We derive the effective Hamiltonian $H - μN$ for open quantum systems with varying particle number from first principles within the framework of non-relativistic quantum statistical mechanics. We prove that under physically motivated assumptions regarding the size of the system and the range of the interaction, this form of the Hamiltonian is unique up to a constant. Our argument relies firstly on establishing a rigorous version of the surface-to-volume ratio approximation, which is routinely used in an empirical form in statistical mechanics, and secondly on showing that the Hilbert space for systems with varying particle number must be isomorphic to Fock space. Together, these findings provide a rigorous mathematical justification for the standard grand canonical formalism employed in statistical physics.
We derive the effective Hamiltonian H−μN for open quantum systems with varying particle number from first principles within the framework of non-relativistic quantum statistical mechanics. We prove that under physically motivated assumptions regarding the size of the system and the range of the interaction, this form of the Hamiltonian is unique up to a constant. Our argument relies firstly on establishing a rigorous version of the surface-to-volume ratio approximation, which is routinely used in an empirical form in statistical mechanics, and secondly on showing that the Hilbert space for systems with varying particle number must be isomorphic to Fock space. Together, these findings provide a rigorous mathematical justification for the standard grand canonical formalism employed in statistical physics.
The celebrated Lindblad equation governs the nonunitary time evolution of density operators used in the description of open quantum systems. It is usually derived from the von Neumann equation for a large system, at given physical conditions, when a small subsystem is explicitly singled out and the rest of the system acts as an environment whose degrees of freedom are traced out. In the specific case of a subsystem with variable particle number, the equilibrium density operator is given by the well-known grand canonical Gibbs state. Consequently, solving the Lindblad equation in this case should automatically yield, without any additional assumptions, the corresponding density operator in the limiting case of statistical equilibrium. Current studies of the Lindblad equation with varying particle number assume, however, the grand canonical Gibbs state a priori: the chemical potential is externally imposed rather than derived from first principles, and hence the corresponding density operator is not obtained as a natural solution of the equation. In this work, we investigate the compatibility of grand canonical statistical mechanics with the derivation of the Lindblad equation. We propose an alternative and complementary approach to the current literature that consists in using a generalized system Hamiltonian which includes a term μN. In a previous paper, this empirically well-known term has been formally derived from the von Neumann equation for the specific case of equilibrium. Including μN in the system Hamiltonian leads to a modified Lindblad equation which yields the grand canonical state as a natural solution, meaning that all the quantities involved are obtained from the physics of the system without any external assumptions.
The object of this thesis is to study the definition and the mathematically rigorous derivation of effective models in many body quantum mechanics. In this field, it is often impossible to compte exactly physically relevant quantities - such as the ground state energy of a system - because of the large number of degrees of freedom. Hence, introducing effective models (depending only on the local density of particles for instance) allows us to compute these quantities. Then, it remains to check that the effective model is indeed an approximation of the exact model in a macroscopic limit. Besides, in some models, some physical quantities (energy, scattering amplitude...), when computed naively, appear to be infinite. One has then to renormalize the theory in order to make these divergences (that are not observed experimentally) disappear. After putting the results of this thesis into context in the introduction, we will focus on the renormalization of a bosonic quadratic Hamiltonian on the Fock space in Chapter 2. This (formal) Hamiltonian is the sum of the second quantization of a one-particle operator and the square of a field operator, and it exhibits an ultraviolet divergence (i.e. coming from local singularities). More accurately, we will try to define a self-adjoint operator, and to show that it is the limit (in the resolvent sense) of the Hamiltonian with ultraviolet cutoff when the cutoff tends to infinity. To do so, we will first diagonalize the Hamiltonian in the regular case, using Bogoliubov transformations. Then, we will prove that the diagonalized Hamiltonian (up to a renormalization of the energy and the charge) still defines a self-adjoint, and we will show the convergence of the Hamiltonian with cut-off. In a second time, we will study the semi-classical limit of a Fermi gas, in two different scalings. In Chapter 3, we will focus on the dilute limite (i.e. with an interaction range very small compared to the typical distance between particles) of an inhomogeneous (trapped with a potential V) Fermi gas, with repulsive interactions and in three dimensions. Due to the Pauli exclusion principle, two fermions of same spin can only rarely be close enough to interact. Thus, we will consider a Fermi gas with two spin components. We will then show that the ground state energy of such a system can be approximated by a Thomas-Fermi energy (in which interactions play the role of a perturbation of the interactionless energy) in the large number of particles limit. Eventually, in Chapter 4, we will study the mean-field limit (with frequent but weak interactions) of an inhomogeneous Fermi gas, with attractive interactions and in one or two dimensions this time. Once more, we will show the convergence of the ground state energy toward a Thomas-Fermi energy. Moreover, we will also prove the convergence of the ground states. More accurately, we will show that the Husimi functions of these ground states converge to a convex combination of minimizers of the Vlasov energy.
For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.
A quantum statistical expression for the entropy of a nonequilibrium system is defined so as to be consistent with Gibbs’ relation, and is shown to corresponds to dynamical variable by introducing analogous to the Heisenberg picture in quantum mechanics. The general relation between system-reservoir interactions and an entropy change operator in an open quantum system, relying just on the framework of statistical mechanics and the definition of von Neumann entropy. By using this formula, we can obtain the correct entropy production in the linear response framework. The present derivation of entropy production is directly based on the first principle of microscopic time-evolution, while the previous standard argument is due to the thermodynamic energy balance.
Abstract We discuss the derivation of an effective Hamiltonian for open quantum many-particle systems. The aim is to define an operator that can be used for (molecular) simulations where, through the exchange of energy and matter with the surrounding environment (reservoir), the number of particles, n, becomes a variable of the problem. The Hamiltonian is formally derived from the Von Neumann equation; specifically, we derive an n-hierarchy of equations for the density matrix, ρ ^ n , for near equilibrium situations. Such a hierarchy, in case of stationary equilibrium, delivers the standard grand canonical density matrix as it would be expected. We report that a similar Hamiltonian was conjectured, from empirical considerations, in the field of superconductivity. Thus our result also provide a formal basis for this long-standing hypothesis. Finally, an application is discussed for Path Integral simulations of molecular systems.