The Partition Function of the Dirichlet Operator on a -Dimensional Rectangle Cavity
We study the asymptotic behavior of the free partition function in the limit for a diffusion process which consists of -independent, one-dimensional, symmetric, -stable processes in a hyperrectangular cavity with an absorbing boundary. Each term of the partition function for this polyhedron in -dimensions can be represented by a quermassintegral and the geometrical information conveyed by the eigenvalues of the fractional Dirichlet Laplacian for this solvable model is now transparent. We also utilize the intriguing method of images to solve the same problem, in one and two dimensions, and recover identical results to those derived in the previous analysis.
Trace formulas for heat kernels of the fractional Laplacian , , and its Schrödinger perturbations in spectral theory  have attracted a lot of attention recently due to the numerous applications to the mathematical physics, mathematical biology, and finance. From a probabilistic point of view the fractional Laplacian on a domain is a nonlocal operator which arises as the generator of a pure jump Lévy process killed upon exiting .
In a recent paper  we investigated the distribution of eigenvalues of the Dirichlet pseudodifferential operator , , on an open and bounded subdomain , which is defined by the principal value integral: The are restrictions of functions that belong to the fractional Sobolev space , , , given by and satisfy in . We have also predicted bounds on the sum of the first eigenvalues, the counting function, the Riesz means, and the first-term asymptotic expansion of the partition function which was found to bewhere the counting function is1
The present paper extends result (3) to all orders in , , in the short-time limit, and provides a novel result for a polyhedron’s partition function which was lacking from the literature even for the ordinary Laplacian. This objective is achieved by using two different methods. The first method relies on the explicit knowledge of the Dirichlet spectrum (20) of the fractional operator and the Euler-Maclaurin summation formula (25). These two ingredients enable us to express the asymptotic behavior of the partition function (29) in terms of the volume of and the th quermassintegral. The second method, commonly known as the method of images, suggests that the heat kernel of the cavity, in the short time limit, receives contributions only from the heat kernels of the unbounded space with virtual source points generated by reflecting anticlockwise (or clockwise) the initial point source through the hyperplanes bounding the cavity. Note that only virtual source points clustering around each vertex of the cavity contribute; see (41). Tracing and integrating the heat kernel over the cavity, we recover results identical to those derived by the first method. It is worth noting that this method, though widely known and well established in classical electromagnetism, has not been used before to solve problems falling into this category.
The paper is organized into five sections. Section 2 reviews fundamental notions of convex geometry and applies them to a -dimensional hyperrectangular parallelepiped in Euclidean space. Section 3 computes the asymptotic behavior of the partition function for the fractional Dirichlet operator by applying the first method. Section 4 uses the alternative approach of images for solving the same problem. Section 5 concludes the work by posing some open problems.
2. Geometric Preliminaries of Convex Bodies in
Let be the family of all convex bodies (nonempty, compact, and convex sets) in the -dimensional Euclidean vector space. We denote by the unit ball and the unit sphere. The Lebesgue measure on is denoted by and the spherical Lebesgue measure on is denoted by . In particular we have and . If , then the parallel body at distance is given by where denotes the Euclidean distance from . According to Steiner’s formula the volume of is given bywhere is the th quermassintegral, or mean cross-sectional measure, introduced by Minkowski, and is the th intrinsic volume2. The in (6) is defined by  where denotes the volume of the convex set of all intersection points of the -plane, passing through the fixed point , with the -planes orthogonal to it and is the invariant volume element of the Grassmannian manifold which is the set of -dimensional planes in . Setting into (7) one can define another functional, the so-called mean breadth of , as follows:
If the boundary of a convex set is a hypersurface of class , the quermassintegrals can be expressed by means of the integrals of mean curvature of . The th integral of mean curvature is defined by where is the th-elementary symmetric function of the principal curvatures and is the area element of . The volume of the parallel body can be then written as and by comparison with (6) we end up with This relation is well defined as long as is . If does not have a smooth boundary, then we compute
In the present paper we focused on the hyperrectangular parallelepiped with edges , . The th intrinsic volume and quermassintegral are given by The mean breadth of can be computed by combining (9) with (8) for the parallel body and taking the limit in the end. A hyperrectangular parallelepiped domain has a total of faces, vertices, and edges. The mean curvature is defined by where are the principal radii of curvature of . The boundary of consists of hyperplanes at the faces, hyperspheres at the vertices, and hypercylinders at the edges of. The values of for are A careful calculation gives and therefore
3. The Partition Function for the Operator on
Let , , be a collection of independent, one-dimensional, symmetric stable processes in and denote by the semigroup on of killed upon exiting . Its transition density satisfies In [3, 6], performing a slight modification, the following proposition has been proved.
Proposition 1. On the open and bounded hyperrectangular parallelepiped of side lengths , , the eigenvalues for the homogeneous Dirichlet problem are given by where forms an orthonormal basis in with , none of the indices vanishes, and is a constant with dimensions .
Arranging the positive, real, and discrete spectrum of in increasing order (including multiplicities), we have The simplicity of (the eigenvalue in the -dimension) is conjectured to hold for , proved for and in , , respectively.
Taking into account the previous proposition, the generator of the semigroup acts on the orthonormal set as The transition density is then given by and the partition function (or trace of the heat kernel), using (23), satisfies Thus from (24) we observe that the total partition function is written as the disjoint product of partition functions for each spatial dimension. The sum will be calculated utilizing the Euler-Maclaurin summation formula  which states the following.
Theorem 2. Suppose is a decreasing function with continuous derivatives up to order . Thenwhere are the Bernoulli polynomials and the Bernoulli numbers.
Applying (25) in one dimension, in the zero time limit, we find for The functions belong to the following spaces: where the subscript indicates that the functions are differentiable with respect to . The case can be summed exactly since it turns out to be a progression. The result is Substituting (26) into (24), for we obtain where , are given by (13).
Remark 3. (1) The term is the -dimensional volume of the hyperrectangle parallelepiped and can be written in terms of the mean breadth of as using (17).
(2) In two dimensions, the trace of the heat kernel for the ordinary Laplacian (), provided that the domain is a rectangle of sides , is given by where is the familiar Riemann theta function3The Poisson summation formula4 can be casted into the form and utilized to determine the asymptotic behavior of the partition function in the limit. Therefore, we reproduce the well-known result where is the surface area of and is the length of the boundary . An independent calculation using (29) gives identical result. Kac, in , extended the dimensionless corner correction for a closed polygon with obtuse angles through a complicated integral. Later it was reported in  that D. B. Ray obtained the correction for arbitrary angles by expressing Green’s function as a Kontorovich-Lebedev transform. It is noteworthy that if we approximate a circle by an inscribed regular polygon then (35) becomes In (36) as the number of edges of the polygon tends to infinity the sum converges to the topological invariant constant . For simply connected, open with compact boundary two-dimensional Riemannian manifolds , where is the metric tensor, the partition function contains the -independent term given by  where is the Euler characteristic and is the Ricci scalar of the manifold .
4. A New Alternative Approach Based on the Image Method
The transition function (or elementary solution) is the solution of the following diffusion problem in described by with initial and Dirichlet boundary conditionsThe physical context of (39a) is the existence of a point source initially described by a generalized Dirac- function while (39b) dictates that the process is killed upon reaching the boundary . The solution of (38) with the initial condition (39a) on is given by where the appearance of the function will be apparent shortly. Note that due to the translational invariance of the integral it can be absorbed in and written in the usual way.
In general the asymptotic behavior of the Dirichlet partition function in the limit is dominated by the diagonal elements of the heat kernel which receive contributions from the heat kernels of the unbounded space for the virtual source points clustering around each vertex of the polytope . The corresponding expression iswhere is the order of the reflection group at vertex . Thus integrating (41) over a suitable subdomain of and then taking the limit we recover the desired result.
In , coincides with the infinite dihedral group with defining relationswhere ’s are involuntary transformations and represent reflections with respect to the boundary points of . For every we generate the following two infinite sequences of virtual source points depending on whether we start reflection from the left or the right fixed point of the isometry (as shown in Table 1). In Figure 1, we graphically depict the virtual domains in which the corresponding image source points belong to.
The solution, using (41), is given byOne can check that both the initial and boundary conditions are fulfilled and moreover the limit of (43) givesand the contribution of each term to the partition function is Therefore, as promised.
The diffusion initial-boundary value problem for with can also be solved applying the method of images in a more general setting . Let be the angle between two intersecting mirrors in . If we require the absence of a virtual mirror between the two given ones, after successive reflections of the fundamental domain, then where . This remark facilitates the enumeration of bounded tessellations of the plane through reflections . One of the admissible polygons apart from the congruent equilateral triangles, the isosceles right triangles and the bisected equilateral triangles, is the rectangle. Generally speaking, if the angle , , then there is a unique, up to isomorphism, group generated by two involutions such that their product has order . The group is denoted by , called the dihedral group of order and has the presentationIn our case and each wedge produces the transition functionIntegrating (48) over a suitable subdomain once, multiplying the result by a factor of four in order to take into account the contribution from all four wedges of the rectangle, and taking the limit we recover (34).
The nonlocal fractional operator , considered in this paper, is the infinitesimal generator of time translations for a symmetric Lévy process with index (or characteristic exponent) , killed upon exiting the cavity. The resulting partition function turns out to depend not only on time and the dimensionality of the space but also on . If one investigates the same problem considering processes with stable law parameters, the skewness , the scale , and the location , then extra contributions should be expected.
In two dimensions, the topological term given by (35), for the ordinary diffusion (), predicts the value . Nevertheless, the calculation of the partition function for the fractional operator suggests the -dependent value . This is an interesting result that signals the connection of the corner corrections of a bounded domain with the -stable Lévy process. In the same spirit, relation (37) requires further investigation in order to reveal a possible similar dependence. Unfortunately, the absence of a relation between (37) and the corresponding term of the partition function, in the case of a symmetric Lévy flight, puts severe constraints in this direction.
Another challenging open problem would be to predict the partition function for an open manifold with convex and compact boundary knowing the sequence of polyhedral partition functions which approximates it. This issue is related to the geometric measure theory and fully resolving it will lead to the discovery of new results in higher dimensions than those presented in .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author would like to thank the anonymous reviewer for constructive critisism that improved the presentation of the paper.
- We denote by the -norm and by the Euclidean norm.
- The word intrinsic here means that the quantity under study does not depend on the dimension of the ambient space.
- The Jacobi theta function is defined by
- The Poisson formula states and can be proved using .
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