Irreversibility and entropy production in transport phenomena, IV: Symmetry, integrated intermediate processes and separated variational principles for multi-currents

The mechanism of entropy production in transport phenomena is discussed again by emphasizing the role of symmetry of non-equilibrium states and also by reformulating Einstein’s theory of Brownian motion to derive entropy production from it. This yields conceptual reviews of the previous papers [M. Suzuki, Physica A 390 (2011) 1904; 391 (2012) 1074; 392 (2013) 314]. Separated variational principles of steady states for multi external fields { X i } and induced currents { J i } are proposed by extending the principle of minimum integrated entropy production found by the present author for a single external field. The basic strategy of our theory on steady states is to take in all the intermediate processes from the equilibrium state to the final possible steady states in order to study the irreversible physics even in the steady states. As an application of this principle, Gransdorff–Prigogine’s evolution criterion inequality (or stability condition) d X P ≡ ∫ d r ∑ i J i d X i ≤ 0 is derived in the stronger form d Q i ≡ ∫ d r J i d X i ≤ 0 for individual force X i and current J i even in nonlinear responses which depend on all the external forces { X k } nonlinearly. This is called “separated evolution criterion”. xplicit demonstrations of the present general theory to simple electric circuits with multi external fields are given in order to clarify the physical essence of our new theory and to realize the condition of its validity concerning the existence of the solutions of the simultaneous equations obtained by the separated variational principles. It is also instructive to compare the two results obtained by the new variational theory and by the old scheme based on the instantaneous entropy production. This seems to be suggestive even to the energy problem in the world.