Introduction
1.1 Thermodynamics and Kinetics
Kinetic processes in a large system are typically rapid over short length scales, so that equilibrium is nearly satisfied locally. (p. 2)
Why are kinetic processes in a large system typically rapid over short length scales?
1.1.1 Classical Thermodynamics and Constructions of Kineetic Theories
Thermodynamics is precise, but is strictly applicable to phenomena that are unachievable in finite systems in finite amounts of time. (p. 3)
Why is thermodynamics unachievable in finite systems in finite amounts of time?
The equilibrium condition, which disallows spatial variations in a potential (e.g., the gradient in chemical potential or pressure), cannot exist in the presence of active physical processes that allow the conjugate extensive density (composition or volume/mole) to change. This implies that a small set of homogeneous potentials can be specified for a heterogeneous system at equilibrium--and therefore the number of parameters required to characterize an equilibrium system is relatively small. (p.3)
Why does the equilibrium condition, which disallows spatial variations in a potential, imply that a small set of homogeneous potentials can be specified for a heterogeneous system at equilibrium?
A quandary arises: general statements regarding the approach to equilibrium that are based on thermodynamic functions necessarily involve extrapolations away from equilibrium conditions. However, useful models and theories can be developed from approximate expressions for functions having minima that coincide with the equilibrium thermodynamic quantities and from assumptions of local equilibrium states. (p. 5)
Is it the application of methods associated with statistical mechanics that are precluded?
Does the ergodic hypotheses not apply to kinetically evolving systems because of a short time scale
In developing useful kinetic theories and models, is there a general approach in making approximations of expressions demonstrating minima that coincide with the equilibrium thermodynamic quantities?
Away from equilibrium, the various parts of a system generally have gradients in potentials and there is no guarantee of the existence of an integrable local free-energy density. (p. 3)
What are the conditions under which free-energy density is integrable?
1.1.2 Averaging
It is possible to construct interatomic potentials and forces between atoms that approximate real systems in a limited number of atomic configurations. (p. 4)
In constructing interatomic potentials and forces, what are the real systems that are approximated?
Why are there a limited number of atomic configurations in which it is possible to construct interatomic potentials and forces between atoms that approximate real systems?
Just as statistical mechanics overcomes difficulties arising from large numbers of interacting particles by constructing rigorous methods of averaging, kinetic theory also uses averaging. However, the application of these methods to kinetically evolving systems is precluded because many of the fundamental assumptions of statistical mechanics (e.g., the ergodic hypothesis) do not apply.
1.2 Irreversible Thermodynamics and Kinetics
However, a chemical potential of a component is the amount of reversible work needed to add an infitesimal amount of that component to a system at equilibrium. Can a chemical potential be defined when the system is not at equilibrium? This cannot be done rigorously, but based on decades of development of kinetic models for processes, it is useful to extend the concept of the chemical potential to systems close to, but not at, equilibrium. (p. 6)
Why can the chemical potential not be defined rigorously when the system is not at equilibrium?
Physically, no such thermometer can exist--nor can a real material be divided infitesimally. However, this does not mean that one's intuition about the existence of such a function T
is wrong; it is reasonable to take a continuum limit of such an idealized measurement and refer to the temperature at a point.
What is the definition of a continuum limit?
1.3 Mathematical Background
1.3.1 Fields
1.3.2 Variations
1.3.3 Continuum Limits and Coarse Graining
This defintion has the correct global behavior for large volumes V because... (p. 9)
Why is it accurate to approximate zeta(r - ri) as zeta(r) in Equation 1.6
In this book, it is assumed that the continuum limits exist and coarse-grained functions can be obtained that do not depend significantly on the choice of zeta. (p. 10)
How are coarse-grained functions obtained?
1.3.4 Fluxes
1.3.5 Accumulation
1.3.6 Conserved and Nonconserved Quantities
1.3.7 Matrices, Tensors, and the Eigensystem
It is often convenient to select the coordinate system for which the only nonzero elements of the property tensor lie on its diagonal.