Near Equilibrium Onsager Transport Matrix

Onsager Reciprocal Relations for Linear Force-Flux Coupling

• corrections/comments: stephenguerin@fas.harvard.edu
• The constitutive laws typically pertain to the diagonal elements of the matrix and are stroked in black
  , where each law relates a specific gradient to its corresponding flux in a linear manner (e.g., Fick's Law relating concentration gradient to mass flux, Ohm's Law relating electric potential gradient to electric current, etc.). These laws describe the linear, "self-coupled" behavior of each type of gradient and flux under near-equilibrium conditions. The off-diagonal elements represent cross-couplings between different types of gradients and fluxes, which are more complex interactions and may not have standard or widely recognized constitutive laws describing them. In the Onsager formalism, these off-diagonal terms are addressed by introducing coupling coefficients, which describe how a gradient in one quantity induces a flux in another. The exact nature of these cross-couplings can be quite complex and may not be fully described by simple linear relations, especially far from equilibrium.
• The individual off-diagonal elements cell(i,j) and cell(j,i) in the matrix are "transpose elements" or "transposed pairs" relative to each other. Two transposed pairs are colored to highlght well-known reciprocal effects: the Soret and Dufour Effects
  and Seebeck and Peltier Effects
• Dufour and Soret effects or the Peltier and Seebeck effects, are often referred to as "reciprocal" phenomena or "reciprocal" effects in the context of the Onsager reciprocal relations. The term "reciprocal" denotes the symmetry or reversibility of these effects with respect to the interchange of driving gradients and resultant fluxes. In the realm of thermoelectric effects, for instance, the Seebeck effect is the generation of a voltage difference in response to a temperature gradient, while its reciprocal, the Peltier effect, is the generation of a temperature difference in response to a voltage difference. This reciprocity is a fundamental feature of the Onsager formalism, which postulates that the coefficients relating gradients to fluxes are symmetric with respect to a specific transformation, reflecting a deep symmetry in the underlying physics of these coupled processes. So, in summary, you could use the term "reciprocal effects" to refer to such dual phenomena in the Onsager matrix.
• The cells marked with an '*' indicate that the term is common or widely accepted for that particular intersection.
• The terms used to fill in the off-diagonal cells are not standard terms and are descriptive of interactions that could potentially occur at these intersections. These terms are creative extrapolations and more research would be needed to determine the most appropriate terms for these intersections.
• Some of the phenomena listed may involve spontaneous symmetry breaking in ways that are not fully captured by this matrix. Therefore, this matrix should be considered as a starting point for further exploration and research rather than a comprehensive list of all phenomena involving spontaneous symmetry breaking.
• See Far From Equilibrium Onsager Transport Matrix for a start at organizing self-organizing phenomena.