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Connecting density fluctuations and Kirkwood-Buff integrals for finite-size systems

Kirkwood-Buff integrals (KBI) connect the microscopic structure and thermodynamic properties of liquid solutions. KBI are defined in the grand canonical ensemble and evaluated assuming the thermodynamic limit (TL). In order to reconcile analytical and numerical approaches, finite-size KBI have been proposed in the literature, resulting in two strategies to obtain their TL values from computer simulations. (i) The spatial block-analysis method in which the simulation box is divided into subdomains of volume $V$ to compute fluctuations of the number of particles. (ii) A direct integration method where a corrected radial distribution function and a kernel that accounts for the geometry of the integration subvolumes are combined to obtain KBI as a function of $V$. In this work, we propose a method that connects both strategies into a single framework. We start from the definition of finite-size KBI, including the integration subdomain and an asymptotic correction to the radial distribution function, and solve them in Fourier space where periodic boundary conditions are trivially introduced. The limit $q\to 0$, equivalent to the value of the KBI in the TL, is obtained via the spatial block-analysis method. When compared to the latter, our approach gives nearly identical results for all values of $V$. Moreover, all finite-size effect contributions (ensemble, finite-integration domains and periodic boundary conditions) are easily identifiable in the calculation. This feature allows us to analyse finite-size effects independently and extrapolate the results of a single simulation to different box sizes. To validate our approach, we investigate prototypical systems, including SPC/E water and aqueous urea mixtures.