An efficient algorithm for the double Legendre–Fenchel transform with application to phase separation

We study the double discrete Legendre–Fenchel Transform (LFT) to approximate the convex hull of a given function. We analyze the convergence of the double discrete LFT in the multivariate case based on previous convergence results for the discrete LFT by Corrias. We focus our attention on the grid on which the second discrete LFT is computed (dual grid); its choice has great impact on the accuracy of the resulting approximation of the convex hull. Then, we present an improvement (both in time and accuracy) to the standard algorithm based on a change in the factorization order for the second discrete LFT. This modification is particularly beneficial for bivariate functions. We also present some situations in which the selection of the dual grid is crucial, and show that it is possible to choose a dual grid of arbitrary size without increasing the memory requirements of the algorithm. Finally, we apply our algorithm to the study of phase separation in ionic solutions where non-ideal effects due to long-range electrostatic and short-range steric correlations between ions play an important role. In many applications, the equilibrium properties of a thermodynamic system can be studied through the minimization of a certain potential under some macroscopic constraints. In ideal situations, the potential is convex, and solving the constrained minimization problem is straightforward. However, complex modelling taking into account non-ideal effects often leads to non-convex potentials. In this situation, the system at equilibrium is not in a state lying in the non-convexity region of the potential, i.e., the region where the potential and its convex hull differ. Depending on the enforced macroscopic constraints, this can lead to phase separation. This behavior was discovered by Maxwell in the study of the Van der Waals equation ([9]); he was able to build the correct potential by applying the so-called Maxwell’s equal area rule to the derivative of the potential, which, for a univariate potential, is equivalent to finding its convex hull. For more complex potentials, e.g., bivariate or multivariate functions, the convex hull cannot be computed analytically. Since the convex hull results ∗Department of Mathematics and Computer Science, University of Udine, Italy †Université Paris-Est, CERMICS, Ecole des Ponts ParisTech, 77455 Marne la Vallée Cedex 2, France. [email protected] 1 ha l-0 08 06 59 7, v er si on 1 2 Ap r 2 01 3 from a double Legendre–Fenchel Transform (LFT), it can be approximated by a double discrete LFT. This approach has been considered in [5] in view of deriving pressure laws in binary mixtures. The main ingredient of the double discrete LFT is obviously the discrete LFT, which can be computed, as in [5], using Lucet’s algorithm [8]. This algorithm consists in reducing the transform by means of dimensional factorization to one-dimensional transforms, and the latter can be computed in linear time. The first part of this paper is centered on the convergence properties of the double discrete LFT in the multivariate case. In the literature, most of the papers focus on the computation of the discrete LFT, which has by itself many applications; on the contrary, there are few theoretical results on the double discrete LFT, mainly dealing with convex functions (see for example [2]). Our convergence results on the double discrete LFT are based on [2] which addresses the convergence of the discrete LFT. However, in order to prove convergence to the convex hull and better understand the behavior of the algorithms, it is important to study the grid on which the second discrete LFT is applied, which we call dual grid. The choice of the dual grid is one of the main issues regarding an accurate approximation of the convex hull, an issue which is often only briefly discussed, as for example happens in [5]. In the second part, we restrict the scope to bivariate functions and present an improvement (both in time and accuracy) of the standard algorithm based on the double application of Lucet’s algorithm. The main idea is to change the factorization order when computing the second discrete LFT. The same change could be made in the multivariate case, but the gain would be marginally inferior. Another improvement consists in merging (through a maximum operation) the results obtained after the two possible changes in the factorization order. We present numerical examples illustrating the benefits of the proposed approach. Moreover, the issue of choosing the dual grid is again highlighted, and an efficient handling of dual grids of arbitrary length is investigated. In the last part, we present a physical application of our algorithm to the study of phase separation in ionic solutions consisting of cations and anions dissolved in a solvent (typically water). We consider the model presented in [7] where non-ideal effects (due to long-range electrostatic and short-range steric correlations between ions) give rise, under certain conditions, to a non-convex free energy. The free energy has unbounded derivatives for vanishing ionic densities, and, owing to steric correlations, becomes unbounded when the total ionic density reaches a certain finite threshold. We present the shapes of the non-convexity regions, as computed by the present algorithm, for various cases concerning the valences of the ions.