1201.6315 (A. N. Gorban)

Thermodynamic Tree: The Space of Admissible Paths    [PDF]

A. N. Gorban

Is a spontaneous transition from a state x to a state y allowed by
thermodynamics? Such a question arises often in chemical thermodynamics and
kinetics. We ask the more formal question: is there a continuous path between
these states, along which the conservation laws hold, the concentrations remain
non-negative and the relevant thermodynamic potential G (Gibbs energy, for
example) monotonically decreases? The obvious necessary condition, G(x)\geq
G(y), is not sufficient, and we construct the necessary and sufficient
conditions. For example, it is impossible to overstep the equilibrium in
1-dimensional (1D) systems (with n components and n-1 conservation laws). The
system cannot come from a state x to a state y if they are on the opposite
sides of the equilibrium even if G(x) > G(y). We find the general
multidimensional analogue of this 1D rule and constructively solve the problem
of the thermodynamically admissible transitions.
We study dynamical systems, which are given in a positively invariant convex
polyhedron and have a convex Lyapunov function G. An admissible path is a
continuous curve along which $G$ does not increase. For x,y from D, x>y (x
precedes y) if there exists an admissible path from x to y and x \sim y if x>y
and y>x. The tree of G in D is a quotient space D/~. We provide an algorithm
for the construction of this tree. In this algorithm, the restriction of $G$
onto the 1-skeleton of $D$ (the union of edges) is used. The problem of
existence of admissible paths between states is solved constructively. The
regions attainable by the admissible paths are described.

View original: http://arxiv.org/abs/1201.6315