Faculty of Science, University of Split, Teslina 12, 21000 Split, Croatia
Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia
Author to whom correspondence should be addressed.
Received: 13 March 2010 / Accepted: 23 April 2010 / Published: 27 April 2010
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.
entropy production; linear nonequilibrium thermodynamics; linearized Boltzmann equation
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 . It states that the rate of entropy production under prescribed forces should be maximum. Paltridge  has formulated empirically MEP principle in order to describe the Earth climate. Apart from details of his model (like albedo or cloudiness) the main features of his model are steady incoming sun radiation, outgoing radiation proportional to the fourth power of the local annual average temperature and horizontal heat flux from equator to poles. There are no boundary conditions on atmosphere. Paltridge has proposed that steady state is the state of maximum entropy production due to the latitudinal heat transport. His predictions of the annually average distribution of the temperature of atmosphere (and some other climatic parameters) fit very well measured values. We note that Paltridge MEP principle is applied to nonlinear process and is different from the linear problems considered in this paper. Chavanis [4,5] has applied MEP principle to the problem of two dimensional turbulence and stellar systems in order to find optimal currents. Connection between MEP principle and Fokker-Planck equation is established, too.
Kohler  starting from the Boltzmann’s transport equation and using the definition of the entropy as the negative integral of product of distribution function and its natural logarithm in molecular velocity space has found that entropy production can be expressed via scattering integral. It comes from this definition that molecular collisions in nonequilibrium state of rare gas are the sources of entropy production. He has shown that the stationary state close to the equilibrium is the state of extremum entropy production. If one requires that entropy production defined via scattering integral is equal to the entropy production defined as the product of the thermal flux and thermal force then this extremum is maximum one. The another constraint of fixed thermal flux, also used by Kohler, that leads to the conclusion that stationary state of rare gas is the state of minimum entropy production is put in question in this paper.
Recently Jaynes’ principle of maximum information entropy (MaxEnt) has been exploited to derive the MEP principle. Dewar [7,8] has introduced paths as possible trajectories of the system in phase space. Then using the MaxEnt procedure he has shown, under some assumptions, that the most probable development of the system is accompanied with MEP. Niven  has applied MaxEnt using the values of fluxes as variables and he found that the stationary state of the system is the MEP state.
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 [10,11]
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.
2. The Principle of the Least Dissipation of Energy and Linear Nonequilibrium Thermodynamics
Onsager’s famous papers [12,13] examine linear nonequilibrium thermodynamics. The first sections of these papers examine reciprocal relations. The following sections are devoted to the formulation of linear nonequilibrium thermodynamics. Onsager has employed the principle of the least dissipation of energy. In the second part of his second paper , Onsager has applied the principle of the least dissipation energy to the general linear nonequilibrium process.
A linear relationship between n independent thermodynamic forces and their conjugated fluxes ,
exists for the system close to equilibrium. The density of the entropy production is equal to the product of the thermodynamic forces and conjugated fluxes , . The total entropy production is
In his second famous paper Onsager  has introduced the dissipation function
and formulated the principle of the least dissipation of energy,
The variational procedure
gives the system of equation (1). In other words, the linear irreversible thermodynamics can be inferred from the principle of the least dissipation of energy.
3. The MEP Principle and Linear Nonequilibrium Thermodynamics
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 [14,15]. Thermodynamic forces and fluxes are not independent quantities. In principle, fluxes determine forces and vice versa. Having in mind this fact and the second postulate of irreversible thermodynamics, we can say that the dynamical state of the system is described only by fluxes, or equivalently by thermodynamic forces.
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 in fluxes up to the second order,
Here is the mean value of the flux. The first term on the right-hand side of this equation vanishes since there is no entropy production in the equilibrium state. Coefficients and are the property of the system in equilibrium state. Entropy production does not depend on the direction of the flux flow, that is to say, it must be invariant under the replacement of . This means that coefficients that multiply the odd power of fluxes vanish, that is, . In the lowest order, entropy production is the bilinear function of fluxes,
Comparison of Equations (7) and (3) shows that the dissipation function is in fact entropy production written in the space of fluxes. When a system is close to equilibrium with locally well defined intensive thermodynamic quantities the entropy production is given by Equation (2), [1,14].
It is pointed in the introductory part of this Section that canonical form of entropy production (2) comes from the law of conservation of energy. Then equation,
is the constraint imposed at the entropy production due to the law of conservation of energy.
We seek the maximum entropy production (7) taking into account the constraint (8). The standard procedure  is to find the maximum of the functional
A standard variational calculus of extremum values combined with the constraint (8) gives and F becomes
This is an equation of a quadratic surface turned upside down. The corresponding extremum of F is maximum. We refer an interested reader to reference  where it is explicitly shown that this extremum is maximum. The system of equations that determines the point of maximum , , is just a system of linearly coupled thermodynamic forces and fluxes (1). Similar approach that relates MEP principle and Onsager’s variational formulation of the linear nonequilibrium thermodynamics can be found in reference .
From Equation (7) follows i.e., Onsager’s reciprocal relations. In short, linear nonequilibrium thermodynamics and Onsager’s reciprocity relations follow from the MEP principle.
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 [19,20] we have considered a linear planar electric network held at a constant temperature. There is only a temperature gradient between network and surroundings. Physically this can be achieved using thermally high and poor conducting material for a network and surroundings, respectively. There is no coupling between electric and heat currents. Only electric currents are coupled via electromotive forces. Assuming that the stationary state is the state of maximum possible generated heat the Kirchhoff loop law is derived [19,20]. The principle of the maximum heat dissipation is closely related to the MEP principles.
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 shell dwell more on this question in the paper II submitted to this special issue.
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  is equivalent to the MEP principle .
4. The Linearized Boltzmann Equation and the Extremum of Entropy Production
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 . The Boltzmann equation is valid for rarefied gas where collisions are very well defined events. One seeks the one-particle distribution function satisfying the equation,
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 F. The right-hand side of the equation describes the net change in the number of molecules in a given element of the velocity space due to molecule collisions.
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, and are the distribution functions of particles before and after collision, respectively. In addition, is the magnitude of relative velocity of particles before collisions, is a unit vector parallel to the velocity change of one particle in collision and is the differential cross section. It is known from collision theory that differential cross section depends on interaction potential, relative velocity q and unit vector e, i.e., on the scattering angle .
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 (10) [6,11].
is the Maxwell-Boltzmann distribution and V is the velocity relative to the barycentric (centre of mass) velocity.
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 (11).
Operator [6,11] is linear and has the following properties
If we designate the left hand side of Boltzmann equation (10) with it becomes
Multiplying the linearized Boltzmann equation with and integrating over velocity space we get
Here is the density of entropy and is the flux of entropy. Entropy looks like a fluid. The left-hand side of Equation (20) is the total time change of the density of entropy, i.e. it is the density of the entropy production. Then, according to the aforementioned equation collisions between molecules described by function Ψ are the sources of entropy production. There are a lot of functions obeying conditions (14). We choose those distribution functions that obey the additional condition
Function Y is not the solution of the Boltzmann equation, i.e.,
Y can be interpreted as a trial distribution function. The corresponding entropy production due to this distribution is
If the temperature gradient in x direction is the only thermodynamic force, the left-hand side of the equation (21) can be written as 
Here, is the heat flux associated with the distribution function Y. This equation gives a physical meaning to condition (21). Varied distribution functions are selected in such a way that the entropy production due to the molecular collisions is equal to the entropy production due to the heat conduction. In other words entropy produced by molecular collisions is equal to the product of the heat flux and conjugated thermodynamic force. This condition is in accordance with the starting assumption of a stationary process.
The multiplication of the Boltzmann equation with the perturbative distribution function Ψ and integration over molecular velocity space V gives
The left-hand side is the entropy production of the rarefied gas. It comes from Equation (29) that solutions of the Boltzmann equation are in accordance with MEP principle.
There is another approach to this problem which has been proposed by Kohler . This approach assumes fixed heat flux,
We note that due to the already fixed temperature field the entropy production is also fixed. In this Kohler’s approach the entropy generated by molecular collisions need not be equal to the entropy production calculated as the product of thermal force and thermal flux. Since the entropy production calculated via scattering integral is not equal, in the variation process, to the entropy production defined as the product of the thermal force and thermal flux, it comes out that the constraint of the constant thermal flux (30) violates the equality between these two definitions of entropy production. If one does not take care of this violation the conclusion can be reached, by using the calculus of variation, that solutions of the Boltzmann equation are those that generate minimum entropy production.
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.
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