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It is shown that Onsager’s principle of the least dissipation of energy is equivalent to the maximum entropy production principle. It is known that solutions of the linearized Boltzmann equation make extrema of entropy production. It is argued, in the case of stationary processes, that this extremum is a maximum rather than a minimum.

It was Ehrenfest (Enzykl. Math. Wissensch, IV, 2(II) fasc.6, p82, note23, 1912) who first asked whether a function exists which, like entropy in the equilibrium state of an isolated system, achieves its extreme value in a stationary non-equilibrium state.

There are various results and formulations of irreversible thermodynamics based on the extremum of entropy production. They are related either to the minimum or maximum entropy production. The well known example based on minimum entropy production is Prigogine theorem [

There are several different results that concern maximum entropy production (MEP). Ziegler applied the principle of maximum entropy production in thermomechanics [

Kohler [

Recently Jaynes’ principle of maximum information entropy (MaxEnt) has been exploited to derive the MEP principle. Dewar [

At first sight, minimum and maximum entropy production results seem to contradict each other. But if one considers starting assumptions one finds that these assumptions are different. Thus, these results are independent of one another [

In this paper we focus on Onsager’s principle of least dissipation of energy and show that this principle is equivalent to MEP principle. Secondly, as we have already noted, we show that Kohler interpretation of stationary state of rarefied gas close to the equilibrium should be restricted only on the constraint that it is consistent with the interpretation of this state as the MEP state.

This is the first of three papers in a series and are referred to as papers I, II and III. In paper II we discuss whether stationary or relaxation processes are suitable for the formulation of principles. We argue in favor of relaxation processes. In paper III we apply MaxEnt formalism to relaxation processes and we derive the MEP principle as its corollary.

Onsager’s famous papers [

A linear relationship between

In his second famous paper Onsager [

The variational procedure

An alternative approach to linear nonequilibrium thermodynamics is based on the phenomenological fact that nonequilibrium processes are characterized by fluxes. Therefore, physical quantities relevant for a description of the time development of the system must be functions of fluxes. The standard approach to the nonequilibrium thermodynamics is based on the laws of conservation of mechanical physical quantities mass, momentum and energy. The next step is to add heat as the additional mechanism of the exchange of energy between systems. Then assuming local equilibrium one comes to the density of entropy production written as the sum of products of heat flux and thermal thermodynamic force and viscous pressure tensor and strain tensor divided by temperature. This result serves as the basis for the canonical form of the second postulate of irreversible thermodynamics that reads: Entropy production can be always written as the sum of products of thermodynamic forces and corresponding fluxes [

Entropy production is a basic, characteristic quantity of a nonequilibrium state. If the system is close to the equilibrium state we can make the Taylor expansion of the density of entropy production

Here

Comparison of Equations (

It is pointed in the introductory part of this Section that canonical form of entropy production (

We seek the maximum entropy production (

A standard variational calculus of extremum values combined with the constraint (

This is an equation of a quadratic surface turned upside down. The corresponding extremum of

From Equation (

A problem analogous to this is the problem of biochemical cycle kinetics close to the equilibrium state. Starting from the assumption that the energy conservation law is valid for a whole network of biochemical reactions we have shown that fluxes are distributed in such a way to produce maximum entropy [

In references [

Using mesh currents the law of charge conservation has been taken implicitly. The energy conservation law is used explicitly as the constraint. If one only takes charge conservation law one comes to the conclusion that stationary state is state of minimum generated heat [

We note that two of us (P.Ž. and D.J.) have considered the heat flow in the anisotropic crystal. It is shown in this special example that the principle of the least dissipation of energy applied by Onsager in reference [

The important influence of constraints at extrema of entropy production can be seen in the case of linearized Boltzmann equation.

Here we follow the elegant approach given in reference [

Here, the two terms on the left-hand side of the equation describe the change in the number of molecules in a given element of the phase space due to the collisionless motion of the molecule in the outer field

Assuming the instantaneous change of molecule velocities in the collisions and taking into account the conservation laws in collisions the integral can be written in the form [

Here,

In the case of local equilibrium, intensive variables (temperature, concentration) are well defined functions of the space. Then one assumes the approximate solution of Equation (

Due to the assumption of local equilibrium the perturbed function must not contribute to the prescribed intensive parameters like temperature, mean velocity and density. Then

Now the collision integral up to the first order of the perturbed term becomes

The designation of Ψ functions is the same as the distribution functions in Equation (

Operator

If we designate the left hand side of Boltzmann equation (

Multiplying the linearized Boltzmann equation with

Here

Function

If the temperature gradient in

Here,

The multiplication of the Boltzmann equation with the perturbative distribution function Ψ and integration over molecular velocity space

Using conditions (

The left-hand side is the entropy production of the rarefied gas. It comes from Equation (

There is another approach to this problem which has been proposed by Kohler [

The Onsager principle of the least dissipation of energy is valid for processes close to the equilibrium state. In this paper, the equivalence between the Onsager principle of the least dissipation of energy and the MEP principle is established. Starting from the fact that fluxes are the main phenomenological characteristic of irreversible processes, we have expanded the density of entropy production as the function of fluxes up to the second order. We have found that the dissipation function introduced by Onsager is the entropy production in the space of the fluxes. Invoking the first law of thermodynamics, the equivalency between the Onsager principle of the least dissipation energy and the MEP principle is established.

It follows from the Boltzmann equation that entropy production is quantitatively related to the collision integral. Collisions between molecules in nonequilibrium state produce entropy. In this paper the solutions of the linearized Boltzmann equations are considered. It is found that these solutions correspond to the extremum of entropy production. The nature of the extremum depends on the constraints. Assuming that the entropy produced by molecular collisions is equal to the entropy productions due to the heat conduction or/and viscosity, we find that the solutions of the Boltzmann equation satisfy the MEP principle.

The principle of minimum entropy production is valid if one grants fixed fluxes. However, the starting assumption of the fixed thermodynamic forces and the additional assumption of fixed fluxes leave no room for variation. Although both approaches are equivalent from the mathematical point of view one has to notice that the assumption of fixed fluxes violates the equality between entropy production defined via collision integral and as the sum of products of thermodynamic forces and conjugated fluxes.

In short, revisiting the linear nonequilibrium thermodynamics and linearized Boltzmann equation shows that both approaches are in accordance with the MEP principle. Generally we can conclude that the MEP principle is valid for processes close to the equilibrium state.

The present work was supported by the bilateral research project of the Slovenia-Croatia Cooperation in Science and Technology, 2009-2010 and Croatian Ministry of Science grant No. 177-1770495-0476 to DJ.