Statistical Physics of the Symmetric Group

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by  Mobolaji Williams

StatisticalPhysics of the SymmetricGroup
Mobolaji Williams
Department of Physics, Harvard University, Cambridge, MA 02138, USA ∗
Eugene Shakhnovich
DepartmentofChemistryandChemicalBiology,HarvardUniversity,Cambridge,MA02138, USA†
(Dated: March7,2017)
7
1 Orderedchains(suchaschainsofaminoacids)areubiquitousinbiologicalcells,andthesechains
0 performspecificfunctionscontingentonthesequenceoftheircomponents. Usingtheexistenceand
2 generalpropertiesofsuchsequencesasatheoreticalmotivation,westudythestatisticalphysicsofsys-
temswhosestatespaceisdefinedbythepossiblepermutationsofanorderedlist,i.e.,thesymmetric
r
a group,andwhoseenergyisafunctionofhowcertainpermutationsdeviatefromsomechosencorrect
M ordering.Suchanon-factorizablestatespaceisquitedifferentfromthestatespacestypicallyconsid-
eredinstatisticalphysicssystemsand consequentlyhas novelbehavior insystemswithinteracting

5 andevennon-interactingHamiltonians. Variousparameterchoicesofamean-fieldmodelrevealthe
systemtocontainfivedifferentphysicalregimesdefinedbytwotransitiontemperatures,atriplepoint,

] andaquadruplepoint.Finally,weconcludebydiscussinghowthegeneralanalysiscanbeextendedto
h
statespaceswithmorecomplexcombinatorialpropertiesandtootherstandardquestionsofstatistical
c
mechanicsmodels.
e
m
- I. INTRODUCTION
t
a
t
s Chains of amino acids are important components of
.
t biological cells, and for such chains the specific order-
a
ingoftheaminoacidsisoftensofundamentaltothere-
m
sulting functionandstabilityof the foldedchainthatif
- majordeviationsfromthecorrectorderingweretooccur,
d
n thefinalchaincouldfailtoperformitsrequisitefunction
o withinthecell,provingfataltotheorganism. FIG.1: Foldingvs. Design(orInverseFolding)
c More specifically, we see the relevance of correct or- problems: Theproteinfoldingproblemisconcerned
[ dering in the study of protein structure, which is of- withdeterminingthethreedimensionalstructure
2 tendividedintothe proteinfoldingandproteindesign producedbyaparticularsequenceofaminoacids. The
v problem. While the protein folding problem concerns proteindesignproblem(whichmotivatesthecurrent
4 findingthethree-dimensionalstructureassociatedwith work)isconcernedwithfindingthesequence(s)of
7 agivenaminoacidsequence,theproteindesignproblem aminoacidswhichyieldagiventhreedimensional
3
(alsotermed the inverse-foldingproblem; see Figure 1) polypeptidestructure. Anumberofapproachestothe
7
concerns finding the correctamino acidsequence asso- designproblemaregivenin[1]
0
ciatedwithagivenproteinstructure.
.
1 Anaspectofonesolutiontotheproteindesignprob-
0
7 lem is to maximize the energy difference between the For example, for a polypeptide chain with N residues,
1 low-energy folded native structure and the higher en- ratherthansearchingovertheentiresequencespace(of
: ergymisfolded/denaturedstructures. Indoing so, one size20N),onesearchesoveraspaceofsequences(ofsize
v
takes native structure asfixed and then determines the N!/n !n !...n !)whicharedefinedbyafixednumber
i 1 2 20
X sequence yielding the minimum energy, under the as- ofeachaminoacid.
sumption (termed the ”fixed amino-acid composition”
r This aspect of the protein design problem alerts one
a assumption) thatonlycertainquantitiesof amino-acids
to agap in the statisticalmechanics literature. Namely,
appear in the chain [2]. In this resolution (specifically
theredonotseemtobeanysimpleandanalyticallysolu-
termedheteropolymermodels[3][4])thecorrectamino
blestatisticalmechanicsmodelswherethespaceofstates
acid sequence is found by implementing an MC algo-
isdefinedbypermutationsofalistofcomponents.
rithminsequencespacegivenacertainfixedaminoacid
We can take steps toward constructing such a model
composition. This entailsassuming the number of var-
by considering reasonable general properties it should
ioustypesofaminoacidsdoesnotchange,anddistinct
have. Ifweassumetherewasaspecificsequenceofcom-
statesinsequencespacearepermutationsofoneanother.
ponentswhichdefinedthelowestenergysequenceand
wasthermodynamicallystableinthemodel,thendevi-
ationsfromthissequencewouldbelessstable. Because
∗[email protected] oftherolesequencesofmoleculesplayinbiologicalsys-
†[email protected] tems, it is worth asking what features we expect such
2
sequences to have from the perspective of modeling in pressedincomponentformas
statisticalmechanics.
InSectionIIweintroducethemodel,andcomputean ω~ =(ω1,...,ωN). (2)
exact partition function which displays what we term
Wewilltakethecollectionofcomponents ω asintrin-
“quasi”-phasetransitions—atransitioninwhichthese- { k}
sictooursystem,andthustakethestatespaceofoursys-
quence of lowest energy becomes entropically disfa-
temtobethesetofallthevectorswhoseorderingofcom-
vored above a certain temperature. In Section III, we
ponentscanbeobtainedbypermutingthecomponents
extendthepreviousmodelbyaddingaquadraticmean
of ~ω, i.e., all permutations of ω ,...,ω . We represent
field interaction term and show that the resulting sys- 1 N
temdisplaystwotransitiontemperatures,atriplepoint, anarbitrarystateinthisstatespaceas~θ = (θ1,...,θN),
andaquadruplepoint. InSectionIV,wediscussvarious wheretheθk aredrawnwithoutrepeatfrom ωk . For-
{ }
ways we can extend this model in theoretical or more mally,wewouldsayourspaceofstatesisisomorphicto
phenomenologicaldirections. thesymmetricgroupon~ω([5]). Wewillthusdenoteour
statespaceas
Sym(ω) := SetofAllPermutationsof(ω ,...,ω ).
1 N
II. SYSTEMANDPARTITIONFUNCTION (3)
andthen anarbitrarystate~θ isjustanelementelement
Ourlargergoalistostudyequilibriumthermodynam- ofthisset.
ics for a system defined by permutations of a set of N Asafirstformulationofthemodel,wewilltake~θ0 =ω~
componentswhereeachunique permutationisdefined (thecorrectsequence)torepresentthezeroenergystate
byaspecificenergy. Ingeneral,weshouldconsiderthe inthesystem,andforeachcomponentθiofanarbitrary
casewherethesetofN componentsconsistsofLtypes vector ~θ which differs from the corresponding compo-
ofcomponentsforwhichifnkisthenumberofrepeated nentωiin~ω,thereisanenergycostofλi >0. TheHamil-
componentsoftypek,then L n = N. Forsimplic- tonianisthen
k=1 k
ity, however, we will take n = 1 for all k so that each
kP N
componentisofauniquetypeandL=N.
( θ )= λ I , (4)
To study the equilibrium thermodynamics of such a HN { i} i θi6=ωi
i=1
systemwithafixedN atafixedtemperatureT,weneed X
to compute its partition function. For example, for a where θ and ω are components of vectors ~θ and ω~ re-
i i
sequence with N components (withno components re- spectively,andI isdefinedby
peated),thereareN!microstatesthesystemcanoccupy
andassumingwelabeleachstatek = 1,...,N! 1,N!, 1 ifAistrue
andassociateanenergyǫkwitheachstate,thent−hepar- IA ≡ 0 ifAisfalse. (5)
titionfunctionwouldbe (cid:26)
N! Wenotethatalthoughwelabelourgeneralstateasθ~ =
Z = e−βǫk, (1) (θ ,...,θ ),thecomponentsθ ,...,θ canonlytakeon
1 N 1 N
kX=1 mutually-exclusivevaluesfromtheset{ωk}.
Wewanttoexploretheequilibriumthermodynamics
whereǫ foreachstatekcouldbereasonedfromamore
k ofasystemwithaHamiltonianofEq.(4). Thisamounts
precisemicroscopictheoryofhowthecomponentsinter-
tocalculatingthepartitionfunction
actwithoneanother. Phenomenologically,Eq.(1)would
be the most precise way to construct a model to study N
theequilibriumpropertiesofpermutations,butbecause Z ( βλ )= exp β λ I , (6)
itbearsnoclearmathematicalstructure,itisunenlight- N { i} ~θ∈SXym(ω) − Xi=1 i θi6=ωi!
eningfromatheoreticalperspective.
Instead we will postulate a less precise, but theoreti- whereSym(ω)isagainthesetofallpermutationsofthe
cally more interesting model. For most ordered chains components of (ω ,...,ω ). To find a closed form ex-
1 N
in biological cells, there is a single sequence of com- pression for the partition function, we group terms in
ponents which is the “correct” sequence for a partic- Eq.(6) according to the number of ways to completely
ular macrostructure. Deviations from this correct se- reorderjcomponentsin~ωwhilekeepingtheremaining
quenceareoftendisfavoredbecausetheyformlessstable componentsfixed. Eachsuchreordering(i.e.,eachvalue
macrostructuresortheyfailtoperformtheoriginalfunc- of j) is associated with a sum over products of e−βλi
tionofthe“correct”sequence. Withthegeneralproper- termswithjfactorsofe−βλi (forvariousi)ineachterm.
tiesofsuchsequencesinmind,wewillabstractlyrepre- Thetotalpartitionfunctionisasumofallsuchreorder-
sentoursystemasconsistingofN siteswhicharefilled ingsforalljsfrom0toN. Ascanbeseenfromadirect
with particular coordinate values denoted by ω . That
k
is,wehaveanarbitrarybutfixedcoordinatevector~ωex-
3
expansionofEq.(6),wehave definedas
N
N
g (j)= d (13)
Z ( βλ )= d Π e−βλ1,...,e−βλN , (7) N j j
N { i} j j (cid:18) (cid:19)
j=0
X (cid:0) (cid:1) isthenumberofwaystoreorderalistofN elementsso
whered ,termedthenumberofderangementsofalistof that j elements are no longer in their original position.
j
j([6]),isthenumberofwaystocompletelyreorderalist Thiscombinatorialdefinitionofg (j)willproveuseful
N
of j elements. The quantity Π (x ,...,x ), termedthe when we explore the phase behavior of more complex
j 1 N
jth elementary symmetric polynomial on n ([7]), is the modelsofpermutations.
sumofallwaystomultiplyjelementsoutoftheN term FromtheformofEq.(10),itisclearthat,physically,its
set x1,...,xN . For example, Π2(x1,x2,x3) = x1x2 + associated Hamiltonian is not realistic as it places dis-
{ }
x2x3+x3x1. Bydefinition tinct permutations (which in any true physical system
mostlikelyhavequitedifferentenergyproperties)inthe
1 dk N same degenerate energy state. Still, from a theoretical
Π (x ,...,x )= (1+qx ) . (8)
k 1 N k! dqk i perspective,thesimplicityofthismodelmakesitasuit-
" #
Yi=1 q=0 ablestartingpointforstudyingthegeneralpropertiesof
systemsofpermutations.
By the definition of the incomplete gamma function as
Γ(x,α)= ∞dttx−1e−tanditsrelationtoderangements
α
(i.e.,d =Γ(j+1, 1)/e,see[8]),wethenfind
j
R −
A. Phase-likeBehaviorof“Non-Interacting”System
∞ N
Z ( βλ )=e−1 dte−t tjΠ e−βλ1,...,e−βλN
N i j
{ } Wecaninvestigatethephase-likebehaviorofthesys-
Z−1 j=0
X (cid:0) (cid:1) tem defined by the Hamiltonian Eq.(4) (for constant λi
∞ N across i), byapplying steepestdescentto Eq.(11) in the
= dse−s 1+(s 1)e−βλℓ , (9)
− N 1limit. Doingso,wefindwecanapproximatethe
Z0 ℓY=1h i fre≫eenergyofthesystemtobe
whichisthedesiredclosed-formexpressionforthepar-
βF = lnZ (βλ )
titionfunctionofthissystem. − N 0
WithEq.(9),theproblemofabstractlystudyingather- Nβλ eβλ0 N 1 +F (N), (14)
0 0
≃ − − −
malsystemofpermutationswithHamiltonianEq.(4)is,
from the perspective of equilibrium statistical mechan- whereF0(N)isaβλ0in(cid:0)dependentcon(cid:1)stant. Notingthat
ics,nowcomplete. However,therearestillsomephysi- j = ∂lnZN(βλ0)/∂(βλ0) = ∂F/∂λ0,wefindtheav-
h i −
calandtheoreticalresultswhichcanbeteasedfromthis erage number of incorrect components satisfies the fol-
formalism. Specifically,wecanaskwhetherthissystem lowingequationofstate:
exhibits phase transitions. To answer this question, it
wouldprovemoreanalyticallytractabletotakeλ = λ j N eβλ0. (15)
i 0 h i≃ −
foralli. Withthiscondition,ourpartitionfunctionsim-
ByEq.(12),wecaninferthat j mustbegreaterthanor
plifiesto
h i
equal to 0. However, the right-hand-side of Eq.(15) ex-
N hibitsnosuchexplicitconstraint. Thuswecaninferthere
Z (βλ )= g (j)e−jβλ0 (10) is a phase-like transition in our system at the tempera-
N 0 N
ture
j=0
X
∞
= dse−s 1+(s 1)e−βλ0 N (11) k T = λ0 . (16)
Z0 − B c lnN
(cid:0) (cid:1)
wherewetransformedourHamiltonianas N( θi ) Belowthistemperature,wemusthave j 0andthus
H { } → h i ≃
(j)=λ0jwithj,definedas the“correctpermutation”hasthelowestfreeenergyand
H
is thermodynamically favored; above this temperature,
N j > 0 and the system is in a disordered phase where
j I , (12) h i
≡ θi6=ωi thepreviouslowestenergy“correct-permutation”isen-
Xi=1 ergeticallydisfavored.
Interestingly, this transition arises from the naively
the number of components of ~θ which are not equal to
non-interactingHamiltonian
the correspondingcomponent in~ω. Wecallj the num-
ber of incorrect components of ~θ, and if j = N we say N
~θ is completely disordered. The factor gN(j) in Eq.(10) is HN({θi})=λ0 Iθi6=ωi. (17)
i=1
X
4
1. NotaTruePhaseTransition
Weclaimthesystemdoesnotexhibittruephasetran-
sition behavior because many of these results are not
consistent with the traditional thermodynamic defini-
tionofphasetransitions. For one, phasetransitionsare
associatedwithdivergencesinthederivativesofthefree
energy, but there is no divergence in the free energy
associated with the partition function Eq.(11) for pos-
sible parameter values. Also, the result Eq.(15) arises
from the steepest descent approximation which makes
j ’stemperaturedependencenear j = 0appearnon-
h i h i
FIG.2: Freeenergyfor“Non-InteractingModel”: For differentiablewhenbyEq.(11)itisactuallydifferentiable
λ0 >0,theLandaufreeenergyofthesystemasa overitsentire domain. Finally, withEq.(10) wecande-
functionofj,thenumberofincorrectcomponentsin~θ, fineaLandaufreeenergyF(j)forthissystemaccording
isalwaysconvexwithasingleglobalminimum. toZ = e−βF(j),andwhatwemayordinarilylabelas
j
Becausej 0,thej <0domainofeachplot(dashed a phase transition (i.e., going from j = 0 to j = 0)
≥ P h i h i 6
section)isinaccessible. Forsufficientlylow arises, not from changes in the functional form of the
temperatures,theminimumisatj =0,butaswe Landaufree energy as we see in real phase transitions,
increasethetemperaturebeyondEq.(16),thefree butfromchangesintheexcludedregionoftheLandau
energycurvemovestotheright(butretainsits freeenergy(SeeFigure 2). Becausethe functionalform
functionalform)andthenewminimumisataj >0 ofthefreeenergyremainsthesameweobservenotrue
value. ActualplotsofEq.(20)forj <0requireusto phasetransition.
replacethecombinatorialargumentofthelogarithm
Alternatively, a heuristic argument for the non-
withitscorrespondinggammafunctionexpression.
existenceofphasetransitionsinourpermutationmodel
ismathematicallyverysimilar totheLandauargument
([9])forthenon-existenceoftransitionsin1dIsingMod-
els. For our permutation system with N lattice sites,
thestateofzeroenergyandzeroentropyconsistsofev-
erysitebeingoccupiedbyitscorrectcomponent. Toin-
creasetheenergyofthissystem,wecanchoosejsitesto
containincorrectcomponents,thusgivingusanenergy
We say “naively non-interacting” because Eq.(17) con- = λ j. The number of wayswe can choose these j
j 0
H
sists of a sum over linear functions of a single index i, componentsisgivenbyEq.(13)Thus,uponintroducing
andthus doesnot suggestanycoupling betweenterms j = 0 incorrect components, the change in the Landau
6
ofdifferingindex. However,statisticalmechanicstellsus freeenergyofoursystemis
thattheenergyofasystemisn’ttheonlythingwhichde-
termines the thermodynamic behavior of a system. In- N
∆F(j)=λ j k T ln d j(λ k T lnN),
deedwehavetoconsiderentropiccontributionsaswell, 0 − B j j ≃ 0− B
(cid:20)(cid:18) (cid:19) (cid:21)
and in this system the entropy (as it is a function of j) (20)
candrivethermodynamic behavior. Inotherwords, al-
though the Hamiltonian is depicted as non-interacting where we took these results in the 1 j N limit
≪ ≪
andcanset-theoreticallyberepresentedas and used d j!/e. In the thermodynamic (N )
j
≃ → ∞
limit, we find that ∆F(j) meaning there is no
system = 1 2 N, (18) non-zeroT atwhichtheen→trop−i∞ccontributionbecomes
H H ⊕H ⊕···⊕H
subdominanttotheenergy. Thusthesystemexhibitsno
our systemreallyexhibits interactions betweencompo-
phasetransition.
nents because our total space of states cannot be fac-
S
torized:
= ... . (19)
system 1 2 N
S 6 S ⊗S ⊗ ⊗S
III. PARTITIONFUNCTIONFORINTERACTING
Thus the “non-interacting” system exhibits a transition
MODEL
atEq.(16)duetothecouplednatureofthestatespace. As
discussedinthesubsequentsection,wetermthistransi-
tiona“quasi-phasetransition”becauseitdoesnotbear Whenwefirstconsideredamodelofthermalpermu-
allofthestandardpropertiesweexpectinphasetransi- tations,webeganwithaHamiltonianwheresitesdidnot
tions. interactwithoneanotherandeachhadasite-dependent
5
energycostforbeingincorrectlyoccupied: A. CalculatingOrderParameter
( θ )= λ I . (21)
H { i} i θi6=ωi
i
X
More generally, we can consider Hamiltonians with
an arbitrary number of multiple-site interaction terms. Our goal is to analyze the “quasi”-phase behavior of
SuchaHamiltoniancouldbewrittenas this system in a way analogous to our analysis for the
non-interactingsystem. TodosowebeginwiththeLan-
1
( θ )= λ I + µ I I + . daufreeenergyfunction
H { i} i θi6=ωi 2 ij θi6=ωi θi6=ωi ···
i i,j
X X λ 1
(22) F (j,β)=λ j+ 2 j2 lng (j). (26)
N 1 N
The first term in Eq.(22) associates an energy cost of λ 2N − β
i
with incorrectly occupying the component at position
Alternative starting points for this derivation are pre-
i. The second term models interactions between sites
sentedinAppendixA.Oursystemisconstitutivelydis-
where the correct (or incorrect) occupation of a single
crete,soitisnotpreciselycorrecttodiscussourfreeen-
site determines the energy of another. The exact val-
ergy in the language of analysis, but given our expres-
ues of λ and µ could be chosen to ensure the “cor-
i ij
sionforEq.(13)wecanmapthissystemtoacontinuous
rect”state(θ = ω foralli)isnon-degenerateasinthe
i i
one which bears the same thermodynamic properties
non-interactingmodel. Theellipsesrepresenthigheror-
andforwhichanalysisisappropriate. Specifically,ifwe
derinteractionsinthisframework.Hamiltonianssuchas
takejtobecontinuousandusetheidentityΓ(x+1)=x!,
Eq.(22)shouldbemorephysicallyrelevantastheywould
wecanwrite
correspond to systems where the energy cost for devi-
atingfromthelowestenergypermutationisnotsimply Γ(N +1) Γ(j+1, 1)
linearbutcouldberepresentedasatensorvaluedfitting gN(j)= − , (27)
Γ(j+1)Γ(N j+1) e
function. −
WiththeapproximationΓ(j+1, 1) Γ(j+1)andthe
− ≃
substitutionEq.(27),Eq.(26)thenbecomes
λ 1
Wecanmakeprogressinstudyingthethermodynam- f (j,β)=λ j+ 2 j2+ lnΓ(N j+1)+f (28)
N 1 0
icsofmoregeneralHamiltonianslikeEq.(22)byfirstonly 2N β −
considering first- and second-order interaction terms
where we defined our approximated free energy as
andtakingthe interactionstobeconstants: λ = λ for
i 1 f (j,β) and collectedthe j independentconstants into
alli;µ = λ /N foralli,j. Thefactorof1/N ischosen N
ij 2 f . Now Eq.(28) is fully continuous and amenable to
sothatthesecondtermmatchestheextensivescalingof 0
analysis. To find the thermodynamic equilibrium of
thefirstterm. Thepartitionfunctionforsuchparameter
this system, we need to find the value of j for which
selectionsisthen
∂f (j,β)/∂j = 0 and ∂2f (j,β)/∂j2 > 0. For the first
N N
N conditionwehave
Z (β;λ ,λ )= exp βλ I
N 1 2 − 1 θi6=ωi− ∂ λ2 1
~θ∈SXym(ω) Xi=1 ∂jfN(j,β)=λ1+ Nj− βψ0(N −j+1)=0. (29)
N
βλ
2 I I , (23) Forx 0.6wehave
2N θi6=ωi θj6=ωj ≥
i,j=1
X ψ (x) ln(x 1/2), (30)
 0 ≃ −
whereλ andλ areinteractionparameterswithunitsof
1 2
ascanbeaffirmed byTaylor expansionor plotsof each
energy. Wecanalsowrite this partition functionin the
side. Since the argument of ψ (N j +1) is bounded
Eq.(12)basisas 0
−
belowby1,theapproximationinEq.(30)canbeapplied
N toEq.(29). Withthissubstitution, andsettingtheresult
Z (β;λ ,λ )= g (j)e−βE(j), (24) tobevalidfortheequilibriumvaluej =j,wethenfind
N 1 2 N
theconstraint
j=0
X
whereg (j)isdefinedinEq.(13)and eβλ2j/N = e−βλ1 j N 1/2 , (31)
N − − −
λ whichhasthesolution (cid:0) (cid:1)
(j)=λ j+ 2 j2 (25)
1
E 2N
j 1 βλ 1
=1 W 2eβλ1+βλ2 + , (32)
istheenergyfunctionforthesystem. N − βλ N O N
2 (cid:18) (cid:19) (cid:18) (cid:19)
6
where W is the (branch unspecified) Lambert W func-
tion,definedby
W(xex)=x. (33)
Tospecifythe branchof the W which corresponds to a
stableequilibriumwecomputethesecondderivativeof
our free energy at this derived critical point. Doing so
yields
∂2 1 1 1
f (j =j,β) λ +
∂j2 N ≃ N 2 β1 j/N
(cid:18) − (cid:19)
λ 1
2
= 1+ . (34)
N  W βλ2eβλ1+βλ2 
N
 (cid:16) (cid:17)
ThusEq.(32)(forλ > 0)yieldsafreeenergyminimum
2
for FIG.3: (Coloronline)Possiblefunctionalformsof
Eq.(28): Wenotethatthestabilitiesthatdefinethej =0
βλ
W 2eβλ1+βλ2 > 1, (35) andj =N pointsarenotthermodynamicstabilities
(cid:18) N (cid:19) − (namelytheydon’tarisefromthef′(j)=0condition).
Rathersincethespectrumofj valuesisboundedbelow
and yields a maximum for the inverse condition. This
by0andabovebyN,owingtotheseboundary
amountstostatingthatthestableequilibriumforjisde-
conditionsthesystemcanbecometrappedinordinarily
finedbytheprincipalbranchoftheLambertWfunction
unstablepartsofthefreeenergycurve. Thecolors
whereW =W 1,andtheunstableequilibriumforj
0 ≥− matchthecoloroftheassociatedregionofparameter
isdefinedbythenegativebranchwhereW =W < 1.
−1 − spaceinFigure4.
Thus,theorderparameterforthissystemis
j 1 βλ 1
0 =1 W 2eβλ1+βλ2 + . (36) tionsinthissystem,thereareinfactfivedistinctregimes
0
N − βλ2 (cid:18) N (cid:19) O(cid:18)N(cid:19) ofparameterspace. Wecanobtainaqualitativesenseof
theseregimesbycreatingschematicplotsofthefreeen-
Wenotethattakingλ 0andusingW(x)=x+ (x2)
2 → O ergyEq.(28)for variousparametervaluesof λ1 and λ2.
for x 1 returns us to the non-interacting result
| | ≪ Thepossibleplotscanbeplacedintofivecategoriesac-
Eq.(15).
cording to the plot’s stable or metastable j values. We
For completeness, we define the value of j which
depictthesepossibleplotsinFigure3. Wenotethatonly
yields a free energy maximum as j ; it is related to
−1 thefreeenergyplotswithvalidvaluesofj containwhat
0
Eq.(36) by replacing the principal branch function W
0 wenormallyconsiderathermodynamicequilibrium;the
withW .
−1 otherplotshave“stable”valuesofjarisingonlyfromthe
j =0and/orj =N boundaryconditions.
Qualitatively,wecannamethestatesaccordingtothe
B. DiscussionofParameterSpace sequence space to which their equilibrium values of j
correspond. We know for j = 0, our system is in a
In the previous section, we found that the order pa- statewithzeroincorrectcomponentsinθ~andhencethe
rameterforthissystemwasgivenbyEq.(36). Wenoted system is “perfectly ordered” or just ”ordered”. Con-
that this solution represented a local minimum of the versely for j = N our system has N incorrect compo-
freeenergyaslongastheLambertWfunctionsatisfied nentsandhence the systemis“completelydisordered”
W = W0 > 1. Thus when this condition is violated, orjust“disordered”.Thein-betweencaseofj =jwhere
−
j is no longer a valid stable equilibrium, and our sys- 0 < j < N can be given the relatedlabel of “partially-
0
temhasundergonea“quasi”-phasetransitionorsimply ordered”. Thus, the regime names associated with our
atransition. possiblevaluesoftheorderparametersare
Moreover,our valuesofj areboundedbelowbyj =
OrderedRegime(j = 0stable):Neitherj orj
0 and bounded above by j = N, neither conditions of • 0 −1
exist;f(N,β)>0.
which are naturallyconstrained byEq.(36). Thus these
twoconditionsareassociatedwithtwoothertransitions. DisorderedRegime(j = N stable):Neitherj or
Inall,then, therearethreeconditions whichdefinethe • j exist;f(N,β)<0. 0
−1
quasi-phaseboundariesinthissystem.
While therearethreeconditions which definetransi- Partially-Ordered Regime (j = j0 stable): Only
•
7
TABLEI:Functionsdefiningboundariesbetween ischaracterizedbythecondition
parameterregimes.
λ =lnN/β and λ = 1/β. (37)
1 2
−
RegimeTransition BoundaryinParameterSpace
Theseconditionscharacterizethesystem’striplepoint.
1
Order/Partial-Order λ1 = lnN Similarly, for N 1, the Partially-Ordered, Dis-
β ≫
ordered, Order-Partially-Ordered Metastability, and
1 N Order-Disorder Metastability coexistence point is char-
Order/Order-Partial- λ = W e−βλ1
2 β −1 − e acterizedbythecondition
OrderMetastability (cid:18) (cid:19)
λ
Order/Disorder λ2 =−1 11/N λ1 =lnN/β ≃−λ2. (38)
−
Thisconditioncharacterizesthe quadruplepointof the
system.
j exists. Figure4alsodepictsthe possible regimesof oursys-
0
tem for a given temperature and various Hamiltonian
Order and Disorder Metastable Regime (j = 0 parametersλ1andλ2. Morephysically,wemaybeinter-
• andj = N stable): Onlyj exists. estedinknowingwhatarethe“quasi”phaseproperties
−1
ofasystemwithafixedλ andλ andavariabletemper-
1 2
Order and Partial-Order Metastable Regime ature. That is, what are the temperatureswhich define
•
(j = 0andj = j0stable): Bothj0andj−1exist. thevarioustransitionsbetweenregimesinthesystem?
AnarbitrarypermutationsystemforafixedN andat
Wenote thatitseemstobe afundamentalfeature(or a avariabletemperatureischaracterizedbyaspecificen-
lack of one) of this system, that the free energy Eq.(28) ergyfunctionEq.(25). Suchasystemisthereforedefined
doesnotadmitametastabilitybetweenpartial-orderand byaspecificλ andλ ,andthesystemcanbeassociated
1 2
disorder. withaparticularpoint(andhenceregion)intheparame-
terspaceofFigure4. Aswevarythetemperatureofthis
system, the temperature dependent regime-coexistence
C. Monte-CarloGeneratedParameterSpace lines change and if theychange in such awayasto ex-
tendtheregionofaregimetonewlyencompassourorig-
inalpointthenoursystemhasundergoneatransition. In
With these regime definitions, we can depict the pa-
thisway,wecandefinethetemperatureswhichcharac-
rameter space graphically. In Figure 3 we showed the
terizevariouspossibletransitionsofthissystem.
possible formsofthe freeenergyforthissystemwhere
First,fromFigure4andEq.(C10)wenotetheregime-
eachwascategorizedaccordingtotheexistenceofthelo-
coexistence line between the partially-ordered and dis-
calminimum criticalpointj , the existenceof the local
0 ordered regime is independent of temperature, and so
maximumcriticalpointj ,andthesignofthequantity
−1 there is no criticaltemperature defining a partial-order
f (j,β). Wecanextrapolatethiscategorizationtoλ λ
N 1 2
− todisordertransition.
parameter space, by determining which regions of pa-
FromEq.(C3),wecaninferthatthepartial-ordertoor-
rameterspacecorrespondtospecificplotsinEq.(28). Do-
dertransitionischaracterizedbymovingbelowthetem-
ingsothroughtheMonteCarloproceduredescribedin
perature
Appendix B, we generated 10,000points of the param-
eter space diagram in Figure 4 for β set to 1. We note
λ
thattheparameterspaceexhibitsfiveregimesseparated k T = 1 . (39)
B c1
lnN
bythreelinescitedinTableIeachofwhichcorrespond
to the three conditions mentioned at the beginning of
Contingentonwhichregionofparameterspacethesys-
this section. These lines can be derived analytically(as
tem lies, this temperature also characterizes the disor-
showninAppendixC)byconsideringtheconditionsin
der to order-disorder metastability transition and the
turnandwhichregimestheyservetoconnect.
partial-ordertoorder-partial-ordermetastabilitytransi-
tion.
AndfromEq.(C6),wecansolvefortheassociatedtran-
D. TripleandQuadruplePointsandTransition sitiontemperaturegivenfixedλ andλ tofind
1 2
Temperatures
N −1
k T =(λ +λ ) W (λ +λ ) (40)
From Figure 4 we see that our system is character- B c2 1 2 0 −eλ 1 2
(cid:20) (cid:18) 2 (cid:19)(cid:21)
ized by two points where there is a coexistence be-
tween multiple regimes. Given that W ( e−1) = wherethisexpressionisonlyrelevantfor λ < λ < 0
−1 1 2
− −
1, we have that the Ordered, Partially-Ordered, and and λ > k T lnN. Moving above this temperature
1 B
−
Order-Partially-OrderedMetastabilitycoexistencepoint leads to the order to order-partial-order metastability
8
FIG.4: (Coloronline)λ λ ParameterSpaceforInteractingMeanFieldSystem: Wesetβ =1andN =100. We
1 2
−
followedtheMonteCarloprocedureoutlinedinthenotesfor10,000points. Inthefigurewealsodenotedthe
analyticlines(i.e.,Eq.(C3),Eq.(C7),andEq.(C10))whichdefinetheseparationbetweenthephases. Thecolors
correspondtothecolorsofthefreeenergycurveinFigure3.
transition. of typical canonical models in statistical mechanics. In
particularwehopetoextendittonon-trivialsitedepen-
dentinteractions. Forexample,anearestneighborinter-
action Hamiltonian of the kind which characterize the
IV. DISCUSSION
IsingModel,
In this work, motivated by an abstraction of a foun- N
dationalprobleminproteindesign,wepositedandana- ( θ )= q I I , (41)
lyzedthebasicpropertiesofastatisticalphysicsmodelof H { i} − θi6=ωi θi+16=ωi+1
i=1
X
permutations. Formally,weconsideredasimplestatisti-
calphysicsmodelwherethespaceofstatesforN lattice would be an alternative physical extreme to the mean-
siteswasisomorphic tothe symmetric group ofdegree fieldinteractionsconsideredinSectionIII.
N [5], and where the energy of each permutation was We could also consider a generalized chain of com-
afunctionofhowmuchthepermutationdeviatesfrom ponentswheretheinteractionsbetweensitesorthecost
theidentitypermutation. foranincorrectlyfilledsiteisnotconstantbutisdrawn
In this model, we found that due to a state space fromadistributionofvalues. Suchasystemofquenched
which could not be factorized in a basis defined by disorderwouldcharacterizeapermutationglasswhich
lattice sites, even the superficially non-interacting sys- maycontaininterestingresultsduetotheuniquenature
tem can exhibit phase-like transitions, i.e., temperature ofthestatespace.
dependent changes in the value of the order parame-
Supposingitispossibletodefinemoreinterestingin-
ter which do note exhibit the properties typically as-
teractions models, a natural investigation would con-
sociated which phase transitions in infinite systems.
cerntherenormalizationgrouppropertiesofthesystem.
When interactions are introduced through a quadratic
Specifically, we would be interested in how would one
meanfieldterm,thesystemiscapableofexhibitingfive
sumoverspecificstates(ascharacteristicofarenormal-
regimesofthermalbehavior,andischaracterizedbytwo
izationgrouptransformation)whenthestatespaceofa
transition-temperaturescorrespondingtovariousquasi-
systemlookslike,
phasetransitions.
The introduced model provides us with a basic ex- N
actlysolublesystemforcertaininteractionassumptions = (42)
system i
S ⊗S
andthusprovidesaconcretemodel-basedunderstand- i=1
Y
ingof asystemwith anon-factorizablestatespace. Be-
i.e.,isnotfactorizablealonglatticesites.
cause of its utility and the type of results obtained, the
modeldeservestobesubjecttothestandardextensions Finally,toconnectthismodelofpermutationstoprob-
9
lemsmorerelevanttoproteindesignitwouldprovenec- AppendixA:AlternativeDerivationsofEq.(31)
essarytoincorporatethepossibilityofrepeatedcompo-
nentsorthebackgroundgeometryofalatticechain. 1. Hubbard-StratonovichApproach
Were-deriveEq.(31)usingtheHubbard-Stratonovich
method. Westartthis derivationassumingλ = λ ;
2 2
−| |
wewilllaterseeourresultingfreeenergycanbeanalyt-
icallycontinuedtotheλ >0case.
2
Forλ = λ ,thepartitionfunctionis
2 2
−| |
N
Z (β;λ ,λ )= g (j)e−βλ1j+β|λ2|j2/2N. (A1)
N 1 2 N
ACKNOWLEDGMENTS Xj=0
Then,applyingtheidentity
MW thanks Verena Kaynig-Fittkau for her unpub- eβ|λ2|j2/2N = N ∞ dxe−Nx2/2β|λ2|−jx, (A2)
lGisihlseodn,wAorbkigtahilatPilnusmpmireedr, tVhiisnoitnhvaenstiMgaatnioonhaarnadn,Aamndy s2πβ|λ2|Z−∞
MichaelBrennerforhelpfuldiscussions. wehave
Z (β;λ ,λ )= N ∞ dxe−Nx2/2β|λ2| N g (j)e−j(βλ1+x)
N 1 2 N
s2πβ|λ2|Z−∞ j=0
X
= N ∞ dxe−Nx2/2β|λ2|Z (βλ +x), (A3)
N 1
s2πβ|λ2|Z−∞
whereZ (x) N g (j)e−jx. FromEq.(11)wefound
N ≡ j=0 N
P ∞
Z (x)= dse−s 1+(s 1)e−x N. (A4)
N
−
Z0
(cid:0) (cid:1)
SoEq.(A3)becomes
Z (β;λ ,λ )= N ∞ds ∞ dx 1+(s 1)e−βλ1−x Ne−s−Nx2/2β|λ2|. (A5)
N 1 2
s2πβ|λ2|Z0 Z−∞ −
(cid:0) (cid:1)
Thefunctiontowhichweapplysteepestdescentisthen Solvingforsinthefirstequationwehave
Nx2 s=N +1 eβλ1+x (A9)
h(s,x)=s+ Nln 1+(s 1)e−βλ1−x . (A6) −
2β λ − −
2
| | andwiththesecondequationwehavethecondition
(cid:0) (cid:1)
Computingtheconditionsfor∂ h(s=s,x=x)=0and
s
x 1
∂xh(s=s,x=x)=0weobtain,respectively, = (s 1). (A10)
β λ −N −
2
| |
e−βλ1−x
1 N =0 (A7) Substitutingthesecondconditionintothefirstyields
− 1+(s 1)e−βλ1−x
−
x (s 1)e−βλ1−x s 1=N eβλ1−β|λ2|(s−1)/N, (A11)
+ − =0. (A8)
β λ2 1+(s 1)e−βλ1−x − −
| | −
10
or Wecanalsodefine
e−β|λ2|(s−1)/N =eβλ1(N (s 1)). (A12) N
− − (j) = (j)e−βλ0j, (A20)
0,N
hO i O
Thus,thesolutionforscanbeexpressedintermsofthe j=0
X
LambertWfunctionas
as the average with respect to our variational Hamilto-
s 1 1 β λ nianEq.(A18). Thus,Eq.(A16)becomes
− =1+ W | 2|eβλ1−β|λ2| . (A13)
N β λ − N
| 2| (cid:18) (cid:19) 1 λ
F[ ] lnZ (βλ )+(λ λ ) j + 2 j2
N 0 1 0 0,N 0,N
For λ > 0, we can employ the complex version of the H ≤−β − h i 2Nh i
2
Hubbard-Stratonovichidentity: f(λ ) (A21)
0
≡
e−βλ2j2/2N = N ∞ dxe−Nx2/2βλ2−ijx. (A14) pDuiffteertehnetiamtianxgimfuwmitohfrtehsipseqctuatontλit0y.allGoiwvesnusjto com=-
s2πβλ2 Z−∞ ∂lnZ (βλ )/∂(βλ ),wethenfind h i0,N
N 0 0
−
Working through an analogous steepestdescent proce-
∂ λ ∂
dure,wefindthattheequilibriumvalueforsis f′(λ )=(λ λ ) j + 2 j2 , (A22)
0 1 0 0,N 0,N
− ∂λ h i 2N ∂λ h i
0 0
s 1 1 βλ j
− =1 W 2eβλ1+βλ2 , (A15) whichifwetaketobezeroatsomeλ0 =λ0givesus
N − βλ N ≡ N
2 (cid:18) (cid:19)
∂ λ ∂
which could have been extrapolated from Eq.(A13) by 0=(λ1 λ0) j 0,N + 2 j2 0,N .
ltiabkriinugm|λc2o|n→diti−oλn,2.aFnrdomasthains eaxnparloesgsyiowniftohr tthheeenqouni-- − ∂λ0h i (cid:12)(cid:12)λ0=λ0 2N ∂λ0h i (cid:12)(cid:12)λ0(=Aλ203)
To compute these deriv(cid:12)atives we make use of (cid:12)various
interactingcase,itturnsouttheorderparameterinthis
identities. Firstwenote
caseisnotsbutrathers 1.
−
1 ∂2
j2 = Z (βλ ), (A24)
h i0,N Z (βλ )∂(βλ )2 N 0
N 0 0
so
∂ ∂2
j = lnZ (βλ )
2. Gibbs-BogoliubovInequalityDerivationofEq.(31) ∂(βλ )h i0,N −∂(βλ )2 N 0
0 0
1 ∂2
= Z (βλ )
−Z (βλ )∂(βλ )2 N 0
We re-derive Eq.(31) using the Gibbs-Bogoliubov N 0 0
Inequality[10]. Theinequalityis 1 ∂ 2
+ Z (βλ )
Z (βλ )2 ∂(βλ ) N 0
F[ ] F[ ]+ . (A16) N 0 (cid:18) 0 (cid:19)
H ≤ H0 hH−H0i0 = j2 + j 2 . (A25)
−h i0,N h i0,N
TheHamiltonianwhichdefinesoursystemis
Thislastequalityimplies
N
λ λ
H=λ1Xi=1Iθi6=ωi + 2N2 Xi,j Iθi6=ωiIθj6=ωj ≡λ1j+ 2N2 j2, ∂(β∂λ0)hj2i0,N =−∂∂2(hβjλi00,)N2 +2hji0,N∂∂h(jβiλ0,0N), (A26)
(A17)
andourvariationalHamiltonianisinstead andsoEq.(A23)becomes
N λ ∂
2
H0 =λ0i=1Iθi6=ωi =λ0j. (A18) 0=(cid:20)λ1−λ0+ Nhji0,N(cid:21)∂λ0hji0,N(cid:12)λ0=λ0
X λ ∂2 j (cid:12)
2 h i0,N (cid:12) (A27)
FromEq.(11)weknow − 2Nβ ∂λ20 λ0=λ0
(cid:12)
(cid:12)
1 Tocomputethesequantities(cid:12)weneedtoapproximatethe
F[ ]= lnZ (βλ )
0 N 0
H −β partitionfunctionforourvariationalsystem. Usingthe
= 1 ln ∞dse−s 1+(s 1)e−βλ0 N .
−β −
(cid:26)Z0 (cid:27)
(cid:0) (cid:1) (A19)