Thermodynamics of Continuous Systems

2003 ◽  
Author(s):  
E. S. Geskin
Keyword(s):  
Boltzmann Equation ◽  
Local Equilibrium ◽  
Continuous System ◽  
Intrinsic Property ◽  
Continuous Systems ◽  
Initial State ◽  
The Boltzmann Equation ◽  
Generalized Variables ◽  
Classical Thermodynamics ◽  
Non Equilibrium

The formalism of the classical thermodynamics, for example Gibbs equations, is routinely and successfully applied to the highly non-equilibrium processes in dynamic systems. Such applications are based on the local equilibrium hypothesis. The presented paper discusses the conditions of the application of this hypothesis. It is shown that the local equilibrium hypothesis is applicable with no limitations to continuous systems. This application is validated by the solution of the Boltzmann equation. This solution was obtained by Enskog, Chapman and Bogolubov. From the Boltzmann equation follows that regardless of the initial state of the system the duration of its approach to the local equilibrium conditions by far less than the time scale of the evolution of the macro properties of the continuous media. This result shows that the local equilibrium is the intrinsic property of this media. Thus it is possible to apply the formalism of the non-equilibrium thermodynamics (the generalized variables, forces and fluxes) to description of the continuous system with no limitations. The derivation of the Carnot theorem is presented to show the effectiveness of such an application.

scholarly journals Simulation of non-equilibrium gas kinetic processes in the multitube Knudsen pump on the basis of the Boltzmann equation

2020 ◽  
Vol 1560 ◽  
pp. 012061
Author(s):  
Kloss Yu Yu ◽  
F G Tcheremissine ◽  
M Yu Shirkin ◽  
I V Govorun ◽  
V Shirokovskaya Yu
Keyword(s):  
Boltzmann Equation ◽  
The Boltzmann Equation ◽  
Kinetic Processes ◽  
Knudsen Pump ◽  
Non Equilibrium

A spectral solution of the Boltzmann equation for the infinitely strong shock

1990 ◽  
Vol 330 (1611) ◽  
pp. 217-252 ◽  
Keyword(s):  
Boltzmann Equation ◽  
Constitutive Relations ◽  
Local Equilibrium ◽  
Nonlinear Dynamical System ◽  
Strong Shock ◽  
Singular Part ◽  
Orthogonal Functions ◽  
The Boltzmann Equation ◽  
Highly Nonlinear ◽  
Background Gas

We formulate and implement a new spectral method for the solution of the Boltzmann equation, making extensive use of the theory of irreducible tensors together with the symbolic notation of Dirac. These tools are shown to provide a transparent organization of the algebra of the method and the efficient automation of the associated calculations. I he power of the proposed method is demonstrated by application to the highly nonlinear problem of the infinitely strong shock. It is shown that the distribution function can in this limit be decomposed into a singular part corresponding to the molecular beam, which represents the supersonic side of the shock, and a regular part, which provides the evolving ‘ background gas and covers the rest of velocity space. Separate governing equations for the singular and regular parts are derived, and solved by an expansion of the latter in an infinite series of orthogonal functions. The basis for this expansion is the same set that was used by Burnett ( Proc. Lond. math. Soc . 39, 385-430 (1935)), but is centred around the (fixed) downstream maxwellian. This basis, because of the presence of spherical harmonics which provide an irreducible representation of the group SO (3), lends itself to the utilization of powerful group-theoretic tools. The present expansion, not being about local equilibrium, does not imply any constitutive relations; instead it reduces the Boltzmann equation to an equivalent infinite-order nonlinear dynamical system. A solution with six modes shows encouraging convergence in the density profile, towards a shock thickness of about 6.7 hot-side mean free paths.

scholarly journals Steepest Entropy Ascent Models of the Boltzmann Equation: Comparisons With Hard-Sphere Dynamics and Relaxation-Time Models for Homogeneous Relaxation From Highly Non-Equilibrium States

2013 ◽  
Author(s):  
Gian Paolo Beretta ◽  
Nicolas G. Hadjiconstantinou
Keyword(s):  
Relaxation Time ◽  
Boltzmann Equation ◽  
Power Law ◽  
Collision Frequency ◽  
Equilibrium States ◽  
Space Distribution ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Far From Equilibrium ◽  
Non Equilibrium

We present a family of steepest entropy ascent (SEA) models of the Boltzmann equation. The models preserve the usual collision invariants (mass, momentum, energy), as well as the non-negativity of the phase-space distribution, and have a strong built-in thermodynamic consistency, i.e., they entail a general H-theorem valid even very far from equilibrium. This family of models features a molecular-speed-dependent collision frequency; each variant can be shown to approach a corresponding BGK model with the same variable collision frequency in the limit of small deviation from equilibrium. This includes power-law dependence on the molecular speed for which the BGK model is known to have a Prandtl number that can be adjusted via the power-law exponent. We compare numerical solutions of the constant and velocity-dependent collision frequency variants of the SEA model with the standard relaxation-time model and a Monte Carlo simulation of the original Boltzmann collision operator for hard spheres for homogeneous relaxation from near-equilibrium and highly non-equilibrium states. Good agreement is found between all models in the near-equilibrium regime. However, for initial states that are far from equilibrium, large differences are found; this suggests that the maximum entropy production statistical ansatz is not equivalent to Boltzmann collisional dynamics and needs to be modified or augmented via additional constraints or structure.

A novel construction of thermodynamically compatible models and its correspondence with Boltzmann-equation-based moment-closure hierarchies

2015 ◽  
Vol 40 (4) ◽  
Author(s):  
Liu Hong ◽  
Zaibao Yang ◽  
Yi Zhu ◽  
Wen-An Yong
Keyword(s):  
Boltzmann Equation ◽  
Irreversible Thermodynamics ◽  
Moment Closure ◽  
State Variables ◽  
Balance Equations ◽  
Equilibrium Thermodynamics ◽  
Macroscopic Modeling ◽  
The Boltzmann Equation ◽  
Novel Approach ◽  
Non Equilibrium

AbstractIn this article, we propose a novel approach to construct macroscopic balance equations and constitutive equations describing various irreversible phenomena. It is based on the general principles of non-equilibrium thermodynamics and consists of four basic steps: picking suitable state variables, choosing a strictly concave entropy function, properly separating entropy fluxes and production rates, and determining a dissipation matrix. Our approach takes advantage of both extended irreversible thermodynamics and GENERIC formalisms and shows a direct correspondence with Levermore's moment-closure hierarchies for the Boltzmann equation. As a direct application, a new ten-moment model beyond the classical hierarchies is constructed and is shown to recover the Euler equations in the equilibrium state. These interesting results may put various macroscopic modeling approaches, starting from the general principles of non-equilibrium thermodynamics, on a solid microscopic foundation based on the Boltzmann equation.

scholarly journals Collective classical motion on hyperbolic spacetimes of any dimensions

2019 ◽  
Vol 34 (21) ◽  
pp. 1950165
Author(s):  
Ion I. Cotăescu
Keyword(s):  
Boltzmann Equation ◽  
Simple Equation ◽  
Conserved Quantities ◽  
De Sitter ◽  
Classical Motion ◽  
Canonical Variables ◽  
The Boltzmann Equation ◽  
Starting Point ◽  
Hyperbolic Spacetimes ◽  
Non Equilibrium

The geodesics equations on de Sitter (dS) and anti-de Sitter (AdS) spacetimes of any dimensions, are the starting point for deriving the general form of the Boltzmann equation in terms of conserved quantities. The simple equation for the non-equilibrium Marle and Anderson–Witting models are derived and the distributions of the Boltzmann–Marle model on these manifolds are written down first in terms of conserved quantities and then as functions of canonical variables.

The Boltzmann Equation and the H Theorem (1872–1875)

2018 ◽  
Author(s):  
Olivier Darrigol
Keyword(s):  
Heat Conduction ◽  
Boltzmann Equation ◽  
Transport Phenomena ◽  
Dilute Gas ◽  
The Boltzmann Equation ◽  
H Theorem ◽  
Irreversible Evolution

This chapter covers Boltzmann’s writings about the Boltzmann equation and the H theorem in the period 1872–1875, through which he succeeded in deriving the irreversible evolution of the distribution of molecular velocities in a dilute gas toward Maxwell’s distribution. Boltzmann also used his equation to improve on Maxwell’s theory of transport phenomena (viscosity, diffusion, and heat conduction). The bulky memoir of 1872 and the eponymous equation probably are Boltzmann’s most famous achievements. Despite the now often obsolete ways of demonstration, despite the lengthiness of the arguments, and despite hidden difficulties in the foundations, Boltzmann there displayed his constructive skills at their best.

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