3.21 Lecture 1 Spring 2006: Fields and Gradients; Fluxes; Continuity Equation

Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics.12 Roughly, heat means "energy in transit" and dynamics relates to "movement"; thus, in essence thermodynamics studies the movement of energy and how energy instills movement. Historically, thermodynamics developed out of the need to increase the efficiency of early steam engines.

In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. The local state of a system at thermodynamic equilibrium is determined by the values of its intensive parameters, as pressure, temperature, etc. Specifically, thermodynamic equilibrium is characterized by the minimum of a thermodynamic potential

  •  The concept of local equilibrium is applicable to real materials on a micro-scale.

It is useful to distinguish between global and local thermodynamic equilibrium. In thermodynamics, exchanges within a system and between the system and the outside are controlled by intensive parameters. As an example, temperature controls heat exchanges. Global thermodynamic equilibrium (GTE) means that those intensive parameters are homogeneous throughout the whole system, while local thermodynamic equilibrium (LTE) means that those intensive parameters are varying in space and time, but are varying so slowly that for any point, one can assume thermodynamic equilibrium in some neighborhood about that point.

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.

In thermodynamics, thermodynamic potentials are parameters associated with a thermodynamic system and have the dimensions of energy. They are called "potentials" because in a sense, they describe the amount of potential energy in a thermodynamic system when it is subjected to certain constraints. The different potentials correspond to different constraints to which the system may be subjected. 

  • A scalar field associates a physical quantity with position--e.g. a composition field

In mathematics and physics, a scalar field associates a scalar value, which can be either mathematical in definition, or physical, to every point in space. Scalar fields are often used in physics, for instance to indicate the temperature distribution throughout space, or the air pressure. In mathematics, or more specifically, differential geometry, the set of functions defined on a manifold define the commutative ring of functions.  A gradient of a scalar field is a vector that quantifies how rapidly the field changes with position

  • The flux of a substance quantifies the rate at which that substance flows through a unit area.  The flux is a vector that is parallel to the local direction of the flow.
  • The rate of accumulation of an extensive quantity is minus the divergence of the flux of that quantity plus the rate of production of the substance.

In the study of transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as the amount that flows through a unit area per unit time, the volumetric flow rate1. Flux in this definition is a vector. 

In physics and chemistry an intensive property (also called a bulk property) of a system is a physical property of the system that does not depend on the system size or the amount of material in the system. By contrast, an extensive property of a system does depend on the system size or the amount of material in the system. However, some of the intensive properties are statistical in nature (e.g. viscosity) and are relevant only in aggregate scales.

In vector calculus, the divergence is an operator that measures a vector field's tendency to originate from or converge upon a given point. For instance, for a vector field that denotes the velocity of air expanding as it is heated, the divergence of the velocity field would have a positive value because the air is expanding. Conversely, if the air is cooling and contracting, the divergence would be negative. 

  • For conserved quantities like the number of moles of a component in a solution there are no sources or sinks and hence no production of the substance.
  • For non-conserved quantities like entropy there can be production of the quantity during the course of a spontaneous process.

In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. Any particular conservation law is a mathematical identity to certain symmetry of a physical system.  

The concept of entropy in thermodynamics is central to the second law of thermodynamics, which deals with physical processes and whether they occur spontaneously. Spontaneous changes occur with an increase in entropy. Spontaneous changes tend to smooth out differences in temperature, pressure, density, and chemical potential that may exist in a system, and entropy is thus a measure of how far this smoothing-out process has progressed. In contrast, the first law of thermodynamics deals with the concept of energy, which is conserved. Entropy change has often been defined as a change to a more disordered state at a molecular level. In recent years, entropy has been interpreted in terms of the "dispersal" of energy. Entropy is an extensive state function that accounts for the effects of irreversibility in thermodynamic systems.

  • Numerous kinetic processes are described by linear equations relating fluxes and driving forces. 

A linear equation is an equation in which each term is either a constant or the product of a constant times the first power of a variable. Such an equation is equivalent to equating a first-degree polynomial to zero. These equations are called "linear" because they represent straight lines in Cartesian coordinates.  

  • Methods from linear algebra are often used to simplify the description by using coordinates parallel to crystal axes, or by finding the principal axes.

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis. Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory. It has extensive applications in the natural sciences and the social sciences, since nonlinear models can often be approximated by a linear one.

  • Matrix eigenvalues, eigenvectors, and similarity transformations are useful tools for describing coupled kinetic processes.

In mathematics, an eigenvector  (help-info) of a transformation1 is a vector which, in the transformation, is multiplied by a constant factor, called the eigenvalue (help-info) of that vector. (See Fig. 1). Often, a transformation is completely described by its eigenvalues and eigenvectors. The eigenspace for a factor is the set of eigenvectors with that factor as eigenvalue.

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