Chapter 12 - Least Action Principle for Dissipative Processes

Abstract

This chapter points out that the least action principle can be applied for dissipative processes. The Lagrange-Hamilton formalism can be completely worked out and this mathematical model of non-equilibrium thermodynamics and the examinations of symmetries lead to the Onsager's reciprocity relations. Based on Poisson structure, the stochastic behavior of processes can be described in the phase space. Classical irreversible thermodynamics deals with such large and continuous media of which, states can be given by the field of equilibrium state variables beside the equilibrium; the hypothesis of local equilibrium is required. Of course, this hypothesis also includes the fact that the relations of thermostatic are valid at a given point ‘r’, and at a given time ‘t’. A famous relation, the Onsager's symmetry relation, can be proved as a consequence of a dynamical symmetry. Physical quantities (entropy, entropy current, and production) can be introduced and this shows that the theory is connected to the theory of non-equilibrium thermodynamics. The concept of a new phase field is based on the space of the calculated canonically conjugated quantities and this enables examination of the stochastic behavior of processes. The constructed Poisson structure gives an opportunity to calculate Onsager's regression hypothesis.