Finite-size effects arise when a sample of particles is not sufficient to provide a statistically satisfactory description of the bulk environment of a physical system. As a consequence, a reliable estimate of finite-size effects in many-particle systems is key to judge the validity of a theoretical model or the accuracy of a numerical simulation. In this context, we propose the use of a theorem on the free-energy cost for separating a system into smaller independent subsystems [J. Stat. Mech.: Theory Exp. (2017) 083201; Lett. Math. Phys. 112, 97 (2022)] to estimate the relevance of finite-size effects in thermodynamic quantities from computer simulations. The key aspect of this study is that for two-body potentials, as mostly occurring in physics, the method requires only two-body distribution functions and the particle number density. The calculation of the involved physical quantities can be done numerically on a three-dimensional grid. In some cases even analytical estimates are possible and as an example the uniform interacting electron gas in the ground state is considered; we derive an approximating scaling law for the finite-size effects.
Finite-size effects arise when a sample of particles is not sufficient to provide a statistically satisfactory description of the bulk environment of a physical system. As a consequence, a reliable estimate of finite-size effects in many-particle systems is key to judge the validity of a theoretical model or the accuracy of a numerical simulation. In this context, we propose the use of a theorem on the free-energy cost for separating a system into smaller independent subsystems [J. Stat. Mech.: Theory Exp. (2017) 083201; Lett. Math. Phys. 112, 97 (2022)] to estimate the relevance of finite-size effects in thermodynamic quantities from computer simulations. The key aspect of this study is that for two-body potentials, as mostly occurring in physics, the method requires only two-body distribution functions and the particle number density. The calculation of the involved physical quantities can be done numerically on a three-dimensional grid. In some cases even analytical estimates are possible and as an example the uniform interacting electron gas in the ground state is considered; we derive an approximating scaling law for the finite-size effects.
In a recent work [Reible et al., Phys. Rev. Res. 5, 023156 (2023)], it was shown that the mean particle-particle interaction across an ideal surface that divides a system into two parts can be employed to estimate the size dependence for the thermodynamic accuracy of the system. In this work we propose its application to systems with finite-range interactions that model dense quantum gases and derive an approximate size-dependence scaling law. In addition, we show that the application of the criterion is equivalent to the determination of a free-energy response to a perturbation. The latter result confirms the complementarity of the criterion to other estimates of finite-size effects based on direct simulations and empirical structure or energy convergence criteria.
In a recent work [Reible et al., Phys. Rev. Res. 5, 023156, 2023], it has been shown that the mean particle-particle interaction across an ideal surface that divides a system into two parts, can be employed to estimate the size dependence for the thermodynamic accuracy of the system. In this work we propose its application to systems with finite range interactions that models a dense quantum gases and derive an approximate size-dependence scaling law. In addition, we show that the application of the criterion is equivalent to the determination of a free energy response to a perturbation. The latter result confirms the complementarity of the criterion to other estimates of finite-size effects based on direct simulations and empirical structure or energy convergence criteria.
Abstract We generalise the two-sided Bogoliubov inequality for classical particles (Delle Site et al. in J Stat Mech Theory Exp 083201, 2017 to systems of quantum particles. As in the classical set-up, the inequality leads to upper and lower bounds for the free energy difference associated with the partitioning of a large system into smaller, independent subsystems. From a thermodynamic modelling point of view, the free energy difference determines the finite size correction needed to consistently treat a small system as a representation of a large system. Applications of the bounds to quantify finite size effects are ubiquitous in physics, chemistry, material science, or biology, to name just a few; in particular, it is relevant for molecular dynamics simulations in which a small portion of a system is usually taken as representative of the idealized large system.
We generalise the two-sided Bogoliubov inequality for classical particles from [L. Delle Site et al., J.Stat.Mech.Th.Exp. 083201 (2017)] to systems of quantum particles. As in the classical set-up, the inequality leads to upper and lower bounds for the free energy difference associated with the partitioning of a large system into smaller, independent subsystems. From a thermodynamic modelling point of view, the free energy difference determines the finite size correction needed to consistently treat a small system as a representation of a large system. Applications of the bounds to quantify finite size effects are ubiquitous in physics, chemistry, material science, or biology, to name just a few; in particular it is relevant for molecular dynamics simulations in which a small portion of a system is usually taken as representative of the idealized large system.
Ab initio quantum Monte Carlo methods, in principle, allow for the calculation of exact properties of correlated many-electron systems but are, in general, limited to the simulation of a finite number of electrons N under periodic boundary conditions. Therefore, an accurate theory of finite-size effects is indispensable to bridge the gap to realistic applications in the thermodynamic limit. In this work, we revisit the uniform electron gas at finite temperature, as it is relevant to contemporary research, e.g., in the field of warm dense matter. In particular, we present a new scheme to eliminate finite-size effects both in the static structure factor S(q) and in the interaction energy v, which is based on the density response formalism. We demonstrate that this method often allows us to obtain v in the thermodynamic limit within a relative accuracy of ∼0.2% from as few as N = 4 electrons without any empirical choices or knowledge of results for other values of N. Finally, we evaluate the applicability of our method upon increasing the density parameter rs and decreasing the temperature T.
To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .