Non-equilibrium Annealed Damage Phenomena: A Path Integral Approach

16 Pages·2017·1.3 MB·English

by  Sergey G. Abaimov

ORIGINALRESEARCH
published:28February2017
doi:10.3389/fphy.2017.00006
Non-equilibrium Annealed Damage
Phenomena: A Path Integral
Approach
SergeyG.Abaimov*
CenterforDesign,Manufacturing,andMaterials,SkolkovoInstituteofScienceandTechnology,Moscow,Russia
We investigate the applicability of the path integral of non-equilibrium statistical
mechanicstonon-equilibriumdamagephenomena.Asanexample,afiber-bundlemodel
with a thermal noise and a fiber-bundle model with a decay of fibers are considered.
Initially, we develop an analogy with the Gibbs formalism of non-equilibrium states.
Later, we switch from the approach of non-equilibrium states to the approach of
non-equilibriumpaths.Behaviorofpathfluctuationsinthesystemisdescribedinterms
of effective temperature parameters. An equation of path as an analog of the equation
of state and a law of path-balance as an analog of the law of conservation of energy
aredeveloped.Also,aformalismofafreeenergypotentialisdeveloped.Forfluctuations
ofpathsinthesystem,thestatisticaldistributionisfoundtobeGaussian.Also,wefind
the “true” order parameters linearizing the matrix of fluctuations. The last question we
Editedby:
discuss is the applicability of the phase transition theory to non-equilibrium processes.
ZbigniewR.Struzik,
UniversityofTokyo,Japan From near-equilibrium processes to stationary processes (dissipative structures), and
Reviewedby: then to significantly non-equilibrium processes: Through these steps we generalize the
BikasKChakrabarti, conceptofanon-equilibriumphasetransition.
SahaInstituteofNuclearPhysics,
India Keywords:damage,pathapproach,fluctuations,fiber-bundlemodel,statisticalphysics
AlexHansen, PACS. 62.20.M—Structural failure of materials—89.75.-k Complex systems—05. Statistical physics,
NorwegianUniversityofScienceand thermodynamics,andnonlineardynamicalsystems.
Technology,Norway
*Correspondence:
INTRODUCTION
SergeyG.Abaimov
[email protected]
Oneofthemostimportantaspectsinthemoderntheoryofdamagephenomenaisthepossibilityto
Specialtysection: describedamageoccurrenceonthebaseoftheformalismofstatisticalphysics.Manyapproaches
Thisarticlewassubmittedto havebeendeveloped[1–24],andsurveysofdevelopmentscanbefoundin[25–30].However,the
InterdisciplinaryPhysics, majorityofthesestudiesaredevotedtoequilibriumornear-equilibriumdamageoccurrence.Inour
asectionofthejournal
study,weconsideranon-equilibriumdamageprocess(acascade,anavalanche)farfromthepoint
FrontiersinPhysics
ofaquasi-staticstate.
Received:07December2016 Once the analogy between damage mechanics and statistical physics had been discovered, it
Accepted:01February2017
immediatelybecameclearthatanydamagephenomenoncanbedescribedasaphasetransition,
Published:28February2017
wheredamagegrowthrepresentstheappearanceofanotherphaseandthepointofmaterialfailure
Citation: isthepointofthephasetransition.Thebestwaytoillustratethissimilarityistocomparethephase
AbaimovSG(2017)Non-equilibrium
diagramsinFigure1.Therearedebatesintheliterature[3–13,21,22,24,29–35]aboutwhether
AnnealedDamagePhenomena:A
thisphasetransitionisofthefirstorderorofthecontinuousorder.Thereasonforthisuncertainty
PathIntegralApproach.
Front.Phys.5:6. isthatspinodalpointsareverysimilartocriticalpointsinthesensethatintheirvicinityasystem
doi:10.3389/fphy.2017.00006 switchesitsbehaviorfromthelawsintrinsiconlytothisparticularsystemtouniversalpower-law
FrontiersinPhysics|www.frontiersin.org 1 February2017|Volume5|Article6
Abaimov Non-equilibriumDamagePhenomena:PathApproach
scaling.However,thereisnodoubtthatanystructureunderload
isinthemetastablestatesincetherealwaysexistsanucleusofthe
criticalsize.Inotherwords,bybreakingenoughofthematerial
with external forces, we can always cause material failure. This
isaclearindicationofmetastability,becausethereisnocritical
nucleus(nomatterhowbigitis)forstablestates.
Todiscussourapproach,weinvolvethefiber-bundle model
(further,FBM)asthemostillustrativerepresentativeofmodels
with damage. However, the applicability of the approach we
developliesbeyondtheone-dimensionalformulationoftheFBM
andcanbeeasilygeneralizedtotwoorthree-dimensionaldamage
phenomena.
For all possible approaches, an analogy between statistical
physicsanddamagephenomenashouldrelyonmappingofone
type of fluctuations on another. In this sense, damage exhibits
more complex behavior than classical thermodynamic systems
of statistical physics due to the presence of two different forms
of disorder: quenched and annealed. Therefore, to perform the
aforementionedmapping,thesetwoformsofdisorderareusually
studiedapartfromoneanother.
The influence of quenched disorder on the model behavior
isgenerallyobservedwherethedynamicaltimescaleoffracture
is much faster than the timescale of fluctuations. In this case,
“frozen” initial disorder in a model plays the crucial role. This
type of disorder, employed in the FBM, has been studied by
many authors [8, 16, 28, 29, 31–85]. In particular, in our
previous studies [21, 22, 30], we introduced quenched disorder
bymeansofthefiberstrengthvariabilityandfortheensembleof
constantstrainwewereabletomapthefluctuationsobservedon
thermodynamicfluctuationsinstatisticalphysics.
Another type of behavior is exhibited when a FBM is
influenced by the annealed disorder. One of two approaches
is generally employed: the disorder is introduced in the model
FIGURE1|Phasediagramsof(A)adamagephenomenon,(B)aliquid-gas
formulation either by means of a phenomenological rate of
system,and(C)amagneticsystem.
fiber failures with time [29, 86–109] or by the direct input of
a stochastic or thermal noise into the state of stress of a fiber
[23,110–118].
Annealed disorder generates fluctuations which are closely the Classical Gibbs Approach of States. In Section Statistical
related to their thermodynamic analogs in statistical physics. Mechanics of the Path Integral Approach, we consider a path
Indeed,addingstochasticnoisetoafiberloadseemstobequite integral, when instead of non-equilibrium states of the system
similartothestatisticalfluctuationsof,forexample,gaspressure we investigate non-equilibrium paths. We map fluctuations of
acting on a piston. However, as it was recently discovered damage on the main concepts of statistical mechanics, such
[119–123], annealed fluctuations in damage phenomena as temperature, entropy, free energy potential, and the law of
exceed thermodynamic fluctuations by orders of magnitude. In conservationofenergy.However,wefindthattheseconceptsare
particular, it was found that the effective temperature required nolongerassociatedwiththeenergycharacteristicsofthestates
toprovidetheannealedbehaviorobservedinexperimentsis10 ofthesystembutaredefinedoverthespaceofallpossiblepaths.
times higher (about 3,000 K) than the real temperature during Also,weinvestigatethebehavioroffluctuationsforthisapproach
thetests.Thisdiscrepancyisexplainedbycomplexinteractions andfind“actual”orderparameterswhichdiagonalizethematrix
of quenched and annealed types of disorder which are present of the probability distribution. In the conclusions, we discuss
simultaneouslyinthemodel[18,113–118,122,124]. howourtheoryleadstothedescriptionofnon-equilibriumphase
In our study, we continue our previous research [23, 125, transitions.
126] and investigate a FBM with annealed damage evolution.
In Section Model, we introduce models which we will use MODEL
to illustrate our approach. In Section The Classical Gibbs
ApproachofStates,wedevelopananalogoftheclassicalGibbs Damage is a complex phenomenon. The basic principles of
approach for non-equilibrium states. However, this approach damage are often completely disguised by the secondary side
has its restrictions which we discuss in Section Restrictions of effects of its appearance. Therefore, to investigate the main
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Abaimov Non-equilibriumDamagePhenomena:PathApproach
conceptsofnon-equilibriumbehavior,itisreasonabletoconsider of infinite number of fibers N is taken first. Therefore,
→ +∞
initiallyasimplemodel. althoughonlythesmallfractionoffibersbreaksateachtime-step,
Inthispaper,asanillustration,weconsideranannealedFBM. thenumberofbrokenfibersismuchhigherthanunity.
WeassumethatthenumberNoffibersinthemodelisconstant Although,weassumethenoisetobeGaussian,thisdoesnot
andinfiniteinthethermodynamiclimit.Intactfibersallcarrythe playanyroleinourmodelduetothefactthats,E,andεarethe
samestrainε whichisidenticallyequaltothestrainεofthetotal sameforallfibersaswellasforallsystemsintheensembleand
f
model:ε ε.Thestressofeachintactfiberisassumedtohave aresomeconstants.Therefore,thenoisevalue1σ s Eε
f break
≡ = −
alinearelasticdependenceonthestrainuntilafiberfailure:σ atwhichfibersbreakisfixedsothatP(s Eε)and1 P(s Eε)
f
− − −
Eε .Inthispaper,weconsiderconstantstrainε constasan aretwoconstantsaswell.
f
= =
externalboundaryconstraintofthemodel.Thecaseofconstant The second modification of the FBM which we consider
stress can be considered in a similar way (the redistribution of (following the original terminology, a model with “breaking
theloadcomplicatestheanalyticalsolutionbutdoesnotchange kinetics”) is an annealed decay fiber-bundle model [86–109;
the applicability of the path integral). The Young modulus E further,DFBM].Thereisnothermalnoiseinthismodel.Instead,
hereisassumedtobethesameforallfibers;however,thestress- each(sofarintact)fiberhasaprioriassignedprobabilityp dtof
f
strain dependence for the total model is non-linear because of failingduringthetimeintervaldt andprobability(1 p dt)of
f
−
fiberfailures.Weintroducean“intact”orderparameterLasthe stayingintact.
fraction of intact fibers (unity minus the damage parameter). Here, p is the decay rate which is constant during model
f
Everywhere further we assume that at the initial time t 0 all evolution. The duration of the time-step dt is considered to be
=
fibersareintact:L 1. fixedaswell.Therefore,p dtand(1 p dt)aresomeconstants,as
t 0 f f
Asafirstmodifi=cat=ionoftheFBM,weconsideranannealed wellas1 P(s Eε)andP(s Eε)−respectively,whichrepresent
− − −
fiber-bundle model with noise [23, 110–118] (further, NFBM). fixedprobabilitiesofafiberbreakingorstayingintactduring a
Eachfiberhasthermalfluctuationsofitsenergycharacteristics. time-step.Inthissense,equatingp dtand1 P(s Eε),wecan
f
− −
Fromstatisticalmechanics,weknowthatapiston,whichworks map the DFBM on the NFBM. However, some care should be
asagasboundaryandissupportedbyaspring,hasoscillations exercisedwhenwesayp dt 1 P(s Eε).Here,weconsiderdt
f
= − −
duetotheequipartitionofenergy.Inparticular,theelasticenergy tobethedurationofthetimeintervalduringwhichtheGaussian
ofthespringfluctuatesasifawhiteGaussiannoisewereaddedto noise loses its correlation with the value of the noise at the
the stress of the spring. Similarly, we assume that the stress of previoustime-stepand“switches”toanewvalue.Anillustrative
eachfiberσ hasanaddition1σ ofawhiteGaussiannoisewith exampleistheoutputoftheGaussianrandomgeneratorwhich
f f
zeromeanandstandarddeviation√k T.Asitwasdiscussedin provides a discrete set of uncorrelated i.i.d. numbers. In this
B
SectionIntroduction,thetemperatureheremaynotcorrespond sense, the duration of dt is not arbitrary but is dictated by the
to the temperature during the experiment but may be several physical processes responsible for the correlations in the noise.
ordershigher[119–123]. Hence,wecannotsayherethattheconsideredprobabilitiesare
Theprobabilitydensityfunctionofthenoiseis proportionaltodt.
Theissuecanbeillustratedwiththeaidofextremestatistics.
1 1σ2 Ifweobserveafiberduringalongtimeintervalτconsistingof
p(1σf) e−2kBT (1) N >> 1time-stepsdt,theprobabilityofthefiberstayingintact
= √2πk T
B duringthistimeintervalisPN(s Eε) edτtlnP(s−Eε),whichis
− =
andthecumulativedistributionfunctionis clearlynotlinearwithtime.Therefore,ourmappingoftheNFBM
onto the DFBM is introduced only for one time-step. In this
1σf sense,wewillconsideronlyoneofthesemodels(theDFBMfor
1 x2
P(1σf) e−2kBTdx. (2) theapproachofstatesatSectionsTheClassicalGibbsApproach
= √2πk T
B Z of States and Restrictions of the Classical Gibbs Approach of
−∞ States, which is more convenient for this as it is a model with
Eachfiberhasanaprioriassignedstrengththresholdswhichwe a well-known solution; the NFBM for the approach of paths
choose to be the same for all fibers and not to change during at Section Statistical Mechanics of the Path Integral Approach
the model evolution. A fiber can break only when its stress σ sinceitismoreillustrativeofthegeneralizationtomorecomplex
f
Eε 1σ exceeds its strength s. We consider discrete time- modelformulations),assuming,however,thatallresultsarevalid
f
= +
stepsdtofthemodelevolution.Ateachtime-step,theprobability fortheothermodelaswell.
of a fiber breaking is 1 P(s Eε) and the probability of a Beforegoingfurther,weshouldclarifythatwedonotconsider
− −
fiberstayingintactisP(s Eε).Hereweassumethatthereare the healing of fibers: Once a fiber has broken, it stays broken
−
nocorrelationsofnoiseamongadjacentfibers.Also,inspiteof forever. However, our formalism easily allows healing to be
thefactthatthetimeintervaldt betweentheconsecutivetime- added to the model formulation. We need only, in addition to
steps is assumed to be small, it is supposed to be much larger P(s Eε) and 1 P(s Eε), to introduce the probability of
− − −
thanthedurationofanynoisecorrelations.Therefore,weassume a broken fiber healing. All formulae can be easily generalized
thattherearenocorrelationsofthenoiseintimeeither.Another for this case as well, but due to the complexity of the results
assumptionweutilizeisthatalthoughthetimeintervalbetween already present for the simpler model, we leave it for further
consecutive time-steps has zero limit, the thermodynamic limit studies.
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Abaimov Non-equilibriumDamagePhenomena:PathApproach
THE CLASSICAL GIBBS APPROACH OF combinatorialnumberofoutcomeswhenwedistributeNLintact
STATES fibersandN(1 L)brokenfibersamongNfibers:
−
N!
The classical Gibbs approach is formulated in terms of states. g . (5)
[L]
To develop its analog for damage phenomena, we need to give = (NL)!(N(1 L))!
−
somedefinitions:whatwerefertoasanon-equilibriumstate(or
Tosimplifythisexpression,weapplyStirling’sapproximation
afluctuation).Inmoredetails,thisapproachisdiscussedin[30],
Chapter2. N N
LetusimagineaFBMwhereweobserveseparatefibers.Asto N! O Nα (6)
≈ e
amicrostate n wewillrefertoaparticularmicroconfiguration (cid:18) (cid:19)
{ } (cid:0) (cid:1)
whenforeachfiberofthemodelweprescribewhetheritisintact and in the limit N >> 1 we discard the slow power-law
orbroken.Forexample,forasystemwithN 3fibers,oneof dependenceO(Nα)onNincomparisonwiththefastexponential
=
themicrostatesis xx wherebytwosymbols“x,”wedenoted multiplier(N/e)N:
{ |}
thatthefirsttwofibersofthemodelarebroken,whilethesymbol
“|”meansthatthethirdfiberisstillintact.Foramodelwithonly N N
N! . (7)
ln
three fibers, it is easy to list all the possible microstates: , ≈ e
{|||} (cid:18) (cid:19)
x , x , x , xx , x x , xx , xxx .
{|| } {| |} { ||} {| } { | } { |} { } Thespecialnotationforthelogarithmicaccuracy,“ ,”stands
Whatistheprobabilityofobservingaparticularmicrostatein ≈ln
heretodenotetheaccuracytotheorderofanarbitrarymultiplier
theensembleofidenticalsystems?FortheDFBM,thesolutionis
withthepower-lawdependenceonN(fordetailedexamples,see
well-knownasthesolutionofradioactivedecay.Theprobability
ofafiberfailingattimetduringtimeintervaldtise−pftpfdtand [30],Chapter2).
Applying Stirling’s approximation with the logarithmic
t
theprobabilityofafiberfailingbeforethetimetis e−pftpfdt accuracytoEquation(5),wefind
=
0
1−e−pft.Therefore,theprobabilityofobservingamRicrostate{n} g[L]≈lnL−NL(1 − L)−N(1−L). (8)
withLintactand(1-L)brokenfibersatthetimetis
LetusnowprovethatZensemble(t) 1 e−pft −N isindeedthe
w{enn}semble = e−pft NL 1−e−pft N(1−L). (3) fpuanrtcittiioonnifsutnhcetisounmooffthexepeonnseenmtiball=ef.uB(cid:0)nyc−tdioenfisnieti(cid:1)oNnL,/Ttheffe(t)paorvteitrioanll
(cid:0) (cid:1) (cid:0) (cid:1) −
Weusedherethesuperscript“ensemble”toemphasizethatthis
microstates:
probabilityisdictatedbytheensembleconsidered.
Wecanrewritethedistribution(Equation3)as Zensemble(t)≡ e−NL{n}/Teff(t) (9)
n
w{enn}semble = Zensem1ble(t)e−NL/Teff(t) (4) GroupingmicrostatesbyfluctuX{at}ions[L],sinceallmicrostatesof
thesamefluctuationhaveequalvaluesofL,wecanwrite:
wherewedenotedTeff(t) ln−1 epft 1 .Formula(4)could
bcaelcloedmeTeiffd(etn)titchael teoffetchteiveG≡itbebmspepr(cid:0)raotbuarbe−.iliTt(cid:1)yhednistZriebnusetmiobnle(ti)f we Zensemble(t)= g[L]e−NL/Teff(t). (10)
[L]
1 e−pft −N plays the role of the partition function of th=e X
ens−embleaswewillprovelaterinEquations(9–11). Substituting here Teff(t) ln−1 epft 1 and the statistical
(cid:0) We see(cid:1) that the probability distribution of microstates is weightfromEquation(5),w≡eeasilyprove−therequiredstatement:
(cid:0) (cid:1)
Gibbsianwiththeeffectivetemperaturechangingwithtime;this
aSlilmowilasrutsotaotchaellrmouordyennasemmicblseysatsembeiwnigth“etffheectliimveiltye”dcsapneocntriucaml. Zensemble(t) = N (NL)!(NN(1! L))! epft1 1 NL1N(1−L)
(like an Ising model), the effective temperature can be both NXL=0 − (cid:18) − (cid:19)
bpeogsiitninveinganodf ntheegamtivoed,elcheovoosluintigoninbtaucttsswtaittcehsinfogrthfiibsecrshoiincethtoe = epft1 1 +1 N = 1−e−pft −N. (11)
(cid:18) − (cid:19)
brokenonceatlatertimes. (cid:0) (cid:1)
TofindtheprobabilityWensemble(t,L)ofobservingafluctuation
As to a fluctuation [L] (a macrostate [L]) we will refer to [L]
[L]intheensemble(toobserveasystemwiththegivenfraction
the union of all microstates corresponding to the given value
Lofintactfibers),wemultiplyprobability(Equations3,4)ofthe
of parameter L: [L] n . For example, in the case
of a model with three≡fib{ne}r:sLS,{nw}=eLo{bs}erve fluctuation [L 1/3] cmoircrreossptoantedsi:ngmicrostatesbythenumber(Equation8)ofthese
=
when we observe one of the three microstates: xx , x x ,
or xx . For the model with arbitrary N, th{e|num}b{er|g[L}] Wensemble g wensemble e−pft NL 1−e−pft N(1−L).
of m{ icro|}states, corresponding to the fluctuation [L], is called [L] = [L] {n}∈[L] ≈ln(cid:18) L (cid:19) (cid:18) 1 − L (cid:19)
the statistical weight of this fluctuation. It can be found as the (12)
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Abaimov Non-equilibriumDamagePhenomena:PathApproach
To attribute a microstate n to a particular fluctuation [L], we whichstatesthatthereareabout√Nfluctuationsunderthewidth
willwriteeither n [L]o{r}n :L L.Thereisnodifference ofthemaximumofWensemblewiththeprobabilitysimilartothe
{ }∈ { } {n} = [L]
betweenthesetwonotations,andwewillutilizetheonethatwill mostprobablefluctuation[L ].Thereby,theprobabilityof[L ]is
0 0
bemoreconvenientatthemoment. oftheorderof1/√NwhichreturnsustoEquation(18).
The dependence Wensemble on NL has a very narrow Let us introduce the partial partition function (of a
[L]
maximum due to the “wrestling” of two exponentially fast fluctuation)
dependences: g and wensemble. The absolute width of this
[L] n [L]
maximum is of the order{o}f∈√N while the relative width is of Z[L] ≡ e−NL{n}/Teff(t) =g[L]e−NL/Teff(t) (20)
the order of 1/√N. The most probable fluctuation (which will n [L]
{X}∈
beobservedmostcommonly)isdeterminedbythemaximumof
∂Wensemble whenwesumtheexponentialfunctionsnotoverallmicrostates
thisprobability.Equatingthederivativetozero, [L∂]L =0 but only over microstates of the particular fluctuation. Then,
(cid:12)0 probability(Equation12)ofobservingthefluctuationisdirectly
∂lnWensemble (cid:12)
or [L] 0,wefindtheequationofstate: (cid:12) relatedtothepartialpartitionfunctionofthisfluctuation:
∂L = (cid:12)
(cid:12)0
(cid:12)
(cid:12)(cid:12) L0 =e−pft (13) W[eLn]semble =g[L]w{enn}s∈em[Lb]le = ZenZs[eLm]ble. (21)
TheprobabilitydistributionW[eLn]sembleisnormalizedbyunity: Next,letusconsideraveragingintheensemble:
1=X[L] W[eLn]semble =X[L] g[L]w{enn}s∈em[Lb]le. (14) (cid:10)f(cid:11)ensemble ≡ X{n} f{n}w{enn}semble =X[L] f{n}:L{n}=Lg[L]w{enn}s:eLm{nb}le=L
Letusmultiplythisequalitybythepartitionfunction: = f{n}:L{n}=LW[eLn]semble. (22)
[L]
X
Zensemble ≡ e−NL{n}/Teff(t) = g[L]e−NL/Teff(t), (15) Asanexample,weconsideraveragingofL:
n [L]
X{ } X
L ensemble LWensemble. (23)
wherewehaveagainsubstitutedthesumovermicrostates n by h i = [L]
thesumoverfluctuations[L].Thenumberofnon-zeroter{m}sin X[L]
this sum is of the order of √N (the width of the maximum of
Wensemble): Here the probability distribution W[eLn]semble is the very fast
[L]
dependencecomparingtoslowdependenceL.Besides,Wensemble
[L]
Zensemble = g[L0]e−NL0/Teff(t)O √N . (16) ifsunncotriomnaδli(zLed bLy0)u,naintdy.wTehiemremfoerdei,atitelpylfiaynsd:the role of a delta-
(cid:16) (cid:17) −
Applying the logarithmic approximation (neglecting slow L ensemble L (24)
0
power-law dependencies and keeping only fast exponential h i ≈
dependencies), we prove the classical result of statistical
The definition [L] n of a fluctuation [L] can be
mechanicsthatapartitionfunctionequalsitsmaximalterm: presentedintermsof≡non{n-S}e:Lq{{un}i=l}ibLriumprobabilitydistributions
Zensemble≈lng[L0]e−NL0/Teff(t). (17) which we developed in details in [30], Chapter 2 as the
generalization of the Leontovich approach of non-equilibrium
Dividing this equation by Zensemble, we find another important states [127, 128]. We observe a fluctuation [L] when in the
relationship: ensembleweobserveasystemwiththegivenfractionLofintact
fibers.Itislikeinstatisticalphysicstoobservefluctuationsofgas
1
w{enn}s:eLm{nb}le=L0≈ln g[L0] (18) dalelnististymwohleecnu,lefosrinextahmeplleef,tthhaelfgaosfhitassvgoaltuhmereedwshpiolenttahneeroeuasrlye
no molecules in another (right) half of the volume. We call it
that the probability of microstates corresponding to the most afluctuation(amacrostate).Buttodeterminethepropertiesof
probable fluctuation [L0] is the inversed number of these this state, it is not enough to observe it for a microsecond as a
microstates (the principle of the equivalence between the microstate.Tostudythisstate,weisolatethegasinthelefthalf
canonicalandmicrocanonicalensembles). ofthevolumebyavirtualboundaryandobserveforsometime
Equality (Equation 18) can be found without multiplying beforereleasingit.Butwhatdoesthisadditionalisolationmean?
(Equation 14) by Zensemble. Only in this case, we arrive at the Itmeansthatthegasvisitsonlythemicrostatescorrespondingto
equation ourfluctuationanddoesnotvisitothermicrostates.
The same purpose can be achieved by considering non-
1=g[L0]w{enn}s:eLm{nb}le=L0O √N ≈lng[L0]w{enn}s:eLm{nb}le=L0 (19) equilibrium probability distributions [30]. Distribution
(cid:16) (cid:17)
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Abaimov Non-equilibriumDamagePhenomena:PathApproach
(Equations3,4)isdictatedbytheensembleboundaryconditions We see that the action of the free-energy is directly related to
similartothepressureandtemperaturedictatedbytheboundary theprobabilitydistributionoffluctuationsanditsminimization
conditions for the gas. However, a system may disregard these is equivalent to the evolution from states of low probability to
orders and behave independently. Imagine students who are highlyprobablestates.
orderedtoreadonepaperaweekonaverage.Onestudentworks Theensembleactionoffreeenergyis:
hardandmayreadfivepapersaweekonaveragewhileanother
onedoesnotstudyatall.Similar,asystemmayfluctuateviolating Aensemble NL /Teff Sensemble lnZensemble. (32)
0
≡ − =−
ordersfromtheensemble.Forthegas,itmeansthatthesystem
stays in the left half of its volume (have non-zero probabilities Allotherresults,generallydevelopedforthecanonicalensemble,
only for those microstates which do not have molecules in the are applicable here as well (for details, see [30], Chapter 2).
right half). For our system with damage, it means that only However, our purpose in this manuscript is to suggest possible
thosemicrostateshavenon-zeroprobabilitieswhichpossessthe approaches to solve an arbitrary system, not as simple as the
correctvalueoftheintactfiberfractionLwhileprobabilitiesof DFBM.Therefore,weprovidedheretheclassicalGibbsapproach
othermicrostatesarezero: of states only to the purpose to highlight differences with the
1/g , n :L L proposedbelowpathintegralapproach.
w{n} = 0[,L]n{:}Ln{n} =L . (25)
(cid:26) { } { } 6= RESTRICTIONS OF THE CLASSICAL
Tofindtheentropyofthefluctuation,weapplyGibbsdefinition GIBBS APPROACH OF STATES
andarriveatBoltzmann’sentropy:
In the previous section, we were lucky that our system had an
S w lnw lng . (26)
[L] ≡− {n} {n} = [L] analyticalsolution.Thiswastheconsequenceofthefactthatwe
X{n} wereabletofindaprioritheprobability te−pftpfdt 1 e−pft
Fortheequilibriumprobabilitydistribution,thistransformsinto: = −
0
ofafiberfailingbeforetimet.ForthecasReofanarbitrarysystem,
Sensemble ≡ − w{enn}semblelnw{enn}semble wegenerallyknowthedistributionofprobabilitiesonlyforone
X{n} time-step.Inotherwords,ifattimeti asystemwithprobability
=−X[L] g[L]w{enn}semblelnw{enn}semble twinh{eannatis}emtmhiebclerporissotbainatebailnimtyicorfowtshtiiatshtesNy{snLtei}mwaiintthttahNcetLnfiiebxientrttsaimcistefi-bsteerps,tiw+e1kbneionwg
i 1 i 1
Wensemblelnwensemble, (27) { + } +
=− [L] {n}
where we have substitutedX[tLh]e sum over microstates n by the w{ennis+em1}ble =w{ennis}emble 1−pft NLi+1 pft N(Li−Li+1). (33)
{ } (cid:0) (cid:1) (cid:0) (cid:1)
sumoverfluctuations[L].Intheright-handsideofEquation(27), dwensemble
thedependencelnwensembleisaslow,power-lawdependenceon Infact,weutilizedherethegeneralGibbsformula {}dt (t)=
NLwhileWensemble i{sn}theexponentiallyfastdependence,acting F wensemble(t′), t′ ≤t thattheevolutionofthedistributionof
[L] {}
like a delta-function due to its normalization Wensemble prhobabilities of microistates is determined by some functional
[L] = dependence on the history of the model. To obtain the
[L]
1. Therefore, utilizing Equation (18), we imPmediately find distribution of probabilities of the microstates at time t we
Boltzmann’sexpressionfortheequilibriumentropyaswell: have to integrate this equation over all possible paths among
allpreviousconfigurations[e.g.,129,130].Wehavetointegrate
Sensemble =−lnw{enn}s:eLm{nb}le=L0 =lng[L0]. (28) aarllepionstseirbcloenninetcetremdedbiyatethecontrfiegmuernatdioounss; nthuemsebecronofifgduirffaetiroennst
The non-equilibrium free energy potential (the effective
paths; each path can have its own probability or be prohibited
Helmholtzenergy)ofafluctuationmaybeintroducedas
(irreversible damage in the case of the FBM, when a broken
fibercannotbecomeintactagain).Therefore,theintegrationof
F NL TeffS Teff lnZ . (29)
[L] ≡ − [L] =− [L] combinatorialsumsbecomescumbersome.
Fortheequilibriumfreeenergywefind This approach corresponds to the classical Gibbs non-
equilibrium statistical mechanics, when microstates of a
Fensemble NL0 TeffSensemble Teff lnZensemble. (30) system are identified with the system’s microconfigurations
≡ − =−
and macrostates of a system are identified with the system’s
Insteadoffreeenergy,wecandefinetheactionoffreeenergyon
macroconfigurations. However, we know that the probabilities
afluctuation(thiswillremovetheasymmetryoftemperature;for
forMarkovprocessesareassociatednotwiththestatesbutwith
details,see[30,Section2.16]):
thepathsamongthesestates(forthefirst-orderMarkovprocess,
the probability of a path A-B-C-D is P(DC)P(CB)P(BA)P(A)
A[L] ≡NL/Teff −S[L] =−lnZ[L] =−ln ZensembleW[eLn]semble . while to find the probability of the s|ystem|arrivin|g after
(cid:16) (31(cid:17)) three steps at state D we should have three sums involved:
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Abaimov Non-equilibriumDamagePhenomena:PathApproach
which all the broken fibers remain broken. This makes our
systemaMarkovprocessoforder1.Forthetotalprocessfrom
t0 0 to tν ν dt, we construct all possible chains of
= = ·
microconfigurations. Each such chain, as a possible sequence
of particular microconfigurations n0 , n1 ,..., nν , will be
{ } { } { }
referredtoasamicropath n0 n1 ... nν (further,
{ } → { } → → { }
wewillabbreviatethisnotationas n0 nν ).Forexample,
{ } → { }
for a system with N 3 fibers, one of possible micropaths is
=
x xx xx xxx where symbol “|”
{|||} → {| |} → { |} → { |} → { }
denotesanintactfiberwhilesymbol“x”denotesabrokenfiber.
Letusassumethatforamicropath n0 nν thesequence
{ }→{ }
of the configurations has the evolution of the intact parameter
L(t) L0,...,Lν.Thentheprobabilityofthismicropathis
≡
w{enn0se}→mb{lneν} = (1 − P)N|1L1|PN(L0−|1L1|)
FIGURE2|ThegraphamongA,B,C,andDstates. ...(1 − P)N|1Lν|PN(Lν−1−|1Lν|)
ν
1 P N(1−Lν) N Li 1
− P i 1 − (34)
= P P=
(cid:18) (cid:19)
P(DX3)P(X3 X2)P(X2 X1)P(X1), see Figure2). astheprobabilityofN|∆L|fibersfailingandofNL N(L
| | | i i i 1
X1,X2,X3 1L ) fibers staying intact, where P P(s – Eε)≡is cons−tan−t.
ThPerefore, it is much easier to find a distribution of the | i| ≡
This probability wensemble is dictated by the ensemble and,
probabilities only for a macrogroup of paths than for all the in particular, by th{ne0}→va{lnuνe} of P. This constraint is a model
pathsleadingtoamacrostate.IntheGibbsapproach,thestates
input and acts similarly to the temperature prescribed in the
ofasystemwerechosenasbases,althoughweseethateverything
canonical ensemble. In the canonical ensemble, an external
points to the fact that as bases we should choose not the states
mediumdictatesthedistributionoftheprobabilitiesfordifferent
but the paths [125, 126, 131–142]. We will see how we can
paths but a system actually can be not in equilibrium with
developsuchanapproachinthenextsection.Thebenefitofthis
the dictated distribution and realizes some other probability
approach will be that its bases will not include integrals of the
distribution w for its paths. Therefore, we used here
states for the previous evolution of the system but, in contrast, {n0}→{nν}
the abbreviation “ensemble” to emphasize that this probability
willbethisevolutionitself.
distributioncorrespondstotherequirementsstatedregardlessof
possiblefluctuations.
STATISTICAL MECHANICS OF THE PATH As to the macropath [L0] [L1] ... [Lν] (further,
→ → →
INTEGRAL APPROACH we will abbreviate this notation as [L0] [Lν]) we will refer
→
toasubsetofallmicropaths n0 nν correspondingtothe
{ } → { }
In the previous sections and in our previous publications [21, specifiedevolutionoftheintactparameterL(t):[L0] [Lν]
→ =
22, 30], we identified microstates of a system with system’s n0 nν .Forexample,forthesystemwithN 3
mmiaccrrooccoonnfifigguurraattiioonnss.anHdomwaecvreor,stattheissofapapsryosatcehm wwiotrhkesydstewmel’sl {finb0}e→rsS{mn{ν}a:cLr{}noi}p=aLtih→[1]{ }[1/3] [1/3] [0]unitesmicropa=ths
→ → →
onlyforequilibriumstatisticalmechanics.Fornon-equilibrium xx xx xxx , x x x x
{|||} → {| } → {| } → { } {|||} → { | } → { | } →
statisticalmechanics,weshoulddevelopanotherapproach[126]. xxx ,and xx xx xxx .Infact,macropath
{ } {|||} → { |} → { |} → { }
As an illustration, we consider the case of the NFBM and statesthatthesystempassesduringitsevolutiontheprescribed
assumethattheprocessconsistsofν timeintervalsofduration valuesofLbutdoesnotspecifyparticularmicrostates.
dt,fromt0 0totν ν dt.Foreachtime-stepti asanorder Theprobabilitiesofmicropathscorrespondingtoamacropath
parameterw=echoose=thev·alueoftheintactparameterLi atthis [L0] → [Lν] are all equal one to another and are given by
time-step.Forthetotalprocesstheorderparameteristhetotal Equation(34);thenumberofthesemicropathsis
historyoftheintactparameterL(t) L0,...,Lν.Weassumethat
Lthewpirlolcneostsablewvaayrsiastbalretsinfrothmezeenrsoemda≡bmlea.gWeLe0a=ssu1m,soetthheatqburaonktietny g[L0]→[Lν] = (N 1L1 )!((NN(LL00)! 1L1 ))!.
0 | | −| |
fibers cannot become intact again; therefore, increments of the (NL1)!
.
intactparameter1Li Li Li 1arealwaysnegative. (N 1L2 )!(N(L1 1L2 ))!
≡ − − | | −| |
At each time-step ti, a particular system in the ensemble (NLν 1)!
has its own value of the intact parameter Li and is in one ...· (N 1Lν )!(N(Lν−1 1Lν ))! (35)
of the microconfigurations n corresponding to this intact | | − −| |
i
{ }
parameter.Thenextpossiblemicroconfigurationforthissystem asacombinatorialchoiceofN|1L|failedfibersamongNL
i i 1
−
at the time-step t can only be a microconfiguration in initial fibers. We can cancel (NL)! in the numerators and
i 1 i
+
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(N(L -|1L|))!inthedenominatorstoobtain Asanexample,weagainconsideraveragingofL:
i 1 i
−
ν
(NL )!
g[L0]→[Lν] = (NLν)!iν1(0N|1Li|)!≈ln(Lν)−NLνiY=1|1Li|−N|1Li|, hLiensemble(t) ≡ {n0}X→{nν}L{n0}→{nν}w{enn0se}→mb{lneν}
Q= (36) L0, t t0
=
L wensemble , t t (42)
wherethenotation“≈ln”meansthatinthethermodynamiclimit ≡({n0}P→{nν} {ni} {n0}→{nν} = i =
N allpower-lawmultipliersareneglectedincomparison L , t t
0 0
→+∞ =
withtheexponentialdependenceonN. g Lwensemble , t t .
Theprobabilityofthesystemhavingamacropath[L0]→[Lν] =([L0]P→[Lν] [L0]→[Lν] i {n0}→{nν} = i
(tomoveamongmacroconfigurationswiththespecifiedL(t))is
g[L0]→[Lν] Of course, in the thermodynamic limit, hLiensemble(t) coincides
Wensemble wensemble g wensemble .(37) withthemostprobablemacropath: L ensemble(t) L(0)(t).
[L0]→[Lν] = n 1 {n0}→{nν} = [L0]→[Lν] {n0}→{nν} Probability distribution (Equatihoni 34) is th≈e distribution
X=
dictated by the ensemble. However, we can also consider
Maximization of Wensemble over all the possible macropaths other distributions w which do not obey the
givesthemostproba[bLl0e]→m[aLcνr]opathL(0)(t): ensemble requirements.{nN0}e→xt{,nνa}s a fluctuation we consider the
Wensemble max Wensemble . (38) cmoarcrerosppoanthdi[nLg0]to→th[iLsνm].aTchroepnauthmbiserggiv[Le0n]→b[Lyν]Eoqfumatiicornop(a3t6h)s.
[L(00)]→[L(ν0)] =L1,...,Lν [L0]→[Lν] Again, to observe this macropath, we need to “isolate” our
system in this macropath. This is achieved by considering the
DependenceL(0)(t)isthedependenceofLwhichwewillobserve
probabilitydistributionw suchthatonlythemicropaths
generallyintheensemble. {n0}→{nν}
The function Wensemble , given by Equation (37), is the belonging to the considered macropath [L0] → [Lν] have
product of g [L0]→an[Ldν]wensemble . Both these functions non-zeroprobabilities:
containexpon[eLn0]t→ial[lLyν]fastdepen{nd0}e→nc{neνs}onN(Nisinfiniteinthe
tfpuhrneorcbmtaiboolndeyWmnaa[emLcn0rsi]eoc→mpbl[aiLlmetνh]it[h)L.a(0sT0)ha]evr→eefroyr[Len,(νa0ir)nr].otwAhesmwspaexaciwmeiluolmfseaelaltbpetahltoehwsm,, ttohhseet w{n0}→{nν} =(cid:26)1/g[L00]→,{n[L0ν}],→{n0{}n→ν}∈/{n[νL}0∈]→[L0[]L→ν] [Lν].(43)
absolutewidthofthismaximumisproportionalto√Nwhilethe
Here we assumed the equiprobability of all micropaths of the
relative width is proportional to 1/√N. Therefore, the number
systemisolatedinthemacropath.Actually,thismaynotalways
of different macropaths [L0] → [Lν] under the width of this be true but depends solely on how we build fluctuations as an
maximumhasapower-lawdependenceonN whilethenumber
instrumentofourinvestigation.
g[L0]→[Lν] ofmicropaths{n0} → {nν}correspondingtoeachof Theentropyoftheconsideredmacropathis
thesemacropaths[L0] [Lν]hastheexponentialdependence
→
onN.Therefore,forthenormalizationofthefunctionWensemble
withthelogarithmicaccuracyweobtain [L0]→[Lν]
S w lnw lng .
[L0]→[Lν] ≡− {n0}→{nν} {n0}→{nν} = [L0]→[Lν]
1 Wensemble Wensemble {n0}X→{nν}
= [L0X]→[Lν] [L0]→[Lν]≈ln [L(00)]→[L(ν0)] (44)
g wensemble . (39)
≡ [L(00)]→[L(ν0)] {n0}→{nν}∈[L(00)]→[L(ν0)] We should emphasize here that the entropy introduced is the
“dynamical” entropy of the distribution of probabilities for the
Thisprovides
pathsandmustnotbeconfusedwiththeclassicalGibbsentropy
g 1/wensemble . (40) associated with the distribution of probabilities for the states
[L(00)]→[L(ν0)]≈ln {n0}→{nν}∈[L(00)]→[L(ν0)] (configurations). In our notations, the classical Gibbs entropy
wouldbeS (t) w (t)lnw (t)astheaverageattime
In other words, the probability of each micropath of the most {n} ≡ − n {n} {n}
probablemacropath[L(00)]→[L(ν0)]hastheorderoftheinverted t over the probabilitieP{s}w{n}(t) of microconfigurations at this
numberofthesemicropaths. time. On the contrary, the path entropy is associated with the
Next, let us consider averagingover the spaceof all possible probabilities w of micropaths of the total process and
{n0}→{nν}
paths: cannot be attributed to system’s characteristics at a particular
timet.
(cid:10)f(cid:11)ensemble(t)≡{n0}X→{nν}w{enn0se}→mb{lneν}f{n0}→{nν}. (41) conIsf,ideinrspteraodbaobiflittyhedisptrroibbuatbiiolinty(Edqiustartiibounti3o4n) dwi{cnt0a}t→ed{nνb}y, twhee
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Abaimov Non-equilibriumDamagePhenomena:PathApproach
ensemble,fortheensemblepathentropywefind: Since our system has two effective temperatures, instead of
a free-energy potential we should define the action of the free-
Sensemble wensemble lnwensemble energy potential to avoid the asymmetry of one temperature
≡ −{n0}X→{nν} {n0}→{nν} {n0}→{nν} chosen in favor to another. The action of the free-energy for a
=− g[L0]→[Lν]w{enn0se}→mb{lneν}lnw{enn0se}→mb{lneν} = macropath[L0]→[Lν]canbedefinedas:
[L0X]→[Lν] ν
=−[L0X]→[Lν]W[eLn0s]e→mb[Lleν]lnw{enn0se}→mb{lneν}. (45) A[L0]→[Lν] ≡NLν/Tν +N iX=1Li−1!/TR −νS[L0]→[Lν] =
Hdeepreen,dethnece ofunncNtioinncolnmwp{eannr0sie}s→mobn{lneνw}ithhatsheafunsclotiwonspgower-law =−ln(g[L0]→[Lν]exp −NLν/Tν −N i 1Li−1!/T !)
aonnd Nw.{enn0sTe}→mheb{rlneeνf}orwe,hicthhehadveepenexdpeonnceentWialelynsemfabslte deapcet[nsL0d]→eans[cLeνas] =−lnZ[L0]→[Lν], X= R (49)
delta-function around the most probab[Ll0e]→m[Laνc]ropath, and where Z[L0]→[Lν] is the partial path partition function [23, 125,
126]ofthismacropath:
wefind:
Sensemble ≈ −lnw{enn0se}→mb{lneν}∈[L(00)]→[L(ν0)] ≈lng[L(00)]→[L(ν0)] Z[L0]→[Lν] ν
=S[L(00)]→[L(ν0)]. (46) = exp(−NLν/Tν −N Li−1!/T )
{n0}→{nν}X∈[L0]→[Lν] iX=1 R
Inotherwords,theentropyofthetotalensembleofpathsequals ZensembleWensemble . (50)
theentropyofthemostprobablemacropath. = [L0]→[Lν]
Wecanrewritetheequilibrium distributionofprobabilities,
Therefore,fortheactionofthefree-energyweobtain
givenbyEquation(34),as
w{enn0se}→mb{lneν} = Zens1emble exp −NLν/Tν −N ν Li−1!/T ! A[L0]→[Lν] =−ln(cid:16)ZensembleW[eLn0s]e→mb[Lleν](cid:17). (51)
i 1 Hence, the action of the free-energy is directly related to the
X= R
(47) probability distribution of the fluctuations and the principle of
itsminimizationisequivalenttotheevolutionfrompathsoflow
N
where Zensemble = 1PP , Tν = ln−1 (1−P)/P , probabilitytohighlyprobablepaths.
and T ln−1P.(cid:16)It−is(cid:17)easy to see that(cid:8)Zensemble (cid:9)is In Gibbs equilibrium statistical mechanics, an equilibrium
= − state is found as a minimum of a free energy potential. For
the path partition function of the system: Zensemble
R the microcanonical ensemble, the free energy potential is the
ν =
{pnl0a}yP→t{hnνe}reoxlpes(cid:26)−ofNeffLeνc/tTivνe−teNmp(cid:18)eiPr=at1uLrie−s.1(cid:19)W/eTsRe(cid:27)e.thTatνwahnedn PTiRs ncaengoatnivicealenentrsoepmyb−le,Sth≡e<freelnewn{enr}gy>p≡oteP{nnt}iwal{nis}ltnhwe{Hn}e;lmforhotlhtez
close to unity (the probability of a fiber breaking is small), the energy F < Hn > T < lnwn > wn [Hn
≡ { } + { } ≡ n { } { } +
temperatureT isinfinitebutpositive. Tlnw ].ForthecaseofgeneralensembleinGibP{b}sequilibrium
The definitRion of the temperature Tν is, to some extent, statisti{cna}l mechanics, the principle of the minimization of the
ambiguous. Indeed, we can rewrite Equation (34) for
free energy potential always works because this potential is
probabilitiesas
always proportional to the minus logarithm of the probability
w{enn0se}→mb{lneν} = (1 − P)N|1L1|PNL1...(1 − P)N|1Lν|PNLν dvaislitdribfourtitohnecoafsefloufcntuoant-ioeqnus.ilWibreiusmeemthecahtatnhiecssaasmweepll,rionncliypnleowis
ν
=(1 − P)N(1−Lν)PNiP=1Li. (48) wbuethfoavrethteopcaotnhsst.ructthefreeenergypotentialnotforthestates
This gives Tν ln−1(1 P) which is different from So, for the path microcanonical ensemble (for the system
the expression Tν= ln−1 (−1 P)/P above. However, the isolatedinamacropath),thenegativedynamicalentropy
= −
differenceisln(P)whichisnegligibleincomparisonwithln(1 P)
(cid:8) (cid:9) − S <lnw > w lnw
intThheelimtietmPp→era1tu.re T = −ln−1P of the system is − ≡ {n0}→{nν} ≡{n0}X→{nν} {n0}→{nν} {n0}→{nν(}52)
complementary to the integral of the evolution of intact
R
ν tν plays the role of the free energy potential. This immediately
parameter:N Li 1 N L(t)dt.Ifatthefinaltime-steptν
i 1 − ∝ 0 providesthatallthemicropathscorrespondingtothismacropath
thetotalsystemP=fails,L(tν) R0,thenthesingleorderparameter must be equiprobable. If we compare this result with our
=
tν hypothesis(Equation43),weseethatwewererightconsidering
leftistheintegralN L(t)dt.
micropathsasequiprobablewithinthemacropath.
0
R
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Abaimov Non-equilibriumDamagePhenomena:PathApproach
For the path canonical ensemble, the role of the free energy at the most probable macropath. These equations could be
potentialisplayedbytheactionofthefree-energy used as definitions of the temperatures. As both the entropy
of a macropath S lng and the ensemble
A ≡ ≡<H{n0}→{nwν}{n>0}→/T{n+ν}[<H{lnn0}w→{n{n0}ν→}/{Tnν+} >lnw{n0}→{nν}] seanmtreopfuynScetniosenmabllede[pL≈0e]n→dS[e[LLnν(0]0c)e]→=o[nL(ν0L)](t)[=La0n]→dln[LLgν(0[]L)((00t))]→re[sLp(ν0e)]cthivaevley,twhee
{n0}X→{nν} obtain
(53)
whereH{n0}→{nν} issupposedtobethedynamicalHamiltonian 1 ∂S[L0]→[Lν] ∂Sensemble,i 1,...,ν 1and
Htwwo{ehannmi0scie}th→imνsltba{olnpeonνpw}iaenna=rhstaeiepnmxppdpein(cid:0)es−rtsartHitobu{urcneto0i,}no→nsHi{sn({tνEno}0q/}fu→Ttaw(cid:1){tnio/oνZ}pn/eanT4rst7esm),beo=laefc.thhTechopisrrNroedLbsyνpan/boaTinmlνidtiiicen+asgl: TT1Rν == ∂S[NNL0∂∂]→LLνi[Lν](cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)00 ≈≈ ∂SNNen∂∂sLLem(i(ν00b))le. = − (58)
(cid:12)
N Li 1 /T , and does not correspond to the classical (cid:12)
(cid:18)i 1 − (cid:19) This resembles the law of conservation of energy dE/T
HamP=iltonian of thRe states because of being determined by the dQ /T dSensemble in the canonical ensemble—an equatio=n
←
rulesofthememoryoftheprocess: of“dynam=ical”pathbalance:
A
≡ ν
{n0}X→{nν} ν N dL(i−0)1!/T +NdL(ν0)/Tν =dSensemble. (59)
w{n0}→{nν}"NLν/Tν +N Li−1!/T +lnw{n0}→{nν}# Xi=2 R
i 1
X= R
(54) Differentials in this equation should be understood as the
response of the most probable macropath to the change of the
For the most probable macropath [L(00)] → [L(ν0)], the action externalboundaryconstraintP.Thisequationcouldbeobtained
of the free-energy is A lnZ which
[L(00)]→[L(00)] ≈ − [L(00)]→[L(00)] directly by differentiating (Equation 44) as the logarithm of
coincideswiththeensemblepathactionofthefree-energy: Equation(36).
ν
Aensemble ≡ (NLν/Tν +N Li−1!/T )w{enn0se}→mb{lneν}−Sensemble =
{n0}X→{nν} iX=1 R
ν
=− (−lnw{enn0se}→mb{lneν}−NLν/Tν −N Li−1!/T )w{enn0se}→mb{lneν} =−lnZensemble ≈
{n0}X→{nν} iX=1 R
lnZ A . (55)
≈− [L(00)]→[L(00)] = [L(00)]→[L(00)]
AtthepointofthemaximumofW[eLn0s]e→mb[Lleν],wehave shoTuoldfifinnddwthheenmtohsetdperroivbaatbivleesmoafctrhoeppartohba[Lb(0i0li)t]yo→fma[Lcr(ν0o)p],atwhes
∂Wensemble ∂lnWensemble (Equation 37) (or of the logarithm of this probability) with
[L∂0L]→i [Lν](cid:12) =0or ∂[LL0i]→[Lν](cid:12) =0 (56) respecttoLiequalzero(Equation56).Thisprovides
(cid:12)0 (cid:12)0
(cid:12) (cid:12)
(where i 1 becau(cid:12)se we assume L to b(cid:12)e non-variable).
(cid:12) 0 (cid:12)
≥
For lνnW[eLn0s]e→mb[Lleν] = lng[L0]→[Lν] + Aensemble − NLν/Tν − L(i0) =Pi,|1L(i0)|=(1 − P)Pi−1,i=1,...,ν. (60)
N L /T ,wecanwritethat
i 1
(cid:18)i 1 − (cid:19)
P= R as an analog of the equation of state in equilibrium statistical
T1 = ∂lngN[L∂0L]→i [Lν](cid:12)(cid:12)0 = ∂lngN[L∂(00L)](i→0)[L(ν0)],i=1,...,ν−1 meqeuTcahotaioinnnivcoes.sftpiBgaeatihtne.gthceobneshisatveinotr,owffleuschtuoautlidonpsr,owbeasbhlyoucldallfinitdtthhee
R (cid:12)(cid:12) (57) secondderivativesoflnW[eLn0s]e→mb[Lleν]:
and
1 ∂lng[L0]→[Lν] ∂lng[L(00)]→[L(ν0)] ∂2lnW[eLn0s]e→mb[Lleν] Ki,j, i,j 1,...,ν, (61)
Tν = N∂Lν (cid:12)0 = N∂L(ν0) ∂Li∂Lj (cid:12)(cid:12)0 =− =
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
FrontiersinPhysics|www.frontiersin.org 10 February2017|Volume5|Article6