Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

Gorban, A.N. and Karlin, I.V. (2004) Uniqueness of thermodynamic projector and kinetic basis of molecular individualism. Physica A: Statistical Mechanics and its Applications, 336 (3-4), 391-432. (doi:10.1016/j.physa.2004.01.039).

Abstract

Three results are presented: First, we solve the problem of persistence of dissipation for reduction of kinetic models. Kinetic equations with thermodynamic Lyapunov functions are studied. Uniqueness of the thermodynamic projector is proven: There exists only one projector which transforms any vector field equipped with the given Lyapunov function into a vector field with the same Lyapunov function for a given anzatz manifold which is not tangent to the Lyapunov function levels.
Second, we use the thermodynamic projector for developing the short memory approximation and coarse-graining for general nonlinear dynamic systems. We prove that in this approximation the entropy production increases. (The theorem about entropy overproduction.)
In example, we apply the thermodynamic projector to derive the equations of reduced kinetics for the Fokker–Planck equation. A new class of closures is developed, the kinetic multipeak polyhedra. Distributions of this type are expected in kinetic models with multidimensional instability as universally as the Gaussian distribution appears for stable systems. The number of possible relatively stable states of a nonequilibrium system grows as 2m, and the number of macroscopic parameters is in order mn, where n is the dimension of configuration space, and m is the number of independent unstable directions in this space. The elaborated class of closures and equations pretends to describe the effects of “molecular individualism”. This is the third result.

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