scholarly journals A Generalized Statement of Highest-Entropy Principle for Stable Equilibrium and Non-Equilibrium in Many-Particle Systems

2016 ◽  
Vol 07 (03) ◽  
pp. 344-357 ◽  
Author(s):  
Pierfrancesco Palazzo
Keyword(s):  
Stable Equilibrium ◽  
Particle Systems ◽  
Entropy Principle ◽  
Generalized Statement ◽  
Non Equilibrium

scholarly journals Non-universal dynamics of staggered non-equilibrium particle systems and Ising chains

1998 ◽  
Vol 31 (2) ◽  
pp. 541-549 ◽  
Author(s):  
R B Stinchcombe ◽  
J E Santos ◽  
M D Grynberg
Keyword(s):  
Particle Systems ◽  
Non Equilibrium

Quantum Thermodynamics for the Modeling of Hydrogen Storage on a Carbon Nanotube

2008 ◽  
Author(s):  
Michael R. von Spakovsky ◽  
Charles E. Smith ◽  
Vittorio Verda
Keyword(s):  
Quantum Mechanics ◽  
Carbon Nanotube ◽  
Carbon Atom ◽  
Stable Equilibrium ◽  
Equation Of Motion ◽  
Second Law ◽  
Quantum Thermodynamics ◽  
Thermodynamic State ◽  
Hydrogen Molecules ◽  
Non Equilibrium

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.

On Singular Closures for the 5-Moment System in Kinetic Gas Theory

2015 ◽  
Vol 17 (2) ◽  
pp. 371-400 ◽  
Author(s):  
Roman Pascal Schaerer ◽  
Manuel Torrilhon
Keyword(s):  
Maximum Entropy ◽  
Maximum Entropy Principle ◽  
Second Order ◽  
Gas Flows ◽  
Moment Equations ◽  
Moment System ◽  
Entropy Principle ◽  
Kinetic Gas Theory ◽  
Closure Theory ◽  
Non Equilibrium

AbstractMoment equations provide a flexible framework for the approximation of the Boltzmann equation in kinetic gas theory. While moments up to second order are sufficient for the description of equilibrium processes, the inclusion of higher order moments, such as the heat flux vector, extends the validity of the Euler equations to non-equilibrium gas flows in a natural way.Unfortunately, the classical closure theory proposed by Grad leads to moment equations, which suffer not only from a restricted hyperbolicity region but are also affected by non-physical sub-shocks in the continuous shock-structure problem if the shock velocity exceeds a critical value. Amore recently suggested closure theory based on the maximum entropy principle yields symmetric hyperbolic moment equations. However, if moments higher than second order are included, the computational demand of this closure can be overwhelming. Additionally, it was shown for the 5-moment system that the closing flux becomes singular on a subset of moments including the equilibrium state.Motivated by recent promising results of closed-form, singular closures based on the maximum entropy approach, we study regularized singular closures that become singular on a subset of moments when the regularizing terms are removed. In order to study some implications of singular closures, we use a recently proposed explicit closure for the 5-moment equations. We show that this closure theory results in a hyperbolic system that can mitigate the problem of sub-shocks independent of the shock wave velocity and handle strongly non-equilibrium gas flows.

Steepest-Entropy-Ascent Quantum Thermodynamic Non-Equilibrium Modeling of Decoherence of a Composite System of Two Interacting Spin-½ Systems

2013 ◽  
Author(s):  
Sergio Cano-Andrade ◽  
Michael R. von Spakovsky ◽  
Gian Paolo Beretta
Keyword(s):  
Equilibrium State ◽  
Composite System ◽  
Stable Equilibrium ◽  
Initial Point ◽  
Weighted Average ◽  
Initial State ◽  
Von Neumann ◽  
Composite Quantum System ◽  
Non Linear ◽  
Non Equilibrium

The equation of motion of steepest-entropy-ascent quantum thermodynamics (SEA-QT) was first postulated in the early 1980s with the intent of modeling the non-linear dynamic behavior encountered in nature, which the unitary (linear) dynamics of the Schrödinger-von Neumann equation cannot. The SEA-QT equation is used here to model the decoherence phenomenon between two distinguishable and indivisible elementary constituents of type spin–½ (e.g., quantum bits or qubits). The resulting set of non-linear, first-order differential equations is solved with a fourth-order-Runge-Kutta routine provided by Matlab®. The time evolution of the state of the composite system as well as that of the reduced and locally-perceived states of the two constituents are traced from an initial non-equilibrium state of the composite along its relaxation towards stable equilibrium at constant system energy. An entangled and generally coherent, initial non-equilibrium state of the composite quantum system is prepared using a heuristic approach, which consists of randomly and homogeneously generating an initial point on the Bloch sphere for each of the constituents and then using a weighted average of their projections to arrive at an initial state for the composite. Results show how the initial entanglement and coherence between the two spin–½ constituents are reduced during relaxation towards a state of stable equilibrium. When the two particles are non-interacting, the initial coherence is lost once stable equilibrium is reached. When they are interacting, the coherence in the final stable equilibrium state is only that due to the interaction.

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