Conﬁgurational degeneracy of a set of dipoles in a quasi-two-dimensional system

The purpose of this paper is to provide an exact evaluation of the conﬁgurational degeneracy when an arbitrary number ( k ) of dipoles are placed in a quasi-two-dimensional space ( Q 2 D ) . This Q 2 D is made up of three contiguous diagonals 3 × N . Our Q 2 D space gives to the central sites of the lattice their full coordination number of nearest neighboring compartments. We are going to determine the exact conﬁgurational degeneracy W ( k , N ) when an arbitrary number k of the above mentioned particles are placed in this 3 × N Q 2 D space. We found that W ( k , N ) is exactly described by

a A 3 × N diagonal array; b Three out of the 116 possible arrangements of two dipoles in a 3 × 3Q2D space role in the structural properties of 2D systems. Representative examples include colloidal particles at an interface, electrorheological fluids, adsorption of molecules on a metal surface, magnetic thin films, nanocrystals deposited on a substrate, amphiphilic molecules adsorbed at an air-water interface, etc. [1][2][3][4][5].
Problems dealing with particles with distinguishable ends (dipoles, spins, magnetic domains, amphiphilic molecules, hetero-diatomic molecules, etc.) placed in a lattice have always been troublesome; unlike simple particles, there is no reciprocity between particles and vacancies. Therefore, as is generally true for problems of this nature, exact solutions are a difficult task.
In recent years we have made a considerable effort to develop analytical methods to find exact solutions to problems dealing with: (i) the kinetics of immobile adsorption of linear molecules on a two-dimensional lattice [6], (ii) a heterogeneous reaction exactly solved on a small lattice [7], (iii) how many Langmuirs are required for monolayer formation [8], (iv) the scaling properties in the average number of attempts until saturation in random sequential adsorption processes [9], (v) the branch counting probability approach to random sequential adsorption [10]. In the present paper we develop an analytical approach to find the configurational degeneracy when a set of particles with distinguishable ends are placed in a quasi-two-dimensional space.
The purpose of this paper is to provide an exact evaluation of the configurational degeneracy when an arbitrary number (k) of dipoles are placed in a quasi-two-dimensional space (Q2D). This Q2D is made up of three contiguous diagonals 3 × N as is shown in Fig. 1a. Our Q2D gives to the central sites of the lattice their full coordination number of nearest neighboring compartments. In Sect. 2 we determine the exact configurational degeneracy W (k, N ) when an arbitrary number k of the above mentioned particles are placed in this 3 × N Q2D space. Our conclusions are summarized in Sect. 3. Therefore #g(k, N ), the number of members of the set g(k, N ), is given by: The compartment of the (N + 2)th column is unoccupied in the set d(k, N ) so that by definition #d(k, N ) is D(k, N ). If that compartment is occupied, then the adjacent one is also occupied. Hence, all other possible arrangements must involve the remaining (k − 1) dipoles in the remainder of the array, which is a λ(N ) array. The number The factor 2 is because the above mentioned compartment can be occupied in two different ways. Therefore we prove Eq (1).

Theorem II
The lower compartment of the N th column is unoccupied in the set d(k, N ), so that by definition, #d(k, N ) is D(k, N − 1). If that compartment is occupied, then the adjacent one is also occupied. Hence, all other possible arrangements must involve the remaining (k − 1) dipoles in the remainder of the array, which is a λ(N − 1) array. The number of elements in l(k, N ) therefore is 2L(k − 1, N − 1), i.e., #l(k, N ) = 2L(k − 1, N − 1). The factor 2 is because the above mentioned compartment can be occupied in two different ways. Therefore we prove Eq. (2).

Corollary 1
Proof From Theorem I, by substituting N by N − 1 in Eq. (1) we obtain  (k, N ). If that compartment is occupied, then the adjacent one is also occupied. Hence, all other possible arrangements must involve the remaining (k −1) dipoles in the remainder of the array, which is a λ(N −1) array. The number of elements in c(k, N ) therefore is 2L(k −1, N −1), i.e., #c(k, N ) = 2L(k −1, N −1). The factor 2 is because the above mentioned compartment can be occupied in two different ways. Therefore, we prove Eq. (5).

Corollary 2
Proof We can evaluate L(k − 1, N − 1) by using Theorem III Substitution of this into the Theorem III yields Repeated use of Eq. (7) gives However, L(0, N − k) = W (0, N − k). Therefore, we prove Eq. (6).  Fig. 3. In other words, the a i (k, N ) are subsets defined on the basis of the manner in which those two compartments are occupied. Since every member of a i (k, N ) differs from any and every member of a i (k, N )(i = j), we conclude that a i (k, N ) a j (k, N ) = , i = j. Also, these four configurations clearly are the only one we can form with the above mentioned compartments, therefore,

Theorem IV
We conclude that The set a 1 (k, N ) contains only those arrangements in which those two compartments are vacant. All k dipoles are then arranged in the remaining λ(N − 1) array; hence #a 1 (k, N ) = L(k, N − 1). The set a 2 (k, N ) contains a dipole occupying both compartments, and the remaining (k − 1) dipoles are arranged in an array composed of the original array minus the two precluded compartments, i.e., in a λ(N −1) array. We may then write #a 2 (k, N ) = 2L (k − 1, N − 1). The factor 2 is because of the two different ways in which a dipole can be placed in the two precluded compartments.
The set a 3 (k, N ) has the upper compartment occupied and the lower one empty. The remaining end of the dipole occupies a compartment of the (N − 1)th column, the remaining (k − 1) dipoles are arranged in an array composed of the original array minus the three precluded compartments, i.e., in a γ (N − 2) array. We may then write #a 3 (k, N ) = 2G(k − 1, N − 2). The factor 2 is because of the two different ways in which the above mentioned dipole can be placed.
The set a 4 (k, N ) has the upper compartment occupied and the lower one empty, the remaining end of the dipole occupies another compartment of the N th column, and the remaining (k − 1) dipoles are arranged in a ω(N − 1) array, i.e., #a 4 (k, N ) = 2W (k − 1, N − 1). The factor 2 is because of the two different ways in which we can place the dipole. Therefore by Eq. (12) we prove Theorem IV.

Conclusions
In the present paper we provide an exact evaluation of the configurational degeneracy when an arbitrary number (k) of dipoles are placed in a quasi-two-dimensional space (Q2D). This Q2D space is made up of three contiguous diagonals 3 × N as is shown in Fig 1a. Our Q2D space gives to the central sites of the lattice their full coordination number of nearest neighboring compartments. We determine the exact configurational degeneracy W (k, N ) when an arbitrary number k of the above mentioned particles are placed in this 3 × N Q2D space. We found that W (k, N ) is exactly described by Table 1 shows the configurational degeneracy W (k, N ) when indistinguishable dipoles are placed in a 3 × N Q2D space for N and K in the range 0-8.
From that table we learn that the configurational degeneracy W (k, N ) shows a maximum when N > 5. Work is in progress in La Plata to determine the characteristic of W (k, N ) when N >> 1, in particular if the dependence of W (k, N ) on k is either a broad or a sharp distribution. The analysis presented in this paper is a first step in order to unravel, through analytical methods, the role played by the configurational entropic contribution to these systems.