A Boltzmann transport equation approach is developed for reaction-diffusion systems which incorporates phonon transport in addition to the traditional approach. Scattering processes up to second order are taken into account. Two forces emerge from this analysis when a spatial gradient exists, one force on reactants and products, the other force on phonons. The forces are equal and opposite and have the tendency for separation of the phonons away from the reactants and products. These forces are capable of creating the types of instabilities that can lead to the formation of Turing patterns. The existence of these forces allows for exergonic conversion where not all of the released energy from reactions and diffusion becomes heat. When applied to homogeneous catalysis, one finds that reactants and products are pushed toward regions of greater catalytic activity. In the realm of high-energy explosions, calculations show that reactants and products can be accelerated laterally to the direction of a TNT reaction front up to speeds near 1000 m/s. This acceleration is in opposition to diffusion and represents active transport. Calculations also show that active transport observed in biological systems such as bacteria, mitochondria, and chloroplasts may be explained by this second-order transport theory. Using reasonable values for key parameters, calculations show that up to one-third of the available chemical energy can be converted toward pumping protons uphill to a potential of 50 mV.  

An electron-phonon Boltzmann transport equation is formulated which accounts for second order collisions with an electron-phonon vertex and a three-phonon vertex. This approach for electronic transport at second order reveals the existence of two forces perpendicular to the primary direction of electrical current, acting on the electrons and phonons. The force on electrons is equal and opposite to that on the phonons. Solutions for stationary states confirm that charge and thermal energy become separated. The force terms include both conservative and dissipative components, which for the phonons, lead to a modified Guyer-Krumhansl equation. The conservative components exist only when there exist explicit transverse gradients in the dissipated energy, and these terms may be incorporated into a Poisson kinematics. The dissipative force terms can cause threshold induced spontaneous symmetry breaking.

Development of thermodynamic induction up to second order gives a dynamical bifurcation for thermodynamic variables and allows for the prediction and detailed explanation of nonequilibrium phase transitions with associated spontaneous symmetry breaking.  By taking into account nonequilibrium fluctuations, long range order is analyzed for possible pattern formation.  Consolidation of results up to second order produces thermodynamic potentials that are maximized by stationary states of the system of interest.  These new potentials differ from the traditional thermodynamic potentials.In particular a generalized entropy is formulated for the system of interest which becomes the traditional entropy when thermodynamic equilibrium is restored.  This generalized entropy is maximized by stationary states under nonequilibrium conditions where the standard entropy for the system of interest is not maximized.  These new nonequilibrium concepts are incorporated into traditional thermodynamics, such as a revised thermodynamic identity, and a revised canonical distribution.   Detailed analysis shows that the second law of thermodynamics is never violated even during any pattern formation, thus solving the entropic coupling problem.  Examples discussed include pattern formation during phase front propagation under nonequilibrium conditions and the formation of Turing patterns.  The predictions of second order thermodynamic induction are consistent with both observational data in the literature as well as the modeling of this data.

A method is described for cooling conductive channels to below ambienttemperature. The thermodynamic induction principle dictates that the electricallybiased channel will cool if the electrical conductance decreases with temperature. Theextent of this cooling is calculated in detail for both cases of ballistic and conventionaltransport with specific calculations for carbon nanotubes and conventional metals,followed by discussions for semiconductors, graphene, and metal–insulator transitionsystems. A theorem is established for ballistic transport stating that net cooling isnot possible. For conventional transport, net cooling is possible over a broad temperaturerange, with the range being size-dependent. A temperature clamping schemefor establishing a metastable nonequilibrium stationary state is detailed and followedwith discussion of possible applications to on-chip thermoelectric cooling in integratedcircuitry and quantum computer systems.