A typical approach for modeling systems at a nanoscale in states of non-equilibrium undergoing an irreversible process is to use an ad hoc mixture of molecular dynamics (linear and nonlinear), i.e. classical mechanics, coupled to assumptions of stable equilibrium which allow one via analogy to incorporate equilibrium thermodynamic state information such as temperature and pressure into the modeling process. However, such an approach cannot describe the actual thermodynamic evolution in state which occurs in these systems since the equation of motion used (Newton’s second law) can only describe the evolution in state from one mechanical state to another. To capture the actual thermodynamic evolution, a more general equation of motion is needed. Such an equation has been proposed, i.e. the Beretta equation of motion, as part of a general theory, which unifies (not simply bridges as is the case in statistical thermodynamics) quantum mechanics and thermodynamics. It is called the unified quantum theory of mechanics and thermodynamics or quantum thermodynamics. This equation, which strictly satisfies all of the implications of the laws of thermodynamics, including the second law, as well as of quantum mechanics, describes the thermodynamic evolution in state of a system in non-equilibrium regardless of whether or not the system is in a state far from or close to stable equilibrium. This theory and its dynamical postulate are used here to model the storage of hydrogen in an isolated box modeled in 1D and 2D with a carbon atom at one end of the box in 1D and a carbon nanotube in the middle of the box in 2D. The system is prepared in a state with the hydrogen molecules initially far from stable equilibrium, after which the system is allowed to relax (evolve) to a state of stable equilibrium. The so-called energy eigenvalue problem is used to determine the energy eigenlevels and eigenstates of the system, while the nonlinear Beretta equation of motion is used to determine the evolution of the thermodynamic state of the system as well as the spatial distributions of the hydrogen molecules in time. The results of our initial simulations show in detail the trajectory of the state of the system as the hydrogen molecules, which are initially arranged to be far from the carbon atom or the carbon nanotube, are seen to spread out in the container and eventually become more concentrated near the carbon atom or atoms.
A typical approach for modeling systems at a nanoscale in states of non-equilibrium undergoing an irreversible process is to use an ad hoc mixture of molecular dynamics (linear and nonlinear), i.e. classical mechanics, coupled to assumptions of stable equilibrium which allow one via analogy to incorporate equilibrium thermodynamic state information such as temperature and pressure into the modeling process. However, such an approach cannot describe the actual thermodynamic evolution in state which occurs in these systems since the equation of motion used (Newton’s second law) can only describe the evolution in state from one mechanical state to another. To capture the actual thermodynamic evolution, a more general equation of motion is needed. Such an equation has been proposed, i.e. the Beretta equation of motion, as part of a general theory, which unifies (not simply bridges as is the case in statistical thermodynamics) quantum mechanics and thermodynamics. It is called the unified quantum theory of mechanics and thermodynamics or quantum thermodynamics. This equation, which strictly satisfies all of the implications of the laws of thermodynamics, including the second law, as well as of quantum mechanics, describes the thermodynamic evolution in state of a system in non-equilibrium regardless of whether or not the system is in a state far from or close to stable equilibrium. This theory and its dynamical postulate are used here to model the storage of hydrogen in an isolated box modeled in 1D and 2D with a carbon atom at one end of the box in 1D and a carbon nanotube in the middle of the box in 2D. The system is prepared in a state with the hydrogen molecules initially far from stable equilibrium, after which the system is allowed to relax (evolve) to a state of stable equilibrium. The so-called energy eigenvalue problem is used to determine the energy eigenlevels and eigenstates of the system, while the nonlinear Beretta equation of motion is used to determine the evolution of the thermodynamic state of the system as well as the spatial distributions of the hydrogen molecules in time. The results of our initial simulations show in detail the trajectory of the state of the system as the hydrogen molecules, which are initially arranged to be far from the carbon atom or the carbon nanotube, are seen to spread out in the container and eventually become more concentrated near the carbon atom or atoms.
Quantum thermodynamics (QT) provides a general framework for the description of non-equilibrium phenomena at any level, particularly the atomistic one. This theory and its dynamical postulate are used here to extend the work reported in previous papers of modeling the storage of hydrogen in an isolated system, by extending the modeling to 3D. The system is prepared in a state with the hydrogen molecules initially far from stable equilibrium after which the system is allowed to relax (evolve) to a state of stable equilibrium. The so-called energy eigenvalue problem, which entails a many-body problem that for dilute and moderately dense gases can be solved using virial expansion theory, is used to determine the energy eigenvalues and eigenstates of the system. This information is then used in the nonlinear Beretta equation of motion of QT to determine the evolution of the thermodynamic state of the system as well as the spatial distributions of the hydrogen molecules in time. The results of our simulations provide a quantification of the entropy generated due to irreversibilities at an atomistic level and show in detail the trajectory of the state of the system as the hydrogen molecules, which are initially arranged to be far from the carbon nanotube, spread out in the system and eventually become more concentrated near the carbon atoms which make up the nanotube
Quantum thermodynamics (QT) provides a general framework for the description of non-equilibrium phenomena at any level, particularly the atomistic one. This theory and its dynamical postulate are used here to model the storage of hydrogen on and in a carbon nanotube. The tube is placed at the center of a tank with a volume of 250 nm 3 . The thermodynamic system of interest is the hydrogen, which is assumed isolated and having boundaries that coincide with the walls of the tank and the carbon nanotube. The hydrogen is initially prepared in a state far from stable equilibrium (i.e., with the hydrogen molecules probabilistically near one of the outer tank walls) after which the system is allowed to relax (evolve) to a state of stable equilibrium. To predict this evolution in state, the so-called energy eigenvalue problem, which entails a many-body problem that for dilute and moderately dense gases can be modeled using virial expansion theory, is first solved for the geometry involved. The energy eigenvalues and eigenstates of the system found are then used by the nonlinear Beretta equation of motion of QT to determine the evolution of the thermodynamic state of the system as well as the 3D spatial distributions of the hydrogen molecules in time. The simulation results provide a quantification of the entropy generated due to irreversibilities at an atomistic level and show in detail the trajectory of the thermodynamic state of the system as the hydrogen molecules, which are initially arranged to be far from the carbon nanotube, spread out in the system and eventually become probabilistically more concentrated near the carbon atoms, which make up the nanotube.
Intrinsic Quantum Thermodynamics (IQT) is a theory that combines Thermodynamics and Quantum Mechanics into a single theory and asserts that irreversibility and the increase of entropy has its origin at the fundamental, atomistic level. The merits and details of IQT are discussed and compared with the well-known theory of Quantum Statistical Mechanics (QSM) and the more recent development of Quantum Thermodynamics (QT). IQT is then used to model in 3D the time evolution of the adsorption of hydrogen on a single-walled carbon nanotube. The initial state of the hydrogen molecules is far from stable equilibrium and over time the system relaxes to a state of stable equilibrium with the hydrogen near the walls of the carbon nanotube. The details of the model are presented, which include the construction of the energy eigenlevels for the system, the treatment of the interactions between the hydrogen and the nanotube along with the interactions of the hydrogen molecules with each other, and the solution of the IQT equation of motion as well as approximation methods that are developed to deal with extremely large numbers of energy eigenlevels. In addition, a new extension to the theory of IQT is proposed for modeling systems that undergo heat interactions with a heat reservoir. The formulation of a new heat interaction operator is discussed, implemented, tested, and compared with a previous version extant in the literature. IQT theory is then further extended to encompass simple mass interactions with a mass reservoir. The formulation, implementation, and testing of the mass interaction operator is also discussed in detail. Finally, IQT is used to model the results of two experiments found in the literature. The first experiment deals with the spin relaxation of rubidium atoms and the second tests the relaxation behavior of single trapped ion that is allowed to interact with an external heat reservoir. Good agreement between experiment and the model predictions is found.
Basing his work on a new formulation of thermodynamics called the Unified Quantum Theory of Mechanics and Thermodynamics first published in a series of four ground breaking papers in 1976 (Hatsopoulos and Gyftopoulos, 1976a, b, c, d), Beretta develops a dynamical postulate (Beretta et al. 1984; Beretta, Gyftopoulos, and Park, 1985) consistent both with the non-dynamical quantum mechanical postulates of the Unified Theory as well as with its thermodynamic ones (the 2nd Law in particular). The theory itself simply and elegantly extends in a unified fashion the concepts of thermodynamics to quantum mechanics and the concepts of quantum mechanics to thermodynamics. It does so without the bridge traditionally used, i.e. statistical mechanics, eliminating a number of the ambiguities, tautologies, and inconsistencies (including a built-in violation of the 2nd Law) inherent in the presentations of both Classical and Statistical Thermodynamics. This new formulation generalizes thermodynamics so that it applies to all systems large or small (including one particle systems) either in a state of thermodynamic (i.e. stable) equilibrium or not in a state of thermodynamic equilibrium. The Beretta equation of motion describes the time evolution of the state of a system via a density operator which is uniquely based on an unambiguous preparation of an ensemble of identical systems, i.e. the so-called homogenous or irreducible ensemble, and does so both for unitary and non-unitary reversible as well as irreversible processes. In this paper, we present a simple application of this general equation of motion to the time evolution of the entropy of a closed system comprised of a Boltzmann type gas consisting of one or of many particles undergoing an irreversible process. A number of different energy eigenlevels and initial states and their effects on entropy generation and the final state of maximum entropy, i.e. stable equilibrium, are examined. A simple time-dependent work interaction is introduced into the formulation to show how this in turn affects the evolution of the state of the system.
A general quantum theory encompassing Mechanics, Thermodynamics and irreversible dynamics is presented in two parts. The first part is concerned exclusively with the description of the states of any individual physical system. It is based on a new nonlinear quantum equation of motion, which reduces to the Schroedinger equation of motion of motion of conventional quantum dynamics only under special conditions. It accounts for the implications of the laws of Thermodynamics as well as for irreversible phenomena, such as the natural tendency of an isolated system to transit from any non-equilibrium state to an equilibrium state of higher entropy. Conversely, the laws of Thermodynamics and irreversibility emerge as manifestations of the fundamental quantum dynamical behaviour of the elementary constituents of any material system. We call this part Quantum Thermodynamics. The second part of the theory, which contains the first as a special case, is concerned with the description of stochastic distributions of states in an ensemble of identical physical systems each of which individually obeys the laws of Quantum Thermodynamics. It is based on a new measure-theoretic description of ensembles. It accounts unambiguously for the essential distinction between two types of uncertainties that are generally present in an ensemble, namely, quantal uncertainties due to the inherent quantal nature of the states of each individual member system and nonquantal uncertainties due to the stochastic distribution of states. We call this part Quantum Statistical Thermodynamics.
The nonlinear equation of motion that accomplishes a self-consistent unification of quantum mechanics (QM) and thermodynamics conceptually different from the (von Neumann) foundations of quantum statistical mechanics (QSM) and (Jaynes) quantum information theory (QIT), but which reduces to the same mathematics for the thermodynamic equilibrium (TE) states, and contains standard QM in that it reduces to the time-dependent Schrodinger equation for zero entropy states is discussed in full mathematical detail. By restricting the discussion to a strictly isolated system (noninteracting, disentangled and uncorrelated), we show how the theory departs from the conventional QSM/QIT rationalization of the second law of thermodynamics, which instead emerges in QT (quantum thermodynamics) as a theorem of existence and uniqueness of a stable equilibrium state for each set of mean values of the energy and the number of constituent particles. To achieve this, the theory assumes Trρ lnρ B −k for the physical entropy and is designed to implement two fundamental ansatzs: (1) that in addition to the standard QM states described by idempotent density operators (zero entropy), a strictly isolated and uncorrelated system admits also states that must be described by non-idempotent density operators (nonzero entropy); (2) that for such additional states the law of causal evolution is determined by the simultaneous action of a Schrodinger-von Neumann-type Hamiltonian generator and a nonlinear dissipative generator which conserves the mean values of the energy and the number of constituent particles, and in forward time drives the density operator in the 'direction' of steepest entropy ascent (maximal entropy generation). The resulting dynamics is well defined for all non-equilibrium states, no matter how far from TE. Existence and uniqueness of solutions of the Cauchy initial value problem for all density operators implies that the equation of motion can be solved not only in forward time, to describe relaxation towards TE, but also in backward time, to reconstruct the 'ancestral' or primordial lowest entropy state or limit cycle from which the system originates. Zero entropy states as well as a well defined family of non-dissipative states evolve unitarily according to pure Hamiltonian dynamics and can be viewed as unstable limit cycles of the general nonlinear dynamics.