Chapter 3
Generalized Onsager Principle
and It Applications
Qi Wang
Abstract We review the established nonequilibrium thermodynamical principle
based on the Onsager linear response theory for irreversible processes, known as the
Onsager principle for dissipative systems, firstly. We present it in two different forms:
(1) the constructive formulation of the Onsager principle encompassing the kinetic
equation, the reciprocal relation on the mobility and the maximum entropy principle; (2) the variational formulation in which the Onsager-Machlup action potential
is maximized, or equivalently the Rayleighian is minimized in the isothermal case.
Then, we generalize the Onsager principle in its constructive form to allow both
reversible and irreversible processes to be modeled as well as mobility coefficient
dependent on thermodynamical variables, which is termed the generalized Onsager
principle (GOP). We carry out derivations of a plethora of thermodynamical and
generalized hydrodynamical theories using the Onsager principle and the generalized Onsager principle to demonstrate their usefulness in establishing new models
for nonequilibrium thermodynamical systems.
Keywords General Onsager principle · Onsager Machlup action potential ·
Generalized hydrodynamics · Non-equilibrium thermodynamical system
3.1 Introduction
Theories for equilibrium thermodynamics have been well developed based on the
three fundamental thermodynamical laws: the first, the second and the third law of
thermodynamics [8, 16, 17]. For nonequilibrium systems, however, theories are not
quite so soundly developed in that there does not exist any physical laws like the
three fundamental thermodynamical laws in nonequilibrium mechanics [3, 11]. For
thermodynamical systems near equilibrium, the second law of thermodynamics in
the form of the Clausius-Duhem inequality and the Onsager linear response theory
Q. Wang (B)
Department of Mathematics,University of South Carolina, Columbia, SC 29208, USA
e-mail: qwang@math.sc.edu
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
X.-Y. Liu (ed.), Frontiers and Progress of Current Soft Matter Research,
Soft and Biological Matter, https://doi.org/10.1007/978-981-15-9297-3_3
101
Generalized Onsager Principle
and It Applications
Qi Wang
Abstract We review the established nonequilibrium thermodynamical principle
based on the Onsager linear response theory for irreversible processes, known as the
Onsager principle for dissipative systems, firstly. We present it in two different forms:
(1) the constructive formulation of the Onsager principle encompassing the kinetic
equation, the reciprocal relation on the mobility and the maximum entropy principle; (2) the variational formulation in which the Onsager-Machlup action potential
is maximized, or equivalently the Rayleighian is minimized in the isothermal case.
Then, we generalize the Onsager principle in its constructive form to allow both
reversible and irreversible processes to be modeled as well as mobility coefficient
dependent on thermodynamical variables, which is termed the generalized Onsager
principle (GOP). We carry out derivations of a plethora of thermodynamical and
generalized hydrodynamical theories using the Onsager principle and the generalized Onsager principle to demonstrate their usefulness in establishing new models
for nonequilibrium thermodynamical systems.
Keywords General Onsager principle · Onsager Machlup action potential ·
Generalized hydrodynamics · Non-equilibrium thermodynamical system
3.1 Introduction
Theories for equilibrium thermodynamics have been well developed based on the
three fundamental thermodynamical laws: the first, the second and the third law of
thermodynamics [8, 16, 17]. For nonequilibrium systems, however, theories are not
quite so soundly developed in that there does not exist any physical laws like the
three fundamental thermodynamical laws in nonequilibrium mechanics [3, 11]. For
thermodynamical systems near equilibrium, the second law of thermodynamics in
the form of the Clausius-Duhem inequality and the Onsager linear response theory
Q. Wang (B)
Department of Mathematics,University of South Carolina, Columbia, SC 29208, USA
e-mail: qwang@math.sc.edu
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
X.-Y. Liu (ed.), Frontiers and Progress of Current Soft Matter Research,
Soft and Biological Matter, https://doi.org/10.1007/978-981-15-9297-3_3
101