Topological extension of the isomorph theory based on the Shannon entropy
Tae Jun Yoon, Min Young Ha, Emanuel A. Lazar, Won Bo Lee, Youn-Woo Lee
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Topological extension of the isomorph theory based on the Shannon entropy
Tae Jun Yoon, Min Young Ha, Emanuel A. Lazar, Won Bo Lee, a) and Youn-WooLee b)1) School of Chemical and Biological Engineering, Institute of Chemical Processes,Seoul National University, Seoul 08826, Republic of Korea Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002,Israel (Dated: 31 July 2019)
Isomorph theory is one of the promising theories to understand the quasi-universalrelationship between thermodynamic, dynamic and structural characteristics. Basedon the hidden scale invariance of the inverse power law potentials, it rationalizesthe excess entropy scaling law of dynamic properties. This work aims to show thatthis basic idea of isomorph theory can be extended by examining the microstructuralfeatures of the system. Using the topological framework in conjunction with theentropy calculation algorithm, we demonstrate that Voronoi entropy, a measure ofthe topological diversity of single atoms, provides a scaling law for the transportproperties of soft-sphere fluids, which is comparable to the frequently used excessentropy scaling. By examining the relationship between the Voronoi entropy and thesolid-like fraction of simple fluids, we suggest that the Frenkel line, a rigid-nonrigidcrossover line, be a topological isomorphic line where the scaling relation qualitativelychanges. a) Electronic mail: [email protected] b) Electronic mail: [email protected] . INTRODUCTION The dynamic behavior of particles in liquids and high-pressure supercritical fluids islargely influenced by the relative local configurations of their neighbors. This strong particle-particle correlation implies that the thermodynamic, dynamic, and structural characteristicsof dense fluids are intimately linked with each other. Hence, it is no wonder that a consid-erable amount of studies were devoted to developing the theory of the liquid state . Oneof the wonderful aspects of the liquid state theory is its simplicity based on the hard-sphereparadigm . The hard-sphere paradigm assumes that the repulsive part of interatomic in-teraction dominates the behavior of the liquid state. Based on the hard-sphere paradigm,the perturbation theory has been advanced to understand the thermodynamic behaviorof dense fluid systems based on the pair correlation function and the hard-sphere potentialas a reference system .Liquid state theory has also been used to relate the thermodynamic properties to trans-port properties. Rosenfeld and Dzugutov proposed the scaling relation that connects thethermodynamic excess entropy ( S exc ) and the scaled transport properties of dense fluids.Rosenfeld discovered that the scaled diffusivities of simple liquids modeled with differentinteratomic potentials collapse to a single line as a function of the thermodynamic excessentropy. Based on these scaling laws, the two-body excess entropy ( S ), which can be di-rectly obtained based on the pair correlation function, has been frequently used when scalingthe dynamic properties based on the structural characteristics. The scaling relation providedby S was fairly good for simple fluid models although the contribution of S on the totalexcess entropy varies depending on thermodynamic conditions .In a more recent approach, Dyre and his coworkers proposed the isomorph theory tounderstand the relationship between thermodynamic, dynamic and structural characteristicsof simple fluid systems in an integrated manner. In the isomorph theory, the two statepoints are defined to be isomorphic if one can find pairs of scaled configurations that havethe same canonical probability . Let two configurations R A and R B ( R i ≡ ( r i , r i , . . . , r Ni ))sampled from two thermodynamic state points ( ρ A , T A ) and ( ρ B , T B ), respectively, have thesame reduced densities, i.e. ρ / A R A = ρ / B R B . The two state points are isomorphic if onehas exp (cid:18) − U ( R A ) kT A (cid:19) = C AB exp (cid:18) − U ( R B ) kT B (cid:19) . (1)2here C AB is a configuration-independent constant. Gnan et al. have shown that thecondition of having a good isomorph is equivalent to having a strong correlation betweenfluctuations of virial and potential-energy, which they term Roskilde-simple (R-simple) liq-uid . The R-simple liquid is defined as fluid models of which the virial potential-energycorrelation ( R ) is higher than 0 .
9. Here, the correlation coefficient R is defined as: R = h ∆ W ∆ U i p h (∆ W ) ih (∆ U ) i , (2)where W is the virial ( W = pV − N k B T ), U is the potential-energy, and ∆ A is a deviation(∆ A = h A i− A ). Schrøder and Dyre demonstrated that the following conditional propositionfor two system A and B is exact when the correlation coefficient R is unity : ρ / A R A = ρ / B R B ⇒ S exc ( R A ) = S exc ( R B ) . (3)In Eqn. (3), ρ i and S exc ( R i ) are the bulk density and the excess entropy of the system i .When two systems satisfy the antecedent, they are regarded to be isomorphic to each other.They showed that the Newton’s second law of motion in reduced units is invariant for theisomorphic states: ρ / A R A = ρ / B R B ⇒ ˜F A = ˜F B , (4)where ˜F is dimensionless force vector ( ˜F ≡ ρ − / F /k B T ). This result explains why theexcess entropy scaling law holds for simple fluids. In subsequent articles, they successfullyshowed that the isomorph theory works as a “good approximation” to different types ofpotentials including Lennard-Jones , Yukawa and exponential pair potentials . Someknown exceptions, that do not follow the isomorph theory and the excess entropy scalinglaw, are the potentials with thermodynamic anomalies .Despite this success of the isomorph theory, it should be noted that a direct connectionbetween the structural definition of isomorphic states and the dynamics scaling was not givenyet. Unlike crystalline systems, it is extremely difficult to discover two liquid configurationsthat exactly satisfy the antecedent of Eqn. (3). A pair correlation function, g ( r ), has beenfrequently used as an indicator of the antecedent, but the details of the local configurationcannot be inferred from the pair correlation function. Moreover, the two-body excess entropydirectly calculated from the pair correlation function cannot work as a robust parameterbecause its proportion varies depending on thermodynamic conditions .3e note that the antecedent of Eqn. (3) can be reformulated from the atomistic pointof view. The scale invariance hypothesized in Eqn. (3) can be expressed as follows. Let ξ ji be a position vector of the neighbor atom j relative to the atom i ( ξ ji = r j − r i ), and Ξ i ≡ ( ξ i , ξ i , ξ i , · · · ) a set of position vectors. Then, two local configurations of particles a and b are isomorphic to each other if the following condition is satisfied: ρ / a,l Ξ a = ρ / b,l Ξ b ⇒ S exc ( R A ) = S exc ( R B ) , (5)where ρ i,l the local density of the atom i , and Ξ i contains all particles of the system exceptthe particle i . This microscopic definition itself does not provide any advantages over themacroscopic description, but this point of view can extend the definition of the isomorphicstates in conjunction with the notion of the Gibbs entropy, which states that the systementropy is given by a distribution on the microstates ( S = − k B P p i log p i where p i is theprobability of the microstate i ). In a similar vein, if a relative configuration ( ρ / i,l Ξ i ) isregarded as a microstate , we can hypothesize that two configurations will have the sameexcess entropy if their distributions of the relative configurations are the same.This viewpoint is related to our works on dense supercritical fluids. We have characterizedthe local structure of an atom with respect to the topological type of its Voronoi polyhedronto develop a theory of structure-dynamics relationship and a notion of quasi-universalityamong simple fluids . In this work, we validate the idea of Eqn. (5) by defining the mi-crostate of an atom as the topological type of its Voronoi polyhedron , and estimating theexcess entropy from the diversity of this topological type in the given thermodynamic con-dition, where the local density ρ i,l in Eqn. (5) is given as the inverse volume of the Voronoipolyhedron. Then, the classical notion about the equivalence of the Shannon entropy andthe thermodynamic entropy is exploited to define Voronoi entropy based on the topologicaltypes observed in the system. By comparing the scaling results of the repulsive n − , can be regardedas a limit of applicability of the exponential scaling relation.4 I. METHODSA. Molecular Dynamics (MD) simulations
We perform the NVT simulations of the soft-sphere fluids of which the interatomicpotentials are modeled with the repulsive n − n − φ ( r ) = φ M ( r ) − φ M ( r cut ) r ≤ r cut r ≥ r cut (6)Here, φ M ( r ) is the Mie n -6 potential, which is given in Eqn. (7). φ M ( r ) = (cid:20) nn − (cid:21) (cid:16) n (cid:17) n/ ( n − ǫ (cid:20)(cid:16) σr (cid:17) n − (cid:16) σr (cid:17) (cid:21) (7)The potential is shifted and truncated at r cut = ( n/ / ( n − σ . The size parameter σ ofargon is used for all potentials ( σ = 3 . ǫ is changed so thatthe coefficient C n ǫ becomes equal to that of the LJ potential ( C n ǫ = 4 ǫ Ar ) where ǫ Ar isthe energy parameter of argon ( ǫ Ar /k B = 119 . K ). The simulation temperatures are T =318 . , . , . , . . n = 8 −
24. Forall simulations, the timestep is 2 fs. To obtain the trajectories for calculating the Shannonentropy, and the thermodynamic and transport properties, the systems are equilibrated for100,000 steps. The details of the production run are given in the following subsections.
B. Evaluation of the virial potential-energy correlation
Virial potential-energy correlation is evaluated for all thermodynamic conditions as fol-lows. In the production run (5 , ,
000 steps), the instantaneous virial ( W ) and thepotential-energy ( U ) are collected every ten steps. Then, the correlation coefficients R are evaluated using Eqn. (2). As shown in the Supplementary Information, the correlationcoefficients R are always higher than 0 .
98 at all conditions. Hence, all repulsive n -6 fluidsdealt with in this work are R-simple. 5 . Topological framework for local structure analysis The topological framework for local structure analysis proposed by Lazar et al. describesthe arrangement of neighbors surrounding a central particle via the Voronoi tessellation, thepartitioning of space into regions, each of which consist of all points closer to a given particlethan to any other. The topology of a Voronoi cell can be described by enumerating thenumber of edges of each of its faces. Although this description provides more informationthan a mere count of faces, it does not completely describe how a particle’s neighbors arearranged relative to the central particle and to one another. A more refined description of theVoronoi cell, and thereby of the arrangement of neighbors, is provided by the isomorphismclass of its edge graph , which identifies two Voronoi cells as the same if pairs of facesare adjacent in one Voronoi cell if and only if corresponding faces in the other are alsoadjacent. This connectivity information can be encoded as a series of integers called theWeinberg vector , which is obtained from a graph-tracing algorithm initially developed tocheck whether two planar graphs are isomorphic . Hence, the Weinberg vector can beviewed as a ‘name’ of the topological type of a Voronoi cell. We use the VoroTop library to gather the statistical data of the topological types discovered in the configurations. D. Characterization of the Frenkel line
Rosenfeld et al. noted that there are two regions where the dependence of transportproperties on the thermodynamic excess entropy are qualitatively different . In the low-density (low excess entropy) region, the diffusivity shows a power-law dependence. Whenthe density is high, it shows an exponential dependence on the excess entropy in the high-density region. This qualitative change of dynamics can also be observed in Monte Carlosimulations. Nezhad and Deiters recently discovered that the excess entropy is an ap-proximately linear function of the reciprocal mean Monte Carlo displacement parameter athigh density. Provided that the Monte Carlo displacement parameter is proportional to thediffusivity, this finding indicates that the collective particle dynamics changes depending onthe bulk density of a system.This qualitative change of the transport properties would be related to the Frenkel lineproposed by Brazhkin et al. They proposed that the Frenkel line of the hard-sphere fluid6orresponds to the crossover density at which the transport properties show a qualitativelydifferent dependence on the bulk density . In recent work, we demonstrated that thisconjecture is quite reasonable based on the topological framework and the two-phase ther-modynamics (2PT) model . In addition, we recently found that the percolation behaviorof solid-like structures of different repulsive n − solid ) was used as an order parameter . Hence, we vali-date this idea that the Frenkel line may be a good candidate to demarcate the fluid regiondepending on the behavior of the transport properties as proposed by Rosenfeld et al.To locate the dynamic crossover conditions, we use the topological classification methodproposed in our earlier works . In this method, the topological types of two dynamic limitsof the fluid phase including the ideal gas and the maximally random jammed state are usedto classify a molecule as either gas-like (diffusive) or solid-like (oscillatory). If a topologicaltype of an atom discovered in a configuration has a higher likelihood to be observed in idealgas, it is classified as gas-like. Otherwise, it is classified as solid-like. A weighted mean-fieldapproximation is then applied to this initial classification result to remove the influence offluctuation. From the finite-size scaling analysis on percolation behaviors, we showed thatthe Frenkel line can be defined as the thermodynamic states where the fraction of solid-likemolecules (Π solid ) reaches the percolation threshold, Π solid = 0 . ± . . In this work,we apply the same procedure and the percolation criterion to determine the Frenkel line ofthe soft-sphere fluids (see Yoon et al. for further details of the algorithm). E. Reformulation of the isomorph definition based on the information theory
The limitation of the hypothesis provided in Eqn. (5) is that the definitions of ρ i,l and Ξ i are incomplete. As a first approximation, we introduce the topological framework proposedby Lazar et al . In the topological framework, the connectivity information of an atom withits nearest neighbors is understood based on the topological type of the Voronoi polyhedron.Since this topological information is invariant under the multiplication of coordinates by aconstant, e.g. ˜ R i = ρ / i R i , the reduced coordinates of the nearest atoms surrounding twoatoms are the same if the topological types of their Voronoi cells are identical. Hence, Eqn.(5) is approximated as: v − / a,l Ξ ′ a = v − / b,l Ξ ′ b (8)7here v i,l is the volume of the Voronoi polyhedron of the particle i and Ξ ′ i is a set of therelative coordinate vectors of the nearest neighbors of which the Voronoi polyhedron sharea face with that of the central particle i . Since the forces exerted on the central atom bythe nearest neighbors usually account for the majority of the total force, we expect that twoatoms would have similar dynamic characteristics if the topological types of their Voronoicells are identical to each other.It is noteworthy that a similar extension of the isomorphism was proposed by Malins, Eg-gers, and Royall . They used the topological classification method proposed by Williams to identify the bicapped square antiprism, which is a locally favored structure in glass for-mers. F. Voronoi entropy
The diversity of the categorical distributions can be measured using the Shannonentropy . The Shannon entropy ( H ) is obtained as: H = − X i p i log p i (9)where p i is the probability of finding a topological type i in the system. The term Voronoientropy was used by Peng, Li and Wang by applying Eqn. (9) to the distribution of theVoronoi types, which were classified based on their Voronoi indices. It was also defined asthe Shannon entropy of the distribution of the Voronoi types based on the number of edges .On the other hand, we classify Voronoi cells based on the Weinberg vectors, a more refineddescriptor than the Voronoi indices, following the philosophy of the isomorph theory.When the probability that an event i occurs ( p i ) is known for all events, we can directlymeasure the Shannon entropy of a system. In real-world problems, however, there are twobottlenecks to apply Eqn. (9) directly. First, p i is only estimated based on observationof the limited samples drawn from a population. The Shannon entropy calculated basedon this limited observation can be heavily biased by rare events. For systems in whichthe number of events is infinite (unbounded), therefore, the Shannon entropy is exactlycalculated only when infinite data are available or an exact mathematical expression for allthe p i ’s is given. For ideal gas, the number of topological types is infinite since the pointparticles can be randomly distributed in a system. Second, Eqn. (9) ignores the correlation8etween events. For ideal gas, this hypothesis holds since no interatomic interaction existsamong the particles. That is, the topological type of an atom has little effect on how aneighbor atom is surrounded by its neighbors in the low-density regime. On the other hand,the probability of finding a topological type is largely influenced by the topological types ofits neighbors in the crystalline state.Several algorithms have been proposed to resolve the problem of infinite sample sizeby estimating the upper bound of the Shannon entropy ( ˆ H ) with an unknown or infinitenumber of samples . This work uses the estimator named Unseen designed by Valiantand Valiant . This algorithm uses a fingerprint of a finite dataset (observed samples), ahistogram of a histogram, to construct a plausible histogram of which the entropy and otherproperties are similar to those of larger population by estimating the “unseen” portion ofthe histogram. Two linear programming (LP) procedures are used to obtain this likelyunderlying histogram. The first LP algorithm finds the plausible histograms as follows.Since the finite data we obtained are the sampled ones from the unbounded population, theprobability of drawing a topological type i exactly k times during n independent trials followsthe binomial distribution B ( n, p i ), which can be approximated as a Poisson distribution( P ( np i , i )). Hence, the first LP algorithm calculates the expected i th fingerprint entry andyields plausible histograms of which the fingerprints are the same as the expected fingerprint.The second LP algorithm selects the simplest distribution among the candidates based onOccam’s razor. To validate the algorithm, we first apply the Unseen estimator to ideal gassystems and compare ˆ H to H . We then build the following procedure to estimate the Voronoientropy of a system based on the ideal gas results (see the Results and Discussion for thedetails). First, we perform five independent simulations for each condition and obtain 500trajectories from each simulation. The number of molecules is 2,000. Second, we randomlyselect 300 trajectories of 2,500 configurations eight times and apply the Unseen algorithmto each set of the trajectories. The estimated entropy data are averaged to obtain ˆ H .Note that this algorithm does not reflect the spatial correlation between neighbors. Sev-eral measures have been suggested to reflect the spatial association to the Shannon entropy,but no algorithm has been proposed to consider both aspects. Later, we will see how thisspatial correlation affects the results. 9 . Thermodynamic excess entropy The integration method of Deiters and Hoheisel is used to calculate the thermodynamicexcess entropy. In this method, the excess Gibbs free energy per particle ( G exc ) is calculatedas: G exc k B T = Z ρ Z − ρ dρ + Z − Z is the compressibility factor ( Z ≡ pV /RT ). Deiters-Hoheisel method constructs afunction Z ( ρ ) by fitting the compressibility factors obtained from a series of NVT simulationswith a smoothing spline curve. ( Z − /ρ at the zero density converges to the second virialcoefficient B , which is computed as: B = lim ρ → Z − ρ = − π Z ∞ ( e − φ ( r ) /k B T − r dr (11)where φ ( r ) is the interatomic potential. The equilibrium pressure and internal energydata are averaged every step during the production run (5 , ,
000 steps). The numberof molecules is 2,000. Then, we use the trapezoidal rule to integrate the smoothing splinefitting function to evaluate the excess Gibbs energy per particle. The excess entropy is thenobtained as: S exc = H exc − G exc k B T (12)where H exc is the excess enthalpy per particle, which is defined as H exc = H − H ig . Theideal gas enthalpy ( H ig ) is given as H ig = (5 / k B T . H. Calculation of transport properties
The transport properties of the soft-sphere fluids are computed as follows. The diffusivityis estimated based on the vibrational density of states Ψ defined as :Ψ( ν ) = 2 k B T N X j =1 3 X k =1 m j ψ kj ( ν ) (13)where ψ kj ( ν ) is the spectral density of atom j in the k direction and m j is the mass of atom j . The spectral density is the square of the Fourier transform of the velocity. ψ kj ( ν ) = lim τ →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z τ − τ v kj ( t ) e − i πνt dt (cid:12)(cid:12)(cid:12)(cid:12) (14)10ere, v kj ( t ) is the k th component of the velocity vector of the j th atom at time t . Thediffusivity ( D ) of a system is then obtained from the intensity of Ψ( ν ) at zero frequency asfollows. D = Ψ(0) k B T mN (15)Note that decomposition of the spectral density into hard-sphere and harmonic oscillatorcontributions leads to another definition of the Frenkel line .The shear viscosity of a system is estimated by integrating the Green-Kubo integral . η = Vk B T Z ∞ h P αβ ( t ) · P αβ (0) i dt (16)where P αβ ( t ) ( α , β = x, y, and z ) is the off-diagonal elements of the pressure tensor, whichis given as: P αβ = N X j =1 m j v αj v βj V + P N ′ j r αj f βj V (17)Here, r αj is the α th component of the position vector r of the j th atom and f βj is the β th component of the force vector f exerted on the j th atom.Unfortunately, calculating η using the Green-Kubo integral is difficult due to the lowsignal-to-noise ratio; the stress autocorrelation function given in Eqn. (16) does not smoothlyconverge to zero. Hence, we combine the methods proposed by Nevins and Zhang as fol-lows. We perform ten independent NVT simulations with different initial configurations andinitial velocities for each thermodynamic condition. The timestep is equal to the equillibra-tion run (2 fs), and the simulation duration is 2 ns. The stress autocorrelation functionsobtained from independent simulations are averaged and truncated at the cutoff time ( t cut )at which the absolute magnitude of the stress autocorrelation function decreases under itsinitial value by a factor of 10 − . Then, a two-term exponential function is fitted to the trun-cated stress autocorrelation function. The viscosity is calculated by integrating the fittedstress autocorrelation function. III. RESULTS AND DISCUSSIONA. Voronoi entropy of the ideal gas
We first estimate the Voronoi entropy of ideal gas to determine the number of samplesand trials for the soft-sphere models. Fig. 1 compares ˆ H and H of the ideal gas. The11 IG. 1. Shannon entropy of ideal gas system estimated from (a) the observed probabilities ( H )and (b) the Unseen algorithm ( ˆ H ). H does not vary significantly ( H ∼ .
04) when the samplesize ( N s ) is larger than 430 , , H becomes close to 14.00 when N s is larger than5 , , ideal gas configurations are generated by distributing N = 2 ,
000 points in a cubic boxrandomly. As shown in Fig. 1a, H ig slowly increases as the sampled number of configurations(molecules) increases. When the sample size ( N s ) is larger than 30 , , H ig does notvary significantly. On the contrary, the Voronoi entropy estimated from the Unseen becomessimilar to ˆ H ig ∼ .
00 when the sample size is larger than 1 , ,
000 (Fig. 1b). The orderof the magnitude of the sample size to obtain ˆ H ig similar to H ig agrees with that proposedby Valiant and Valiant (30 , , / log(30 , , ∼O (10 )). Since the population of thetopological types of the ideal gas system is larger than those of any other system, O (10 )trajectories of N = 2 ,
000 molecules are enough to estimate the Voronoi entropy at allthermodynamic conditions studied in this work. The estimated Voronoi entropy of theideal gas ( ˆ H ig ∼ .
04) can be used to define the Voronoi excess entropy, which is given asˆ H exc ≡ ˆ H − ˆ H ig . 12 IG. 2. Dependence of ˆ H on the bulk density ( ρ ). (a) n = 8 and (b) n = 16. It shows a power-lawdependence on the bulk density. The ˆ H , the upper bound of the Voronoi entropy, is calculatedfrom the Unseen algorithm. B. Voronoi entropy of repulsive n -6 fluids Fig. 2 shows ˆ H of the fluids modeled with the repulsive 8 − − H slowly decreases in the low-density region, but | d ˆ H /dρ | increases as the density increases;it shows the power-law dependence on the density (Eqn. (18)).ˆ H = aρ b + c (18)where a , b , and c are fitting parameters. The power-law equations fitted to different isothermsconverge to ˆ H ∼ . . As the system density increases, thedistances between neighbor atoms surrounding the central atom become shorter. When theyare so close that they hinder each others’ diffusive motions, the number of ways to placeneighbors around a central atom without an increase of the potential-energy decreases. As13 IG. 3. Dependence of the solid-like fraction (Π solid ) on the bulk density ( ρ ). For the definitionof the solid-like and gas-like states, see Sec. II D. (a) n = 8 and (b) n = 16. It shows a sigmoidaldependence on the bulk density for all temperatures and repulsive exponents. a result, the Voronoi entropy drastically decreases when the bulk density is high. A largediscrepancy between H and ˆ H in the low-density region reflects this phenomenon. Since theset of possible topological types are drastically large and cannot be sufficiently sampled inlow-density systems, H of the low-density fluid is much lower than ˆ H , whereas that of thehigh-density fluid is similar to ˆ H .Fig. 3 shows Π solid of the repulsive 8-6 fluids (For the definition of Π solid , see Sec. II D.).As shown in our earlier works , it starts to steeply increase near the dynamic crossoverdensities and reaches unity near the freezing densities at constant temperatures. The de-pendence of Π solid on the bulk density is well expressed by the sigmoidal function for allconditions based on the theory of fluid polyamorphism .Π solid = 11 + a exp( bρ ) (19)Both the Voronoi entropy and the fraction of solid-like molecules at different isothermsbecome close to each other when the repulsive exponent increases. They become closeto each other and ultimately collapse to a single line when the repulsive exponent ( n ) is14 IG. 4. A relation between the Voronoi entropy ( ˆ H ) and the fraction of solid-like molecules (Π solid ).ˆ H has a one-to-one correspondence with Π solid . This correspondence relation enables to redefinethe Frenkel line as a set of isomorphic states, which demarcate the fluid region into the non-rigidand the rigid regions. infinite .Since both Voronoi entropy and the fraction of solid-like molecules are defined from thetopological types of the Voronoi cells, it can be expected that both parameters are deeplyrelated. Fig. 4 shows that a one-to-one correspondence relation exists between ˆ H and Π solid .This one-to-one correspondence relation has two implications. First, it substantiates thatthe topological framework captures the isomorphism observed along the freezing line. SinceΠ solid reaches unity at the freezing densities, ˆ H becomes approximately ˆ H ∼ . solid over the interval of10 ≤ ˆ H ≤
12. Considering that the Frenkel line is an iso-Π solid line (Π solid = 0 . , thisresult implies that the Frenkel line can be related to the qualitative different regimes of thecollective particle dynamics as we expected. This expectation is further discussed in thenext section. C. Isomorph theory based on the Voronoi entropy
We test the validity of the Rosenfeld scaling law for repulsive n -6 fluids. Fig. 5 shows thatthe Rosenfeld diffusivity ( ˜ D ) and viscosity (˜ η ) of all simple fluids modeled with repulsive n -6potential collapse to single lines as the isomorph theory for R-simple fluids predicts. Theyare defined as: ˜ D = Dρ / (cid:18) mk B T (cid:19) / ; ˜ η = ηρ − / (cid:18) mk B T (cid:19) − / (20)15 IG. 5. Thermodynamic excess entropy scaling of (a) diffusivity ( ˜ D ) and (b) shear viscosity (˜ η ).For all repulsive n − R − ˆ H exc = − pS exc /k B + q ). Regardless of the repulsive exponents,the coefficient of determination ( R ) are higher than 0 .
99, and the intersects ( q ) are close to zero. Both collapsing lines show a similar dependence on − S exc observed by previous works .˜ D curve steeply decreases as − S exc increases to a certain extent. It shows an exponentialdependence on − S exc when S exc decreases. ˜ η shows more complex dependence on − S exc . Itdecreases to its minimum in the middle-density region, and increases as − S exc increases.Fig. 6 shows the dependence of ˜ D and ˜ η on the Voronoi entropy. Similar to the thermo-dynamic excess entropy, they collapse to their own single lines to a good approximation. In16 IG. 6. Voronoi entropy scaling of (a) diffusivity ( ˜ D ) and (b) shear viscosity (˜ η ) of fluids modeledwith repulsive n -6 potentials. The dotted lines denote the dynamic crossover Voronoi entropies( ˆ H = ˆ H cr ), which are estimated based on the Π solid - ˆ H curve. addition, the shapes of the collapsing lines are similar to those obtained when − S exc /k B isused to scale the transport properties. Simultaneously, they substantiate that the Frenkelline is a set of isomorphic states ( ˆ H ∼ .
0) where the collective particle dynamics qual-itatively changes. When ˆ H is greater than the topological crossover diversity, ˜ D shows apower-law dependence on ˆ H . As the density increases, ˆ H drastically decreases and expo-nential decay of ˜ D is observed. ˜ η also shows an exponential dependence on − ˆ H when ˆ H is lower than the topological crossover diversity. All these results support that the Frenkelline, a rigid-nonrigid transition line, is a good candidate that demarcates the fluid regionconsidering the collective particle dynamics.Meanwhile, it should be noted that the scaling results from ˆ H show a slight but systematicdiscrepancy compared to those from − S exc . For the same repulsive n − D and ˜ η curves from different temperatures agree with each other. In contrast, the extent ofthe data collapse is low for different repulsive exponents compared to the thermodynamicexcess entropy scaling result.Fig. 7 shows the parity plot of − S exc and − ˆ H exc . As the repulsive exponent increases, the17 IG. 7. A linear relation between the thermodynamic excess entropy and the Voronoi excessentropy. Although all models show the linear relationship, ˆ H exc ( S exc ) curves of repulsive n − slope of − ˆ H exc ( − S exc ) decreases. This small discrepancy between slopes of different modelswould occur because the algorithm ignores the mutual dependence between the topologicaltypes of neighbors. At the same thermodynamic excess entropy, the bulk density of a systemincreases as the repulsive exponent ( n ) decreases. Bearing in mind that the cutoff radiusincreases as n decreases, the spatial correlation between neighbors will be high when n islow.Despite these limitations, − S exc and − ˆ H exc have a linear relation, and the intersects ofthe fitted equations are close to zero (Table I). This linearity indicates that the influence ofthe spatial correlation is little compared to the total entropy contribution. Hence, we can seethat the classical idea about the equivalence of the Shannon entropy and the thermodynamicentropy holds . IV. CONCLUSION
This work demonstrates that the isomorph theory can be extended to the molecularlevel in conjunction with the topological framework and the information theory. In thisframework, two systems are regarded to be isomorphic if their topological diversities (Voronoientropies) are equal. The Voronoi entropy of the Bernoulli distribution, in which the numberof categories is infinite, can be estimated based on the fingerprint of the distribution. Similarto the thermodynamic excess entropy, the Voronoi entropy can work as a scaling parameter to18ollapse the transport properties of soft-sphere fluids. The Voronoi entropy scaling results aresatisfactory but also show slight but noteworthy deviations compared to the thermodynamicexcess entropy scaling. These systematic deviations come from the limit of the proposedmethod; it ignores the entropic contribution of the particles which are not nearest neighborsbut influence the net force exerted on the central particle. Lastly, a qualitatively differentdependence of the transport properties on the ˆ H and S exc can be understood based onthe rigid-nonrigid dynamic crossover across the Frenkel line. Since the isomorph theory isquasi-universal for various types of potentials, it would be interesting to understand thequasi-universal characteristics of broader classes of fluid models based on the topologicalframework in conjunction with the information theory. V. SUPPLEMENTARY MATERIAL
Supplementary material includes numerical data that can help understand and reproducethe results obtained in this work.
VI. ACKNOWLEDGEMENTS
M.Y.H. and W.B.L. acknowledge the support by Creative Materials Discovery Programthrough the National Research Foundation of Korea (NRF) funded by Ministry of Scienceand ICT (2018M3D1A1058624). E.A.L. gratefully acknowledges the generous support of theUS National Science Foundation through Award DMR-1507013.
REFERENCES H. Eyring and J. Hirschfelder, J. Phys. Chem. , 249 (1937). J.-P. Hansen and I. R. McDonald,
Theory of simple liquids , 3rd ed. (Academic Press, SanDiego, 2006). J. C. Dyre, J. Phys. Condens. Matter. , 323001 (2016). L. Verlet and J.-J. Weis, Phys. Rev. A , 939 (1972). J. A. Barker and D. Henderson, J. Chem. Phys. , 4714 (1967). N. F. Carnahan and K. E. Starling, J. Chem. Phys. , 635 (1969). Y. Rosenfeld, Phys. Rev. A , 2545 (1977).19 M. Dzugutov, Nature , 137 (1996). Y. Ding and J. Mittal, Soft Matter , 5274 (2015). A. Baranyai and D. J. Evans, Phys. Rev. A , 849 (1990). R. Chopra, T. M. Truskett and J. R. Errington, J. Phys. Chem. B , 10558 (2010). J. J. Hoyt, M. Asta and B. Sadigh, Phys. Rev. Lett. , 594 (2000). N. Jakse and A. Pasturel, Sci. Rep. , 20689 (2016). I. Yokoyama and S. Tsuchiya, Mater. Trans. , 67 (2002). J. C. Dyre, Phys. Rev. E , 042139 (2013). N. Gnan, T. B. Schrøder, U. R. Pedersen, N. P. Bailey and J. C. Dyre, J. Chem. Phys. , 234504 (2009). T. B. Schrøder and J. C. Dyre, J. Chem. Phys. , 204502 (2014). A. K. Bacher, T. B. Schrøder and J. C. Dyre, Nat. Commun. , 5424 (2014). L. Bøhling, T. S. Ingebrigtsen, A. Grzybowski, M. Paluch, J. C. Dyre and T. B. Schrøder,New J. Phys. , 113035 (2012). A. A. Veldhorst, T. B. Schrøder and J. C. Dyre, Phys. Plasmas , 073705 (2015). A. K. Bacher and J. C. Dyre, Colloid Polym. Sci. , 1971 (2014). Y. D. Fomin, V. N. Ryzhov and N. V. Gribova, Phys. Rev. E , 061201 (2010). T. J. Yoon, M. Y. Ha, E. A. Lazar, W. B. Lee and Y.-W. Lee, J. Phys. Chem. Lett. ,6524 (2018). T. J. Yoon, M. Y. Ha, W. B. Lee, Y.-W. Lee and E. A. Lazar, Phys. Rev. E , 052603(2019). E. A. Lazar, J. Han and D. J. Srolovitz, Proc. Natl. Acad. Sci. , E5769 (2015). T. J. Yoon, M. Y. Ha, W. B. Lee and Y.-W. Lee, J. Phys. Chem. Lett. , 4550 (2018). S. Plimpton, J. Comput. Phys. , 1 (1995). E. A. Lazar, J. K. Mason, R. D. MacPherson and D. J. Srolovitz, Phys. Rev. Lett. ,095505 (2012). L. Weinberg, IEEE Trans. on Circuit Theory , 142 (1966). E. A. Lazar, Modelling Simul. Mater. Sci. Eng. , 015011 (2017). J. Mittal, J. R. Errington and T. M. Truskett, J. Phys. Chem. B , 10054 (2007). S. Y. Nezhad and U. K. Deiters, Mol. Phys. , 1074 (2017). V. V. Brazhkin, Y. D. Fomin, A. G. Lyapin, V. N. Ryzhov and K. Trachenko, Phys. Rev.E , 031203 (2012). 20 V. V. Brazhkin, Y. D. Fomin, V. N. Ryzhov, E. N. Tsiok and K. Trachenko, Physica A , 690 (2018). A. Malins, J. Eggers and C. P. Royall, J. Chem. Phys. , 234505 (2013). S. R. Williams, arXiv preprint arXiv:0705.0203, (2007). L. Jost, Oikos , 363 (2006). L. Masisi, V. Nelwamondo and T. Marwala,
The use of entropy to measure structuraldiversity , in , IEEE(2008). T. M. Cover and J. A. Thomas,
Elements of information theory , (John Wiley & Sons,2012). H. L. Peng, M. Z. Li and W. H. Wang, Appl. Phys. Lett. , 131908 (2013). E. Bormashenko, M. Frenkel and I. Legchenkova, Entropy , 251 (2019). E. Archer, I. M. Park, and J. W. Pilow,
Bayesian Estimation of Discrete Entropy withMixtures of Stick-Breaking Priors , edited by F. Pereira, C.J.C. Burges, L. Bottou, and K.Q.Weinberger (Neural Information Processing Systems Foundation, Lake Tahoe, 2012). A. Chao and T.-J. Shen, Environ. Ecol. Stat. , 429 (2003). P. Valiant and G. Valiant, Estimating the Unseen: Improved Estimators for Entropy andOther Properties, edited by C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, andK. Q. Weinberger (Neural Information Processing Systems Foundation, Lake Tahoe, 2013). B. Boots, Ecoscience , 168 (2002). C. Hoheisel and U. Deiters, Mol. Phys. , 95 (1979). P. H. Berens, D. H. J. Mackay, G. M. White and K. R. Wilson, J. Chem. Phys. , 2375(1983). M. S. Green, J. Chem. Phys. , 398 (1954). R. Kubo, J. Phys. Soc. Japan , 570 (1957). D. Nevins and F. J. Spera, Mol. Simul. , 1261 (2007). Y. Zhang, A. Otani and E. J. Maginn, J. Chem. Theory Comput. , 3537 (2015). M. A. Anisimov, M. Duˇska, F. Caupin, L. E. Amrhein, A. Rosenbaum and R. J. Sadus,Phys. Rev. X , 011004 (2018). M. Pfleger, T. Wallek and A. Pfennig, Ind. Eng. Chem. Res.54