Incorporation of the intensive and extensive entropy contributions in the disk intersection theory of a hard disk system

The one-body free volume, which determines the entropy of a hard disk system, has extensive (cavity) and intensive (cell) contributions. So far these contributions have not been unified and considered separately. The presented theory incorporates both contributions, and their sum is shown to determine the free volume and partition function. The approach is based on multiple intersections of the circles concentric with the disks but of twice larger radius. The result is exact formulae for the extensive and intensive entropy contributions in terms of the intersections of just two, three, four, and five circles. The method has an important advantage for applications in numerical simulations: the formulae enable one to convert the disk coordinates into the entropy contribution directly without any additional geometric construction. The theory can be straightforwardly applied to a system of hard spheres.

considered separately.The presented theory incorporates both contributions, and their sum is shown to determine the free volume and partition function.The approach is based on multiple intersections of the circles concentric with the disks but of twice larger radius.The result is exact formulae for the extensive and intensive entropy contributions in terms of the intersections of just two, three, four, and five circles.The method has an important advantage for applications in numerical simulations: the formulae enable one to convert the disk coordinates into the entropy contribution directly without any additional geometric construction.The theory can be straightforwardly applied to a system of hard spheres.

I. INTRODUCTION.
Spheres and their arrangement in space has been playing a very notable role in practical life and, in particular, in mathematics and physics.Recently Maryna Vjazovska received Fields medal for solving the problem of dense hard sphere packing in 8 [1] and 24 [2] dimensions.In physics, however, the analytical achievements are more modest.In statistical physics we are interested in a random packing problem of hard spheres in a macroscopic volume.Over more than a century, the idea to model molecules as hard spheres has been widely used in the theory of liquids [3][4][5].The model of spheres, interacting only by their hard cores, plays the role similar to that of Ising's model in the theory of magnetism, but despite apparent simplicity, the behavior of hard sphere systems is so complex mathematically that no exact analytical result has been obtained in the physical dimensions 3 and 2.
Under these circumstances, the numerical Monte Carlo and molecular dynamics approaches have become the main tools in the study of 3D hard sphere and 2D hard disk (HD) systems (see review [6] and numerous references therein).However even using the modern powerful numerical methods one encounters the fundamental theoretical problem of computing the main thermodynamic potential of hard sphere and HD systems, the entropy.Although the problem of hard spheres is very similar to that of HDs, for simplicity, in this paper the presentation will be mainly related to HDs.
The potential energy of a HD system is zero and the entropy gives the total thermodynamic information and, in particular, equation of state and possible nuances of the phase behavior as a function of density which is the single parameter of the system state.The numerical simulations consist of producing different independent configurations of disks' coordinates which is the task input, and then certain related theory must provide calculation of the entropy, equation of state, and other quantities of interest which are the task outcome.
In principle, the ultimate theory must give the outcome directly from the disks' coordinates (with the consequent averaging over different configurations), but actually the available theoretical methods relate disks' coordinates with the expected outcome only through intermediate and quite sophisticated geometrical constructions.The main elements of such constructions are the so-called free volume, cavity, "private" one-disk cell, and the surface thereof.For a brief review and presenting the main idea of this paper, I first introduce these quantities.
A HD of radius σ/2, the core, is supplemented with a concentric circle of radius σ which is called here σ circle.The cores cannot overlap, but their σ circles can and are transparent for cores.A configuration of N HDs in a 2D volume V consists of N nonoverlapping cores and N connected σ circles which can overlap, Fig. 1.The free volume V N of a disk in a given configuration of an equilibrium N HD system is the volume accessible for its center in this configuration, which is the total V minus union of the rest N − 1 σ circles; V N can comprise more than one disconnected pieces (in Fig. 1, a single piece is shown).The cavity C N in a HD system of N HDs is the area where an additional HD can be inserted which is total V minus union of all N σ circles.The private cell c N of a disk in a HD system of N HDs is the free volume of this disk V N without the cavity C N (stroked area), Fig. 1; if the cavity is zero and no additional HD can be inserted, then private cell is the total free volume of the disk.The free volume, cavity, and private cell in the case of a single free area are illustrated in Fig. 1.It is seen that the division on cavity and private cell of the dashed disk depends on the position of this disk, but the total free volume V N = C N + c N does not depend on its coordinate.The average free volume V N N , average cavity C N N , and average private cell per disk c N N are those for a single configuration of N disks averaged over the configurations of the equilibrium system of same N disks.The perimeters (surfaces in 3D) of all the three volumes introduced are complex lines (surfaces) whose shapes and lengths are not in a one-to-one relation with the volume size.In the above definition, we emphasized that the quantities related to an N disk system are defined for the equilibrium system of same N disks.
In 1977 Speedy introduced the spare volume and defined it like that: "The spare volume V s of an assembly of N spheres of diameter σ in a volume V is defined as the average over configurations of the volume which is not within σ of a sphere center,..., the probability that another sphere can be placed at a random point in the assembly" [7].Clearly, this is equivalent to a cavity available for an additional, N + 1 st disk in a system of N disks, and which is averaged over configurations of N HDs.As a result, Speedy related the partition function (PF) of a HD system with the product of cavities C N ′ N ′ −1 in the systems of number of disks reduced by one, i.e., of N − 1, N − 2, ..., 1, 0 disks, which are averaged over the equilibrium systems of respectively N − 1, N − 2, ...1, 0 disks in the same volume V [7].
The way this PF was obtained was going back to the earlier results by Adams [8] and Anrews [9] which had in turn been inspired by Widom's approach [10].But Speedy was the first to address the calculation of the spare volume in a HD system in terms of disks' intersections [11] which has greatly influenced the further development of this area [12].Later Speedy explicitly shifted from the nomenclature of spare volumes to cavities [11].Thus, Speedy approach does not incorporate the free volumes but only their parts, cavities.Actually, however, the formula for the PF has not been further employed.Instead, in 1980 [11,13] Speedy proposed the equation of state which relates the pressure with the ratio (average cavity volume)/(average cavity surface area).Since then different geometrical methods of finding the cavities and their surface have become the main emphasis in the ongoing studies of hard particle systems [14][15][16][17][18][19][20][21].However, the complex shape and connectedness of cavity space make precise measurement of the quantities, characterizing them, very difficult which resulted in new and new geometrical constructions that are highly nontrivial to implement [14-16, 19, 20].The main problem of this approach has been that, even for densities far from the crystallization density, cavities become so rare that finding them was sometimes called a task futile [16].The root of this problem is that a cavity in an N HD system has been mainly computed as that of N th disk in an equilibrium system of N − 1 HDs.But whereas in an equilibrium system of N HDs the place for N th disk is guarantied, in a dense equilibrium system of N − 1 HDs, a place for an additional N th disk is a very rare event.The paradox is that relating cavity with a system of a smaller number of disks when considering an N disk system, one finds no place for N th disk.This problem was pointed out by Schindler and Maggs who had to invent a modified numerical algorithm for finding cavities and distinguishing them from free volumes [20].
The cavity is an extensive quantity that scales with the number of disks and volume.
At the same time, even when an additional, N + 1 st disk cannot be inserted, the original N disks can vibrate in their cages created by their neighbors.This implies that the total entropy is nonzero and the volumes of such cages are its source even in the absence of cavities which is the case of densities approaching that of crystallization.These cages are what is called private disk cells in the cell models [22][23][24][25].Even before Speedy's publication [7], Hoover and coworkers [22][23][24] correctly argued that along with the extensive cavity volume, there must be an intensive one which scales as V /N and consists of individual single-disk cells.The free volume is the sum of these two terms, and when the cavity is getting smaller and smaller, the total free volume reduces to the volume of individual cells.Based on this important idea, as early as in 1972, Hoover, Ashurst, and Groover showed that the pressure can be expressed via the ratio of the average of the free volume to the average of its surface which incorporates the cell contribution [23].The cell model can quantitatively describe the HD equation of state near freezing density in numerical simulations [22][23][24] and even allows one to obtain qualitatively accurate results analytically [25].The cell models also have the problem of describing the cell distribution with a strong geometric component, but the main problem is to connect the one-body cell approach with a many-body one, i.e., to incorporate the intensive and extensive free volumes and entropy contributions in a unified theory.
In this paper I present such a theory in which the extensive and intensive terms have the same status and are computed in the framework of the same approach.I develop the method of multiple intersections of σ circles and, in terms of their intersection volumes, express the free volume, its extensive and intensive parts, and the PF of a system of N HDs.
Due to the fact that only up to five σ circles can intersect without overlapping of their cores, the theory needs only four quantities, i.e., the intersection volumes of two, three, four, and five σ circles.These four quantities are fully specified by the disks coordinates and can be calculated analytically using formulae obtained in [26][27][28].The theory does not resort to a system of reduced number of HDs and gives the values of a cavity, private cell, and total free volume in a system of N HDs only in terms of this very N HD system.No geometrical or any other intermediate constructions appear between the input, disks' coordinates, and the output, quantities of interest, and the only source of inaccuracy is that of the numerical simulations.
The paper is organized as follows.In Sec.2, the method of multiple disk intersections is introduced and the general formula for the free volume of a single disk is derived.Sec. 3 is devoted to the connection between the single disk volume and many-body description.
First Speedy-Widom's approach is used to derive the PF.It is shown that this PF is exactly Speedy's PF [11] in the form of product of cavities in the equilibrium systems of the reduced number of disks.Next it is shown that, in the thermodynamic limit, the correct PF is the product of the free volumes averaged in the proper systems.In Sec.4, the formulae relating the extensive and intensive free volume contributions with intersections of σ circles are obtained and their application to the average values is explained.In Sec.5, the analytical computation of all the intersections of σ circles and the intensive and extensive terms for the densely packed triangular HD lattice is presented in detail.It shows that both terms in this state vanish identically.Final Sec.6 is a brief conclusion.The white area in between is the free volume of N th disk as its center can be anywhere in this area.Only the stroked fraction of the white area is the cavity in the N HD system.The clear fraction of the white area is not cavity, but is available for the center of dashed disk center.This clear white area is equal to the area of dashed σ circle, πσ 2 , minus its fraction overlapped by other σ circles.

II. HARD DISK INTERACTION AND MULTIPLE DISK INTERSECTIONS.
The configuration PF of a 2D system of N particles in the 2D volume V with paiwise interaction U ij is the following integral: where x i are the two component vectors of disks' coordinates, dx N = dx 1 ...dx N , and we separated the x N integral.For HDs of the radius σ/2, the potential U ij = ∞ for x j within the circle of radius σ centered at x i , and U ij = 0 for x j outside this circle.We introduce a circle B i , i = 1, ..., N − 1 : By definition, the indicator τ i (B i ) of the set of points x N ∈ B i is The product of n indicators of n different sets is the indicator of the intersection set shared by all of them.Such product of two indicators, which is nonzero only if the two related circles intersect, can be defined as Then, by definition, where n k=1 B i k is the set of points x N shared by all circles B i , i.e., their intersection.Now the utmost right exponential in Z N (1) for the HD interaction can be presented in terms of τ 's.It is easy to see that the HD interaction is equivalent to the following formula: This formula shows that the center of any other HD j = i cannot enter the circle B i centered at x i which has the radius σ twice the HD radius σ/2.The product of two exponentials is Similarly, exp − 1 2 It is well-known that more than five circles of a radius σ cannot intersect without intersection of their cores of radius σ/2 (six HDs intersect at a single point).As a result, all the products of six and more τ 's do not contribute to the above sum.Thus, the last term in the sum ( 8) is τ ijklm which corresponds to the intersection of five circles B, and this formula greatly simplifies: exp − 1 2 Denote by µ the volume (i.e., the measure in the 2D space, surface area) of a set : Then the last integral in Z N (1) reduces to the following form: where To exclude configurations x 1 , ..., x N −1 in which there are disks whose hard cores overlap, we introduced the hard core indicator θ N (x 1 , ..., x N −1 ) in a configuration x 1 , ..., x N −1 : ) is the integral over all possible locations x N of N th disk in the system of N HDs for given fixed positions of the rest N − 1 HDs.In other words, this is the integral over the volume accessible to the N th disk in the N HD system, the N th disk's free volume.It is expected to be the total area V minus the total area covered by the circles B i , i.e., V minus the union ∪ N −1 i=1 B i .And the formulae ( 10) and ( 11) do describe the integral over exactly this area.The formula (11) can be presented in the form Indeed, the expression (11) for V excl is exactly the volume of the union ∪ N −1 i=1 B i known in the set theory, which is restricted to the intersections of maximum five sets B i and implicitly excludes core overlapping.Thus, we have expressed the single N th particle integral in terms of the intersections of the circles B connected to the other N − 1 HDs.In the next section we show how this result can be incorporated in the many-body theory.In order to simplify formulae, in what follows we omit the explicit presence of the indicator θ N and assume that only acceptable configuration are considered whereas all configurations with any core overlap give zero contribution to V N .

III. MANY-BODY PROBLEM
A. The Speedy-Widom approach: PF is the product of cavities To consider implementation of the obtained result in Z N we first follow Widom's idea [10] and transform the PF like that.Divide and multiply Z N by the PF Z N −1 for N − 1 HDs and introduce the distribution function (DF) f N −1 of N − 1 disks' coordinates in the equilibrium system of N − 1 HDs: Then one has: where V N N −1 is the average volume accessible for N th disk in the equilibrium system of N − 1 disks, i.e., the cavity C N in a system of (N − 1) HDs.Continuing along this line by introducing the equilibrium distribution functions of a lower and lower number of HDs, one arrives at the following formula for the PF: The result is the product of cavities, the average empty voids in the equilibrium systems of N − N ′ − 1 HDs into which the (N − N ′ ) th HD can be inserted, for all N ′ from 0 to N − 1.This is the second Speedy's result [11] which can rightfully be called Speedy-Widom PF.
This result is behind the idea which has been the pivot of practically all of the search for the equation of states based on the notions of cavity and spare volume.The problem that has been encountered in these studies is that, already at liquid densities, cavities become so rare and finding them in computer simulations so difficult that it was even dubbed a task futile [16].This practically means that in sufficiently dense HD and hard sphere systems that are still far from their close packing, Z N is zero, the entropy is minus infinity, and higher densities are inaccessible because the N th hard core particle cannot be inserted in such dense systems.This situation is paradoxical as we started to study a system of N particles but found that one particle has no space in this system.Below I shall resolve this paradox and derive the consistent theory in which all HDs have guarantied space in an N HD system.

B. PF is the product of single-disk free volumes
How could it happen that the rightful guarantied space of N th HD got lost in a system of N HDs?To answer let us compare the averaging ( 14) of V N (x 1 , ..., x N −1 ) with the DF f N −1 (x 1 , ..., x N −1 ) of an equilibrium system of N − 1 HDs, eq.( 13), with the integral over V N (x 1 , ..., x N −1 ) in the second line of eq. ( 1).The PF f N −1 (x 1 , ..., x N −1 ) is that in a system of N − 1 disks and is established without any effect of an additional disk N of which f N −1 (x 1 , ..., x N −1 ) never knew.Hence, the factor V N does not influence f N −1 , its role is passive and reduces to guiding the external disk along the maze formed by the N − 1 "naitive" HDs.In particular, if the maze leaves no place for an external disk N, the integral V N N −1 = 0.In a dense system, this situation is most probable as the most probable distribution of N − 1 disk is uniform.In contrast, the integral (1) over V N is that over x N in a system of N HDs.Now any collection of N − 1 disks does know about the presence of another disk which has the same "naitive" status, and is therefore adjusted as to accomodate it with the probability one.In such system, the most probable situation is also a uniform distribution, but now of all N disks (so that N + 1 st disk could have found no place, but now this is irrelevant).Thus, f N −1 has "no idea" of the N th disk whereas all the N − 1 coordinates, the arguments of V N in (1), do keep knowledge of the N th disk to which they cannot approach to a distance below σ.To summarize, in the PF (15), the average of V N in the original statistical integral was replaced by a different average (14) with the equilibrium

IV. RELATION BETWEEN SINGLE-DISK FREE VOLUME AND MULTIPLE DISKS' INTERSECTIONS
A. The total excluded and free volume for a single HD.
The free volume of a disk in an N disk system is the cavity left by the rest N − 1 disks.
We see that the practical definition of the free volume is related to removing N th disk from the N HD system to which it belongs.However, as we showed above, dealing with such objects one should be careful.Therefore, it is both convenient and essential to relate the expression for the free volume in a system of N disks in terms of the equilibrium system of this very number of disks N.Moreover, the PF ( 16) makes this task crucial.
The formula (10) for the free volume of N th disk in a system of N disks is correct but inconvenient because it is related to the reduced distribution function of N − 1 disks which is highly nontrivial.Let us consider instead directly the system of N disks.Assume that center of disk N is at x ′ and let us find the free volume for this disk.Above we noted that the division on cavity and private cell depends on x ′ but their sum, which is what we actually need, does not.Our task is thus to find this sum in terms of all N disks' intersections.This sum is represented by the total white area in Fig. 1, i.e., the cavity left by the rest N − 1 disks, but in the N disk system!The excluded area of points x N ∈ V created by all N disks, N i=1 B i , exceeds the excluded area due to the rest N − 1 disks by the area of the σ circle B N , but without all its areas B N ∩ B i already covered by other N − 1 disks (because these areas should not be counted twice), Fig. 1.Thus, the single disk excluded volume V N,excl , where x ′ cannot enter, is the integral (12) over this inaccessible area: Making use of eq.( 10) and (11), one obtains the formula which expresses the free volume of a single disk i 0 in a system of N HDs via multiple intersections of N σ circles: This formula shows that the total free volume for the disk i 0 is equal to the cavity in the N disk system plus πσ 2 minus the area of intersection of disk i 0 with the rest N −1 disks.Both formulae ( 17) and ( 18) do not refer to any system of N − 1 HDs: the upper summation limit N − 1 in the excluded volume in the form ( 12) is replaced by N, the second term in (17) also determines the intersection areas in the N HD system as indicated by the presence of B N .In the thermodynamic limit the above V N tends to the constant V N = lim N,V →∞ V N which is the function of N/V.It can be expressed in terms of the following average values v N,n which are different intersections of an individal σ circle averaged over all σ circles: where index N indicates that the average is computed in an N HD system and another index indicates the number of intersecting σ circles.We remember that for a fixed i, the maximum number of terms in the above sums is five so that the summations are actually not extensive.In terms of simulation results, the procedure of finding v N,n consists of computing all intersection of each disk with other n − 1 σ circles and then averaging over the results.
It is essential that the areas of all intersections of our interest are uniquely determined by the coordinates of the participating disks and can be computed analytically making use of the formulas derived in [26][27][28].The detailed computation of the intersection volumes v N n in the densely packed triangular HD lattice is presented in section 5.

B. Extensive and intensive free volume terms: the cavity and the private cage cell
Separating the extensive and intensive terms in (18) in the context of (19), one finally obtains: where V The denominators in V N,exten reflect the fact that in the sum over all disks, any intersection of n circles is counted n times; at the same time, the intersections of a single circle in V N,inten are counted only once.The free volume contains two contributions, the extensive V N,exten = C N N , which is the average cavity volume in the N HD system, and the intensive V N,inten = c N N , which is the guarantied volume of a cell connected to or, better to say, containing a single HD.This cell of size c N N is available for any single disk even if the cavity practically vanishes, which is the case of high densities.We say practically because in an infinite system a fluctuation in the form of cavity, however small its probability be, still must exist.This situation is expected to be similar to that with the so-called windowlike defects in a quasi-one-dimensional HD system [33]: in the thermodynamic limit, the probability to have such a "window" in the crystalline zigzag vanishes only at the close packing [32][33][34].Coming back to our two-dimensional HD system, we see that at densities close to crystallization densities, the intensive term, which is fully negligible at lower densities where cavity dominates, becomes the only source of entropy.In that case V N is the volume of a cell, c N N , in which a single HD is caged by its neighbors.Note that the size of this cell, eq.( 22), is determined not only by the central disk's next neighbors but also by next next neighbors that can contribute to the intersection if their centers are within the σ circle of the central disk.Such intersections with the next and next next neighbors has been taken into account, at a phenomenological level, in the cell models of the equation of state for HD systems [22][23][24][25].It is this cell's size that was the main task of the cell models: the counterpart of V N,inten was computed in configurations that were assumed to contribute the most (usually these were the symmetric configurations related to the triangular lattice).Our formulas show the way to find the cell size as thermodynamic average.Moreover, formulas (21) and (22) give the total free volume in an N HD system so that a) the cavities and cells are not considered separately, b) they can be computed directly from disks' coordinates even analytically, and c) only the original, the very same N HD system needs to be considered.
Thus, the idea of both extensive and intensive free volume contributions put forward by Hoover and Ree [22] and Hoover, Ashurst, and Grover [23] has been embodied in our theory

FIG. 1 :
FIG.1: Fragment of a system of N HDs.The N − 1 HDs of radius σ/2 are represented by dark circles and the connected concentric σ circles by light circles.The N th disk and circle are shown by dashes.The white area in between is the free volume of N th disk as its center can be anywhere