Another resolution of the configurational entropy paradox as applied to hard spheres
AAIP/123-QED
Another resolution of the configurational entropy paradox as applied to hardspheres
Vasili Baranau a) and Ulrich Tallarek Department of Chemistry, Philipps-Universit¨at Marburg, Hans-Meerwein-Strasse 4, 35032 Marburg,Germany (Dated: 12 September 2018)
Recently, Ozawa and Berthier [M. Ozawa and L. Berthier,
J. Chem. Phys. , 2017, , 014502] studied theconfigurational and vibrational entropies S conf and S vib from the relation S tot = S conf + S vib for polydispersemixtures of spheres. They noticed that because the total entropy per particle S tot /N shall contain themixing entropy per particle k B s mix and S vib /N shall not, the configurational entropy per particle S conf /N shall diverge in the thermodynamic limit for continuous polydispersity due to the diverging s mix . They alsoprovided a resolution for this paradox and related problems—it relies on a careful redefining of S conf and S vib .Here, we note that the relation S tot = S conf + S vib is essentially a geometric relation in the phase space andshall hold without redefining S conf and S vib . We also note that S tot /N diverges with N → ∞ with continuouspolydispersity as well. The usual way to avoid this and other difficulties with S tot /N is to work with the excessentropy ∆ S tot (relative to the ideal gas of the same polydispersity). Speedy applied this approach to therelation above in [R. J. Speedy, Mol. Phys. , 1998, , 169] and wrote this relation as ∆ S tot = S conf + ∆ S vib .This form has flaws as well, because S vib /N does not contain the k B s mix term and the latter is introduced into∆ S vib /N instead. Here, we suggest that this relation shall actually be written as ∆ S tot = ∆ c S conf + ∆ v S vib ,where ∆ = ∆ c + ∆ v while ∆ c S conf = S conf − k B N s mix and ∆ v S vib = S vib − k B N (cid:104) (cid:0) V Λ d N (cid:1) + UNk B T (cid:105) with N , V , T , U , d , and Λ standing for the number of particles, volume, temperature, internal energy,dimensionality, and de Broglie wavelength, respectively. In this form, all the terms per particle are alwaysfinite for N → ∞ and continuous when introducing a small polydispersity to a monodisperse system. We alsosuggest that the Adam–Gibbs and related relations shall in fact contain ∆ c S conf /N instead of S conf /N . I. INTRODUCTIONA. The paradox
When studying glasses and glass-like systems, like col-loids, it is typical to separate the total entropy of a sys-tem into the configurational and vibrational parts: S tot = S conf + S vib . Here, S conf enumerates the statesaround which the system vibrates and S vib correspondsto the average volume in the phase space for vibrationsaround a single such state—i.e., vibrations in a basin ofattraction of a configuration.The total entropy in the canonical ensemble S tot isexpressed in the standard way through the Helmholtzfree energy A tot , internal energy U and temperature T as A tot = U − T S tot , while A tot is expressed throughthe total partition function Z tot as A tot = − k B T ln Z tot .Thus, S tot = U/T + k B ln Z tot . In turn, Z tot is ex-pressed through the configurational integral as Z tot = Mt =1 N t ! 1Λ dN (cid:82) V N e − U N ( (cid:126)r ) /k B T d (cid:126)r , where Λ denotes the deBroglie thermal wavelength Λ = h/ √ πmk B T (giventhat all particles have the same mass m ). The term t N t ! accounts for “indistinguishability” of constituentparticles: there are in total N particles with M particlespecies. It does not necessarily stem from quantum in-distinguishability, but rather from our choice which par- a) Electronic mail: [email protected] ticles we consider interchangeable to still be able to saythat switching a pair of particles leaves the configurationunchanged.
For example, in colloids of sphere-likeparticles it is typical to consider particles with the sameradius to be of the same type, though surface featuresapparently can allow to distinguish any pair of parti-cles. This “colloidal” indistinguishability term is neededto prevent the Gibbs paradox and define entropy in areasonable way (so that the entropy is extensive).
Forhard spheres, U N ( (cid:126)r ) = 0 if there are no intersections be-tween particles and U N ( (cid:126)r ) = ∞ otherwise.If we consider entropy per particle S tot /N k B (in unitsof k B ), it contains the term N ln (cid:16) t N t ! (cid:17) . With the helpof the Stirling approximation ln( N ) = N ln( N ) − N , oneobtains for the thermodynamic limit N → ∞ N ln (cid:18) t N t ! (cid:19) = 1 + s mix − ln( N ) , (1)where s mix = − (cid:80) Mt N t N ln N t N is the mixing entropy perparticle (in units of k B ) or the information entropy of theparticle type distribution. This quantity diverges in thethermodynamic limit in the case of a continuous parti-cle type distribution — for example, if spherical col-loidal particles have a continuous radii distribution f ( r ).Indeed, if we discretize the distribution with the step δ ,then N t = N f ( r t ) δ in the limit δ → s mix = − (cid:90) f ( r ) ln( f ( r ))d r − ln( δ ) , δ → . (2) a r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec The mixing entropy per particle diverges due to the di-verging ln( δ ) term. In information theory, it is typical towork with the differential entropy when dealing with con-tinuous probability distributions, where the differen-tial entropy is the right hand side of Eq. (2) without the − ln( δ ) term, s dif = − (cid:82) f ( r ) ln( f ( r ))d r . Given that for auniform distribution in the interval [ a, b ] s dif = ln( b − a ), s dif of an arbitrary function is its information entropy( s mix ) with respect to the uniform distribution in a unitinterval [0 , N ) term in Eq. (1) does not posea problem in the thermodynamic limit, because it is infact incorporated into the ln( V / Λ d N ) term in S tot /N k B ( cf . Eq. (6) below). Of course, to ensure applicabil-ity of the Stirling approximation, limits N → ∞ and δ → N δ min f ( r ) (cid:29)
1. If f has an infinite support, e.g. [0 , + ∞ ), we have to sample radii from an interval[0 , R ( N )) (where R → ∞ with N → ∞ ), still imposing N δ min [0 ,R ) f ( r ) (cid:29)
1. To ensure that the Stirling approxi-mation becomes precise in the thermodynamic limit, wecan choose a certain scaling for a = N δ min f ( r ) as well.For example, for f ( r ) with a finite support (and a fixed f min = min f ( r )), we can select a ( N ) = √ N or in gen-eral a ( N ) = N γ , γ <
1, so that δ = N γ − f − → N → ∞ .Now, if we restrict the phase space only to a cer-tain basin of attraction V basin , we can write similar to S tot S vib = U/T + k B ln Z vib . Z vib is often written as Z vib = t N t ! 1Λ dN (cid:82) basin e − U N ( (cid:126)r ) /k B T d (cid:126)r . But for everybasin of attraction there are exactly Π t N t ! equivalentbasins due to particle permutations. Because the Π t N t !terms are compensated, Z vib shall actually be expressedas Z vib = dN (cid:82) basin e − U N ( (cid:126)r ) /k B T d (cid:126)r . This fact was re-alized for monodisperse systems long ago.
If we now consider the equation S tot /k B N = S conf /k B N + S vib /k B N , S tot /k B N contains the s mix termwhile S vib /k B N does not. Ozawa and Berthier pointedout that there are several problems with this. Firstly, itmeans that S conf /k B N shall diverge in the thermody-namic limit for a continuous particle type distributionwith diverging s mix . Secondly, if we take a colloid withspherical particles of equal size and introduce a slightpolydispersity, s mix will exhibit a jump (from zero to anon-zero value, e.g. ln 2 in the case of a 50 : 50 binarymixture, however similar particle radii are). S conf /k B N will exhibit the same jump. These are the two basicproblems that constitute the paradox. B. Resolution of Ozawa and Berthier
Ozawa and Berthier suggested to carefully redefineentropies: roughly, to “merge” (besides the Π t N t ! merg-ing) those basins that have high overlap, i.e. , that looksufficiently similar to each other due to similar parti-cle types and constituent configurations. This proceduredecreases the effective number of configurations around which the system is considered to vibrate and compen-sates the jump in S conf /k B N when particles are madeonly slightly different from each other. In other words, S conf /k B N will behave continuously when particles aremade only slightly different from each other. This pro-cedure also essentially decreases the number of particlespecies and keeps s mix finite. Ref. assumes that origi-nal basins (before redefinitions) are defined in some sortof a free energy landscape ( e.g. , emerging from thedensity-functional theory, where a state is a particularspatial density profile). C. Motivations for another resolution
The resolution of Ozawa and Berthier is perfectly valid,but it relies on “merging” the basins with high over-lap and is thus not applicable if for some reason we donot want to do any redefinition or “merging” of basins(except for the Π t N t ! merging), even if they have highoverlaps. One popular definition of basins uses steepestdescents in the potential energy landscape (PEL). S conf is then defined through the number of lo-cal PEL minima or inherent structures (that stillhave to be merged due to the “colloidal” indistinguisha-bility of particles). The vibrational entropy is then de-fined through basins of attraction in the PEL. For hardspheres, one has to use a pseudo-PEL, where inher-ent structures correspond to jammed configura-tions. These definitions are mathematically precise andallow splitting the phase space into basins even for theideal gas. For simple systems, decomposition of thephase space into basins can be done numerically by doingsteepest descents from many starting configurations. The resolution of Ozawa and Berthier is not applica-ble to these definitions ( i.e. , if we require keeping thebasins from these definitions unchanged), but the rela-tion S tot = S conf + S vib shall be valid, because it is essen-tially a geometrical relation that tells us how the phaseor configuration space is split into volumes around somepoints. It shall be valid for an arbitrary decomposition ofthe configuration space into basins, the only requirementbeing the saddle point approximation. D. Other previous resolutions
We mentioned that S tot /N k B contains s mix . It meansthat S tot /N k B alone has all the problems that Ozawaand Berthier were solving: (i) discontinuity with intro-duction of a small polydispersity into a monodisperse sys-tem and (ii) divergence with a continuous particle typedistribution. This is not a severe problem, because inexperiments only entropy differences or entropy deriva-tives matter ( S conf is such a difference). But wheneveran equation containing S tot /N k B is valid in the polydis-perse case, there shall be other terms that cancel s mix exactly. Thus, it makes sense to write such equationsthrough non-diverging terms, when all equivalently di-vergent terms are omitted. Hence, a lot of well-knownpapers on hard-sphere fluids and glasses, including theclassical ones by Carnahan and Starling, work solely withthe excess entropy ∆ S tot (with respect to the ideal gas ofthe corresponding particle size distribution), which doesnot contain the unpleasant term s mix . For ex-ample, the equation that connects S tot , the chemical po-tential µ and reduced pressure Z = pV /N k B T in equi-librium polydisperse hard-sphere systems is written as∆ S tot /N k B = Z − − (cid:104) ∆ µ (cid:105) /k B T . Speedy used the same approach of working with excessquantities when studying glassy systems of hard spheresand the relation S tot = S conf + S vib as early as in 1998. He wrote this relation in the form ∆ S tot = S conf + ∆ S vib .As we explained above, this form seems to be not thecomplete resolution of the problems with these quantitiesas well, because S vib /N k B does not actually contain the s mix term, so it is rather introduced into the ∆ S vib /N k B instead of being removed. E. Overview of our resolution
We believe that the general strategy of working withexcess quantities is the one to follow, but the approach ofSpeedy shall be slightly revised. We show that insteadof writing ∆ S tot = S conf +∆ S vib one has to write ∆ S tot =∆ c S conf +∆ v S vib , where ∆ c is an operator that subtracts k B N s mix and ∆ v is an operator that subtracts k B N [1 +ln (cid:0) V Λ d N (cid:1) + UNk B T ]. Together, they are equivalent to ∆(in the operator sense, if all the operators are applied toa common variable, ∆ = ∆ c + ∆ v ).We start our discussion with an example of 1 D hard“spheres” (rods) in a non-periodic system and then intro-duce the general case. We always assume that basins aredefined through steepest descents in the (pseudo-)PELand focus the discussion on hard spheres for simplicity.Essentially, the main idea of the paper is how exactlywe have to distribute the terms from the ideal gas entropy(subtracted from S tot to get ∆ S tot ) between S conf and S vib . They are distributed in an uneven way, similar tothe relation ∆ S tot /N k B = ∆ Z − (cid:104) ∆ µ (cid:105) /k B T = Z − −(cid:104) ∆ µ (cid:105) /k B T . F. The Adam–Gibbs and related relations
As pointed out by one of the reviewers, this workwould be incomplete without discussing which form ofthe configurational entropy shall be present in the Adam–Gibbs relation (or in general any relation thatconnects the relaxation time of a system and S conf , e.g. , the one from the Random First Order Transitiontheory ). The Adam–Gibbs relation expresses therelaxation time of the system τ R through S conf /N as τ R = τ exp (cid:16) AT S conf /N (cid:17) , where τ and A are constants. FIG. 1. Configuration space of N = 3 one-dimensional par-ticles with non-periodic boundary conditions. Numbers like“231” denote the order of particles in a corresponding jammedconfiguration. Reproduced with modifications from Ref. As explained above, S conf /N diverges for systems withcontinuous polydispersity for N → ∞ , which means that τ R ≡ τ R shall be continuous with respect to adding a slightpolydispersity to a monodisperse system, because relax-ation dynamics will remain almost unchanged. S conf /N has a jump in such a case, and τ R from the Adam–Gibbsrelation will have it as well. These two unphysical proper-ties of the Adam–Gibbs relation indicate that it shall beamended. We show below that our ∆ c S conf /N is alwaysfinite and continuous and is thus a natural candidate forthe Adam–Gibbs and similar relations for the relaxationtime. We discuss this aspect in more detail in SectionIV. II. EXACTLY SOLVABLE EXAMPLE: 1D RODS IN ANON-PERIODIC INTERVAL
Let us look at first at a very simple 1D system: severalrods (1D hard spheres) in a non-periodic interval. For 3rods, one can visualize the entire configuration space.The phase space of 1D hard rods is never ergodic (but asingle basin is), but we don’t currently require ergodicity,because we only try to split the total configuration spaceinto basins of attraction and look how different quantitiesscale with the number of particles.The complete configuration space is presented in Fig.1. It is a variant of Fig. 2 in Ref. , but these authorsassumed periodic boundary conditions.If all 3 particles can be distinguished from each other,there are 6 jammed configurations: 123 132 213 231312 321 or N ! in the general case. If the particles aremonodisperse (of diameter A), it is essentially one con-figuration of indistinguishable particles: AAA. Let us as-sume now that the particles are bidisperse, N !Π t N t ! in the general case.The configurational entropy has a jump after switchingto the bidisperse system, but this is natural, while thereare more distinct jammed configurations now. The totalentropy has an equivalent jump.Let us examine the vibrational, total, and configu-rational entropies of such a system. We assume forsimplicity zero solid volume fraction of a system, ϕ = V spheres /V box = 0. The total volume of the configurationspace is in this case simply I tot = V N . The total parti-tion function is Z tot = t N t ! 1Λ dN V N , where we write Λ d with d for dimensionality for the general case.If all particles are treated as distinguishable, the num-ber of jammed configurations is N dist J = N !(= 6) and theaverage volume of a basin of attraction if all particles aretreated as distinguishable is I distvib = V N N ! (each green sim-plex in Fig. 1). We can trivially write I tot = N dist J I distvib or V N = N ! V N N ! .If we treat particles as indistinguishable, we have tomerge some jammed configurations to treat as a singleone (divide N dist J by Π t N t !) and have to merge somebasins of attraction to treat as a single one (multiply I distvib by Π t N t !). The number of jammed configurations ifparticles are treated as indistinguishable is thus N ind J = N !Π t N t ! (= 3). The average volume of a basin of attrac-tion if particles are treated as indistinguishable is thus I indvib = Π t N t ! V N N ! (= V N ). In Fig. 1, we have to merge thegreen tetrahedra in pairs. We can write similar to the dis-tinguishable case I tot = N ind J I indvib . The vibrational parti-tion function is then Z indvib = t N t ! 1Λ dN I indvib = dN V N N ! . Asmentioned in the introduction, the multiplication by thenumber of permutations and the division by this numberdue to basin multiplicity always cancel out.The total entropy per particle is expressed for our sys-tem as S tot /k B N = U/N k B T + N ln Z tot = U/N k B T +1 + s mix + ln (cid:0) V Λ d N (cid:1) . The vibrational entropy perparticle (of indistinguishable particles) is expressed as S vib /k B N ≡ S indvib /k B N = U/N k B T + N ln Z indvib = U/N k B T +1+ln (cid:0) V Λ d N (cid:1) . The configurational entropy perparticle (of indistinguishable particles) is by definition S conf /k B N ≡ S indconf /k B N = N ln N ind J = N ln N !Π t N t ! = s mix .Naturally, these results conform to the equation S tot /k B N = S conf /k B N + S vib /k B N , which is just an-other expression for the relation I tot = N ind J I indvib . Thefollowing result is surprising, though: the s mix term from S tot /k B N is consumed on the right side of the equationby S conf /k B N and the terms 1 + ln (cid:0) V Λ d N (cid:1) + UNk B T from S tot /k B N are consumed on the right side of the equa-tion by S vib /k B N . S vib /k B N does not contain the mix-ing contribution, because t N t ! stemming from indistin- guishability is exactly compensated by Π t N t ! stemmingfrom basin multiplicity. Thus, one can write∆ S tot = ∆ c S conf + ∆ v S vib , (3)where ∆ c S conf k B N = S conf k B N − s mix and∆ v S vib k B N = S vib k B N − − ln (cid:18) V Λ d N (cid:19) − UN k B T . (4)All the terms in Eq. (3) if taken per particle are finitein the thermodynamic limit even for continuous particlesize distributions and continuous with introduction of asmall polydispersity to a monodisperse system.
III. GENERAL THEORY, ARBITRARY d At first, we routinely derive the relation for the totalentropy and demonstrate that S tot /k B N contains s mix +ln (cid:0) V Λ d N (cid:1) in the general case.Then, we show that the relation S tot = S conf + S vib istruly a geometrical one and requires only a saddle pointapproximation. This approximation is actually exact inthe thermodynamic limit.Our next and main aim is then to show that thevibrational entropy per particle shall contain the termln (cid:0) V Λ d N (cid:1) , but not s mix in the general case as well, if “col-loidal” indistinguishability of particles is treated care-fully. It will mean that s mix is contained in S conf /k B N .To investigate the volume of basins of attractionat arbitrary ϕ , we use a variant of thermodynamicintegration. We use the same superscripts as before: “dist” as ifall particles are distinguishable and “ind” as if particlesof the same type (radius) are indistinguishable, implyingthat S conf ≡ S indconf and S vib ≡ S indvib . A. Total entropy
For the ideal gas, the integral (cid:82) V N e − U N ( (cid:126)r ) /k B T d (cid:126)r = V N . Thus, the entropy of the ideal gas S ◦ tot is expressedwith the help of S ◦ tot = U/T + k B ln Z ◦ tot as S ◦ tot N k B = UN k B T + 1 + s mix + ln (cid:18) V Λ d N (cid:19) , (5)where U/N k B T = d/ / d = 3). We assumefor the ideal gas the same relative particle radii distribu-tion f ( r/ (cid:104) r (cid:105) ), but with (cid:104) r (cid:105) →
0. The total entropy perparticle can then be expressed as S tot N k B = S ◦ tot N k B + ∆ S tot N k B == UN k B T + 1 + s mix + ln (cid:18) V Λ d N (cid:19) + ∆ S tot N k B , (6)where ∆ S tot /N k B = ( S tot − S ◦ tot ) /N k B = Nk B ln (cid:0) V N (cid:82) V N e − U N ( (cid:126)r ) /k B T d (cid:126)r (cid:1) is the excess en-tropy per particle in units of k B . It can be shownthrough the definition of pressure p = − ( ∂A tot /∂V ) N,T that for hard spheres∆ S tot N k B = − ϕ (cid:90) Z ( ϕ (cid:48) ) − ϕ (cid:48) d ϕ (cid:48) , (7)where ϕ = V spheres /V box is the solid volume fraction(packing density) and Z ( ϕ ) = pV /N k B T is the reducedpressure ( cf . Appendix 2 a). Eq. (7) can be re-garded as a special case of thermodynamic integration.The quantity ∆ S tot /N k B does not share problems for S tot /N k B mentioned above (divergence and discontinu-ity). Additionally, it does not contain the term ln (cid:0) V Λ d N (cid:1) .Thus, ∆ S tot /N k B depends on ϕ only and does not de-pend on the temperature T (which is indirectly presentin Λ). B. Saddle point approximation: separation of the totalentropy
Even in the monodisperse case, the total volume ofthe configuration space I tot and the available volume ofa basin of attraction I indvib depend on ϕ . The quantity I indvib is expressed (in the general—polydisperse—case) as I indvib = Π t N t ! I distvib , where I distvib = (cid:82) basin e − U N ( (cid:126)r ) /k B T d (cid:126)r .With d >
1, there are jammed configurations at dif-ferent densities.
Hence, there is a “density of jam-ming densities” N ind J ( N, ϕ J ). We assume that propertiesof I indvib depend only on N , ϕ , and ϕ J (not on a particularbasin of attraction), so we write I indvib ( N, ϕ, ϕ J ). Thus,we express the volume of the configuration space as I tot ( N, ϕ ) = (cid:90) ϕ N ind J ( N, ϕ J ) I indvib ( N, ϕ, ϕ J )d ϕ J . (8)For any fixed N , N ind J ( N, ϕ J ) and I indvib ( N, ϕ, ϕ J ) shalldepend on ϕ J as follows: N ind J shall decrease rapidlywith the increase of ϕ J (and more rapidly with larger N ), as indicated by numerous results on the configu-rational entropy and relaxation times, while I indvib ( N, ϕ, ϕ J ) increases rapidly with increasing ϕ J (if ϕ is fixed as well). The last statement is just an-other formulation of the fact that basins of attractiondecrease in volume when ϕ approaches ϕ J for a fixed N . Thus, the integrand in Eq. (8) has a sharp maxi-mum and we can replace the integral with I tot ( N, ϕ ) = N ind J ( N, ϕ DJ ) I indvib ( N, ϕ, ϕ DJ ) w ( N, ϕ ), where ϕ DJ is the“dominant” jamming density, given by the maximum ofthe integrand, and w ( N, ϕ ) represents the “width” of thepeak in the integrand.
It is usually believed thatit is subexponential, so when we switch to entropies perparticle (take the logarithm and divide by N ), the term with w ( N, ϕ ) disappears. Thus, we can just write I tot ( N, ϕ ) = N ind J ( N, ϕ DJ ) I indvib ( N, ϕ, ϕ DJ ) . (9)It means that we essentially have to discuss the sameform of the separation into configurational and vibra-tional parts as in the 1D case: S tot ( N, T, ϕ ) = S conf ( N, ϕ DJ ) + S vib ( N, T, ϕ, ϕ DJ ) . (10)We make some remarks on the function ϕ DJ ( ϕ ) in Ap-pendix 2 c. C. Vibrational entropy through thermodynamicintegration
Our aim here is to find whether S vib /k B N containsthe s mix and ln (cid:0) V Λ d N (cid:1) terms. Though we state as earlyas in the introduction that s mix shall not be present in S vib /k B N due to compensating pre-integral terms, herewe show explicitly that s mix is not hidden in S vib /k b N through the integral (cid:82) basin e − U N ( (cid:126)r ) /k B T d (cid:126)r either. Wechoose a certain variant of thermodynamic integrationfor our purpose, but the possibility to practically imple-ment it for real systems is of no concern to us. It is avariant of a tether method of Speedy or of a cell methodof Donev et al. We want to find the volume of the available part ofa certain basin of attraction (with the jamming density ϕ J ) if the current volume fraction of the system is ϕ .At first, we do not multiply this volume by Π t N t !; thus,we want to find the vibrational partition function of asingle basin Z singlevib = t N t ! 1Λ dN (cid:82) basin e − U N ( (cid:126)r ) /k B T d (cid:126)r = t N t ! 1Λ dN I distvib (while Z vib = Π t N t ! Z singlevib ).We imagine that this basin of attraction in the PELcan somehow be ideally determined ( e.g. , by perform-ing steepest descents in the PEL for all possible startingpoints ). Then, we restrict the phase space to thisparticular basin. If a hard-sphere system during its dy-namics reaches the boundary of this basin, it is elasticallyreflected from the boundary.Now, we apply the tether method of Speedy. Weimagine that the center of each sphere is attached with atether to a point where this center is located in the corre-sponding jammed configuration. Alternatively, the cen-ters of particles are surrounded with imaginary sphericalcells, where the radius of a cell equals the tether lengthfor this sphere L i . When a sphere center reaches its cellwall during molecular dynamics, it is elastically reflected.For such a system, the vibrational Helmholtz free energy A singlevib = − k B T ln( Z singlevib ) is additionally parameterizedby radii of cells L i . If we imply L i = λR i , A singlevib is afunction of λ . A singlevib ( λ = ∞ ) coincides with the vibra-tional free energy without cells.Thermodynamic integration over λ implies that thechange in A singlevib ( λ ) is equal to the work that particlecenters perform on the walls of their cells during the cellexpansion, i.e. A singlevib = A singlevib ( λ min ) − N (cid:104) (cid:90) ∞ λ min p c d υ ( λ ) (cid:105) cells , (11)where p c is the pressure on the cell walls and υ ( λ ) is thevolume of a cell, υ i ( λ ) = (4 / πL i = λ V sp ,i , where V sp ,i is the volume of the i th particle.We can express the work on the wall ofthe cells through dimensionless quantities as N k B T (cid:104) (cid:82) ∞ λ min Z c d υ ( λ ) υ ( λ ) (cid:105) cells , where Z c = p c υ/k B T isthe reduced pressure on the cell walls. Reduced pressureis expressed in the general case as Z = pV /N k B T , butthe pressure on each cell wall is counted from exactlyone particle.If λ min is sufficiently small and spheres locatedin minimal cells can never intersect, Z singlevib ( λ min )can be expressed trivially as t N t ! 1Λ dN Π Ni (4 / πL i = t N t ! 1Λ dN Π Ni V sp ,i λ d min = t N t ! 1Λ dN λ dN min Π Ni V sp ,i . Thesame result can be obtained from the fact that Z c = 1for λ ∈ [0 , λ min ).Now, if we switch to entropies per particle S singlevib /N k B = U/N k B T − A singlevib /N k B T , we get (us-ing Eq. (1)) S singlevib /N k B = UNk B T + 1 + s mix − ln( N ) − ln(Λ d ) + ln( λ d min ) + (cid:104) ln( V sp ) (cid:105) + (cid:104) (cid:82) ∞ λ min Z c d υ ( λ ) υ ( λ ) (cid:105) .We would like to switch now to ln( (cid:104) V sp (cid:105) ). We do thisby simply writing (cid:104) ln( V sp ) (cid:105) = α + ln( (cid:104) V sp (cid:105) ) , (12)where α = (cid:104) ln( V sp / (cid:104) V sp (cid:105) ) (cid:105) is some dimensionless quantitythat characterizes the particle radii distribution. Con-trary to s mix , it remains finite for all but very exoticdistributions ( cf . Appendix 1). For the monodispersecase, α = 0.After switching to (cid:104) V sp (cid:105) , we can introduce the densityterm V /N given that ϕ = N (cid:104) V sp (cid:105) /V and ln( (cid:104) V sp (cid:105) ) =ln( V ϕ/N ). We finally write S singlevib N k B =1 + s mix − ln( N )+ UN k B T + ln (cid:18) V Λ d N (cid:19) + (cid:104) ln (cid:18) V sp (cid:104) V sp (cid:105) (cid:19) (cid:105) + ln( ϕλ d min ) + (cid:104) (cid:90) ∞ λ min Z c d υ ( λ ) υ ( λ ) (cid:105) . (13)As mentioned, we are really interested in S vib /N k B ≡ S indvib /N k B = S singlevib /N k B + N ln(Π t N t !), where eachbasin of attraction is counted Π t N t ! times. The terms1 + s mix − ln( N ) in Eq. (13) cancel out exactly and wewrite S vib N k B = UN k B T + ln (cid:18) V Λ d N (cid:19) + (cid:104) ln (cid:18) V sp (cid:104) V sp (cid:105) (cid:19) (cid:105) + ln( ϕλ d min ) + (cid:104) (cid:90) ∞ λ min Z c d υ ( λ ) υ ( λ ) (cid:105) . (14) Eq. (14) shows that S vib /N k B contains ln (cid:0) V Λ d N (cid:1) anddoes not contain s mix , which means that s mix shall beconsumed by S conf /N k B to make the relation S tot = S conf + S vib hold. The choice of λ min is not particu-larly important: any changes in the choice of λ min willbe incorporated by (cid:104) (cid:82) ∞ λ min Z c d υ ( λ ) υ ( λ ) (cid:105) . In Appendix 1, wedemonstrate that the free volume equation of state for hard spheres immediately follows from Eq. (14). Itis an approximate equation of state, but in the the limit ϕ → ϕ J it asymptotically equals the “polytope” equa-tion of state by Salsburg and Wood, which can bederived from first principles for the limit ϕ → ϕ J . Wealso demonstrate in Appendix 1 that α from Eq. (12)can not contain s mix , even indirectly. D. Our resolution of the paradox
Eq. (14) shows that S vib /N k B does not contain the s mix term but contains the ln (cid:0) V Λ d N (cid:1) term, which meansthat s mix from S tot /N k B (Eq. (6)) shall be consumed by S conf /N k B to make the relation S tot = S conf + S vib hold.The only remaining question is which entropy containsthe unity from the ideal gas entropy per particle, Eq.(5). This unity has to be removed from one of the terms( S vib /N k B or S conf /N k B ) if we subtract the ideal gasentropy S ◦ tot from S tot .The answer to this question is not provided by Eq.(14) directly, but we expect that this unity is containedin S vib /N k B , because that is what happens in the one-dimensional case. Also, s mix = 0 for the monodispersecase and we assume as usual that S conf ( N, ϕ J ) /N k B decreases with the increase in ϕ J and reaches zero atsome ϕ J . If unity shall be subtracted from S conf ( N, ϕ J ) /N k B , S conf ( N, ϕ J ) /N k B decreases to unityin the monodisperse case, not to zero, and this is physi-cally unrealistic.Thus, we suggest to write the relation S tot = S conf + S vib as ∆ S tot = ∆ c S conf + ∆ v S vib , (15)where∆ c S conf k B N = S conf k B N − s mix and∆ v S vib k B N = S vib k B N − − ln (cid:18) V Λ d N (cid:19) − UN k B T . (16)All the quantities from Eq. (26) if taken per particle are • finite in the thermodynamic limit even for a con-tinuous particle type distribution (polydispersity), • continuous when introducing a small polydispersityto a monodisperse system,Additionally, ∆ c S conf ( N, ϕ J ) /k B N is supposed to de-crease to zero with the increase of ϕ J . For hard spheres,all the terms from Eq. (26) if taken per particle are alsoindependent of the temperature T and depend only on ϕ and ϕ J .It may seem surprising that the terms from the idealgas entropy S ◦ tot are distributed between S conf and S vib ,but exactly the same situation occurs for the well-knownrelation ∆ S tot /N k B = ∆ Z − (cid:104) ∆ µ (cid:105) /k B T (which stemsfrom the expressions for the Gibbs free energy). Here, the unity from Eq. (6) is consumed in the excessreduced pressure ∆ Z = Z −
1, while s mix + ln (cid:0) V Λ d N (cid:1) are consumed in the average excess chemical potential (cid:104) ∆ µ (cid:105) /k B T . Indeed, the ideal gas chemical potential for asingle particle type µ ◦ i is expressed as − µ ◦ i k B T = ln (cid:16) V Λ d N i (cid:17) and −(cid:104) µ ◦ (cid:105) k B T = (cid:80) i ln (cid:16) V Λ d N i (cid:17) N i N = s mix + ln (cid:0) V Λ d N (cid:1) . The U/N k B T term from Eq. (6) is actually consumed by theomitted ∆ U/N k B T term, which is always zero for hardspheres. IV. THE ADAM–GIBBS AND RANDOM FIRST ORDERTRANSITION THEORIES
The Adam–Gibbs (AG) relation connects the re-laxation time of a (glassy) isobaric system to the config-urational entropy per particle: τ R = τ exp (cid:18) AT S conf /N (cid:19) , (17)where τ and A are constants. Eq. (17) corresponds toEq. (21) in the original paper of Adam and Gibbs. Eq.(21) in the original paper does not contain the numberof particles, but S c in the original notation is the molarconfigurational entropy. The relaxation time is often as-sociated with the asymptotic alpha-relaxation time or the inverse diffusion coefficient. As explained in Section I A, S conf /N diverges (alongwith s mix ) in the thermodynamic limit for systems withcontinuous polydispersity. This alone indicates that theform (17) shall be amended (otherwise, τ R ≡ S conf /N exhibits a jump when introducing asmall polydispersity into a monodisperse system, and thecorresponding jump would be induced into Eq. (17).This is unphysical, because the relaxation dynamics shallremain almost unchanged if a small polydispersity is in-troduced into a monodisperse system. We demonstratedthat ∆ c S conf /N is always finite and continuous when in-troducing a small polydispersity into a monodisperse sys-tem. We thus believe that any relation connecting therelaxation time or an equivalent quantity to the config-urational entropy per particle shall actually depend on∆ c S conf /N : τ R = f (∆ c S conf /N ) , (18)Now, we provide a more elaborate explanation for theclassical AG theory. The AG theory assumes that a system is composedof relatively independent “cooperatively rearranging re-gions” (CRR), i.e. , portions of the system that can un-dergo structural changes relatively independent of otherregions, neighboring or not. A structural change (or co-operative rearrangement) of a region means a transitionbetween different states (different basins of attraction inthe potential energy landscape of this region). Some ofthe regions (subsystems) may not be able to performstructural changes because they have only one state avail-able. Adam and Gibbs assume that the relaxation rateof the entire system is proportional to the fraction of re-gions that can in principle undergo a structural change.They arrive at the following equation (Eq. (11) in theiroriginal paper ): τ R ( T ) = τ exp (cid:18) z ∗ ∆ µk B T (cid:19) , (19)where τ and ∆ µ are approximately independent of T ,while z ∗ is the minimal number of constituent elements ina subsystem that can undergo a structural change (con-stituent elements are molecules, monomeric segments incase of polymers, or hard spheres in case of colloids).Then, Adam and Gibbs demonstrate that the config-urational entropy of a cooperatively rearranging region s CRR is related to the number of its constituent elementsas z S conf N = s CRR , which is quite a natural result (to get s CRR , we multiply the configurational entropy per parti-cle by the number of particles in a region). They writethis relation directly for z ∗ : z ∗ S conf N = s ∗ CRR , (20)which is Eq. (20) in the original paper. s ∗ CRR here isthe critical entropy corresponding to the minimum regionsize. Note that the original notation is slightly different:the authors write the Avogadro number N A instead of N (which is because they denote with S conf the molarconfigurational entropy), while N in their paper denotesthe number of cooperatively rearranging regions.Next, Adam and Gibbs write the following: “theremust be a lower limit z ∗ to the size of a cooperative sub-system that can perform a rearrangement into anotherconfiguration, this lower limit corresponding to a criticalaverage number of configurations available to the sub-system. Certainly, this smallest size must be sufficientlylarge to have at least two configurations available to it... For the following, however, we need not specify thenumerical value of this small critical entropy [ s ∗ CRR ]”.We know now that for systems with continuous poly-dispersity both sides of Eq. (20) contain the divergingterm z ∗ k B s mix . After canceling this diverging term onboth sides we arrive at a relation where all the terms arewell-behaving (finite and continuous): z ∗ = ∆ c s ∗ CRR ∆ c S conf /N . (21)After substituting this result into Eq. (19), we obtainthe modified AG relation τ R = τ exp (cid:18) AT ∆ c S conf /N (cid:19) . (22)We note that z ∗ can be too small to apply the Stir-ling approximation to z ∗ ! or at least to one of the con-stituent particle types to arrive at s mix as in Eq. (1).In this case one can imagine taking a large number ofCRRs of size z ∗ , N CRR , and writing Eq. (20) for allof them, N CRR z ∗ S conf N = N CRR s ∗ CRR . For a sufficientlylarge N CRR , the Stirling approximation applies and wearrive at Eq. (21) after canceling N CRR on both sides ofthe resulting equation.The following example demonstrates why we have tosubtract k B s mix from S conf /N in the AG relation. Sup-pose we have a monodisperse hard-sphere system ( s mix =0) where minimum CRRs have a certain size z ∗ corre-sponding to s ∗ CRR = k B ln(2). Thus, the number of lo-cal PEL minima of a minimum CRR (if all particles aretreated as distinguishable) is N dist J = 2 z ∗ !. Next, supposethat we introduce a small polydispersity to the system.In the extreme case, we can just color the particles andpostulate that we distinguish particles by color as well.For a sufficiently small polydispersity (or for coloring), N dist J shall remain unchanged, N dist J = 2 z ∗ !, because thestructure of basins remains almost unchanged. On thecontrary, s ∗ CRR shall be counted as if particles with equalradii (or color) are treated as indistinguishable. Thus, s ∗ CRR = k B z ∗ !Π t z ∗ t ! , where z ∗ t is the number of particles oftype t among z ∗ . After applying the Stirling approxima-tion to the nominator and Eq. (1) to the denominator,we obtain s ∗ CRR = k B ln(2) + k B z ∗ s mix . After canceling k B z ∗ s mix on both sides of Eq. (20), we once again ob-tain Eq. (21). If z ∗ is too small to apply the Stirlingapproximation to the nominator or denominator, we canimagine analyzing many CRRs simultaneously and thencancel N CRR , as suggested in the previous paragraph.Now, we briefly justify the usage of ∆ c S conf /N in therelaxation time prediction from the Random First Or-der Transition (mosaic) theory. The equation forthe relaxation time from this theory looks similar to theoriginal AG relation (17): τ R = τ exp (cid:32) C Y ( T ) dd − θ T [ T S conf /V ] θd − θ (cid:33) , (23)where Y ( T ) is the generalized surface tension coefficient, θ is the parameter of the theory, and S conf V = S conf N NV .Due to the presence of S conf N , Eq. (23) possesses the sameproblems as Eq. (17): τ R ≡ τ R is discontinuous when intro-ducing a small polydispersity to a monodisperse system.Similarly to the AG relation, we suggest that S conf /N shall be replaced by ∆ c S conf /N . Indeed, the theory as-sumes that a (glassy) system consists of a patchwork (mo-saic) of different metastable regions, while transitions be-tween different system states occur via nucleation of such metastable regions (entropic droplets). Their growth ishindered by the surface tension with neighboring regions,and the free energy loss at a droplet radius R due to thistension is ∆ F loss ∼ Y R θ . For the usual surface tension, θ = d −
1, but the theory only implies that θ ≤ d − F gain ∼ − T S conf V R d : when adroplet transitions from an “unstable” state with manyavailable configurations into a metastable state with asingle (on an experimental timescale) available configu-ration (up to permutations), the free energy correspond-ing to the configurational entropy of the droplet is re-leased. Thus, the droplet final radius R is obtainedfrom ∆ F loss = ∆ F gain and the free energy barrier of nu-cleation ∆ equals the maximum value of ∆ F loss + ∆ F gain for R ≤ R , which leads to ∆ ∼ Y ( T ) dd − θ [ T S conf /V ] θd − θ , whichafter substitution into τ R = τ exp(∆ /k B T ) producesEq. (23). As already noted, S conf /N or S conf /V arepoorly-behaving quantities, divergent and discontinuous.Hence, we suggest that the entropic gain shall be cal-culated through the well-behaving quantity ∆ c S conf /V .Indeed, the k B s mix part of S conf /N (if calculated per par-ticle) is always present in any part of the system just dueto system composition and can not be released as the freeenergy during the growth of metastable entropic droplets(and in any other process if a system remains uniform incomposition). The “entropic” free energy gain can thushappen only up to k B s mix (per particle) and shall in factbe expressed as ∆ F gain ∼ − T ∆ c S conf V R d . Eq. (23) shallthus be written as τ R = τ exp (cid:32) C Y ( T ) dd − θ T [ T ∆ c S conf /V ] θd − θ (cid:33) , (24)Even if one follows the resolution of Ozawa andBerthier and redefines the configurational entropy byessentially redefining the mixing entropy, the redefinedmixing entropy is still inaccessible to the entropic freeenergy gain during the growth of metastable droplets.Thus, the redefined mixing entropy shall still be sub-tracted from the configurational entropy in Eq. (24) (aswell as in Eq. (22)).Finally, we specify how Eq. (21) shall look for a hard-sphere system (following Ref. but accounting for ∆ c ).For a system of hard spheres, we can express the reducedpressure as Z = pV /N k B T = p (cid:104) V sp (cid:105) /k B T ϕ , where (cid:104) V sp (cid:105) is the average sphere volume. The isobaric assumptionof the AG theory ( p = const) implies that in Eq. (21) A/T = CϕZ ( ϕ ), where C = Ap/k B V sp = const. Weconsequently write for hard spheres τ R ( ϕ ) = τ exp (cid:18) C ϕZ ( ϕ )∆ c S conf ( ϕ ) /N (cid:19) , (25)where S conf ( ϕ ) = S conf ( ϕ DJ ( ϕ )) is the equilibrium com-plexity ( cf . Eq. (10)). V. CONCLUSIONS
In this paper, we suggest that a natural way to writethe relation S tot = S conf + S vib is∆ S tot = ∆ c S conf + ∆ v S vib , (26)where∆ c S conf k B N = S conf k B N − s mix and∆ v S vib k B N = S vib k B N − − ln (cid:18) V Λ d N (cid:19) − UN k B T . (27)All the quantities from Eq. (26) if taken per particle are • finite in the thermodynamic limit even for a con-tinuous particle type distribution (polydispersity), • continuous when introducing a small polydispersityto a monodisperse system,Additionally, ∆ c S conf ( N, ϕ J ) /k B N is supposed to de-crease to zero with the increase of ϕ J .This resolution does not require any redefinition ofbasins of attraction and is in line with usual treatmentof S tot , when only ∆ S tot is discussed instead. One mayargue that this is merely a technical rewriting of theequation for entropies, but we think that working withsome sorts of delta-quantities lies in the nature of en-tropy. Information entropy for an arbitrary distribution(basically, s mix ) diverges when switching to continuousdistributions. Thus, information entropy for continu-ous distributions is represented in the information the-ory through the differential entropy, which for an arbi-trary function is its information entropy with respect tothe uniform distribution in a unit interval [0 , Instatistical physics, we can use even a more natural ap-proach: measure entropies with respect to the ideal gasof a corresponding particle size distribution. This paperessentially discusses how exactly we have to distributethe terms from the ideal gas entropy S ◦ tot between S conf and S vib .We also demonstrated that the Adam–Gibbs and theRandom First Order Transition theory relations for therelaxation time of (glassy) systems shall be writtenthrough ∆ c S conf /N or ∆ c S conf /V instead of S conf /N or S conf /V , respectively: τ AG R = τ exp (cid:18) AT ∆ c S conf /N (cid:19) ,τ RFOT R = τ exp (cid:32) C Y ( T ) dd − θ T [ T ∆ c S conf /V ] θd − θ (cid:33) . (28)In general, we suggest that any relation that expressesthe relaxation time through S conf shall in fact depend on∆ c S conf /N : τ R = f (∆ c S conf /N ) . (29) Our final remark is on how to interpret previous papersthat rely on the separation of entropies. If a paper writesout the expression for entropies as ∆ S tot = S conf + ∆ S vib or implies it, one has to read this relation rather as∆ S tot = ∆ c S conf + ∆ v S vib . If the authors used the rela-tion S conf /N = 0 to define the density of the ideal glasstransition or of the glass close packing limit, this rela-tion just has to be reinterpreted as ∆ c S conf /N = 0 andthe estimated location of either the ideal glass transitionor the glass close packing limit shall be kept unchanged,though special care shall be taken of course on how ex-actly the calculations were performed. Similarly, if theauthors used the Adam–Gibbs or Random First OrderTransition (mosaic) theories for validating the values of S conf /N against measured relaxation times or for fittingsome unknown parameters, it can well be that these re-sults hold, but one has to read ∆ c S conf instead of S conf everywhere in the paper, including the AG or mosaic re-lations.The presented results can be useful in understandingthe Edwards entropy for polydisperse systemsin granular matter studies. For frictionless particles, theEdwards entropy is equivalent under some definitions tothe configurational entropy. ACKNOWLEDGMENTS
We thank Misaki Ozawa and Ludovic Berthier for help-ful and insightful discussions as well as comments on themanuscript. We thank Patrick Charbonneau for readingthe manuscript. We thank Sibylle N¨agle for preparingFig. 1. We are also grateful to the two anonymous re-viewers of the manuscript for their comments and sug-gestions.
APPENDIX: REMARKS ON THE VIBRATIONALENTROPY AND STATISTICAL PHYSICS OF GLASSESAND HARD SPHERES1. Remarks on the vibrational entropy: free volume theoryand polytopes
One may ask whether the α = (cid:104) ln( V sp / (cid:104) V sp (cid:105) ) (cid:105) termfrom Eq. (12) somehow contains s mix indirectly. It doesnot, because it remains finite for all but very exotic con-tinuous distributions, contrary to s mix . Indeed, whendiscretizing a particle radii distribution f ( r ) with a step δ , α = (cid:82) f ( r ) ln (cid:16) r d (cid:104) r d (cid:105) (cid:17) d r when δ →
0, which is funda-mentally different from s mix in Eq. (2), which containsthe diverging ln( δ ) term. Additionally, α is continuouswhen introducing a small polydispersity to a monodis-perse system, contrary to s mix . In general, s mix can bemade arbitrary different from α —for example, by intro-ducing particle types with radii infinitely close to someexisting particle types. Then, α will remain almost un-0changed, while s mix can be changed arbitrarily. As anextreme example, one can introduce particle types bycoloring colloidal particles and postulating that we dis-tinguish particles by color as well as by radii. Then, α will remain exactly the same, while s mix will change. Thisexample shows the difference in the nature of s mix and α : s mix stems from our conventions on indistinguishabilityand α —from geometrical radii.We can easily determine the largest possible λ min inEq. (14). We can take a jammed configuration at ϕ J ,then scale particle radii linearly by a factor ( ϕ/ϕ J ) /d toensure the density ϕ . Now, to maintain the original par-ticle radii, we scale the entire system (particle radii anddistances between particles) by ( ϕ J /ϕ ) /d . The densityof such a system is still ϕ , the shrunk particles possessthe original particle radii R i , and the original particlesare enlarged and possess the radii ( ϕ J /ϕ ) /d R i . Theseenlarged particles can be treated as initial cells for thetether/cell method. Such cells will be “jammed”, becausethe original particles were jammed at ϕ J . The lengthsof tethers are then ( ϕ J /ϕ ) /d R i − R i . Thus, a naturalchoice for λ min = ( ϕ J /ϕ ) /d − ϕλ d min ) termfrom Eqs. (13) and (14) looks likeln( ϕλ d min ) = d ln( ϕ /dJ − ϕ /d ) . (30)If ϕ approaches ϕ J , particles are hardly able to movefurther away from tether centers than prescribed by λ min from Eq. (30). It means that we can assume Z c ≈ λ > λ min and thus write (cid:104) (cid:90) ∞ λ min Z c d υ ( λ ) υ ( λ ) (cid:105) = 0 . (31)in Eqs. (13) and (14), making these equations completelyanalytical. Eqs. (13) and (14) will then essentially rep-resent the free volume theory for the polydispersecase, because Eq. (30) essentially describes such free vol-umes. Eqs. (13), (14), and (30) show that S vib /N k B (or S singlevib /N k B ) behave with ϕ in exactly the same wayas for the monodisperse case in the free volume approx-imation (up to the size distribution-dependent constant α = (cid:104) ln (cid:16) V sp (cid:104) V sp (cid:105) (cid:17) (cid:105) ).In the same way as we write p = − ( ∂A tot /∂V ) N,T = k B T ( ∂ ln Z tot /∂V ) N,T for the total free energy and theentire phase space, we can introduce the glass pressure p g if we assume that only a particular basin of attractionis left in the phase space ( cf . Appendix 2 b). Glass pres-sure in this formulation has been studied, among otherworks, in the papers of Speedy and Donev, Stillinger,and Torquato. We write p g = k B T ( ∂ ln Z vib /∂V ) N,T .Repeating the steps from Appendix 2 a, we write in thesame way Z g ( ϕ, ϕ J ) = p g V /N k B T = 1 − ϕ ∂ ∆ S vib /Nk B ∂ϕ ,where ∆ S vib = S vib − S ◦ tot . One can also use S singlevib ,depending on the context— Z g does not depend on thischoice. Using Eqs. (14), (30), and (5), we obtain for thepolydisperse case Z g = 1 + ϕ J /ϕ ) /d − , which is a well-known free volume glass equation of state (previouslyderived for the monodisperse case, though). When Speedy and Donev et al. applied the origi-nal tether/cell methods, they could not ideally determinebasins of attraction (the tether method would actuallybe quite useless in that case). Still, it is natural to as-sume that up to a certain λ the system is not be ableto (quickly) leave the original basin of attraction. Thus,these authors performed the integration in Eqs. (13) or(14) up to a certain λ max . λ max was determined by ajump in the measured cell pressure. Such a jump in-dicates that the system starts to explore other basins ofattraction. Some more advanced corrections, like extrap-olating Z c ( λ ), can also be utilized.Finally, we note that it is known that the basin ofattraction approaches a polytope when ϕ → ϕ J . For the monodisperse case, a glass equation of state hasbeen derived long ago for polytopes and slightly latera complete form of the polytope free energy A vib wasobtained. The polytope glass equation of state is equiv-alent to the free volume one for ϕ → ϕ J and looks like Z g = 1 + d ( ϕ J /ϕ ) − . We found that it was easier for ourpurposes to amend the tether/cell method to the poly-disperse case than to amend the complete computationof A vib through polytope geometries.
2. Remarks on statistical physics of glasses and hardspheres
In this section, we use entropies per particle s tot = S tot /k B N , s vib = S vib /k B N , and s conf = S conf /k B N .It is convenient, because ∆ s tot = s tot − s ◦ tot is truly afunction of ϕ only, ∆ v s vib = s vib − − ln (cid:0) V Λ d N (cid:1) − UNk B T is truly a function of ϕ and ϕ J only (as well as ∆ s vib = s vib − s ◦ tot ), and s conf and ∆ c s conf = s conf − s mix are trulyfunctions of ϕ J only. a. Total entropy through pressures Equilibrium fluid pressure p is routinely defined in thecanonical ensemble through the Helmholtz free energy A tot = − k B T ln Z tot as p = − ( ∂A tot /∂V ) N,T . This re-lation essentially defines pressures through the partitionfunction: p = k B T ( ∂ ln Z tot /∂V ) N,T . We use the re-duced pressure (compressibility factor) Z = pV /N k B T .For hard spheres, it is possible to express Z through theexcess entropy per particle ∆ s tot . Specifically, by usingthe relations A = U − T S tot and U = N k B T , we write p = T ( ∂S tot /∂V ) N,T . After utilizing S tot = ∆ S tot + S ◦ tot ,we get p = T ( ∂ ∆ S tot /∂V ) N,T + T ( ∂S ◦ tot /∂V ) N,T . Thelast term is the ideal gas pressure
N k B T . If we switchto the reduced pressure Z = pV /N k B T , we obtain Z = V (cid:0) ∂ ∆ s tot ∂V (cid:1) N,T + 1. By replacing V with ϕ through ϕ = N (cid:104) V sp (cid:105) /V (where (cid:104) V sp (cid:105) is the average sphere vol-ume), we finally write: Z ( ϕ ) = 1 − ϕ d∆ s tot ( ϕ )d ϕ . (32)1Integration of Eq. (32) leads to ∆ s tot ( ϕ ) = ∆ s tot ( ϕ ) − ϕ (cid:82) ϕ Z ( ϕ (cid:48) ) − ϕ (cid:48) d ϕ (cid:48) . If we use the ideal gas as the referencestate, we get∆ s tot ( ϕ ) = − ϕ (cid:90) Z ( ϕ (cid:48) ) − ϕ (cid:48) d ϕ (cid:48) . (33) b. Glass pressure and vibrational entropy In this subsection, we study the relationships be-tween the glass pressure and the vibrational entropy.Glass pressure in the present formulation has been stud-ied, among other works, in the papers of Speedy andDonev, Stillinger, and Torquato, but without the cor-rections in the vibrational entropy needed in the poly-disperse case. In the same way as we write p = − ( ∂A tot /∂V ) N,T = k B T ( ∂ ln Z tot /∂V ) N,T for the totalfree energy and the entire phase space, we can intro-duce glass pressure p g if we assume that only a par-ticular basin of attraction is left in the phase space. We write p g = k B T ( ∂ ln Z vib /∂V ) N,T . Repeating thesteps from Appendix 2 a, we write in the same way Z g ( ϕ, ϕ J ) = p g V /N k B T and Z g ( ϕ, ϕ J ) =1 − ϕ ∂ ∆ s vib ( ϕ, ϕ J ) ∂ϕ =1 − ϕ ∂ ∆ v s vib ( ϕ, ϕ J ) ∂ϕ , (34)One can also use s singlevib , depending on the context— Z g does not depend on this choice.If one wants to measure Z g , this definition assumesthat one has to track during the system evolution (molec-ular dynamics) time points when the system crosses theboundary of a basin of attraction and to elastically reflectthe velocity hypervector from this boundary. For exam-ple, one can perform a steepest descent in the pseudo-PEL at each particle collision during the event-drivenmolecular dynamics simulation. If the basin is changedbetween collisions, one has to find with the binary searchthe time between the last collisions when the basin isswitched from one to another. This procedure is compu-tationally expensive, but presumably tractable for smallsystems. The basin of attraction does not have to dom-inate the phase space at a given density to define andmeasure Z g , what is required is that a system can beequilibrated inside this particular basin, if only this basinis left in the phase space. It is usually assumed that non-ergodicity at high densities stems from hindered move-ment of a system between basins, not inside basins, sowe assume that it is always possible to equilibrate a sys-tem inside a single basin.Note that one can use either ∆ or ∆ v in Eq. (34), itproduces the same Z g (while ∂s mix ∂ϕ = 0). The purpose of using ∆ or ∆ v in Eq. (34) as well as in Eq. (32) is to re-move the ln (cid:0) V Λ d N (cid:1) term to be able to use derivatives over ϕ correctly. Eq. (33) uses the ideal gas state once again(along the ∆ usage)—as a starting configuration for thethermodynamic integration, but these are two “indepen-dent” usages. Similarly, writing ∆ s vib in Eq. (34) doesnot mean that we use the ideal gas as a starting point forthe thermodynamic integration—it just means that wemeasure s vib with respect to the ideal gas. Indeed, inte-gration of Eq. (34) shall rather start from the jammedconfiguration and produces∆ v s vib ( ϕ, ϕ J ) = ∆ v s ( ϕ J )+ ϕ J (cid:90) ϕ Z g ( ϕ (cid:48) , ϕ J ) − ϕ (cid:48) d ϕ (cid:48) . (35)By comparing Eqs. (35) and (14) we conclude that∆ v s ( ϕ J ) = (cid:104) ln (cid:16) V sp (cid:104) V sp (cid:105) (cid:17) (cid:105) + A , where A represents thegeometry of the polytope. If we use Z g from thepolytope theory, we get ∆ v s vib ( ϕ, ϕ J ) = ∆ v s ( ϕ J ) + d ln( ϕ J − ϕ ), slightly different from the free volume the-ory, ∆ v s vib ( ϕ, ϕ J ) = ∆ v s ( ϕ J )+ d ln( ϕ /dJ − ϕ /d ), where∆ v s ( ϕ J ) = (cid:104) ln (cid:16) V sp (cid:104) V sp (cid:105) (cid:17) (cid:105) + (cid:104) (cid:82) ∞ λ min Z c d υ ( λ ) υ ( λ ) (cid:105) (Eqs. (14) and(30)). Note that Speedy wrote the equation for “poly-tope” vibrational entropies as ∆ s vib ( ϕ, ϕ J ) = ∆ s ( ϕ J ) + d ln( ϕ J − ϕ ), which is technically correct but conceals thepoint that s mix is extra-removed from s vib ( ϕ, ϕ J ). c. Dominant jamming densities Here, we make some remarks on the jamming densitythat dominates the phase space at a given ϕ , the domi-nant jamming density ϕ DJ ( ϕ ) from Eq. (10). Its value isdetermined by the maximum of the integrand in Eq. (8)and thus by (cid:20) d∆ c s conf ( ϕ J )d ϕ J + ∂ ∆ v s vib ( ϕ, ϕ J ) ∂ϕ J (cid:21) ϕ J = ϕ DJ = 0 . (36)It is useful to investigate the behavior of the “dominantglass reduced pressure” Z g ( ϕ, ϕ DJ ( ϕ )). It was done byDonev, Stillinger, and Torquato, but the necessity towork with ∆ v s vib instead of ∆ s vib in the polydispersecase was not realized at that time, so we repeat theirderivation with the corresponding changes.According to Eq. (34), we need to investigate thebehavior of ∂ ∆ v s vib ( ϕ,ϕ DJ ) ∂ϕ . To do this, we subtract S ◦ tot from Eq. (10), divide it by k B N , and fully differentiate it: d∆ s tot ( ϕ )d ϕ = (cid:104) d∆ c s conf ( ϕ J )d ϕ J (cid:105) ϕ J = ϕ DJ d ϕ DJ d ϕ + ∂ ∆ v s vib ( ϕ,ϕ DJ ) ∂ϕ + (cid:104) ∂ ∆ v s vib ( ϕ,ϕ J ) ∂ϕ J (cid:105) ϕ J = ϕ DJ d ϕ DJ d ϕ = ∂ ∆ v s vib ( ϕ,ϕ DJ ) ∂ϕ + d ϕ DJ d ϕ (cid:104) d∆ c s conf ( ϕ J )d ϕ J + ∂ ∆ v s vib ( ϕ,ϕ J ) ∂ϕ J (cid:105) ϕ J = ϕ DJ . Accord-ing to Eq. (36), the term in square brackets shall bezero and thus d∆ s tot ( ϕ )d ϕ = ∂ ∆ v s vib ( ϕ,ϕ DJ ) ∂ϕ . By comparing2this result with Eqs. 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