Table Of ContentStatistical
Mechanics and
Thermodynamics
of Liquids and
Crystals
Edited by
Santi Prestipino
Printed Edition of the Special Issue Published in Entropy
www.mdpi.com/journal/entropy
Statistical Mechanics and
Thermodynamics of Liquids
and Crystals
Statistical Mechanics and
Thermodynamics of Liquids
and Crystals
Editor
SantiPrestipino
MDPI•Basel•Beijing•Wuhan•Barcelona•Belgrade•Manchester•Tokyo•Cluj•Tianjin
Editor
SantiPrestipino
UniversityofMessina
Italy
EditorialOffice
MDPI
St.Alban-Anlage66
4052Basel,Switzerland
ThisisareprintofarticlesfromtheSpecialIssuepublishedonlineintheopenaccessjournalEntropy
(ISSN1099-4300)(availableat: https://www.mdpi.com/journal/entropy/specialissues/statistical
mechanicsliquidscrystals).
Forcitationpurposes,citeeacharticleindependentlyasindicatedonthearticlepageonlineandas
indicatedbelow:
LastName,A.A.;LastName,B.B.;LastName,C.C.ArticleTitle. JournalNameYear,VolumeNumber,
PageRange.
ISBN978-3-0365-1551-9(Hbk)
ISBN978-3-0365-1552-6(PDF)
©2021bytheauthors. ArticlesinthisbookareOpenAccessanddistributedundertheCreative
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Contents
AbouttheEditor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
SantiPrestipino
StatisticalMechanicsandThermodynamicsofLiquidsandCrystals
Reprintedfrom:Entropy2021,23,715,doi:10.3390/e23060715 . . . . . . . . . . . . . . . . . . . . 1
MarianoLo´pezdeHaro,Andre´sSantosandSantosB.Yuste
EquationofStateofFour-andFive-DimensionalHard-HypersphereMixtures
Reprintedfrom:Entropy2020,22,469,doi:10.3390/e22040469 . . . . . . . . . . . . . . . . . . . . 5
YikunWei,PingpingShen,ZhengdaoWang,HongLiangandYuehongQian
TimeEvolutionFeaturesofEntropyGenerationRateinTurbulentRayleigh-Be´nardConvection
withMixedInsulatingandConductingBoundaryConditions
Reprintedfrom:Entropy2020,22,672,doi:10.3390/e22060672 . . . . . . . . . . . . . . . . . . . . 23
SantiPrestipinoandPaoloV.Giaquinta
EntropyMultiparticleCorrelationExpansionforaCrystal
Reprintedfrom:Entropy2020,22,1024,doi:10.3390/e22091024 . . . . . . . . . . . . . . . . . . . . 39
VeraGrishina,VyacheslavVikhrenkoandAlinaCiach
Structural and Thermodynamic Peculiarities of Core-Shell Particles at Fluid Interfaces from
TriangularLatticeModels
Reprintedfrom:Entropy2020,22,1215,doi:10.3390/e22111215 . . . . . . . . . . . . . . . . . . . . 61
SantiPrestipino
UltracoldBosonsonaRegularSphericalMesh
Reprintedfrom:Entropy2020,22,1289,doi:10.3390/e22111289 . . . . . . . . . . . . . . . . . . . . 79
LuisM.Sese´
Real Space Triplets in Quantum Condensed Matter: Numerical Experiments Using Path
Integrals,Closures,andHardSpheres
Reprintedfrom:Entropy2020,22,1338,doi:10.3390/e22121338 . . . . . . . . . . . . . . . . . . . . 99
SergiiD.Kaim
TheMolecularTheoryofLiquidNanodropletsEnergeticsinAerosols
Reprintedfrom:Entropy2021,23,13,doi:10.3390/e23010013 . . . . . . . . . . . . . . . . . . . . . 127
NkosinathiDlamini,SantiPrestipino,GiuseppePellicane
Self-AssembledStructuresofColloidalDimersandDisksonaSphericalSurface
Reprintedfrom:Entropy2021,23,585,doi:10.3390/e23050585 . . . . . . . . . . . . . . . . . . . . 153
v
About the Editor
SantiPrestipino(Professor), bornin1966, isatheoreticalandcomputationalphysicistatthe
Universita`degliStudidiMessina(Italy).Hislong-termresearchtopicsincludethethermodynamics
and statistical mechanics of liquids, crystals, and their interfaces. In the last few years, he has
beeninvolvedinproblemsrelatedwithcrystalnucleation, colloidalself-assembly, andthephases
ofultracoldbosonicgases.HeisontheEditorialBoardofEntropy(MDPI)andFoundations(MDPI).
vii
entropy
Editorial
Statistical Mechanics and Thermodynamics of Liquids
and Crystals
SantiPrestipino
DipartimentodiScienzeMatematicheedInformatiche,ScienzeFisicheeScienzedellaTerra,
UniversitàdegliStudidiMessina,VialeF.Stagnod’Alcontres31,98166Messina,Italy;[email protected]
Thermodynamicphasesarethemostprominentmanifestationofemergentbehavior.
Amongthem,crystalsandliquidstraditionallyepitomizetheantagonisticconceptsoforder
anddisorder(i.e.,thepresenceorabsenceofasymmetry).Accordingtocommonwisdom,
thecompetitionforstabilitybetweensolidandliquidreflectsthestrugglebetweenthe
oppositetendenciesofenergyminimizationandentropymaximization,whichisregulated
bytemperature.Itgoeswithoutsayingthatthepreviousstatementisinfacttoosimplistic,
sinceinequilibriumtheguidingprincipleisrathertheminimizationofthethermodynamic
potential(maximizationoftheMassieufunction)appropriatetothegivencontrolparame-
ters.Forinstance,crystallizationsometimesoccurswiththepurposetomaximizeentropy
(asforhardspheresunderpressure).Afurtherexampleisprovidedbysuperfluid,which
canbeoneofthephasesofminimumenergy/enthalpyatzerotemperature.
Therelevanceofthesolid/liquiddichotomyforstatisticalphysicscannotsimplybe
overstated.Itwasunderthepressureofaccuratelylocatingthesolid–liquidtransitionin
(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9) simple-fluidmodelsthat“exact”free-energymethodswereinitiallydeveloped[1,2].Since
(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)
then,variationsonthesemethodshavebeenemployedtocomputethephasediagramof
Citation: Prestipino,S.Statistical manycomplexfluids,suchasliquidcrystals[3],clustercrystals[4],andfluidsofpatchy
MechanicsandThermodynamicsof particles[5].Generallyspeaking,thefullcontrolofphasebehaviorcanhelpinthesynthesis
LiquidsandCrystals.Entropy2021, ofartificialmaterialswiththedesiredspecifications.
23,715. https://doi.org/10.3390/ Turning to theory, while the exact determination of the partition function for a
e23060715 non-trivialsystemwithsolid,liquid,andvaporphaseswillprobablyneverbeaccom-
plished,thereareneverthelessvariationaltreatmentsofthesolid–liquidtransition(density
AcademicEditor:AntonioM. functionaltheories)thathavebynowreachedahighdegreeofsophistication(see,e.g.,
Scarfone Refs.[6,7]).However,aschematicphasediagramwiththestandardthreephasescanbe
obtainedwithlesseffort,seeforinstancethemean-fieldanalysisofthePottslatticegas[8].
Received:2June2021 Itisusuallythoughtthat(classical)liquidisauniquephase, whilecrystalsarea
Accepted:3June2021 multitude.Infact,thisisonlypartiallytrue,sinceliquidscomposedofhydrogen-bonded
Published:4June2021 moleculesexhibitanumberofso-calledwater-like“anomalies”thatmakethemdifferent
in many respects from conventional (rare-gas) fluids (see, e.g., [9]). The relationship
Publisher’sNote:MDPIstaysneutral betweensuchanomaliesandsolidpolymorphism/polyamorphismisanimportanttopic
withregardtojurisdictionalclaimsin
ofstatisticalphysics.Phasediagramswithmanysolidpolymorphsaretheruleforsimple
publishedmapsandinstitutionalaffil-
substances(e.g., Na[10])underhugepressure. Here, exactfree-energymethodsface
iations.
a serious limitation, since the crystalline structures—Bravais and non-Bravais—being
potentiallyrelevantarecountless. Metadynamics[11]andevolutionaryalgorithms[12]
areapossiblewayout,sincetheydonotinvolveanyassumptiononthetopologyofthe
energylandscape.
Copyright: © 2021 by the author. Condensed-matterphysicsalsofeaturesawidevarietyofphaseswithmixedsolid
Licensee MDPI, Basel, Switzerland. andliquidcharacteristics.Hexaticfluids[13],liquid-crystalsmectics[14],crystallinemem-
Thisarticleisanopenaccessarticle
branes[15],andquantumsupersolids[16]arejustafewexamplesofhybridstatesof
distributed under the terms and
aggregation,withothersyettobediscovered.Inthelastfewyears,anewcategoryofsys-
conditionsoftheCreativeCommons
tems,i.e.,self-assemblingmaterials,hasfallenunderthescrutinyofthestatistical-physics
Attribution(CCBY)license(https://
community(see,e.g.,[17]).Here,somekindoforderatthemesoscopicscalespontaneously
creativecommons.org/licenses/by/
emergesfromtheinteractionbetweensimplemicroscopicunits. Forthesesystems,the
4.0/).
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Entropy2021,23,715.https://doi.org/10.3390/e23060715 https://www.mdpi.com/journal/entropy
