Enhancement of kinetic energy fluctuations due to expansion

0.17 MB·English

by  A. Chernomoretz

Enhancement of kinetic energy fluctuations due to expansion
A. Chernomoretz,1,∗ F. Gulminelli,2,† M.J. Ison,1 and C.O. Dorso1
1 Departamento de F´isica, Universidad de Buenos Aires,
Buenos Aires, Argentina.
2LPC Caen (IN2P3-CNRS/Ensicaen et Universit´e) F-14050 Caen C´edex. France
(Dated: February 8, 2008)
Globalequilibriumfragmentation insideafreezeoutconstrainingvolumeisaworkinghypothesis
widelyusedinnuclearfragmentationstatisticalmodels. IntheframeworkofclassicalLennardJones
moleculardynamics,westudyhowtherelaxationofthefixedvolumeconstraintaffectstheposterior
evolutionofmicroscopiccorrelations, andhowanon-confinedfragmentationscenarioisestablished.
A study of the dynamical evolution of the relative kinetic energy fluctuations was also performed.
Wefoundthatasymptoticmeasurementsofsuchobservablecanberelatedtothenumberofdecaying
channelsavailable to thesystem at fragmentation time.
4
0 PACSnumbers: 25.70-z,25.70.Mn,25.70.Pq,02.70.Ns
0
2
n I. INTRODUCTION motionbehaveslike aheatsink,andprecludesthe disor-
a deredkineticenergyassociatedwiththesystemtempera-
J tureto increasewithoutbounds[13]. Theselastremarks
The multifragmentation phenomenon has attracted the
6 rise some reasonable concerns regarding the sharp con-
attention of the intermediate/high energy nuclear com-
1 straining volume working hypothesis that are worth to
munitysincemorethanadecade. Startinginthe1980’s,
a lot of attention has been paid to the identification and be analyzed.
1
v characterizationofthe occurrenceof anuclear liquid-gas In recent contributions the study of the behavior of
7 phasetransition. Ononehand,indicationsofcriticalbe- the relative kinetic energy fluctuations, AK ≡ NσK2 / <
3 havior were obtained assuming a Fisher-like scaling re- K >2, has received considerable attention. It has been
0 lation for fragment mass distributions [1, 2, 3, 4]. On shown[7,14]thatpartialenergyfluctuationscanbeused
1
the other, it has been claimed that the observed critical to measure the EOS, and to extract information of the
0
behavior in static observables is compatible with a first possible phase transition occurring in finite systems like
4
order phase transition [5, 6, 7, 8, 9]. multifragmenting nuclei. Negative specific heats are ex-
0
/ As in nuclear fragmentation experiments one only pectedtosignalfirstorderphasetransitionsinthesesys-
h
has access to asymptotic observables, the fragmentation tems [9, 15]. Taking into consideration that kinetic en-
t
- stage of the dynamic evolution must be reconstructed ergy fluctuations and the system’s specific heat are re-
l
c using simplified models. To accomplish this task, the lated by [16]:
u
mostwidelyusedstatisticalmodelsoftheMMMS[10]or
n
SMM [11] type, replace the intractableproblem ofquan- 1 3 3
: <σ2 > = (1− ) (1)
v talN-bodycorrelationswithanoninteractinggasoffrag- N K E 2β2 2C
i ments confined with sharp boundary conditions [10, 11].
X
A distinctive feature of these equilibrium fragmentation
r (where β = 1/kT) negative values of the specific heat
a models is that a vapor-branch can be recognized in the areexpectedtoappearwheneverσ2 getslargerthanthe
system caloric curve (CC). Moreover, in the statistical K
corresponding canonical expectation value: 3N.
sampling of configurations,the observedasymptotic flux 2β2
is exclusively associated with the mean Coulomb energy Usingaclassicalmoleculardynamicsmodelithasbeen
of partitions inside the constraining volume. Collective already shown in [17] that large relative kinetic fluctua-
motionsarecompletelydisregardedbeforeandduringthe tions dotake placein confinedclassicalisolatedsystems,
fragment formation stage. providedthat the constrainingvolumes are large enough
to accommodate non-overlapping configurational frag-
On the contrary, when fragmentation is considered in
a non-constrained free-to-expand scenario qualitatively ments. Inthispaperwewanttopursuewiththesystem’s
different features are found. In this case a local equilib- characterizationby studying its temporal evolutiononce
rium picture replaces the global one, and a characteris- we remove the constraining walls within which it was
tic flattening of the caloric curve at high energy can be equilibrated. Our aim is to understand how much mem-
found [12]. This happens as a direct consequence of the ory of the initial configuration is kept in the asymptotic
presence of a collective degree of freedom: the expansive fragmentation patterns which are accessible experimen-
tally. To that end we will focus on the dynamical evolu-
tionoffragmentmassdistributionfunctions(MDF),and
the relative kinetic energy fluctuation, A , that turns
K
∗Electronicaddress: [email protected] out to be a useful tool to probe the unconstrained frag-
†MemberoftheInstitutUniversitairedeFrance. mentation scenario.
2
II. THE MODEL in the center of mass frame of the cluster which contains
particle j, and V stands for the inter-particle poten-
ij
The system under study is composed by excited drops tial. ItcanbeshownthatclustersbelongingtotheMBP
made up of 147 particles interacting via a 6-12 Lennard are related to the most-bound density fluctuation in r-p
Jones potential with a cut-off radius r = 3σ. Ener- space [20]. The algorithm that finds the MBP is known
c
gies are measured in units of the potential well (ǫ), σ as the “Early Cluster Recognition Algorithm” (ECRA).
characterizes the radius of a particle and m is its mass. In a previous contribution we studied the EOS of a
We adopt adimensional units for energy,length and time finite,constrained,andclassicalsystemusingclusterdis-
such that ǫ = σ = 1, t = σ2m/48ǫ. Our particles tributions properties in phase space and in configura-
0
are uncharged and the inclusipon of Coulomb is certainly tional space [17]. Figure 1 summarizes our findings. In
a necessary step to insure that our results are pertinent the upper panel we show the density dependence of the
to the nuclear fragmentationproblem[18]. However,the system energy Ecrit at which transition signals (power-
fact that the the nuclear force andthe van der Waals in- law mass spectrum, maxima in the second moment of
teractionbothsharethesamegeneralfeatures(i.e. short themassdistribution,andnormalizedmeanvariance)are
range repulsion plus a longer range attraction) supports detected. Given a confined system with a density value
the use of this simplified classicalmodel to obtain quali- ρ, for E < Ecrit(ρ) a big ECRA cluster can be found,
tativelymeaningfulresultsinthenuclearcase,andserves whereas for E > Ecrit(ρ) a regime with high multiplic-
alsotoexplorephenomenaofgreatergenerality. Theini- ity of light ECRA clusters dominates. The lower panel
tial condition is chosen as a spherical confining ‘wall’, shows the same transition line in the most usual (T,ρ)
introducing an external potential V ∼ (r−r )−12 plane (see [17] for details).
wall wall
with a cut off distance r = 1σ. The set of classi- Accordingtothecalculatedcriticalexponentsτ,andγ,
cut
cal equations of motion were integrated using the well three density regions could be identified. One (very low
knownvelocityVerletalgorithm,whichpreservesvolume density limit, region labeled A in Figure 1) in which the
in phase space[19]. boundary conditions are almost ineffective, and a back-
Using this system, we have performed the following bendingofthecaloriccurvecanconsequentlybeobserved
molecular dynamics (MD) experiments. First, we let evenintheisochoreensemble[21]. Anotherone(regionC
the system equilibrate inside the constraining volume inFigure1),correspondingtothehighdensityregime,in
at a chosen density. We call this initial condition which fragments in phase space display critical behavior
the constrained state. Afterwards, we remove the of 3D-Ising universality class type. And an intermediate
constraining walls, and let the system evolve for a very density region (region B in Figure 1), in which power-
long time, such that the fragment composition reaches lawsaredisplayedbutcannotbeassociatedtotheabove
stability. Thiswillbereferredtoastheasymptotic stage. mentioneduniversalityclass(seeRef.[17]). Thethermo-
dynamic critical point of the liquid-gas phase transition
Oneofthe keyobservablesto studyphasetransforma- (ρc ≈ .35σ−3,Tc ≈ 1.12ǫ0) corresponds to the meeting
tions from a morphologicalpoint of view is the fragment point of these two last regions.
mass distribution. The simplest and more intuitive clus- Having these results in mind, we chose the follow-
ter definition is based on correlations in configuration ing density values, ρ1 = 0.026σ−3, ρ2 = 0.10σ−3,
space: a particle i belongs to a cluster C if there is an- ρ3 = 0.6σ−3 (i.e. one from each relevant region of the
other particle j that belongs to C and |r − r | ≤ r , EOS) in order to study the effect of the dynamic evo-
i j cl
where r is a parameter called the clusterization radius. lution on the observables that characterize the initially
cl
In this work we took r = r = 3σ. The algorithm equilibrated confined systems.
cl cut
that recognizes these clusters is known as the “Mini-
mum Spanning Tree” (MST). The main drawback of
this methodis thatonly correlationsinr-space areused.
MST clusters give no meaningful information about the III. TIME EVOLUTION OF PARTICLE
fragmentation dynamics when the system is dense. CORRELATIONS
Amorerobustalgorithmisbasedontheanalysisofthe
so called “Most Bound Partition” (MBP) of the system,
In order to explore the effects of constrain removal on
introduced in Ref. [20]. The MBP is the set of clusters
thefragmentspectraofthesystemsunderstudy,wecom-
{C } for whichthe sum ofthe fragmentinternalenergies
i pare the confined ECRA-fragment mass distribution of
attains its minimum value:
the constrained system, against the corresponding MST
asymptotic ones.
{Ci} = argmin{Ci}hE{Ci} =PiEiCniti InFigure2we showMDF calculationsperformedover
ECi = Kcm+ V (2) systems initially constrained at the density values above
int X j X jk mentioned (from left to right: ρ ,ρ , and ρ ) with dif-
j∈Ci j,k∈Cij≤k 1 2 3
ferent total energies (increasing E from bottom to top).
where the first sum in (2) is over the clusters of the par- Foragivensetofvalues(E,ρ)theconfinedstateECRA-
tition, Kcm is the kinetic energy of particle j measured fragmentmassspectraandthecorrespondingasymptotic
j
3
0.6 A B C ρ=0.026 (σ-3) ρ=0.1 ρ=0.6
100
0.4
ε)E (0crit 0.20 01.011 E=0.4 (ε
-0.2 0)
0.01
-0.24
10
εature ()0 1.51 0.11 0E=0.1 ()ε
er 0.01
p
m 0.5 1 10 100
te E
MST(asymptotic) =
00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ECRA(constrained) -0.4
ρ (σ-3) (ε
0)
FIG. 1: Upper panel: energy E at which transition 1 10 1001 10 100
crit
signals are detected in a confined, finite size fragment-
ing system, as a function of the system density. Lower FIG. 2: Confined ECRA vs asymptotic MST
panel: same transition line in the temperature-density mass spectra calculated for 3 different initial den-
plane (see text for details). The arrows indicate the sities: ρ = 0.026σ−3,0.1σ−3, and 0.6σ−3 shown
density values considered for the present calculations. in first, second, and third columns respectively.
MST-fragmentmassspectra can be compared. It should expansion dynamics leads to a fragmentation pattern
be kept in mind that for ρ and ρ initial states present essentially determined by the total energy. On the
2 3
just one big fragment according to the MST algorithm. other hand the distribution originated from the initially
For the case of an initially extremely diluted system most diluted systemis sensibly different fromthe denser
(first column in Figure 2) confined states are already cases. Indeedthe size of the heaviestfragmentis smaller
fragmented in configurational space. The corresponding and the IMF distribution is steeper in this case. This
asymptotic fragment structure remains essentially unal- means that a fragmentation pattern originated from an
tered, aside of the evaporation of light clusters from the expansion process is not equivalent to an equilibrated
heaviest fragment. freeze out with sharp boundary conditions. It has been
Ontheotherhand,itcanbeseenthatforinitialhigher proposed[22]thatsucha fragmentationpatternmaystill
densities (ρ , and ρ ) the q−p correlations of the con- be described in statistical terms through information
2 3
theory with the constraints of a fluctuating volume and
fined state, probedby ECRApartitions,are modified by
a radial flow. The possible pertinence of such a scenario
the dynamics that follows the walls removal. For these
densities,thesizeofthebiggestclustersdetectedinboth, is left to future investigations.
the confined and asymptotic stages, seem to agree to a
high degree. However noticeable differences are found in
relative abundances, specially for intermediate size frag-
ments (IMF). IV. KINETIC ENERGY FLUCTUATIONS
The development of an expansive flux modifies the
correlation pattern observed in phase space for initially As was stated in the introduction, fragmenting finite
dense confined configurations acting as a memory systems at microcanonical equilibrium display negative
loosing mechanism of their initial fragment structure. specific heats whenever relative kinetic energy fluctua-
This result implies that even a subcritical density tions surpass the canonical expected value. This is the-
relatively diluted as the ρ = 0.1σ−3 case cannot be oretically expected in the liquid-gas phase transition of
taken as a freeze out configuration, since freeze out by small systems if the volume is not constrained by sharp
definition implies a persistency of the information up to boundary conditions[23]. Consistently it has been sug-
asymptotic times. It is then clear that between confined gested that the magnitude A can be used to identify
K
states and corresponding asymptotic configurations, the the occurrence of first order phase transitions in such
intermediate stage, characterized by the development of systems[7]. Ifthe equiprobabilityofthe differentconfig-
expansive collective motion and the consequent expan- uration microstates is assumed, as it is by construction
sion of the system, plays a crucial role in the process of the case in our initial conditions, K =K(T), and
fragment formation. The relative similarity between the
asymptotic distributions originated from the ρ=0.1σ−3 σ2 σ2
and ρ = 0.6σ−3 initial conditions, suggests that the AK ≡N<KK>2 ∝ TK (3)
4
E=-1.7 ε E=0.1 ε E=0.8 ε
100
10 t=20 t0
t=30 t
d 1 0
el t=40 t0
yi 0.1 t=300 t
0
0.01
(a) (b) (c)
1 10 100 1 10 100 1 10 100
mass mass mass
1
0.8
0.6
0.4
size /N
0.2 biggest
(d) AK (e) (f)
15 K (x 0.02)
σ
K
10
5
(g) (h) (i)
0
0 100 200 300 0 100 200 300 0 50 100150200250300
time (t ) time (t ) time (t )
0 0 0
FIG. 3: Upper panels (3.a, 3.b, 3.c): MST fragment spectra calculated at t = 20,30,40 and,
300t for a ρ = 0.6σ−3 system at three different energies. Middle panels (3.d, 3.e, 3.f): A (t) (red
0 K
circles) and MST-biggest cluster mass (dashed lines) as a function of time. Lower panels (3.g, 3.h,
3.i): Total kinetic energy (solid line) and Kinetic energy fluctuations (red circles) as a function of time.
In this way, the system specific heat can be calculated panel of Figure 3 we show MST spectra calculated at
through Eq. 1 knowing both, <K >, and σ2 . different times. It can be noticed that at t ∼ 40t a
K 0
Sincethedynamicsoftheexpansiontendstodrivethe significativeproportionofthe mid-sizeasymptoticMST-
system into a rather rarefied state, and allows event by fragments are already produced, i.e. internal surfaces
event volume fluctuations which are suppressed by the alreadyappeared. Forlatertimes the bigresidueevolves
sharp boundary, it is interesting to see whether such ab- just by light particles evaporation.
normal fluctuations can be reached in a fully dynamical As a result of the expansion, physical observables like
simulation. In this case the average kinetic energy can- A change in time during a relatively short lapse af-
K
not be associated to a temperature any more, and the ter which they attain almost constant values. A freeze-
relationship expressed in Eq. 1 can no longer be invoked out time, τ , can then be introduced, after of which
fo
to estimate the system specific heat. However meaning- the evolution becomes almost irrelevant. Moreover, Fig-
ful information can still be obtained from A when we ures 3.d, 3.e, and 3.f also indicate that asymptotic mea-
K
resortto dynamicalquantities only. InFigure 3 weshow surements allow to reconstruct quantities as early as
the time evolutionofMST massspectra(panels 3.a, 3.b, τ ∼40t .
fo 0
and3.c),AK (panels3.d,3.e,and3.f),K,andσK (panels Thisobservationagreeswithpreviousresults. Inrefer-
3.g,3.h,and3.i)calculatedforasysteminitiallyconfined ence [24] it was shown that for unconstrained systems a
atadensityρ=ρ3 =0.6σ−3,forthreedifferentenergies. timeoffragmentformation(τff),relatedtothestabiliza-
We also show in panels 3.d, 3.e, and 3.f, the correspond- tion of microscopic clusters composition, can be defined.
ing time evolution of the normalized mass of the biggest For the cases studied we found that τ ∼ τ , meaning
ff fo
MST fragment (dashed line). thattheinformationisfrozenoncethefragmentsurfaces
From this figure we can see that the time at which have developed. From the lower panels we see that the
A attains its asymptotic value ( panels d, e, and f) varianceofthekineticenergy,σ ,alsoattainsitsasymp-
K K
signals the time at which, in average, the fragmentation toticvalueatthistime. Itisimportanttostressthatthis
pattern is settled for each event of a given energy and definition of a freeze out time corresponds to a MaxEnt
initial density values. To support this idea, in the upper principle within the constraints imposed by the dynami-
5
calevolution. Assuch,thisdefinitiondoesnotimplythat 1.2 (a)
the freeze out configuration corresponds to an absolute 1
entropymaximum,i.e. to athermodynamicequilibrium.
0.8
For the lowest considered energy, σK does not change AK 0.6
significantly in time and this stays true if we change the
0.4
initial density. This is the expected behavior as the sys-
tem, in almost every realization, only evaporates light 0.2
clusters in its early evolution, and a big fragment is al- 0
1.2
ways present almost independent of the system volume.
1 (b)
On the other hand, for the more energetic cases the ini-
tial decrease of σ can be related to the preequilibrium 0.8
K
K
dynamics that involves a flux development (expansion) A 0.6
and prompt emission of light particles. Afterwards, the 0.4
development of internal surfaces gives rise to a large set
0.2
of possible decaying channels (opening of phase space)
0
that is responsible for the σ enhancement. 1.2
K (c)
A asymptotic values characterize the number of dif- 1
K
ferent fragmentation configurations compatible with the 0.8
totalenergyconstraint. Largekineticenergyfluctuations K
A 0.6
reflectlargepotentialenergyfluctuations thatcanbe di-
0.4
rectlyrelatedtoeventbyeventfinalclusterdistributions.
In this sense K fluctuations can be associated with the 0.2
availablenumberofdecayingchannelsforagivenenergy, 0
-2 -1 0 1
i.e. with the microcanonical entropy.
energy (ε)
0
V. DISCUSSION FIG. 4: Top: Relative kinetic energy fluctua-
tions as a function of the total energy for densi-
ties ρ (circles), ρ (squares), and ρ (diamonds).
We summarize in Fig.4 the results of A calculations 1 2 3
K
Middle: same as top, but evaluated at the time
obtained so far. In the upper panel we show the fluc-
of σ stabilization (t(ρ ) ∼ 5t , t(ρ ) ∼ 30t ,
tuations in kinetic energy for the three confined states K 1 0 2 0
t(ρ ) ∼ 40t ). Bottom: same as top, but evalu-
analyzed at densities ρ (circles), ρ (squares), and ρ 3 0
1 2 3
ated for asymptotic configurations. See text for details.
(diamonds). The solid lines are not a fit, but an A es-
K
timation using information from the respective system’s
caloric curve. The nice agreement between the lines and
the symbols demonstrates the quality of our numerical of the kinetic energy fluctuations are always below the
sampling. In the middle and lower panels we report the canonical value.
same quantity calculated at the time of σK stabilization Ifwenowturnourattentiontothe resultsofthe same
(aswasmentionedabove,thistimeiswell-correlatedwith calculation but now performed on the states resulting
the time of fragment formation) and for the correspond- from removing the confining walls and allowing the sys-
ing asymptotic configurations respectively. In the upper temtoexpandandfragment,the resultsdisplayedinthe
panel the horizontal dotted-dashed line depicts the ex- middle and lower panel of Fig.4 are obtained. Almost
pected canonical value for AK. We included it also in independent ofthe initialcondition, the normalizedfluc-
the middle and bottom panels, only as a referencevalue. tuations fall on a curve which is mainly determined by
In the fluctuation analysis performed on experimental the total energy and fairly close to the thermal response
heavy ion data, the average kinetic energy is approxi- of an initially diluted system in microcanonical equilib-
mately corrected for evaporation and for the Coulomb rium. The kinetic energy fluctuations that were below
boost, while fluctuations are assumed to be correctly the threshold for all energies have been clearly enhanced
given by the asymptotic measurement [9]. This means in the energy region defined by 0 ≤ E ≤ 0.5ǫ . This
0
that the middle panel is the most relevant for a compar- feature, as was already mentioned, reflects the existence
ison with experimental data. ofquitedifferentavailabledecayingchannelsthatareex-
The confined state with density ρ , that belongs to plored by the systems dynamics. In particular, at these
1
region A in the phase diagram of Fig.1 (full circles) dis- energies, partitions with big residues on one hand, and
plays a maximum that is well above the canonicalvalue. high multiplicity partitions with no such big structures
As such there is a regionfor which the thermal response on the other, can be produced giving rise to power-law
function is negative. On the other hand for the confined like distributions. If we look at the asymptotic stage
states ρ and ρ , that belong to region B and C respec- (lower panel), the picture is somewhat blurred by sec-
2 3
tively of the phase diagram, the corresponding values ondary evaporation that continuously decreases the av-
6
erage kinetic energy without affecting th fluctuation. It ters.
is worth noticing that at time of fragmentformation the In addition, the analysis of the dynamical evolution
average system volume is approximately independent of of A turned out to be fruitful. On one hand it al-
K
the initial density: this means that for all initial condi- lowedustogiveaphysicallymeaningfulestimationofthe
tions the information is frozen when the system reaches freeze-out time, which turned out to be in good agree-
the region of the phase diagram (region A of figure 1) ment with the corresponding fragment formation time.
whereabnormalfluctuationsareexpectedinequilibrium, Ontheother,itwasshownthatalinkcanbeestablished
i.e. in the case of a full opening of the accessible phase between asymptotic values of that observable and mass
space. This observation may suggest that the collective distribution functions. In isolated systems, like the one
motion leads the system to a low density freeze out con- studiedhere,a maximumofA as afunction ofthe sys-
K
figuration which is not far away from a thermodynamic temenergycanberelatedtoanoticeablyenlargementof
equilibrium. However,to makeanyconclusionaboutthe thenumberofpossibledecayingchannels. Thisenhance-
possibledegreeofequilibrationandthe characteristicsof ment of the relative kinetic energy fluctuation can thus
therelevantstatisticalensemble,amoredetailedanalysis be associated with an opening of phase space that is ex-
is needed. pected in phase transitionstaking place in finite isolated
systems.
Itisworthnotingthatevenifthenon-constrainedfrag-
VI. CONCLUSION
ment formation stage seems to provide a quite effective
memory loosing mechanism (see Fig. 2), preliminary re-
In conclusion, in this work we have shown that for sultssupporttheideathatinformationontheinitialsys-
freetoexpandfragmentingsystems,initiallyequilibrated temdensitycanstillberetrievedfromcertainasymptotic
withinsharpboundaryconditions,anintermediatestage clusterconfigurationalfeatures. Thisanalysis,alongwith
exists where new microscopic correlations arise. This a thorough characterization of the system equilibration
happens in the presence of a collective expansive motion properties at τfo is currently under progress.
that induce internal surfaces to appear in the system, This work was done with partial financial supports
preceding the formation of well defined asymptotic clus- from UBA and CONICET.
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