is limited. They were found to provide
symmetric and asymmetric FPU potentials, respectively. Fi-
i
X inaccurate predictions on the sound velocity of a nonlinear nally,webrieflysummarizetheworkinSec.VI.
lattice with an asymmetric inter-particle potential [14]. Re-
r
a cently,anumericalmethodwhichaimstojustifythevalidity
of phonon concept in nonlinear lattices was proposed [15].
II. THEVARIATIONALAPPROACH
The existence of phonon modes in nonlinear lattices with
asymmetricpotentialsisconfirmedforawiderangeofparam-
A. Thevariationalprinciples
eters.Hence,aunifiedquasi-harmonictheorywhichextendto
coverthegeneralcasesbeyondharmonicisdesirable.
In statistical mechanics, variational approaches are often Ourcurrentinvestigationaimstodevelopaquasi-harmonic
theory applicable for nonlinear lattices with asymmetric po-
tentials. Oneofthe attractivefeaturesofsuchsystemsis the
existenceofnonzerointernalpressureduetotheasymmetryof
∗Electronicaddress:[email protected] thepotential.Thenfromastatisticalmechanicspointofview,
†Electronicaddress:[email protected] it’sconvenienttousethelanguageoftheisothermal-isobaric
2
ensemble which maintains constant temperature T, constant F ,respectively,theenthalpygoestotheenergy,theprobabil-
0
particlenumberN andconstantpressureP (so-calledNPT itymeasureEq. (3)shouldalsobereplacedbythecanonical
ensemble) to describe them [18]. Their equilibriumthermo- measure
dynamicpropertiescanthusbewelldeterminedbytheGibbs
freeenergy ρc(q,p)= e−βTH e−βT(H−F). (8)
e−βTHdqdp ≡
G = −βT−1ln e−βT(H+PL)dqdp , (1) Then we found that tRhe inequality Eq. (6) recoversthe well
(cid:20)Z (cid:21) knownGibbs-Bogoliubov(GB)inequality[21]
where H, P and L denote the Hamiltonian, pressure and
F F + H H , (9)
volume (or length in 1D cases) of the system, respectively, ≤ 0 h − 0iρc0
β 1/T is the inverse temperature (we set k = 1), q
T B
andp≡areshortfortheproductsofallthecoordinatesandmo- where F0 is the Helmholtz free energy of the reference sys-
mentaofthesystem,respectively.ThevolumeLisafunction temandh·iρc0 standsfortheensembleaveragewithrespectto
ofqonly.Introducing the canonical measure ρc(q,p) [ e−βT(H0−F0)]. And the
0 ≡
inequalityEq.(7)goesto
= H +PL, (2)
H F F + H H (10)
≥ 0 h − 0iρc
whose ensemble average is just the enthalpy of the system
[19]. The correspondingprobabilitymeasure in phase space with ρc theensembleaveragewithrespecttotheprobabil-
reads itymhe·aisureρc(q,p) [c.f. Eq. (8)]. Asa lowerboundofthe
HelmholtzfreeenergyF [16,22],it’slittleappliedsincethe
e−βTH ensembleaverageinitcannotbeevaluatedanalytically.How-
ρ(q,p)= e−βT(H−G). (3)
e−βTHdqdp ≡ ever,itismoreaccuratethantheupperboundindetermining
the free energy of solids [23, 24]. As can be seen later, in
AvariationalprincRipleisaninequalitysatisfiedbythephys- our case for determining the sound velocity for anharmonic
ical quantityin which we may be interested[16]. Hence we lattices, itisstillthelowerboundthatgivesbetterprediction
shouldlookfor inequalitiesforthe Gibbsfree energyG. To comparingtotheupperbound.
do this, we introducea referencesystem with a Hamiltonian
H andpreparetheoriginalsystemandthereferencesystem
0
atsamepressure.Thenfollowingthisspirit,weintegrateboth B. Thenonlinearsystems
sidesofthefollowingequality
We consider 1D momentum conserving nonlinear lattices
ρ(q,p) = ρ (q,p)e−βT(H−H0)−βT(G0−G) (4) describedbythegeneralHamiltonian[25]
0
overthewholephasespacetoyield N p2
H = n +V(q q ) , (11)
n n−1
2m −
1 = e−βT(G0−G) e−βT(H−H0) , (5) nX=1(cid:20) (cid:21)
· ρ0
D E whereN istheparticlenumber,pndenotesthemomentumof
whereG0istheGibbsfreeenergyofthereferencesystemand n-thparticle,qn =xn−nrdenotesthedisplacementofn-th
tHhe0p=roHba0bi+litPymL0e,ashu·irρe0ρde(nqo,tpes)theee−nβsTe(mHb0−leGa0v)e.rageunder lpuatretipcolesiftiroonmanitdsreqthueileibqruiuilmibrpiuomsitdioisntannrcewfoitrhthxeninttheeraacbtisoon-
0
Takinglogarithmoverbothside≡softheaboveequationand bond,andV representstheinter-particlepotential.Forbrevity
using the Jensen’s inequality for exponential functions, i.e., andwithoutloss ofgenerality,we takem = 1andr = 1 as
ex >ehxi(following[20]),wehave theunitofmassandlength,respectively.
h i The average lattice spacing a is then given by 1+ q
n
G ≤ G0+hH−H0iρ0. (6) qfonr−a1niρN, -apnadrttihcelealvaettriacgee. WlaettiscaeyltehnagttthheL¯la≡tticheLiisρateiqtsuanhlastuNr−aal
By switching the role of the nonlinear system and the refer- length if a equals r [=1], namely, L¯ = N for an N-particle
encesysteminEq.(4),weobtain lattice. Theaveragelengthcanbechangedtoothervaluesby
applyingpressure.
G ≥ G0+hH−H0iρ (7) satFisufirethseVrm(δore),=weVin(troδdu)c,ewδenre≡feqrnit−toqna−s1y.mIfmtheetripcopteontetina-l
n n
−
with ρ theensembleaveragewithrespecttotheprobability tial, otherwise we call it an asymmetric one. In terms of δn
meashu·rieρ(q,p) [c.f. Eq. (3)]. These two inequalities[Eqs. andpn,theequationsofmotion(EOMs)read
(6)and(7)]givetheupperandlowerboundofG,respectively.
If the pressure vanishes, i.e., P = 0, the Gibbs free ener- δ˙n = pn+1 pn, (12)
−
gies G and G reduce to the Helmholtz free energiesF and p˙ = V′(δ ) V′(δ ), (13)
0 n n n−1
−
3
wherethedotandtheprimedenotethetimeandspacederiva- D. Determiningtheoptimalharmonicsystems
tive,respectively.
Specifically,inthiswork,wefocusontheFPUlatticeswith Notethatdq dq dq = dδ dδ dδ inthelargeN
1 2 N 1 2 N
theinter-particlepotential[17] limit(theJacobianis··u·nity),thenforas·y·s·temwiththeproba-
bilitymeasureEq. (3),wehave
1 α β
V(δ ) = δ2 + δ3 + δ4, (14)
n 2 n 3 n 4 n N
ρ(q,p)= ρ (δ ,p ), (18)
s n n
which has become an archetype1D nonlinearsystem in sta-
n=1
Y
tisticalmechanics[26,27]. Wecallthelatticewithα=0the
whereρ denotesthemarginalphasespacedistributionforthe
FPU-β lattice. Otherwise, we refer it to the FPU-αβ lattice. s
singlesitevariablesδ andp ofn-thparticle[28,29]
For simplicity but without loss of generality, we only study n n
the case ofβ = 1 in thispaper. Itis evidentthatthe FPU-β 1 p2
latticehasasymmetricpotentialwhiletheFPU-αβlatticehas ρs(δ,p)= exp βT +V(δ)+Pδ (19)
z − 2
anasymmetricone. Therefore,anFPU-β latticewithnatural (cid:26) (cid:20) (cid:21)(cid:27)
lengthhaszeropressure,whileanFPU-αβ latticewithnatu- withzthecorrespondingpartitionfunction
rallengthhasa nonvanishingpressuredueto theasymmetry
ofthepotential.Nevertheless,ourvariationalprinciples[Eqs. p2
z = exp β +V(δ)+Pδ dδdp. (20)
(6)and(7)]enableustostudythemwithinthesametheoreti- − T 2
ZZ (cid:26) (cid:20) (cid:21)(cid:27)
calscheme.
SucharesultcanbereadilytestedusingMDsimulations,see
Sec. III.
ForasystemwithsufficientlylargeparticlenumberN,the
C. TheharmonicreferencesystemH0 Gibbsfreeenergycanbeexpressedas
Phonons bear a solid basis only in harmonic systems, in G=Ng = Nβ−1lnz, (21)
− T
ordertodevelopaneffectivephonontheoryfornonlinearlat-
tices, we should choose a quadratic H0 as a reference sys- whereg =−βT−1lnzistheGibbsfreeenergyperparticle.So
tem. Moreover, the present work investigates nonlinear lat- theGibbsfreeenergyofthereferencesystemH0reads
tices with asymmetric inter-particle potentials. Taking those
G = Nβ−1lnz (22)
intoconsideration,we considera harmonicreferencesystem 0 − T 0
inthefollowingform with z the partition function determined by Eqs. (15) and
0
(20). Itcanbeevaluatedanalytically,whichgives
N p2 K
H = n +V (δ ) ; V (δ ) = (δ d)2, 2π P2
0 0 n 0 n n
2 2 − z = exp β Pd . (23)
nX=1(cid:20) (cid:21) (15) 0 βT√K (cid:20)− T (cid:18) − 2K(cid:19)(cid:21)
in which K and d are variational parameters with K being Usingthesingle-sitedistributionEq.(19),andnoticingthat
the effectiveelastic constantandd quantifyingthe degreeof thelengthLandL haveasameexpression,namely, (1+
0 n
asymmetryofthe potentials. Notethatbysettingd = 0, we δ ),theaveragesinEqs. (6)and(7)read
n
can use H to deal with nonlinear systems with symmetric P
0
potentialsaswell. hH−H0iρ0 = N ·hV(δ)−V0(δ)iρ0s; (24)
Thedispersionrelationofthisreferencesystemisgivenby = N V(δ) V (δ) , (25)
hH−H0iρ ·h − 0 iρs
ωk = 2√Ksinka, (16) whereh·iρ0s andh·iρs denotesingle-siteaverageswithrespect
2 to the probabilitymeasure Eq. (19) with the potential being
V andV,respectively. Wewillusethesesymbolsintherest
wherekisthewavenumber.Itcanberegardedasanapproxi- 0
ofthepaper.
matedispersionrelationforther-phsinnonlinearlattices[15].
Thecorrespondingsoundvelocityreads
1. Theupperboundharmonicsystem
c =√Ka. (17)
s
Firstly,wefocusontheupperboundofG[Eq. (6)]. Min-
The variationalmethod is then to select the optimal refer-
imizingitwithrespecttothevariationalparametersK andd,
encesystemswithparameterizedHamiltonianH thateither
0 i.e.,
minimizethe righthandside (r.h.s.) of Eq. (6) or maximize
ther.h.s. ofEq. (7). Thisstrategythenprovidesapproxima- ∂
[G + ] = 0, (26)
tionsforthe Gibbsfreeenergyandallowsustofind optimal ∂K 0 hH−H0iρ0
H0 forthesystem H. The processis goingto be detailedin ∂
thenextsubsection. ∂d[G0+hH−H0iρ0] = 0, (27)
4
weobtainthefollowingcoupledequations:
1.5
FPU−β
FPU−αβ
(V V )(δ+ P )
d = − 0 KU ρ0s, (28) 1.25
U D E
V V
h − 0iρ0s N
/ 1
K = hV −V0iρ0s . (29) ¯L
U
β (V V )(δ d + P )2 0.75
T − 0 − U KU ρ0s
D E
It can be readily tested that the second order derivatives are 0.5
−1.8 −1.2 −0.6 0 0.6 1.2
positiveat (d ,K ) so thatthis set ofsolutionindeedgives P
U U
theminimumoftheupperbound.ThusK andd determine
U U
the optimalupperboundof the Gibbsfree energyGand the
corresponding optimal Harmonic reference system (denoted FIG.1: Average latticelength L¯ as afunction of pressure P. The
astheUHsystem). dashed-dottedlineindicatesthatforthe1DFPU-βlatticewithT =
1.Thesolidlinedenotesthatforthe1DFPU-αβlatticewithT =1
andα=1.
2. Thelowerboundharmonicsystem
average length L¯ equals N(1 + δ ), so a stressless sys-
NowweturntothelowerboundofG[Eq. (7)]. Similarly, tem with an asymmetric potentialhcoiρrsrespondsto an average
awnedmd,axthimeoizpetiimtawlistholuretisopnecrteatodsthe variationalparametersK ldeinstgrtihbuNtio(1n+,i.heδ.i,ρρcs),[wEqh.er(e1ρ9cs)]mweiatnhsPthe=ca0n.oPnaicratilcsuilnagrllye,-sfioter
s
the FPU-αβ lattice with zero pressure, the average length is
P
d = δ + , (30) smallerthanN ascanbeseenfromthefigure.
L h iρs K
L AnNPT systemwithpressureP canbepreparedeitherby
1 applyinganexternalpressureP totheendparticleswithfree
K = . (31)
L βT [hδ2iρs −(hδiρs)2] wbohuenredaL¯ry(Pco)nadsitiaonfuonrctbioynfioxfinPg tihsejtuosttaldelepnicgttehdLintoFiL¯g(.P1),.
We have also checkedthatKL and dL indeedgivethe max- In the present work we adopt the second approach. Specif-
imum of the lower bound,thus them uniquelydeterminethe
ically, a modified periodic boundary condition is thus used
optimallowerboundoftheGibbsfreeenergyGandalsothe to fix the total length at a certain value, namely, q q =
N 0
correspondingoptimalharmonicreferencesystem(denotedas L¯ N = N δ . For such systems, a symplectic−integra-
theLHsystem).NotethatKLdoesnotrelyondLandiscom- tor−SABA whithiρas correctorSABA C [30] is adoptedto in-
2 2
pletelydeterminedbyEq. (31).
tegrate the EOMs with a time step h = 0.02 to ensures the
TheLHsystem andtheUH systemcanbe regardedasef-
conservationofthetotalsystemenergyandmomentumupto
fectivedescriptionsoftheoriginalnonlinearsystem.Ifweuse highaccuracy10−14inthewholeruntime,atransienttimeof
thesetwooptimalsystemstoconstructtheeffectivetheoryof order107isusedtoequilibratethesystem.
r-phsintheoriginalnonlinearsystem,thenthedispersionre-
lation should follow Eq. (16) with K given by K or K .
L U
Theaccuracyoftheresultscanbeevaluatedbycomparingthe
A. Single-sitedistributions
calculatedsoundvelocitywiththepredictionsusingEq. (17).
Inthefollowing,wepresentnumericaldetailsandthenapply
Wefirstcheckthesingle-sitedistributionsinEq.(19).After
thisstrategytotwo1Dnonlinearlattices,namely,theFPU-β
thesystemreachesitsthermalequilibriumstate,wecanobtain
latticeandtheFPU-αβ lattice.
timeseriesofv (equalsp )andδ ofthen-thparticle(ncan
n n n
be an arbitraryintegerfrom1 to N). We thenplottheir his-
togramsandcomparetheenvelopesagainstρ =ρ ρ with
III. NUMERICALDETAILS s v δ
1 v2
In this work, we consider systems either with or without ρ = exp β , (32)
v T
pressure P. The average lattice length L¯ can be controlled √2π (cid:18)− 2 (cid:19)
by adjusting the pressure, as demonstrated in Fig. 1. It can ρ = z−1exp β 1δ2+ αδ3+ 1δ4+Pδ ,(33)
be clearly seen that for the FPU-β lattice once the pressure δ δ − T 2 3 4
(cid:20) (cid:18) (cid:19)(cid:21)
isnonzero,theaveragelengthofthelattice isawayfromthe
natural length, i.e., L¯ = N. It further indicates that we can wherez = dδexp[ β (1δ2+ αδ3+ 1δ4+Pδ)].
6 δ − T 2 3 4
applya positiveor negativepressureto this lattice system in Toillustratethecomparison,wesimulatetheFPU-βlattice
order to compress or elongate its total length. But for sys- with N = 2R048, T = 1, and L¯ = N or L¯ = 1.2N, the
temswithasymmetricpotentials,i.e.,theFPU-αβ lattice,the correspondingpressureis0or-0.4309accordingtoFig. 1,re-
pressurewouldbenonzeroatitsnaturallength. Notethatthe spectively.AsimilarsimulationiscarriedoutontheFPU-αβ
5
lattice with N = 2048, T = 1, α = 1, and L¯ = 0.8023N or stronganharmonicitywill affectits accuracy. The second
or L¯ = N, with the pressure equals 0 or -0.3904, respec- oneisto lookatthe frequencyofthe lowestphononpeakof
tively. The results are depicted in Figs. 2 and 3. ρ and ρ thepowerspectrumofanN-particlelattice[14]
v δ
areshownasgreenandredsolidcurvesinthefigures,respec-
tively. Perfectagreementsbetweentheoreticalcurvesandthe
c π
MD resultsare presented, whichindicatesthe validityof the ω = 2 s sin , (34)
1
single-sitedistributioninEq. (19). a N
0.8 wherec isthesoundvelocity. Thislowestphononpeakcan
P =0 (a) P =0 (b) s
ability00..46 a(slweeaFyisgb.e4d),ettheuctsetdheinstohuenpdovweleorcsiptyecctarnumbewwitehllhdigetherrmesionleudtiboyn
b
Pro0.2 ω1. Thethirdoneobtainsthesoundvelocityfromthedisper-
sion relation[15]. Thismethod,althoughfollowsthedefini-
0.8
P 6=0 (c) P 6=0 (d) tionofthesoundvelocity,ismostcomputationallyexpensive
bility0.6 comparingwiththeothertwo, becausethe calculationofthe
ba0.4 dispersionrelationistimeconsuming.
o
Pr0.2
Therefore, in this study, we choose the second method to
0
−2 0 2 −4 v0 4 obtainMDresultsofthesoundvelocityofnonlinearlattices.
δ
According to the Wiener-Khinchin theorem [32], the power
spectrumP(ω)canbeobtainedbydoingFouriertransformof
the velocity autocorrelationof a single particle v (t)v (0)
n n
FIG.2:Single-sitedistributionsforthe1DFPU-βlatticewithP =0 with t = 0,h,2h, ,t . In orderto getsmhoothphononi
max
(upper panel) or P = 0 (lower panel). The blue parts in the left peaksin the power··s·pectrum, the maximumcorrelationtime
6
andrightcolumndenotethehistogramofδandv,respectively. The
t whoseinverse1/t determinesthefrequencyresolu-
green and red solid lines are ρv and ρδ [c.f., Eqs. (32) and (33)], timonaxissettobe224hundmearxtheconditionthatN =1024.Fig.
respectively.
4 presents the power spectrum of the FPU-β lattice and the
FPU-αβ lattice withα = 1attheirnaturallength. T = 1is
used for both models. It is apparentthat phononpeaks with
0.8 thelowestfrequenciesarewelldistinguishedfromthepower
P =0 (a) P =0 (b)
bility0.6 sppeeacktrburmoa.dTenhiengn.onIlninteeraersittyinrgelny,dewrseitfsouenffdectthaint tthhee pphhoonnoonn
a0.4
b
o peak of the FPU-αβ lattice is broader than the one of the
Pr0.2
FPU-β lattice,whichindicatesthattheasymmetricpotentials
0.8
P 6=0 (c) P 6=0 (d) providestrongerphonon-phononinteractioncomparedtothe
bility0.6 symmetricones.
a0.4
b
o
Pr0.2
0 −2 0 2 −4 v0 4 6x 10−4
δ FPU−β
FPU−αβ
)4
FIG.3:Single-sitedistributionsforthe1DFPU-αβlatticewithwith u.
P = 0(upperpanel)orP = 0(lowerpanel). Thebluepartsinthe a.
(
6 )
leftandrightcolumndenotethehistogramofδ andv,respectively. ω
(
Thegreenandredsolidlinesareρvandρδ[c.f.,Eqs.(32)and(33)], P2
respectively.
0
0 0.01 0.02 0.03
ω
B. Soundvelocity
FIG. 4: Power spectrum P(ω) in the low frequency regime. The
Nowweturntothecalculationofthesoundvelocity.Sofar,
dashed-dottedlinestandsfortheFPU-βlatticeatitsnaturallength.
there are mainly three numerical methods which can obtain
ThesolidlinerepresentstheFPU-αβlatticewithα=1andnatural
thesoundvelocityofthenonlinearlattices. Thefirstmethod
length. Forcomparison, thelowestharmonicfrequencyisdepicted
regardsthemovingvelocityoffrontpeaksoftheequilibrium
asblacksolidline.
spatiotemporalcorrelationfunctionasthesoundvelocity[12,
31].However,abroadeningoffrontpeaksathightemperature
6
IV. THEFPU-βLATTICE 5
8
4
6
Inthefollowing,wewillgiveadetailedstudyoftheFPU- a %
β lattice by using our variational approach. Being a special ty/3 σ/24
case of the FPU-αβ lattice, i.e., α = 0, it has a symmetric oci 0
inter-particlepotential. Vel2 10−2 10−1 1T00 101 102
d
n
u
o
A. P =0 S MD
1 LH
UH
Theexistingquasi-harmonictheoriesofthismodelallfocus
10−2 10−1 100 101 102
on thisparticularcase [4, 5, 7–11]. In thispart, we notonly
T
presentthecomparisonbetweenourapproachandMDresults,
butalsopointouttheconnectionsbetweenourapproachand
FIG.5:Soundvelocityasafunctionoftemperatureforthe1DFPU-
someoftheexistingquasi-harmonictheories.Thefirstobser-
β lattice with zero pressure. The circles are MD results obtained
vation is d = d = 0 [c.f., Eqs. (28) and (30)] by noting
U L fromthepowerspectrum,thesolidlinethepredictionoftheLHsys-
that the potentialof the FPU-β lattice is an evenfunctionof tem(√KL [Eq. (37)]),andthedashed-dottedlinethepredictionof
δ. Theninthe following,we onlyneedto concernaboutthe the UH system (√KU [Eq. (36)]). The inset presents the relative
parameterK. deviationofthesoundvelocity:σ =(cTsH cMsD)/cMsD 100%with
Firstly, we focus on the UH system. The corresponding cTsHandcMsDthetheoreticalresultandtheM−Dresultofth·esoundve-
parameterKU [Eq.(29)]canbecalculatedtoyield locity,respectively. ThesquaresdenotecTsH =√KL[Eq. (37)]and
√KU [Eq.(36)]inredandblue,respectively.
K2 K 3β−1 =0, (35)
U − U − T
whichhasapositivesolution
specific situation, i.e., the lattice length is fixed to be 1.2N.
1 Since L¯ = N(1+ δ ), we have δ = 0.2. Therefore,
KU = 2(1+ 1+12βT−1). (36) thepressureP canbheioρbstainedbysohlviinρgs
q
Thisresultcoincideswiththepredictionofthesocalledself- δe−βT(V+Pδ)dδ
consistentphonontheory(SCPT) [10] forthe FPU-β lattice, δ = =0.2. (38)
h iρs e−βT(V+Pδ)dδ
sincetheSCPTbasedontheGBinequalityEq. (9)aswell. R
Then we turn to the LH system. The parameter KL [Eq. ThepressureistemperaRturedependent,whichisplottedinthe
(31)]reducestothesimpleform
inset(b)ofFig. 6. Sincethelatticeisslightlyelongated,the
1 pressureshouldbenegative.
KL = βThδ2iρcs. (37) ageWsiitnhEthqo.s(e31v)aflouresKofnpurmesesruircea,llwyeancdanthcuaslocubltaatientthheevaavleure-
L
The correspondingsound velocity √KL is exactly the same ofKL. AsforKU, we insertthe valuesofpressureintoEq.
asthatpredictedbytheeffectivephonontheory(EPT)[8,12] (28) andEq. (29), thensolvethe coupledequationstogether
based on the generalized equipartition theorem [33] as well togetthevalueofKU. ThesoundvelocityoftheLHsystem
as the nonlinearfluctuating hydrodynamics(NFH) using the andtheUHsystemcanbeobtainedby√KLaand√KUa,re-
hydrodynamicapproximation[29,34,35],althoughtheNFH spectively. WepresentresultsofthesoundvelocityinFig. 6.
isnotaneffectivetheoryforphonons. ItisclearlyshowsthattheLHsystemstillgivesbetterresults
ResultsofthesoundvelocityareillustratedinFig. 5. The thantheUHsystem,asignificantdeviationexistsbetweenthe
excellentagreementbetweentheLHsystem’spredictionsand latter and MD results. So the LH system can also describe
MDresultsisobvious.WhilefortheUHsystem,thedeviation r-phsintheFPU-β latticeswithpressure.
fromMDresultsbecomeslargerandlargerasthetemperature The prediction of the sound velocity given by the NFH is
increases.Thereasonbehindthisfactisclear,sincehighorder alsoshowninFig. 6. Ittakesacomplexform
nonlineartermsignoredbytheUHsystembecomeimportant
athightemperature. HencetheLHsystemcandescriber-phs 1β−2+ V +Pδ;V +Pδ 1/2
intheFPU-β latticeswithoutpressure. cs/a = βT [hδ;2δiThV;Vhi−hδ;Vi2]+ 2β1iThδ;δi!
K , (39)
NFH
B. P =0 ≡
6 p
in which all the is short for , i.e., averages under
h·i h·iρs
Now we begin to apply our variational approach to the the single site probability measure Eq. (19), and A;B =
h i
FPU-β lattice with pressure. The method used to obtain AB A B . NoticingtheEPTandtheSCPTareunable
h i−h ih i
nonzero pressure has been discussed in Sec. III. For sim- todealwithsystemswithpressure,theybecomeinvalidinthis
plicity, but without loss of generality, here we only study a case.
7
A. P =0
MD
8 (a)
6 LH
/a79 σ/%24 UNHFH We first apply the UH system to the FPU-αβ lattice. The
elocity35 −1020−2 1T00 102 corresp2odn3din+gdp4aramete1rsdU andKU nowsatisfy
V U U + (d +αd2 +d3)
nd1 −01 (b) 4 βTKU U U U
u 1
So P−−32 +β2K2 (α+3dU) = 0, (40)
−4 T U
10−2 100 102 1 1 3
T (1+2αd +3d2) = 0.(41)
10 −2 10−1 100 101 102 βT − βTKU U U − βT2KU2
T
We notethatd isnolongerzeroduetotheasymmetricpo-
U
tential V. The two equations are coupled, we should solve
FIG.6:Soundvelocityasafunctionoftemperatureforthe1DFPU-
themtogethertogetthevalueofK .
βlatticewithnonzeropressure. ThecirclesareMDresultsobtained U
WethenutilizetheLHsystemtoinvestigatethelattice.The
from the power spectrum, the solid line the prediction of LH sys-
tem (√KL [Eq. (31)]), the dashed linethe prediction of the NFH parametersoftheLHsystemaregivenby
(√KNFH [Eq. (39)]),andthedashed-dottedlinethepredictionof
d = δ , (42)
theUHsystem(√KU [Eq. (29)]). Inset(a)presentstherelativede- L h iρcs
viationofthesound velocity: σ = (cTsH cMsD)/cMsD 100% with 1
cvTseHlocaintdy,crMsesDpethcteivtehleyo.reTthicealsqrueasureltsadnednotthee−cMTsHD=re√suKltLo·f[Ethqe. s(o3u1n)]d, KL = βT hδ2iρcs −(hδiρcs)2 . (43)
√KU [Eq. (31)], and √KNFH [Eq. (39)]inred, blue, and cyan, Similarly, d is nonzero f(cid:2)or an asymmetric(cid:3)potential. Note
L
respectively.Inset(b)showspressureP asafunctionoftemperature thattheEPTstillpredictstheeffectiveforceconstantas[8]
T for1DFPU-βlatticewithL¯ =1.2N.
1
K = (44)
EPT βThδ2iρcs
V. THEFPU-αβLATTICE
withV nowtheFPU-αβ potentialEq. (14). Itisevidentthat
thedenominatorofK andK taketheformofthevari-
L EPT
WhenacubictermisaddedtotheFPU-β potential,weget anceandthesecondmomentofδ,respectively. Thisdiscrep-
the asymmetric FPU-αβ potential [Eq. (14)]. The potential ancyresultsfromtheasymmetryofthepotential,sincehδiρcs
with different α is plotted in Fig. 7. It is clearly seen that vanishesforsymmetricpotentials. Ascanbeseenlater,such
thepotentialbecomesmoreasymmetricasαincreases.When acorrectionimprovestheaccuracyoftheresultssignificantly.
α exceeds 2, dV/dδ = 0 begins to have two solutions, the We consider lattices at a fixed temperature T = 0.5 and
potentialtends to become a double-wellone which is out of varyαfrom0.2to2. Resultsofthesoundvelocityareillus-
the scope of the present investigation, we can see that from tratedinFig. 8. ThedeviationofpredictionsoftheEPTfrom
the form of the potential with α = 2.1 in the figure. The the measuredvalue is clearly revealed as α increasing. This
demonstrates the invalidity of the EPT for the strong asym-
metriccases. While the LH systemgivesaccurateresultsno
0.4
matterhowlargetheasymmetryis. Itthusshowsthesignif-
0.3 icanceof the correctionterm in the denominatorof KL [Eq.
(43)]. Still,weobservethattheLHsystemgivesmuchbetter
0.2 resultsthantheUHsystem. SowecanregardtheLHsystem
δ) asaneffectivedescriptionofr-phsintheFPU-αβlatticewith-
( 0.1
V outpressure.Meanwhile,anagreementbetweentheNFHand
0 the LH system rendersa fact thathydrodynamicapproxima-
tionscanalsobeappliedtolong-wavelengthr-phsinlattices
−0.1 α=0.5
α=1 withasymmetricpotentials.
α=2
α=2.1
−0.2
−2 −1 0 1
δ
B. P =0
6
FIG.7:PotentialV(δ)ofthe1DFPU-αβlatticewithvaryingα.
Now we consider the FPU-αβ lattice with pressure. For
simplicity,Weonlyinvestigatethesystematitsnaturallength.
inherentasymmetrycaninducethermalexpansion,whichhas Thevalueofthepressurecanbeobtainedbysolving
beenshowntoplayasignificantroleinheatconduction[36–
39]. In the following,we willstudy thislattice byusing our δe−βT(V+Pδ)dδ
δ = =0. (45)
variationalapproach. h iρs e−βT(V+Pδ)dδ
R
R
8
1.7
MD 12 MD
Velocity/a1.3 LUNEHPHFHT Velocity/a3579 σ/%−104840−(2a) 1T00 102 LUNHHFH
nd 10 nd1 −01 (b)
Sou0.9 σ/%−100 Sou P−−−432
−5
−20 10−2 100 102
0.5 1α 1.5 2 T
0.50 0.5 1α 1.5 2 10 −2 10−1 1T00 101 102
FIG. 8: Sound velocity as a function of α for the 1D FPU-αβ FIG.9:Soundvelocityasafunctionoftemperatureforthe1DFPU-
lattice with T = 0.5 and zero pressure. The circles are MD re- αβlatticewithα=1andnonzeropressure. ThecirclesareMDre-
sultsobtainedfromthepowerspectrum,thedashedlinethepredic- sultsobtainedfromthepowerspectrum,thesolidlinetheprediction
tion of the NFH (√KNFH with P = 0 [Eq. (39)]), the dashed- ofLHsystem(√KL[Eq.(31)]),thedashedlinethepredictionofthe
dottedlinethepredictionoftheUHsystem(√KU [Eq. (41)]),the NFH(√KNFH[Eq.(39)]),andthedashed-dottedlinetheprediction
red solid line the prediction of the LH system (√KL [Eq. (43)]), oftheUHsystem(√KU [Eq. (29)]). Inset(a)presentstherelative
andthemaroonsolidlinethepredictionoftheEPT(√KEPT [Eq. deviationofthesoundvelocity:σ =(cTsH cMsD)/cMsD 100%with
(44)]).Theinsetpresentstherelativedeviationofthesoundvelocity: cTsH and cMsD the theoretical result and the−MD result o·f the sound
σ=(cTsH cMsD)/cMsD 100%withcTsHandcMsDthetheoreticalresult velocity,respectively. ThesquaresdenotecTsH = √KL [Eq. (31)],
− ·
andtheMDresultof thesound velocity, respectively. Thesquares √KU [Eq. (31)], and √KNFH [Eq. (39)] inred, blue, andcyan,
denote cTsH = √KL [Eq. (43)], √KU [Eq. (41)], √KNFH with respectively. Inset(b)showspressureP asafunctionofthetemper-
P = 0[Eq. (39)],and√KEPT [Eq. (44)]inred, blue,cyan, and atureT fortheFPU-αβlatticewithα=1atitsnaturallength.
maroon,respectively.
2.4
−0.4
Results of pressure as a function of temperature are plotted ty/a G/N−0.5
in the inset (b) of Fig. 9. The absolute value of pressure ci Exact
o Lowerbound
increases as the temperature increases because of the inter- Vel1.6 −0.60 .2 0.8 α 1.4 2
particleinteractionisstronger.Atthelowtemperatureregime d
n
where thermal excitations dwell the very bottom of the po- u
o
tential, particles can not feel the asymmetry of the potential S
strongly, so the pressure approacheszero as the temperature MD
LH
tendstozero.
0.8
0.2 0.8 1.4 2
α
With these values, K can be easily calculated from Eq.
L
(31). ThevalueofK canbeobtainedbysolvingEqs. (28)
U
and (29) together. Since the existing quasi-harmonic theo- FIG.10: Soundvelocityasafunctionofαforthe1DFPU-αβ lat-
riesfailinthismodel,onlyourresultsandtheNFH’swillbe ticewithT =0.5andnonzeropressure. ThecirclesareMDresults
shown. obtained from the power spectrum, the solid line the prediction of
√KL [Eq. (31)]. TheinsetpresentstheGibbsfreeenergyperpar-
Theresultforthecaseα = 1isillustratedinFig. 9. From ticle. ThedashedlineistheexactGibbsfreeenergyg accordingto
Eq.(21),thesolidlineisthepredictionofthelowerboundaccording
this figure, we see that the LH system works better than the
toEq.(7).
UHsystemasusual.
We furtherinvestigatetheFPU-αβ latticeswithvaryingα
byusingtheLHsystem. ThetemperatureT isfixedtobe0.5. VI. SUMMARY
FromtheFig. 10,itisapparentthatthediscrepancybetween
thetheoreticalpredictionandMDresultstendstoincreaseas In summary, we have presented a variational approach to
αincreases. OnepossiblereasonisthattheGibbsfreeenergy study renormalized phonons in momentum conserving non-
givenbythelowerbounddepartfromtheexactresultsignif- linearlattices. Specifically, we obtaintwo optimalharmonic
icantlyatlargeα(wecanseethatfromtheinsetinFig. 10). reference systems, namely, the LH system and the UH sys-
This fact indicates that the first cumulant approximation we tem, by optimizingthe two boundsof the Gibbs free energy
adoptedinderivingtheinequalitiesisnotenough. Higheror- ofthenonlinearsystem. Theseoptimalsystemswhichdeter-
dercontributionstotheGibbsfreeenergymustbeconsidered minethepropertiesofrenormalizedphononscanberegarded
inordertoimprovethismethod[40,41]. asoptimalharmonicapproximationsofthenonlinearsystem.
9
This method has been applied to lattices with either sym- asymmetric potentials. We owe this ability to two aspects.
metricorasymmetricpotentials,withorwithoutpressure.For One is the choice of an parameterizedasymmetric harmonic
latticeswithsymmetricpotentials,suchastheFPU-β lattice, potentialwith a parameterd whichcan quantifiesthe degree
our variational approach gives two optimal and symmetrical ofasymmetryofthepotentials.Theotheristheconsideration
reference harmonic lattices. In the zero pressure case, it is oftheGibbsfreeenergyinsteadoftheHelmholtzfreeenergy,
very interesting to find that the sound velocity derived from whichenablesustodealwithsystemswithpressure.
theLHsystemreproducetheprevioustheoreticalresultsfrom We also foundthat the NFH givessatisfactory predictions
the effectivephonontheoryand nonlinearfluctuatinghydro- of sound velocity in all those cases. The theory focuses on
dynamictheory,andthesoundvelocityderivedfromtheUH a hydrodynamicdescription of nonlinear lattices, not a con-
system recover the previous theoretical results from a self- struction of an effective theory of the renormalizedphonons
consistentapproach. innonlinearlattices. Thoseagreementsmeansthatthehydro-
For lattices with asymmetric potentials, such as the FPU- dynamic approximation really captures essential features of
αβ latticewheretheexistingeffectivephonontheoryfailsto the systems in the long-wavelengthlimit. However, for sys-
predictthe sound velocity, our variationalapproachcan also tems withoutmomentumconservation,the conceptof sound
giveaccuratepredictions. Inparticular,ourapproachreveals velocity is invalid, the NFH may not be able to give the in-
thereasonwhytheeffectivephonontheorycannotpredictthe formation of the renormalized phonons, although it can still
correctsoundvelocity. describe correlation functions of those systems [42]. While
In all the cases, the LH system works better than the UH our variational approach can be uesd to investigate phonon
system, since the ensemble averages in the former are eval- bandgapandphonondispersionofthosesystems[43].
uatedundertheprobabilitymeasureofthenonlinearsystem,
it simultaneously takes nonlinear contributions into account
compared with the latter. Therefore, the LH system can be
Acknowledgments
treatedasaneffectivetheoryofrenormalizedphononsinvar-
ious momentum conserving nonlinear lattices. A deviation
between the prediction of our variational approach and the We acknowledge the financial supports from the Na-
MD results has also been foundfor stronger asymmetryand tional Nature Science Foundation of China with Grant No.
nonzero pressure situation where the underlying mechanism 11334007(N.LiandB.Li),theNationalBasicResearchPro-
needsfurtherinvestigations. gram of China(2012CB921401) and the Program for New
Compared with the existing quasi-harmonic theories, this Century Excellent Talents of the Ministry of Education of
approach can be used to investigate nonlinear systems with ChinawithGrantNo. NCET-12-0409.
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