We present a method aimed at a stochastic derivation of the equilibrium distribution of a classical/quantum ideal gas in the framework of the canonical ensemble. The time evolution of these ideal systems is modelled as a series of transitions from one system microstate to another one and thermal equilibrium is reached via a random walk in the single-particle state space. We look at this dynamic process as a Markov chain satisfying the condition of detailed balance and propose a variant of the Monte Carlo Metropolis algorithm able to take into account indistinguishability of identical quantum particles. Simulations performed on different two-dimensional (2D) systems are revealed to be capable of reproducing the correct trends of the distribution functions and other thermodynamic properties. The simulations allow us to show that, away from the thermodynamic limit, a pseudo-Bose–Einstein condensation occurs for a 2D ideal gas of bosons.
We present a method aimed at a stochastic derivation of the equilibrium distribution of a classical/quantum ideal gas in the framework of the canonical ensemble. The time evolution of these ideal systems is modelled as a series of transitions from one system microstate to another one and thermal equilibrium is reached via a random walk in the single-particle state space. We look at this dynamic process as a Markov chain satisfying the condition of detailed balance and propose a variant of the Monte Carlo Metropolis algorithm able to take into account indistinguishability of identical quantum particles. Simulations performed on different two-dimensional (2D) systems are revealed to be capable of reproducing the correct trends of the distribution functions and other thermodynamic properties. The simulations allow us to show that, away from the thermodynamic limit, a pseudo-Bose–Einstein condensation occurs for a 2D ideal gas of bosons.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
We present a new exact method to numerically compute the thermodynamical properties of an interacting Bose gas in the canonical ensemble. As in our previous paper (Carusotto I, Castin Y and Dalibard J 2001 Phys. Rev. A 63 023606), we write the density operator ρ as an average of Hartree dyadics | N :ϕ 1 ⟩⟨ N :ϕ 2 | and we find stochastic evolution equations for the wavefunctions ϕ 1,2 such that the exact imaginary-time evolution of ρ is recovered after averaging over noise. In this way, the thermal equilibrium density operator can be obtained for any temperature T . The method is then applied to study the thermodynamical properties of a homogeneous one-dimensional N -boson system: although Bose-Einstein condensation cannot occur in the thermodynamical limit, a macroscopic occupation of the lowest mode of a finite system is observed at sufficiently low temperatures. If k B T >>µ, the main effect of interactions is to suppress density fluctuations and to reduce their correlation length. Different effects such as a spatial antibunching of the atoms are predicted for the opposite k B T ⩽µ regime. Our exact stochastic calculations have been compared with existing approximate theories.
The quantum dynamics and grand canonical thermodynamics of many-mode (one-, two-, and three-dimensional) interacting Bose gases are simulated from first principles. The model uses a lattice Hamiltonian based on a continuum second-quantized model with two-particle interactions, external potential, and interactions with an environment, with no further approximations. The interparticle potential can be either an (effective) delta function as in Bose-Hubbard models, or extended with a shape resolved by the lattice. Simulations are of a set of stochastic equations that in the limit of many realizations correspond exactly to the full quantum evolution of the many-body systems. These equations describe the evolution of samples of the gauge P distribution of the quantum state, details of which are developed. Conditions under which general quantum phase-space representations can be used to derive stochastic simulation methods are investigated in detail, given the criteria: 1) The simulation corresponds exactly to quantum mechanics in the limit of many trajectories. 2) The number of equations scales linearly with system size, to allow the possibility of efficient first-principles quantum mesoscopic simulations. 3) All observables can be calculated from one simulation. 4) Each stochastic realization is independent to allow straightforward use of parallel algorithms. Special emphasis is placed on allowing for simulation of open systems. In contrast to typical Monte Carlo techniques based on path integrals, the phase-space representation approach can also be used for dynamical calculations. Two major (and related) known technical stumbling blocks with such stochastic simulations are instabilities in the stochastic equations, and pathological trajectory distributions as the boundaries of phase space are approached. These can (and often do) lead to systematic biases in the calculated observables. The nature of these problems are investigated in detail. Many phase-space distributions have, however, more phase-space freedoms than the minimum required for exact correspondence to quantum mechanics, and these freedoms can in many cases be exploited to overcome the instability and boundary term problems, recovering an unbiased simulation. The stochastic gauge technique, which achieves this in a systematic way, is derived and heuristic guidelines for its use are developed. The gauge P representation is an extension of the positive P distribution, which uses coherent basis states, but allows a variety of useful stochastic gauges that are used to overcome the stability problems. Its properties are investigated, and the resulting equations to be simulated for the open interacting Bose gas system are derived. The dynamics of the following many-mode systems are simulated as examples: 1) Uniform one-dimensional and two-dimensional Bose gases after the rapid appearance of significant two-body collisions (e.g. after entering a Feshbach resonance). 2) Trapped bosons, where the size of the trap is of the same order as the range of the interparticle potential. 3) Stimulated Bose enhancement of scattered atom modes during the collision of two Bose-Einstein condensates. The grand canonical thermodynamics of uniform one-dimensional Bose gases is also calculated for a variety of temperatures and collision strengths. Observables calculated include first to third order spatial correlation functions (including at finite interparticle separation) and momentum distributions. The predicted phenomena are discussed. Improvements over the positive P distribution and other methods are discussed, and simulation times are analyzed for Bose-Hubbard lattice models from a general perspective. To understand the behavior of the equations, and subsequently optimize the gauges for the interacting Bose gas, single- and coupled two-mode dynamical and thermodynamical models of interacting Bose gases are investigated in detail. Directions in which future progress can be expected are considered. Lastly, safeguards are necessary to avoid biased averages when exponentials of Gaussian-like trajectory distributions are used (as here), and these are investigated.
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].
Based on the canonical ensemble, we suggested the simple scheme for taking into account Gaussian fluctuations in a finite system of ideal boson gas. Within framework of scheme we investigated the influence of fluctuations on the particle distribution in Bose -gas for two cases - with taking into account the number of particles in the ground state and without this assumption. The temperature and fluctuation parameter dependences of the modified Bose-Einstein distribution have been determined. Also the dependence of the condensation temperature on the fluctuation distribution parameter has been obtained.
We derive a stochastic process that describes the kinetics of a one-dimensional Bose gas in a regime where three-body collisions are important. In this situation the system becomes nonintegrable, offering the possibility of investigating dissipative phenomena more simply than in higher-dimensional gases. Unlike the quantum Boltzmann equation describing the average momentum distribution, the stochastic approach allows a description of higher-order correlation functions in a canonical ensemble. As will be shown, this ensemble differs drastically from the grand-canonical one. We illustrate the use of this method by determining the time evolution of the momentum-mode particle-number distribution and the static structure factor during the evaporative cooling process.