Usual approach to the foundations of quantum statistical physics is based on conventional micro-canonical ensemble as a starting point for deriving Boltzmann-Gibbs (BG) equilibrium. It leaves, however, a number of conceptual and practical questions unanswered. Here we discuss these questions, thereby motivating the study of a natural alternative known as Quantum Micro-Canonical (QMC) ensemble. We present a detailed numerical study of the properties of the QMC ensemble for finite quantum systems revealing a good agreement with the existing analytical results for large quantum systems. We also propose the way to introduce analytical corrections accounting for finite-size effects. With the above corrections, the agreement between the analytical and the numerical results becomes very accurate. The QMC ensemble leads to an unconventional kind of equilibrium, which may be realizable after strong perturbations in small isolated quantum systems having large number of levels. We demonstrate that the variance of energy fluctuations can be used to discriminate the QMC equilibrium from the BG equilibrium. We further suggest that the reason, why BG equilibrium commonly occurs in nature rather than the QMC-type equilibrium, has something to do with the notion of quantum collapse.
Usual approach to the foundations of quantum statistical physics is based on conventional micro-canonical ensemble as a starting point for deriving Boltzmann-Gibbs (BG) equilibrium. It leaves, however, a number of conceptual and practical questions unanswered. Here we discuss these questions, thereby motivating the study of a natural alternative known as Quantum Micro-Canonical (QMC) ensemble. We present a detailed numerical study of the properties of the QMC ensemble for finite quantum systems revealing a good agreement with the existing analytical results for large quantum systems. We also propose the way to introduce analytical corrections accounting for finite-size effects. With the above corrections, the agreement between the analytical and the numerical results becomes very accurate. The QMC ensemble leads to an unconventional kind of equilibrium, which may be realizable after strong perturbations in small isolated quantum systems having large number of levels. We demonstrate that the variance of energy fluctuations can be used to discriminate the QMC equilibrium from the BG equilibrium. We further suggest that the reason, why BG equilibrium commonly occurs in nature rather than the QMC-type equilibrium, has something to do with the notion of quantum collapse.
The statistical properties of non-interacting bosons and fermions confined in trapping potentials are most easily obtained when the system may exchange energy and particles with a large reservoir (grand-canonical ensemble). There are circumstances, however, where the system under consideration may be considered as being isolated (micro-canonical ensemble). This paper first reviews results relating to micro-canonical ensembles. Some of them were obtained a long time ago, particularly by Khinchin in 1950. Others were obtained only recently, often motivated by experimental results relating to atomic confinement. A number of formulas are reported for the first time in the present paper. Formulas applicable to the case where the system may exchange energy but not particles with a reservoir (canonical ensemble) are derived from the micro-canonical ensemble expressions. The differences between the three ensembles tend to vanish in the so-called Thermodynamics limit, that is, when the number of particles and the volume go to infinity while the particle number density remains constant. But we are mostly interested in systems of moderate size, often referred to as being mesoscopic, where the grand-canonical formalism is not applicable. The mathematical results rest primarily on the enumeration of partitions of numbers.
For a small system at a low temperature, thermal fluctuation and quantum effect play important roles in quantum thermodynamics. Starting from micro-canonical ensemble, we generalize the Boltzmann–Gibbs statistical factor from infinite to finite systems, no matter the interactions between particles are considered or not. This generalized factor, similar to Tsallis’s q-form as a power-law distribution, has the restriction of finite energy spectrum and includes the nonextensivities of the small systems. We derive the exact expression for distribution of average particle numbers in the interacting classical and quantum nonextensive systems within a generalized canonical ensemble. This expression in the almost independent or elementary excitation quantum finite systems is similar to the corresponding ones obtained from the conventional grand-canonical ensemble. In the reconstruction for the statistical theory of the small systems, we present the entropy of the equilibrium systems and equation of total thermal energy. When we investigate the thermodynamics for the interacting nonextensive systems, we obtain the system–bath heat exchange and “uncompensated heat” which are in the thermodynamical level and independent on the detail of the system–bath coupling. For ideal finite systems, with different traps and boundary conditions, we calculate some thermodynamic quantities, such as the specific heat, entropy, and equation of state, etc. Particularly at low temperatures for the small systems, we predict some novel behaviors in the quantum thermodynamics, including internal entropy production, heat exchanges between the system and its surroundings and finite-size effects on the free energy.
The description of the non-equilibrium dynamics of isolated quantum many-body systems within the framework of statistical mechanics is a fundamental open question. Conventional thermodynamical ensembles fail to describe the large class of systems that exhibit nontrivial conserved quantities, and generalized ensembles have been predicted to maximize entropy in these systems. We show experimentally that a degenerate one-dimensional Bose gas relaxes to a state that can be described by such a generalized ensemble. This is verified through a detailed study of correlation functions up to 10th order. The applicability of the generalized ensemble description for isolated quantum many-body systems points to a natural emergence of classical statistical properties from the microscopic unitary quantum evolution.
We study how the grand-canonical density matrix arises in macroscopic quantum systems. ``Canonical typicality'' is the known statement that for a typical wave function Ψ from a micro-canonical energy shell of a quantum system S weakly coupled to a large but finite quantum system B, the reduced density matrix ρ^ΨS=trB∣Ψ⟩⟨Ψ∣ is approximately equal to the canonical density matrix ρ^can=Zcan−1exp(−βH^S). Here, we discuss the analogous statement and related questions for the \emph{grand-canonical} density matrix ρ^gc=Zgc−1exp(−β(H^S−μ1N^1S−…−μrN^rS)) with N^iS the number operator for molecules of type i in the system S. This includes (i) the case of chemical reactions (which requires some novel considerations) and (ii) that of systems S defined by a spatial region which particles may enter or leave. It includes statements about how ρ^gc arises from the density matrix of the appropriate (generalized micro-canonical) Hilbert subspace Hgmc⊂HS⊗HB (defined by a micro-canonical interval of total energy and suitable particle number sectors) or from typical Ψ in Hgmc, as well as statements about the distribution of the (conditional) wave function ψS of S, which turns out to be a so-called GAP or Scrooge measure. That is, we discuss the foundation and justification of both the density matrix and the distribution of the wave function in the grand-canonical case. To this end (particularly for the chemical reactions), we also need to extend these considerations to the so-called generalized Gibbs ensembles, which apply to systems for which some macroscopic observables are conserved.
We develop a regularization of the quantum microcanonical ensemble, called a Gaussian ensemble, which can be used for derivation of the canonical ensemble from microcanonical principles. The derivation differs from the usual methods by giving an explanation for the, at the first sight unreasonable, effectiveness of the canonical ensemble when applied to certain small, isolated, systems. This method also allows a direct identification between the parameters of the microcanonical and the canonical ensemble and it yields simple indicators and rigorous bounds for the effectiveness of the approximation. Finally, we derive an asymptotic expansion of the microcanonical corrections to the canonical ensemble for those systems, which are near, but not quite, at the thermodynamical limit and show how and why the canonical ensemble can be applied also for systems with an exponentially increasing density of states. The aim throughout the paper is to keep mathematical rigour intact while attempting to produce results which are both physically and practically interesting.
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.