Statistical mechanics of fluids confined by polytopes: The hidden geometry of the cluster integrals
This paper, about a fluid-like system of spatially confined particles, reveals the analytic structure for both, the canonical and grand canonical partition functions. The studied system is inhomogeneously distributed in a region whose boundary is made by planar faces without any particular symmetry. This type of geometrical body in the -dimensional space is a polytope. The presented result in the case of gives the conditions under which the partition function is a polynomial in the volume, surface area, and edges length of the confinement vessel. Equivalent results for the cases are also obtained. Expressions for the coefficients of each monomial are explicitly given using the cluster integral theory. Furthermore, the consequences of the polynomial shape of the partition function on the thermodynamic properties of the system, away from the so-called thermodynamic limit, is studied. Some results are generalized to the -dimensional case. The theoretical tools utilized to analyze the structure of the partition functions are largely based on integral geometry.
Thermodynamic properties of fluids are relevant to biology, chemistry, physics and engineering. In most cases of interest the fluid system is inhomogeneously distributed in the space, confined to a region of finite size and constituted by a bounded number of molecules. A typical example of this kind of systems is that of fluids confined in pores. They constitute a prototypical inhomogeneous system which occupies a small region of the space and may involve a small number of particles. It is well known that the spatial distribution of a fluid confined in a small pore may follow or not the symmetry of the cavity Sartarelli and Szybisz (2010); Szybisz and Sartarelli (2011), and that its properties may be strongly influenced by the geometry of the container. To study the thermodynamic properties of these systems it is customary to introduce several strong assumptions or approximations that simplify the analysis. An usual approach is to treat the system as it were homogeneous (which necessarily implies that the system completely fills the space, and thus, involves infinitely many particles). A second usual approach assumes that the inhomogeneous fluid is spatially distributed in several regions, each one with homogeneous properties, while the inhomogeneous nature of the system is concentrated in regions with vanishing size (surfaces, lines and points). Frequently, this scenario is complemented with the assumption that the spatial distribution of the fluid involves continuous translational and/or rotational symmetries. Finally, a third approach only assumes that the inhomogeneous fluid takes spatial configurations with those symmetries. In general, the assumed symmetric distribution of the fluid could be attained spontaneously (as in the case of free drops or bubbles) or could be induced by an external potential as it is the case of confined system that are constrained to regions with simple symmetry: a semi-space, a slit, an infinite cylinder, or a sphere (for example in wetting and capillary condensation, phenomena). Anyway, the continuous symmetries (translations and/or rotations) play an important role providing the bases to identify the extensive and intensive magnitudes which enable the development of thermodynamic theory Callen (1985).
Moreover, thermodynamics only provides an incomplete set of relations between intensive and extensive magnitudes which must be complemented with other sources of information to obtain the thermodynamic properties of a given fluid system. In any case, these sources, that may be experimental, theoretical, or based on numerical simulation, also involve assumptions or approximations related to the existence of continuous symmetries. In this sense, our statistical mechanical and thermodynamical approach to the study of inhomogeneous fluids appears to be intrinsically entangled with some hypothesis about the constitution and behavior of the system under study 111See Chapter 6 in Ref.Hansen and McDonald (2006) an also Ref.Henderson (1992).. Particularly, I refer to three hypothesis concerning the fluid system: the symmetry of its spatial distribution, the large volume occupied and the large (and unbounded) number of particles involved. Of course, one may ask about if those hypothesis are or not a central part of the theories. What happens with inhomogeneous fluid-like systems which do not necessary attain spatial distribution with simple symmetry and/or occupy small regions of the space and/or are constituted by a small number of particles? Can we apply statistical mechanics and thermodynamics to study their equilibrium properties? How can we do that? The analysis presented below attempts to advance in the understanding of these questions. From a complementary point of view, this work also deals with the long standing objective of finding fluid-like systems that are exactly solvable, i.e., its partition function integral can be integrated and thus transformed into an analytic expression. Each of these systems provides the unique opportunity of testing some of the fundamental hypothesis of the theoretical framework.
This paper is devoted to analyze, using an exact framework, few- and many-body fluid like systems confined in cavities without continuous symmetry. In particular, we focus on their partition function and thermodynamic magnitudes as functions of the spatial set that defines the region in which particles are allowed to move. In fact, it is shown that under certain conditions the partition function of the fluid-like system is a polynomial function of certain geometric measures of the confining cavity, like its volume and surface area. The adopted approach is theoretical and exact, being our core results largely based on integral geometry. The rest of the manuscript is organized as follows: In Section II it is presented the general statistical mechanics approach to canonical and grand canonical ensemble with two modifications: the cutoff in the maximum number of particles which enables the analysis of few-body open systems and the generalization of cluster integrals to inhomogeneous fluids. Sections III and IV are devoted to study the structure of cluster integrals with particular emphasis in the case of a polytope-type confinement. There, two theorems and their corollaries are demonstrated, which constitute the main result of the present work (PW). Extensions to other type of confinements are discussed in Section V, while the consequences of the cluster integral behavior in the thermodynamical properties of the system are studied in Sec. VI. The final discussion is given in Section VII.
Ii Partition function of an inhomogeneous system
In this work we consider an open system of at most particles that evolves at constant temperature in a restricted region of the -dimensional euclidean space . The region is the set of points where the center of each particle is free to move and has a boundary . From here on the system will be shortly referred to as the fluid while will be referred to as the region or set where the fluid is confined. In fact, the system is an inhomogeneous fluid due to the existence of . The restricted grand canonical partition function of the fluid is
where is the maximum number of particles that are accepted in . The absolute activity is , being the chemical potential while , and are the inverse temperature, Boltzmann constant and temperature, respectively. is a -degree polynomial in and its -th coefficient is the canonical partition function of the closed system with exactly particles confined in . The grand canonical partition function of the fluid without the cutoff in the maximum number of particles can be obtained from . On the other hand, the canonical partition function of the system with particles is
where is the indistinguishably factor, is the thermal de Broglie wavelength, is the mass of each particle and is the Planck’s constant. Therefore, we can transform Eq. (1) into
where the activity is . Finally, the configuration integral (CI) of a system with particles is
being [alternative dimensionless definitions for and may be obtained by introducing the volume of the system or the characteristic volume of a particle in Eq. (4)]. Here, is the Boltzmann factor related to the spherically symmetric pair potential between the and particles, which are separated by a distance . It will be assumed that has a finite range , being and if . This assumption is not very restrictive because any pair interaction potential can be approximated by truncation at a finite range, e.g. it is frequent to study the Lennard Jones fluid cut and shifted at Shaul et al. (2010); Horsch et al. (2012). The indicator function
is the Boltzmann factor corresponding to the external potential produced by a hard wall confinement. Note that, the integration domain in Eq. (4) is the complete space due to the spatial confinement of the particles in is considered through . The CI is a function of the set and a functional of . itself is given in its full generality by Hill (1956)
where the sum is over all sets of positive integers or zero such that . Eq. (6) shows that is a polynomial in which are essentially the (reducible) Mayer cluster integrals for inhomogeneous systems. This equation is obtained through the following procedure: replace in Eq. (4) each using the identity (this Eq. defines ), distribute the products, and collect all terms of the integrand which consist of groups of particles that conform a cluster, in the sense that they are at least simply connected between them by -functions. Note that our assumption about for implies if showing the spatial meaning of the cluster term. The procedure gives Eq. (6) with
where is a sum of products of functions with that involves all the products of functions, which can be represented as a connected diagrams (clusters) with nodes and bonds. Clearly, depends on , and . If one assume that is very large the usual homogeneous system approximation gives , where is the volume of and s are the Mayer cluster integrals which depend on and .
where the first sum is over all sets with non-negative integers such that and with . From Eq. (9) it is apparent that depends on with , and thus, the s with and the s with involve the same physical information. On the other hand, one can return to the Eqs. (1) - (4) to observe that they can be re-written in this alternative form: replace in Eq. (1) but assume with . In this context Eq. (9) shows that cluster integrals with are not affected by the restriction , being the first affected (because it depends on , which is zero).
Before ending this section we wish to focus on a relevant characteristic of , and functions. Let us define (the set of all the subsets of ), for fixed , and one can write which implies that may depend on the shape of . Clearly, the same argument applies to and which may also depend on the shape of . Here we anticipate the principal result of PW, related with this non-trivial shape dependence, that will be demonstrated in the Secs. III and IV. Thus, we turn the attention to a fluid confined by a polytope . For simplicity we focus in the three dimensional case, i.e. a fluid confined by a polyhedron. If a system of particles that interact via a pair potential of finite range is confined in a polyhedron such that its characteristic length [see Eq. (25)] is greater than (being C a constant) for some integer . Then the -th cluster integral with is a linear function in the variables , and (the length of the edges of ). In fact
where the coefficients and are independent of the shape of while and are functions of the dihedral angles involved. Besides, all the coefficients depend on the pair interaction potential and temperature. Expressions similar to Eq. (10) are also found for the euclidean space with dimension and , while they are conjectured for dimension larger than . Two non-trivial consequences derive from the Eq. (10). On one hand it implies that, if the system of particles confined by involves particles and , then is polynomial on , and . On the other hand it implies that, if the system of particles confined by is open, involves at most particles and , then is a polynomial in , , and .
Iii The properties of some functions related to
In this section we analyze the properties of some many-body functions related to the partial integration of the cluster integral [Eqs. (7) and (8)]. This analysis will be complemented in the next section where we will reveal the linear behavior of . In the following paragraphs several definitions are introduced and two different proofs of the locality and rigid invariance of functions related to the partial integration of are presented. Both proofs are necessary for clarity. In the first approach the mentioned properties of those functions are demonstrated and it is found a cutoff for its finite range, while in the second approach (which is more complex than the first one) the properties are proved and a better bound of the finite range is obtained.
We define, is a local function in over with range (from hereon local with range ) if its value is entirely determined by the set that is where is the ball centered at with radius . This definition can be generalized to functions of several variables in the following way: is said to be a local function in over with range if for and when exist .
Let be a local function with range . We say that is invariant under rigid transformations (the elements of the euclidean group, i.e., any composition of translations, rotations, inversions and reflections) if rigid transformation , with implies . The generalization to functions with many variables is: let be a local function in with range , we say that is invariant under rigid transformations if rigid transformation , implies , with . A direct consequence of the locality with range and rigid transformation invariance of a bounded function is that it attains a constant value for all such that .
is a local function with finite range invariant under rigid transformations.
First Proof: Consider the following -body function
with the position of an arbitrarily chosen particle, and the rule to obtain the -body function from the -body one given by
where and is the coordinate of particle with respect to particle . has range
being if for at least one pair of particles and in the cluster. This property derives from the fact that contains the open simple-chain cluster term that enables that two particles reach the maximum possible separation of all the cluster terms in (there are terms of this type). Even more, one can show that is local with range . Naturally, Eq. (12) shows that is also local with range . The integration in Eq. (13) applied to implies that is local in with range . We proceed by induction. Let us assume that for some , is local with range , then by Eq. (13) is also local with the same range. The procedure continue until the function is reached. Therefore, we find that the functions with are local in with range [even more, we have obtained that are local in with range , for any ].
It is convenient to express the coordinates in terms of the rigid transformed coordinates (related each other by and with the inverse of ) and to change the integration variable to (note that the Jacobian is one). Thus we find
where the integration domain is equal to by hypothesis. Given that is invariant under any rigid transformation applied to the coordinate of the particles one can drop each in Eq. (16) and return to the original form introducing
which is the definition of . This shows that is invariant under rigid transformations. Again, we proceed by induction. Let us assume that , a local function in over with range , is invariant under rigid transformations. Taking into account that for the right hand side of Eq. (13) we can write an expression similar to Eq. (15) but replacing by and by . Turning to transformed coordinates we find
where the integration domain is equal to and we used that . Given that is invariant under rigid transformations we can drop each in the arguments of in Eq. (18), use the finite range of to split the argument into and introduce to obtain
which is the definition of . Therefore, is local of range and rigid transformation invariant. The procedure continue until the function is reached. Therefore, returning to the original coordinates, we obtain that with is local in any () with range and rigid transformation invariant. In particular, for it implies that is local in with range and rigid transformation invariant.
Second Proof: This second proof is based on a more subtle consideration about the particle labeled as in the first proof. We introduce the coordinate of the central particle of the cluster, , and the relative coordinates of the particle with respect to . Again, consider the cluster integral and its simple open-chain cluster term. For we take with (if is an odd number is the position of the middle-chain particle while if is an even number is the position of one of the pair of particles that are at the middle of the chain). We note that is zero if for at least one particle in the cluster. The procedure to obtain for all the other terms in is as follows: for a given cluster of particles take iteratively each pair of particles, separate them to find the maximum possible elongation distance under the condition . Let the more stretchable chain of particles (that could be non-unique) be a chain of particles with end-particles and . Thus, with and for this cluster we can define with . By using this second approach we find that has finite range
Once is identified for each cluster term in we can rename as and follow the procedure developed in the first proof of the theorem.
Based on Eq. (20) along with the assumption that the range of must be a unique function of the maximum elongation length of independently of the parity of we have the following guess
We may mention that Eq. (21) can be demonstrated for the case of being a convex body [by virtue of Eq. (20) one must focus on the case of even ]. The demonstration follows a procedure similar to that used to obtain the Eq. (20), but for the case of even one define . The development made in this section enable to write Eq. (7) as
with two different definitions for . Both definitions and any other possible approach must give mathematically equivalent expressions for .
Iv Integration over a polytope-shaped domain
In order to analyze the implications of the domain’s shape on certain type of integrals it is necessary to introduce some notions about sets . They are summarized in this and the next paragraphs. The closure of is . It is said that is a closed set if . Besides, the interior of is . It is said that is an open set if . There are sets that are neither closed nor open. is a partition of the non-empty set if and, for all with , and . A set in is connected if it cannot be partitioned in two non-empty sets and such that , otherwise it is disconnected. We also introduce the concept of connectedness-based partition of a set; a partition of is the connectedness-based partition of if, for all with , is connected and is disconnected. The following notions of distances are adopted: the distance between the point and the set is with the usual Euclidean distance between points, while the distance between the sets and is . Finally, a set is said to be convex if every pair of points and in are the endpoints of a line segment lying inside .
An affine -subspace is a linear variety of rank in , whether it contains the origin or not. For example, the -dimensional affine subspace is a hyperplane. The affine hull of a set , , is the affine subspace with the smallest rank in which every point of is contained. Given that is a fixed parameter from hereon we will refer to the rank of as the dimension of the affine space of , denoted or simply . For , the sets , and are the relative closure, the relative interior and the relative boundary of , i.e., the closure, interior and boundary of within its affine hull, respectively Schneider (2008).
Even though a polytope is essentially a mathematical entity, in the current work it also has a physical meaning because it serves to characterize the shape of the vessel where particles are confined. This duality makes difficult to start with a simple and satisfactory geometrical description of the kind of polytopes that are relevant for our purposes. We begin a somewhat indirect approach that ends in the definition of the set of polytopes that have simple boundary (a well behaved one), in the sense that the boundary can be dissected in its elements or faces. To make further progress it is convenient to introduce the convex polytopes which, from a geometrical point of view, are simpler than the general polytopes. A convex polytope in is any set with non-null volume given by the intersection of finitely many half-spaces Schneider (2008) (this definition includes unbounded, closed and non-closed polytopes). On the other hand, a connected and closed (CC) polytope is the connected union of finitely many closed convex polytopes. We define, the CC polytope is a simple-boundary (SB) polytope if for each such that , both sets and are topologically equivalent (homeomorphic) to an open ball. It is clear that a closed convex polytope is also a CC SB polytope. In this sense, we say that the boundary of a CC SB polytope and the boundary of a convex polytope are locally equivalent. The SB condition excludes some degenerate or pathological cases e.g. a polytope with two vertex or two edges, in contact. On the other hand, the definition of SB polytopes includes bounded and unbounded polytopes, non-convex polytopes, polytopes with holes or cavities and many kind of faceted knots embedded in . Let be the class of CC SB polytopes in .
For a given polytope we focus on the partition of its boundary based on its faces. As before, we treat first the convex polytope case. Let be a convex polytope and a hyperplane. If , and with , then we say that is a closed -face of Grünbaum and Shepard (1969). One can demonstrate that the set whose elements are all the closed -faces and relopen -faces with , of is a partition of . Other polytopes, non-necessarily convex, may admit a similar face-decomposition of its boundary. Given a polytope with face-decomposable boundary, we introduce the notation () for the -th -dimensional relopen face of (from here on an open -face), while is the -th closed -face. The closed -faces are the vertex of the polytope. The face-based partition of is with for every . Note that is , i.e., the -element of . Besides, we define
as the set of points in that lies in some of the -faces of , with a partition of . Furthermore,
with going from to , where is a partition of . Given any open face, the corresponding closed face is its closure. Furthermore, given any closed -face with , its relint is the corresponding open -face (an open face relative to its affine hull). Some -faces are designed by their names: a -face is a cell (polyhedron), a -face is a facet (polygon) and a -face is an edge (line).
The next step is to demonstrate that every CC SB polytope is face-decomposable. To find the -faces of a CC SB polytope one can exploit that locally they are equal to the -faces of a convex polytope. Let , belongs to an open -face of if no matter how small is with the set is the interior of a half-ball (a ball cut by a hyperplane through its center). In this case, belongs to a given open -face and we introduce the tangent affine space of this face. Let with and assume that exist a path that connects and such that for every , then , and are in the same open -face of . Furthermore, the connectedness-based partition of is the set of open -faces of , i.e. . Given that , thus every belongs to the relative boundary of two -faces (may be more than two). Let and be two different open -faces of with , then is a closed -face of with . By analyzing each pair of mutually intersecting closed -faces one find each of the remaining and elements with . Now, given we can obtain a non-closed SB polytope by removing from one or several of its faces. However, for PW purposes the polytope represents the integration domain of a well behaved function. Given that the value of the integral is not modified by removing one or several faces, we will only consider the case of closed .
Let , we say that the two elements and of are neighbors if or , for . On the other hand, they are adjacent if . It is simple to verify that if two elements are neighbors they are adjacent, and the distance between two adjacent elements is zero. We define the characteristic length of the polytope as
i.e. the minimum distance between two non-adjacent elements of . It is clear that if , then such that .
In the following lines we state the general proposition for positive integer values of . The cases of are demonstrated each one in a separated proof. The remaining cases that concern every positive integer value are not demonstrated here and thus, the proposition is for these cases a conjecture. We note that Theorem 2 and its the conjectured generalization to given in the following Eq. (26) strongly resemble the combination of Hadwiger’s characterization theorem and The general kinematic formula (see Theorems 9.1.1 and 10.3.1 in pp. 118 and 153 of Klain and Gian-Carlo (1997), respectively) and transpires analogies with several results of integral geometry. These points will be discussed at the end of Sec. VII.
General conjecture (cases d): Consider and let be a well behaved function in (with fixed ), and a local function with range invariant under rigid transformations with [for fixed function is simply ]. Then
where is the -dimensional measure (Lebesgue measure) of (i.e. its volume ), is the -dimensional measure of , and is the -dimensional measure of the -th -element of (in particular ). Besides, and are constant coefficients which are independent of the size and shape of , while is a function of the angles that define the geometry of near its -th -element far away from its relative boundary.
Theorem 2 (case d=3): For the case the Eq. (26) is
where is the volume of , is the surface area of , and is the length of the -th edge. Besides, and are constant coefficients which are independent of the size and shape of , while is a function of the dihedral angle in the -th edge, and is a function of the set of dihedral angles between the adjacent planes that converge to the -th vertex.
Proof (case d=3): Consider the set and introduce a partition of given by with the principal part (the inner parallel body of ) and the boundary or skin part characterized by a thickness . We have
Here is the region where takes a constant value. This partition may be obtained by wrapping through sliding the center of a ball of radius on . Let be the set of points that are inside of any of this balls, then . Besides, it may also be obtained in terms of a Minkowski sum. Let be the unit ball centered at the origin, then the Minkowski sum is the outer parallel body of Schneider (2008) and . In Fig. 1 a picture representing this partition for is shown. There, one can observe on continuous line and several balls of radius (in dotted lines) corresponding to the wrapping procedure. For the case of being the shaded region (both the darker and brighter ones), the darker shaded region represents while the brighter shaded one corresponds to . On the contrary, if is the non-shaded or white region then corresponds to the white region between and the dashed line while the rest of the white region represents . For both cases a dashed line separates regions and .
The integral in Eq. (27) gives
with and for all . Naturally, is a bounded and local function with range invariant under rigid transformations which implies that and its value is independent of the position and orientation that the set takes on the space, only depends on the shape of . Besides if then .
Note that in the context of a planar face is simply a face. Consider the set partitioned in terms of ,
Here, is the surface-type region where each is related to a unique face. Let us define thus is a partition of . Since then for some face. Hence, the shape of is determined by the one dimensional position variable , and then, . In Eq. (30) is the edges-type region where each is related to a unique pair of faces. Let and be such that and