Exact bulk correlation functions in one-dimensional nonadditive hard-core mixtures
aa r X i v : . [ c ond - m a t . s o f t ] D ec Exact bulk correlation functions in one-dimensional nonadditive hard-core mixtures
Andr´es Santos ∗ Departamento de F´ısica, Universidad de Extremadura, E-06071 Badajoz, Spain (Dated: October 27, 2018)In a recent paper [Phys. Rev. E , 031202 (2007)], Schmidt has proposed a Fundamental Mea-sure Density Functional Theory for one-dimensional nonadditive hard-rod fluid mixtures and hascompared its predictions for the bulk structural properties with Monte Carlo simulations. The aimof this Brief Report is to recall that the problem admits an exact solution in the bulk, which isbriefly summarized in a self-contained way. PACS numbers: 61.20.Gy, 61.20.Ne, 64.10.+h, 05.20.Jj
Perhaps the most successful class of density functionaltheories are based on Rosenfeld’s Fundamental MeasureTheory (FMT) [1]. In a recent paper [2], Schmidt hasproposed a FMT for the excess free energy of inhomoge-neous one-dimensional nonadditive hard-rod fluid mix-tures. As a test of the theory, the FMT predictionsfor the pair correlation functions in the bulk region arecompared with Monte Carlo simulations, a general goodagreement being found. On the other hand, notwith-standing the merits of the FMT constructed in Ref. [2],it presents some limitations that become more importantas the density and/or the nonadditivity increase. For in-stance, it yields non-zero values of the pair correlationfunctions inside the core and predicts a spurious demix-ing transition.It seems to have been overlooked in Ref. [2] the factthat the one-dimensional nonadditive hard-rod problemadmits an exact solution in the bulk. Actually, any one-dimensional homogeneous system is exactly solvable, pro-vided that every particle interacts only with its nearestneighbors [3, 4, 5]. The aim of this Brief Report is to fillthe gap in Ref. [2] by presenting a brief and self-containedsummary of the exact solution, particularizing to binarynonadditive mixtures, and comparing with the bulk FMTpredictions for one of the cases considered in Ref. [2].Let us consider an m -component one-dimensional fluidmixture with constant (bulk) number densities { ρ i ; i =1 , . . . , m } and interaction potentials φ ij ( x ) = φ ij ( − x )acting only on nearest neighbors. Given a particle ofspecies i at the origin, the probability that its ℓ th neigh-bor belongs to species j and is located at a point between x and x + dx is given by p ( ℓ ) ij ( x ) dx , what defines the (con-ditional) probability density distribution p ( ℓ ) ij ( x ). In par-ticular, p (1) ij ( x ) is the nearest-neighbor distribution. Thedistributions p ( ℓ ) ij ( x ) verify the normalization condition m X j =1 Z ∞ dx p ( ℓ ) ij ( x ) = 1 (1) ∗ Electronic address: [email protected];URL: and obey the recurrence relation p ( ℓ ) ij ( x ) = m X k =1 Z x dx ′ p ( ℓ − ik ( x ′ ) p (1) kj ( x − x ′ ) . (2)Its solution in Laplace space is P ( ℓ ) ( s ) = h P (1) ( s ) i ℓ , (3)where P ( ℓ ) ( s ) is the m × m matrix whose elements P ( ℓ ) ij ( s )are the the Laplace transforms of p ( ℓ ) ij ( x ).The total probability density of finding a particle ofspecies j , given that a particle of species i is at the origin,is obtained as ρ j g ij ( x ) = p ij ( x ) = ∞ X ℓ =1 p ( ℓ ) ij ( x ) , (4)where g ij ( x ) is the pair correlation function. In Laplacespace, G ij ( s ) = 1 ρ j P ij ( s ) , P ( s ) = P (1) ( s ) · h I − P (1) ( s ) i − , (5)where use has been made of Eq. (3). Therefore, theknowledge of the nearest-neighbor distributions { p (1) ij ( x ) } suffices to obtain the pair correlation functions { g ij ( x ) } .Note that the Fourier transform e h ij ( k ) of the total cor-relation function h ij ( x ) ≡ g ij ( x ) − G ij ( s ) of g ij ( x ) by e h ij ( k ) = G ij ( ık ) + G ij ( − ık ), where ı is the imaginary unit.It can be proven that the nearest-neighbor distributionpossesses the following explicit form [4, 5]: p (1) ij ( x ) = ρ j K ij e − βφ ij ( x ) e − ξx , (6)where β = 1 /k B T and ξ = βp , k B , T , and p being theBoltzmann constant, the temperature, and the pressure,respectively. The Laplace transform of Eq. (6) is P (1) ij ( s ) = ρ j K ij Ω ij ( s + ξ ) , (7)where Ω ij ( s ) denotes the Laplace transform of e − βφ ij ( x ) .To close the problem, one needs to determine the am-plitudes K ij = K ji and the damping coefficient ξ . A con-venient way of doing so is by enforcing basic consistencyconditions. Note first that the normalization condition(1) for ℓ = 1 is equivalent to m X j =1 P (1) ij (0) = 1 . (8)Next, since lim x →∞ g ij ( x ) = 1, one must havelim s → sG ij ( s ) = 1 . (9)A subtler consistency condition [4] dictates thatlim x →∞ p (1) ij ( x ) /p (1) ik ( x ) must be independent of thechoice of species i . From Eq. (6) this implies that K ij K ik = independent of i. (10)Equations (8)–(10) are sufficient to obtain K ij and ξ .To be more specific, let us consider the case of a binarymixture ( m = 2). Thus, Eq. (5) yields G ( s ) = Q ( s ) [1 − Q ( s )] + Q ( s ) ρ D ( s ) , (11) G ( s ) = Q ( s ) [1 − Q ( s )] + Q ( s ) ρ D ( s ) , (12) G ( s ) = Q ( s ) √ ρ ρ D ( s ) , (13)where Q ij ( s ) ≡ q ρ i /ρ j P (1) ij ( s ) = √ ρ i ρ j K ij Ω ij ( s + ξ ) , (14) D ( s ) ≡ [1 − Q ( s )] [1 − Q ( s )] − Q ( s ) . (15)The behavior of Q ij ( s ) for small s is Q ij ( s ) = √ ρ i ρ j K ij (cid:2) Ω ij ( ξ ) + Ω ′ ij ( ξ ) s + O ( s ) (cid:3) , (16)where Ω ′ ij ( s ) is the first derivative of Ω ij ( s ). Applicationof Eq. (8) yields K = 1 − ρ K Ω ( ξ ) ρ Ω ( ξ ) , (17) K = 1 − ρ K Ω ( ξ ) ρ Ω ( ξ ) . (18)Next, Eq. (9) implies ρ K Ω ′ ( ξ ) + ρ K Ω ′ ( ξ ) + 2 ρ ρ K Ω ′ ( ξ ) = − . (19) Finally, Eq. (10) becomes K K = K . (20)Equations (17)–(20) constitute a set of four independentequations whose solution gives K , K , K , and ξ .Inserting Eqs. (17) and (18) into Eqs. (19) and (20) onegets K = 1 ρ ρ Ω ( ξ ) 1 + ρ L ( ξ ) + ρ L ( ξ ) L ( ξ ) + L ( ξ ) − L ( ξ ) , (21)1 − ρK Ω ( ξ ) + ρ ρ (cid:2) Ω ( ξ ) − Ω ( ξ )Ω ( ξ ) (cid:3) K = 0 , (22)where we have called L ij ( s ) ≡ Ω ′ ij ( s ) / Ω ij ( s ) and ρ = ρ + ρ is the total density. Substitution of Eq. (21)into Eq. (22) yields a single equation for ξ , which in gen-eral is transcendental. Once solved, the coefficients K ij are obtained from Eqs. (17), (18), and (21). The exactpair correlation functions are then entirely determined inLaplace space through Eqs. (11)–(15).In the particular case of nonadditive hard rods, onehas e − βφ ij ( x ) = Θ( x − σ ij ), where Θ( x ) is Heaviside’sstep function, so thatΩ ij ( s ) = e − σ ij s s , L ij ( s ) = − σ ij − s − , (23) Q ij ( s ) = √ ρ i ρ j K ij e − σ ij ( s + ξ ) s + ξ . (24)The constraint to nearest-neighbor interactions impliesthat σ ij ≤ σ ik + σ jk for all { i, j, k } . In the binary case thisamounts to 2 σ > max( σ , σ ). The recipe describedby Eqs. (17), (18), (21), (22), and (23) for the thermo-dynamic quantity ξ = βp and the amplitudes K ij , andby Eqs. (11)–(15) and (24) for the structural quantities G ij ( s ) are easy to implement. In order to go back to realspace and obtain the pair correlation functions g ij ( x ) onecan use any of the efficient numerical schemes describedin Ref. [6]. On the other hand, the simplicity of Eq. (24)allows one to get a fully analytical representation. Notefirst that1 D ( s ) = ∞ X m =0 (cid:2) Q ( s ) + Q ( s ) + Q ( s ) − Q ( s ) Q ( s ) (cid:3) m . (25)When Eq. (25) is inserted into Eqs. (11)–(13), one canexpress G ij ( s ) as linear combinations of terms of the form Q n ( s ) Q n ( s ) Q n ( s ) = e − a ( s + ξ ) ( s + ξ ) n ( ρ K ) n + n / × ( ρ K ) n + n / , (26)where a ≡ n σ + n σ + n σ and n ≡ n + n + n . The inverse Laplace transforms g ij ( x ) = L − [ G ij ( s )] are readily evaluated by using the property L − (cid:20) e − a ( s + ξ ) ( s + ξ ) n (cid:21) = e − ξx ( x − a ) n − ( n − x − a ) . (27) g ( x ) FMT Exact g ( x ) g ( x ) x/ FIG. 1: (Color online) Bulk pair correlation functions g ij ( x ) for a one-dimensional binary hard-rod mixture with σ /σ = 2, σ /σ = 15 /
8, and ρ = ρ = σ − /
4. Thesolid lines are the exact results and the dashed lines are theFMT predictions of Ref. [2].
It is important to realize that if one is interested in dis-tances x smaller than a certain value R , only a finite numbers of terms contribute to g ij ( x ), namely those with { n , n , n } such that n σ + n σ + n σ < R .In particular, for the most nonadditive case considered inRef. [2], i.e., σ /σ = 2 and σ /σ = 15 /
8, only those terms satisfying 8 n + 16 n + 15 n <
80 are neededfor x < σ . Moreover, g ij ( x ) = ρ − j p (1) ij ( x ) = K ij e − ξx in the first shell, i.e., for σ ij < x < σ ij + ∆ ij , where∆ = min( σ , σ − σ ), ∆ = min( σ , σ − σ ),and ∆ = min( σ , σ ).Let us consider a specific system with σ /σ = 2, σ /σ = 15 /
8, and ρ = ρ = σ − /
4. The cor-responding solution of the transcendental equation for ξ is ξ ≃ . σ − , so that βp/ρ ≃ . K ij and the con-tact values g ij ( σ + ij ) are K ≃ . K ≃ . K ≃ . g ( σ +11 ) = g ( σ +22 ) ≃ . g ( σ +12 ) ≃ . g ( σ +11 ) = g ( σ +22 ) iscommon to all the equimolar cases ( ρ = ρ ), since thenEqs. (17) and (18) imply that K Ω ( ξ ) = K Ω ( ξ ).Figure 1 compares the three exact bulk correlation func-tions g ij ( x ) with those predicted by the FMT proposedin Ref. [2]. The discrepancies are similar to those foundin Ref. [2] between Monte Carlo simulations and FMT.It must be emphasized that the scheme (5)–(10) pro-vides the exact bulk correlation functions for a one-dimensional mixture in the absence of external fields.The more general problem addressed in Ref. [2], namelythe excess free energy as a functional of the inhomoge-neous densities, is much more complicated and, to thebest of my knowledge, its exact solution is not known.On the other hand, the exact density profiles ρ j ( x ) in-duced by external potentials V j ( x ) can be obtained undercertain conditions. The trick consists of assuming thatone of the species (here labeled as i = 0) has a vanish-ing concentration ( ρ = 0) and interacts with the otherspecies via the potentials φ j ( x ) = V j ( x ). The knowledgeof the bulk correlation functions g ij ( x ) (with ρ j → ρ bulk j )can then be exploited to get ρ j ( x ) = ρ bulk j g j ( x ). Theimportant limitation, however, is that V j ( x ) must repre-sent the potential exerted by a wall that acts only on itsnearest particles.To conclude, it is expected that the exact solutions forone-dimensional homogeneous systems derived elsewhere[3, 4, 5] and summarized in this paper can be useful asbenchmarks to construct, test, and refine approximatetheories like the FMT of Ref. [2]. This would allow oneto gain some illuminating insight into the subtleties anddifficulties of the problem of interest, which can be help-ful in its extension to the more realistic case of three-dimensional systems. Acknowledgments
I am grateful to M. Schmidt for kindly providingthe FMT values represented in Fig. 1. This work hasbeen supported by the Ministerio de Educaci´on y Cien-cia (Spain) through Grant No. FIS2007–60977 (partiallyfinanced by FEDER funds) and by the Junta de Ex-tremadura through Grant No. GRU07046. [1] Y. Rosenfeld, Phys. Rev. Lett. , 980 (1989).[2] M. Schmidt, Phys. Rev. E , 031202 (2007).[3] Z. W. Salsburg, R. W. Zwanzig, and J. G. Kirkwood, J.Chem. Phys. , 1098 (1953).[4] J. L. Lebowitz and D. Zomick, J. Chem. Phys. , 3335 (1971).[5] M. Heying and D. S. Corti, Fluid Phase Equil. , 85(2004).[6] J. Abate and W. Whitt, Queuing Systems10