Fluctuations, Trajectory Entropy, and Ziegler's 
Maximum Entropy Production 
V.D. Seleznev, L.M. Martyushev 
Institute of Industrial Ecology, Russian Academy of Sciences, 20A  S. Kovalevskaya St., 
620219 Ekaterinburg, RUSSIA,  
Ural Federal University, 19 Mira St., Ekaterinburg, 620002, RUSSIA, 
LeonidMartyushev@gmail.com 
We consider relaxation of an isolated system to the equilibrium using detailed 
balance condition and Onsager’s fluctuation approximation. There is a small 
deviation from the equilibrium in two parameters. For this system, explicit 
expressions both for the dependence of trajectory entropy on random 
thermodynamic fluxes and for the dependence of entropy production on the most 
probable thermodynamic fluxes are obtained. Onsager’s linear relations are 
obtained for the considered model using two methods (maximization of trajectory 
entropy and Ziegler’s maximization of entropy production). Two existing 
interpretations of the maximum entropy production principle - as a physical 
principle and as an effective inference procedure - are discussed in the paper. 
1 Introduction 
There are quite a lot of recent examples of successful application of the 
maximum entropy production principle (MEPP) in physics (kinetic theory of 
gases, hydrodynamics, theories of crystallization and radiation etc.), biology, and 
chemistry. Some results of similar investigations can be found, for instance, in 
reviews [1-3] and the present book. The interest in theoretical grounding of MEPP 
and in connecting MEPP with other principles [1-12] has become a natural 
consequence of the achieved success.  
MEPP has been considered as a natural generalization (expansion) of the 
second law of thermodynamics starting from Kohler’s and Ziman’s papers [13, 
14] that used maximization of entropy production for solving the Boltzmann 
equation. Indeed, if the second law stated the entropy increase in an isolated 
system for nonequilibrium processes, MEPP stated that this increase would occur 
to the maximum possible extent. Furthermore, whereas the second law enabled to 
obtain the basic thermodynamics relations and describe phase transitions in the 
case of equilibrium (quasistatic) processes, MEPP led to the basic laws of 
nonequilibrium thermodynamics and enabled to describe nonequilibrium (kinetic) 
phase transitions [15-17]. Thus, on the basis of the foregoing, MEPP was 
considered as an independent postulate on entropy substantially complementing 

2  
and generalizing the second law of thermodynamics. Such interpretation was 
supported particularly by H. Ziegler, who independently developed one of the 
MEPP wordings and specified an effective procedure of entropy production 
variation in the case of fixed thermodynamic forces, which allowed him to obtain 
the explicit form of thermodynamic fluxes (and particularly, Onsager’s linear 
relations [1,15,16]). Thus, there is a large group of researchers who consider 
MEPP to be a new and important principle of the physics of nonequilibrium 
processes; furthermore, a number of scientists (for example, M. Kohler, J. Ziman) 
proceeded from statistic (kinetic) considerations and the other scientists (for 
example, H. Ziegler) proceeded from thermodynamic considerations. 
However, there is another, a little different view on MEPP. The 
macroscopic state entropy is associated with the number of microscopic states in 
the phase space of coordinates and momenta of molecules and atoms satisfying 
such macroscopic state. In the equilibrium, the number of these microscopic states 
turns out to be a maximum, and correspondingly the entropy has a maximum 
value. For certain models, these studies represent a statistical justification of the 
second law of thermodynamics. Due to С. Shannon and E. Jaynes researches, this 
kind of view on entropy was generalized: by analogy with the Boltzmann-Gibbs 
entropy, they introduced the so-called informational entropy applicable to the 
description of objects of any nature [18,19]. Maximization of the informational 
entropy enabled to determine the probability of a particular state of the system. 
Within this trend, the papers (see e.g. [4-11], [20-25]) develop an idea about the 
relation between the probability of a nonequilibrium process and the number of 
microscopic trajectories implementing it, about the introduction of the 
informational entropy as a measure of the number of such microscopic trajectories 
with the subsequent maximization of that entropy for defining the most probable 
way of the nonequilibrium system evolution. As consequence, the following 
opinion was formed: the maximization of the informational entropy written in the 
phase space of microscopic states for equilibrium is similar to the Boltzmann-
Gibbs maximization, and the maximization of the informational entropy in the 
space of microscopic trajectories for a nonequilibrium process should apparently 
lead to MEPP. This area of investigations has not reached the required level of 
rigor so far, though it is very promising for understanding and illustration (using 
specific models) of the microscopic properties of MEPP and its relations with the 
other statements existing in the non-equilibrium physics. However, studies in this 
area sometimes give rise to an idea that MEPP is more or less a simple 
consequence of the Jaynes information entropy maximization and that MEPP is 
not an important physical law/principle, but only “an inference algorithm that 
translates physical assumptions into macroscopic predictions”1 [4-11]. We will 
refer to such interpretation of MEPP as an informational interpretation for brevity, 
as opposed to the above statistical and thermodynamic one. 
Thus, there are two opinions about MEPP. In order to obtain a better 
understanding of these two approaches and be able to compare them it is desirable 
                                                          
1 The criticism of such ideas will be set forth in the conclusion hereof. 

3 
to find a problem whose solving would enable to independently apply both 
approaches within the same approximations (formalism): for instance, Ziegler’s 
maximization formalism and an approach through the trajectory entropy 
maximization. The first approach is purely thermodynamical and the second one is 
informational (i.e. it is applicable to any scale of description, particularly to the 
mesoscopic one). Therefore, such investigation is basically possible; however, it 
has not yet been conducted. The objective of this study is to consider an 
elementary transfer problem using two methods and to determine the relations and 
differences between the two approaches maximizing either the trajectory entropy 
or the entropy production based on Ziegler's procedure.  
The paper consists of four parts. The second part introduces a model, its 
basic assumptions, and the fundamental equations obtained within the scope 
thereof. This part has primarily a methodological purpose. The third and fourth 
parts consider the problem through the trajectory entropy maximization method 
and using Ziegler’s procedure, respectively. This is the central section of the paper 
achieving the objective set forth herein. The conclusion of the article contains, in 
addition to the main results, a brief reasoning on the various approaches to the 
interpretation of MEPP. 
 
2 Onzager’s model and linear thermodynamic relations 
 
Let us consider an isolated system with possible fluctuations. An 
arbitrary (nonequilibrium) macroscopic state of the system will be described by 
the set of parameters 
)
(t
Ai
 that acquire the values 
eq
iA
 in the equilibrium state. 
Let us designate the difference between the parameter values and their equilibrium 
values as 
eq
i
i
i
A
t
A


)
(

. In the nonequilibrium case, 
0

i

. For brevity of 
further description, it would be convenient to introduce the vector α with the 
macroscopic parameters 
i
serving as its components. Herein we shall assume 
that the observation time  is much smaller than the time of the system relaxation 
to the equilibrium, i.e. we shall consider the “momentary” response of the system 
to the nonequilibrium state generated therein.  
Any state of the system with α can be characterized by the number of 
microscopic states 
)
(α

, which, as is known [26,27], is a maximum in the 
equilibrium. As usual, let us assume that all microscopic states corresponding to 
that macroscopic state are equiprobable. Let us define the probability of finding 
the equilibrium system in the state with α as: 
 
)
(
)
(
α
α


W
 
(1) 

4  
The process of the system relaxation to the equilibrium will be 
characterized by the conditional probability that the system being in the state α0 at 
the initial moment of time will get to the state α at the moment of time  [27]. 
Let us designate this probability as 
)
,
(

α
α0
P
 (Fig.1). As is in the previous case, 
we shall assume that this conditional (trajectory) probability is proportional to the 
number of microscopic trajectories of the transition from the state α0 to the state α  
for the time , i.e. 
 
)
,
(
)
,
(


α
α
α
α
0
0


P
. 
(2) 
Let the transition from α0 to α*(α0) for the time  be the most probable 
(according to (2), it means that such a transition occurs with the maximum number 
of microscopic trajectories). 
              Let us use the classical method [26] to determine the entropy in the state  
α and in the equilibrium (α=0): S(α)=lnГ(α) and S(0)=lnГ(0), then: 
 
)
0
(
)
(
ln
)
0
(
)
(
ln
S(0)
)
(
)
(
W
W
S
S











 
(3) 
By 
analogy, 
let 
us 
introduce 
the 
trajectory 
entropies 
)
,
(
ln
)
,
(


α
α
α
α
0
0


tr
S
 and 
)
),
(
(
ln
)
),
(
(
0
0




*
0
*
0
α
α
α
α


tr
S
, 
then: 
 
 
)
),
(
(
)
,
(
ln
)
),
(
(
)
,
(
ln
)
),
(
(
)
,
(
)
,
(
0
0
0










*
0
0
*
0
0
*
0
0
0
α
α
α
α
α
α
α
α
α
α
α
α
α
α
P
P
S
S
S
tr
tr
tr








 
(4) 
As a result: 
 
)
,
(
0
)
),
(
(
)
,
(




α
α
*
0
0
0
α
α
α
α
tr
S
e
P
P



 
(5) 
  
Processes for which 
)
),
(
(
0 

*
0 α
α
P
is a constant independent of α0 
will be considered below. Such approximation is quite common [27, 28].  
In the equilibrium, the number of system transitions in the forward 
α0→α and reverse α→α0 directions during the macroscopic time  should be 
equal, correspondingly it can be shown [27,29] that the so-called principle of 
detailed balance (equilibrium) relating the probability of finding the system in a 

5 
certain macroscopic state to the conditional probability of a transition therefrom is 
true: 
 
)
,
(
)
(
)
,
(
)
(


0
0
0
α
α
α
α
α
α
P
W
P
W

. 
(6) 
Condition (6) is proved for fluctuations of the equilibrium system 
[27,29].  Following the classical studies of L. Onsager [29] and the monograph 
[27] where this approach is stated in the most complete form, let us suppose 
(Onsager’s hypothesis) that this relation is also true for near equilibrium 
conditions. In other words, the evolution of the fluctuating equilibrium system 
found itself in the state α0 will be similar to the evolution of the specifically 
prepared (close to equilibrium) system brought to the same state α0, and then left 
for spontaneous relaxation. 
   
  
Further, in order to avoid cumbersome calculations, let us consider that 
α has only two components. It should be noted that the calculations below can be 
generalized for any number of components. According to (6), we have: 
 
)
,
,
,
(
/)
,
,
,
(
)
,
(
/)
,
(
20
10
2
1
2
1
20
10
20
10
2
1














P
P
W
W

. (7) 
Using (3)-(5), Eq. (7) can also be written in the form: 
 
)
,
(
)
,
(
)
(
)
(


0
0
0
α
α
α
α
α
α
tr
tr
S
S
S
S







. 
(8) 
In the case of L. Onsager’s classical treatment [29], the principle of 
detailed balance together with the supposition of a linear relation between the 
mean change of α for the time  and the quantity α itself leads to the proof of the 
so-called reciprocal relations and to the Gaussian form of 
)
(α
W
and
)
,
(

α
α0
P
. 
Here we consider an inverse problem: the Gaussian form of 
)
(α
W
and
)
,
(

α
α0
P
 
is postulated and the linear relations for the change of α are obtained using the 
principle of detailed balance. Such a statement of the problem does not pretend to 
essential originality (in fact, such possibility seems quite obvious); however, in 
our opinion, such treatment is the simplest and the shortest way to explicitly 
express the trajectory entropy, the entropy production, as well as other required 
quantities using the model parameters. It enables, in the easiest way, to achieve the 
objective set forth in the introduction (see sections 3 and 4). 
So, minor deviations from the equilibrium will be assumed. In this case, 
the so-called Gauss distribution is a frequently used approximation for the 
equilibrium deviation probability [26, 27]: 
 
)
2
exp(
 
)
(
2
1
12
2
2
22
2
1
11












α
W
, 
(9) 

6  
and for the trajectory probability [27-29]: 
 
)
)
)
,
(
(
exp{
)
,
(
2
1
*
1
11










0
0
α
α
α
P
)}
)
,
(
)(
)
,
(
(
2
)
)
,
(
(
2
*
2
1
*
1
12
2
2
*
2
22
















0
0
0
α
α
α
, 
(10) 
where ,  are normalization constants (independent of α,α0, but dependent, 
particularly, on 
eq
iA
); ij is a coefficient that is inversely proportional to the 
distribution variance of the random quantity α relative to the equilibrium value 
)
0
(

α
; ij is a coefficient2 that is inversely proportional to the distribution 
variance relative to the average (most probable) value 
)
,
(
*


0
α
i
 during the 
transition from the point α0 for the time . It should be noted that the variances in 
this approximation are assumed independent of α0, α [27-29]. It should be also 
emphasized that ij does not dependent on the time   (as it characterizes the 
fluctuation in the equilibrium state); ij, on the contrary, depends on the time and, 
moreover, substantially increases with the decrease of  (for 
0


, 
0
*
i
i


 
and the distribution tends to a delta function). Let us accept a simple supposition 
that 



/
0
ij
ij 
 [27-29], where 
0
ij

 is some constant.  
 
By inserting (9) and (10) into (7), we will obtain: 
 







)
2
exp(
)
2
exp(
20
10
12
2
20
22
2
10
11
2
1
12
2
2
22
2
1
11














))
)
,
(
)(
)
,
(
(
2
)
)
,
(
(
)
)
,
(
(
exp(
))
)
,
(
)(
)
,
(
(
2
)
)
,
(
(
)
)
,
(
(
exp(
20
*
2
10
*
1
12
2
20
*
2
22
2
10
*
1
11
2
*
2
1
*
1
12
2
2
*
2
22
2
1
*
1
11













































α
α
α
α
α
α
α
α
0
0
0
0
. (11) 
Here 
)
,
(
*


α
i
 is the most probable value in the case of the transition from α 
during the time . 
By finding the logarithm of the latter, we will obtain: 
 
)).
)
,
(
)(
)
,
(
(
)
)
,
(
)(
)
,
(
((
2
)
)
)
,
(
(
)
)
,
(
((
)
)
)
,
(
(
)
)
,
(
((
)
(
2
)
(
)
(
2
*
2
1
*
1
20
*
2
10
*
1
12
2
2
*
2
2
20
*
2
22
2
1
*
1
2
10
*
1
11
2
1
20
10
12
2
2
2
20
22
2
1
2
10
11



























































0
0
0
0
α
α
α
α
α
α
α
α
 (12) 
                                                          
2 According to their definition [27-29]: 
0

ii

 and 
0
2
12
22
11




. It meets the 
requirement of a non-negative power exponent for the two-dimensional Gauss (normal) 
distribution. 

7 
If the definitions of the state entropy and the trajectory entropy introduced above, 
as well as Eq. (8) are recalled, then the taking of logarithm of Eq.(11) allows 
obtaining: 
 
20
10
12
2
20
22
2
10
11
2
)
(












0
α
S
, 
(13) 
 
2
1
12
2
2
22
2
1
11
2
)
(












α
S
, 
(14) 
 
))
)
,
(
)(
)
,
(
(
2
)
)
,
(
(
)
)
,
(
(
)
,
(
2
*
2
1
*
1
12
2
2
*
2
22
2
1
*
1
11

























0
0
0
0
0
α
α
α
α
α
α
tr
S
, 
(15) 
 
))
)
,
(
)(
)
,
(
(
2
)
)
,
(
(
)
)
,
(
(
)
,
(
20
*
2
10
*
1
12
2
20
*
2
22
2
10
*
1
11

























α
α
α
α
α
α
0
tr
S
. 
 (16) 
In order to find the relationship between the forward and reverse trajectories, let us 
transform (12); for this purpose, we will consider very small times  and apply 
the Taylor expansion3: 
 
 
...
)
,
(
)
,
(
0
*
0
*














0
0
α
α
i
i
i
 
We will apply two consecutive expansions for the quantity below: first, in 
terms of  near zero, then in terms of 
i
 near 
0
i

: 
 
...
)
(
)
,
(
)
,
(
...
)
,
(
)
,
(
0
0
2
1
*
0
*
0
*
*
0






















i
i
i
i
i
i
i
i
i
i
i
























i
,
0
α
α
α
α
 
                                                          
3 The following uses the fact that the initially specified value is the most probable value for 
the deviation from the equilibrium at the initial moment of time 
0
*
)
0,
(
i
i



0
α
. 

8  
By neglecting the second orders of smallness (
)
(
0
i
i





), we will 
obtain: 
 
i
i
i
i









)
,
(
)
,
(
*
0
*
α
α0
. 
(17) 
Let 
us 
introduce 
the 
following 
notations: 
i
i
i
i
i
i













0
*
0
*
);
,
(
0
α
. These quantities characterize the most 
probable and actual (random) change of values of the parameters 
)
(
iA
 relative 
to the original 
)
0
(
iA
 during the time τ. By using them jointly with (17), we will 
obtain: 
 
i
i
i
i
i
i
i
i
i
i
i
i






























*
0
0
*
0
*
*
*
)
)
,
(
(
)
,
(
;
)
,
(
0
0
α
α
α
 
(18) 
Using two last expressions, Eq. (12) can be reduced to the form4: 
 
).
(
4
)
(
4
)
(
2
)
(
2
*
1
12
*
2
22
2
*
2
12
*
1
11
1
10
12
20
22
2
20
12
11
10
1




































 
(19) 
Since the deviations (
1


,
2


) from the initial state are independent, 
then the following can be obtained from equality Eq.(19): 
 
.
4
4
)
(
2
4
4
)
(
2
*
1
12
*
2
22
10
12
20
22
*
2
12
*
1
11
20
12
10
11


























 
(20) 
Let us introduce a number of important quantities. Since the system is 
considered as isolated, then its entropy change rate proves to be equal to the 
entropy production  [1, 27]. Let us use the classical method [26, 27, 29] to 
introduce the thermodynamic forces Xi and fluxes Ji: 
 
2
2
1
1
2
2
1
1
)
(
J
X
J
X
dt
d
S
dt
d
S
dt
dS














α
,             (21) 
                                                          
4 For small τ values: 
*
j
i
ij
j
i
ij











  (because 



/
0
ij
ij 
). 
 

9 
where:  
 
 
 
i
i
S
X




/
 
 
  
            (22) 
 
 
dt
d
J
i
i
/



 
 
  
            (23) 
According to (13) and (22), the thermodynamic forces acting in the 
system at the initial moment of relaxation to the equilibrium, are equal to:  
 
 
)
(
2
)
(
2
10
12
20
22
2
20
12
10
11
1












X
X
 
 
            (24) 
 
Using (23), (24)5, and the relation between 
ij

 and 
0
ij
, expression (20) 
can be rewritten in the form: 
 
,
4
4
4
4
*
2
0
22
*
1
0
12
2
*
2
0
12
*
1
0
11
1
J
J
X
J
J
X








 
(25) 
or by transforming, we will obtain: 
 
,
2
22
1
21
*
2
2
12
1
11
*
1
X
L
X
L
J
X
L
X
L
J




 
(26) 
where the following kinetic coefficients are introduced: 
 
),
)
(
(
4
/
2
0
12
0
11
0
22
0
22
11






L
 
),
)
(
(
4
/
2
0
12
0
11
0
22
0
12
21
12







L
L
 
 
            (27) 
).
)
(
(
4
/
2
0
12
0
22
0
11
0
11
22






L
 
 
Based on the properties of 
0
ij
 (see footnote 2): 
0

ij
L
 and 
0
2
12
22
11

L
L
L
. 
Thus, it is shown that the assumption of Gaussianity (9), (10) and the 
principle of detailed balance (6) result in the known linear Onsager relations [26, 
27, 29]. This relation links the most probable flux in the system with the 
thermodynamic force existing within the time interval . The reasoning set forth 
herein is the study of the problem opposite to the problem considered by Onsager 
[29].  
 
                                                          
5 For small  values: 



/
/
*
*
i
i
dt
d



. The minus sign has resulted from the fact that 
*
i


 
was introduced as a difference between the initial and final value.  

10  
We conclude the present section with a number of useful relations 
following from the above equations. According to Eqs. (17), (18), let us rewrite 
Eqs. (15), (16) in the form: 
 
)
)(
(
2
)
(
)
(
)
,
(
*
2
2
*
1
1
12
2
*
2
2
22
2
*
1
1
11






























α
α0
tr
S
             (28) 
 
)
)(
(
2
)
(
)
(
)
,
(
*
2
2
*
1
1
12
2
*
2
2
22
2
*
1
1
11






























0
α
α
tr
S
 
            (29) 
 
According to (8) and (21), the following can be written for small τ values: 
 
)
,
(
)
,
(
)
(
)
(



0
0
0
α
α
α
α
α
α
tr
tr
S
S
S
S







  
 
            (30) 
 
or according to (21)-(24): 
 
)
(
2
)
(
2
10
12
20
22
2
20
12
11
10
1
2
2
1
1

























X
X
  
            (31) 
   
Now let us proceed to extreme properties of the trajectory entropy and the 
entropy production for the model under consideration.  
 
 
 
3 Informational approach: trajectory entropy maximization 
 
 
When describing nonequilibrium processes using the informational 
approach, first some entropy (for example, trajectory entropy) is extremized in 
order to define the trajectories distribution function in the phase space. Such a 
procedure is performed subject to the existing (or supposed) constraints (relations 
between the quantities more or less evident for the process under study). Then the 
found distribution function is used for calculation of the needed nonequilibrium 
properties of the process. This classical procedure was repeatedly described in the 
literature (see, e.g. [4]). The unfalsifiability (according to K. Popper) of that 
procedure is its fundamental disadvantage [7,8,10]. Indeed, if the calculation 
results for nonequilibrium properties fail to match the experimental data, it is 
considered that the constraints used for maximization were incorrectly selected. 
These constraints (relationships) are adjusted and the procedure is repeated. As a 
result, the informational approach appears to be some kind of method aimed at 
selecting the result not contradicting the known experiments. Obviously, the 
unfalsifiability could be a serious disadvantage of the mentioned approach (casting 

11 
serious doubt on the scientific nature of the method); however, this method is 
considered to be only a procedure (mathematical algorithm) for passive translation 
of physical assumptions (constraints) into predictions without introducing any 
additional physical assumptions [7]. Thus, such a method is some kind of 
mathematical device. However, this mathematical procedure entails problems of 
purely mathematical nature. The first problem of such a procedure is its 
mathematical unreasonableness. Thus, the “liberty of action" (when selecting 
constraints) may either yield no desired results at all or allow several solutions that 
meet the selected criteria but substantially differ in terms of both used constraints 
and predictions (outside the scope of the selected criteria). The second problem is 
connected with the choice of the measure of information and, correspondingly, the 
formula of informational entropy. From the logical viewpoint, there is no best 
option. There are multiple variants besides the Shannon formula; and many of 
them prove to be useful in different applications [30].  
Despite the mentioned fundamental disadvantages, we will maximize the 
trajectory entropy for the problem under consideration. This is justified by the fact 
that the studied model is fairly simple and, as consequence, the existing 
constraints are accurately set.  
In the model at hand, the trajectory entropy as a function of random 
deviation of α  has the form (15) or (28). The explicit form of this entropy was 
obtained using the Gaussian form of the distribution function 
)
,
(

α
α0
P
 and the 
supposition of detailed balance. As consequence, the trajectory entropy depends 
not on the distribution function but on other variables. In this case, the trajectory 
entropy maximization should lead to the finding of equations linking these 
variables. The maximization can only formally be considered as unconstrained 
because all constraints have already been introduced to the expression of 
trajectory entropy when obtaining its explicit form.  
According to Eq.(4), the trajectory entropy for the forward 
trajectory
)
,
(

α
α0
tr
S
 is related to 
)
,
(

α
α0
tr
S

 and to the trajectory entropy 
for the most probable forward trajectory 
)
),
(
(
*
*

0
0
α
α
α
tr
S
 as:  
 
i
tr
i
tr
tr
i
tr
S
S
S
S





















)
,
(
))
,
(
)
),
(
(
(
)
,
(
0
*
*
α
α
α
α
α
α
α
α
α
0
0
0
0
    (32) 
 
By inserting the explicit form 
)
,
(

α
α0
tr
S

 (Eq. (28)), we will set (32) 
equal to zero. It can easily be obtained that the trajectory entropy maximum 
conforms to 
*
i
i





.  According to (10) and (18), the most probable process 
trajectory also conforms to the condition 
*
i
i





.  
Since the maximum of the trajectory entropy deviation is obtained in the 
case 
*
i
i





, then using (29)-(31) it is easy to obtain:  

12  
 
,0
4
))
,
(
(
))
,
(
(
)
,
(
*
4
*
*
*
*


































j
ij
i
ii
i
i
tr
i
i
i
tr
i
tr
X
S
X
S
S
i
i
i
i
i
i


















0
0
0
α
α
α
α
α
α
 
 
*
4
*
4
j
ij
i
ii
i
X








. 
(33) 
This expression coincides with Eq.(25). Thus, Onsager’s linear relations 
correspond to the trajectory entropy maximum. 
The results obtained from the maximization can be interpreted in two 
ways. On the one hand, they indicate internal consistency of the used 
informational method because any other obtained result different from (33) could 
be deemed, at best, the consequence of false transformations and, at worst, another 
(in this case, logical) disadvantage of the informational approach6. On the other 
hand, the obtained result certainly indicates that for the model under study the 
trajectory entropy maximization with a number of constraints allows finding the 
most probable macrotrajectory satisfying the valid law of relation between 
thermodynamic fluxes and forces (33). This points to the possibility of 
generalizing the conventional method of equilibrium entropy maximization.  
 
  
 
 
4 Thermodynamic approach: Ziegler’s maximization 
 
 
         As opposed to the above method, this approach focuses immediately on 
the search for relationships between the most probable quantities. Random 
quantities and their distribution functions are beyond the scope of the approach. 
The entropy production is considered to be a known function of thermodynamic 
fluxes [1, 15,16]. The relationship between the thermodynamic fluxes and forces 
is searched through the maximization of entropy production in the space of 
independent fluxes at the fixed thermodynamic forces [1, 15,16]. As opposed to 
the informational approach, Ziegler's method is falsifiable. Indeed, the entropy 
production is a well-defined macroscopic property of the system connected with 
the energy dissipation to heat. Thermodynamic forces and fluxes also have a clear 
physical meaning and are measurable in the experiment. As consequence, if the 
entropy production maximization based on Ziegler’s method yields predictions 
                                                          
6 Indeed, as a logical consequence, as it is shown in section 2, the considered model contains 
the linear relations of forces and fluxes, as well as the form of the trajectory entropy. The 
disagreement between the result obtained within the trajectory entropy maximization and (25) 
would therefore represent a serious problem.  

13 
different from the experiment (particularly, relationships between the fluxes and 
forces), then it will be possible to disprove the principle (or at least limit the scope 
of its validity).  The fact that for specific systems the finding of the entropy 
production as a function of fluxes is not always an easy task can be reckoned 
among the disadvantages of this thermodynamic principle:  there are no standard 
and formal procedures, each specific case requires an individual approach. The 
fact that H. Ziegler has developed his method only for the systems with biunique 
correspondence between the thermodynamic force and flux represents another 
disadvantage. 
      Formally, Eqs. (25) or (26) were many times obtained using Ziegler’s 
procedure [1, 15,16]. For this purpose, the entropy production as a bilinear 
function of thermodynamic fluxes was postulated and then the maximization was 
carried out with the fixed forces. In the present study, it is possible, within the 
scope of the model under consideration, to explicitly obtain both the form of the 
entropy production and the constraint for the maximization. We will show it. Let 
us rewrite the detailed balance relation (30) for the most probable trajectory 
)
(
*
i
i





. In this case, 
0
)
,
(



α
α0
tr
S
 (see Eq.(28)), and as a result (see 
Eq.(29)): 
 
*
2
*
1
12
8
2
*
2
22
2
*
1
11
4
4
)
,
(




















0
α
α
tr
S
  
or using Eqs. (23), (31) we will obtain:  
,
4
4
*
2
*
1
12
8
2
*
2
22
2
*
1
11
*
2
2
*
1
1



















X
X
 
 
.
4
4
*
2
*
1
0
12
8
2
*
2
0
22
2
*
1
0
11
*
2
2
*
1
1
J
J
J
J
J
X
J
X







 
(34) 
Obviously, the left-hand side of the last expression contains the entropy 
production 
)
( *
J

 through thermodynamic forces, whereas the entropy 
production on the right-hand side is written in the space of thermodynamic fluxes 
only. It is easy to check that the constrained entropy production maximum 
*
2
*
1
0
12
8
2
*
2
0
22
2
*
1
0
11
*
4
4
)
(
J
J
J
J
J







 subject to (34) (where  is the Lagrange 
multiplier): 
 

0
)
)
(
(
)
(
*
2
2
*
1
1
*
*
*








J
X
J
X
J
J
Ji

, 
(35) 
also leads to Onsager’s linear relations (25). This procedure is often given in the 
literature [1, 15,16] and will not be repeated here. 

14  
5 Conclusion 
The present paper considers the simplest model of a slightly 
nonequilibrium system with two thermodynamic forces that satisfy the detailed 
balance condition and with the equilibrium and the transition probabilities 
distributed according to Gauss. Within the scope of this model, we have managed 
to:  
- Explicitly write a formula for the trajectory entropy as a function of 
random deviations from the equilibrium. For the first time, it enabled to show that 
the maximization of the entropy of random microtrajectories leads to Onsager's 
linear relations.  
- Explicitly obtain a formula for the entropy production as a function of 
thermodynamic fluxes, which is a necessary procedure for the entropy production 
maximization according to Ziegler. The obtained expression agrees with the 
expression that was previously postulated and used by H. Ziegler for obtaining 
Onsager's linear relations. 
  
Thus, it is shown that the phenomenological transfer equations 
(Onsager’s linear relations) can be independently obtained using at least two 
methods: Ziegler’s procedure, or the trajectory entropy maximization. Within the 
scope of the considered problem, it can not be stated that the method using the 
trajectory entropy maximization is more general or that Ziegler's maximization 
procedure follows from this method (and the opposite is not true). It can be rather 
concluded that these are two different methods7 leading to the same result. The 
model within the scope of which the solutions are given is the only link between 
the considered methods. The model contains the linear relationship between fluxes 
and forces as shown in Sect. 2. That is what may suggest excessiveness of the 
suppositions used in Sections 3 and 4. It is, however, certainly not the case. Here 
the following analogy from mechanics can be drawn. Newton’s laws allow solving 
any mechanical problem; however, these laws can be used for developing new 
ways to the solution of mechanical problems using the variational methods 
(Hamilton, Lagrange, etc.). These new methods will not represent excessive 
formulations for solution of the original mechanical problem; in a number of 
cases, they just prove to be more convenient for solution of problems.  
 
In conclusion, let us take this opportunity to express our view about 
MEPP and two versions of MEPP interpretation mentioned in the Introduction.  
1) Maximum entropy production principle is an important physical 
principle following from the generalization of experimental data. This is 
particularly confirmed by multiple examples given in the papers [1-3]. In this 
context, that principle resembles the second law of thermodynamics, essentially 
                                                          
7 For one approximation, extremization is carried out for a random deviation from the 
equilibrium, whereas for the other approximation, it is carried out for a thermodynamic flux, i.e. 
the most probable deviation. 

15 
generalizing it.  Its possible wording is as follows: at each level of description, 
with preset external constraints, the relationship between the cause and the 
response of a nonequilibrium system is established in order to maximize the 
entropy production density [17, 31]. This wording substantially generalizes the 
formulation previously proposed by H. Ziegler and discussed herein in detail 
within the scope of the particular model. Specifically, the requirement for 
biunique correspondence between the thermodynamic fluxes and forces can be 
reckoned among the disadvantages of Ziegler’s formulation. This requirement has 
substantially limited Ziegler’ principle, depriving it of important and interesting 
fields of application (particularly, study of nonequilibrium phase transitions [17, 
31]).  
2) It will be recalled that the second law of thermodynamics (essentially 
equivalent to the statement of positivity of entropy production) allows identifying 
"one-way direction" or "asymmetry" of time [32]. As it is known, TIME is the 
most complex and versatile physical quantity that still lacks a universally 
acknowledged definition [32-34]. Despite having learned to measure time, we still 
fail to understand its nature. In this regard, we have a conviction that the 
consideration of MEPP as a new and important addition to the second law of 
thermodynamics8 will enable to progress in the understanding of new properties of 
time. Here the papers [12, 31] where the origin of MEPP is associated with the 
hypothesis of independence of the entropy production sign when transforming the 
time scale (that influence the reference system of fluxes), can be mentioned.  
 
3) MEPP is a relatively new principle. Therefore, the range of its 
applicability is not fully understood. The constraints of the principle should be 
searched for based primarily on the experiment. The experiment can also falsify 
MEPP (this allows considering MEPP as a physical principle). MEPP is to be 
tested using unambiguously interpreted, well-studied and fairly simple 
experimental systems (in this regard, at the present development stage of MEPP 
science, climatic, biological and similar systems are certainly not suitable for 
confirmation or falsification due to their complexity and  ambiguous 
interpretation)9. Mathematical models are also absolutely unsuitable for 
falsification. So, a model is only some more or less crude and often one-sided 
reflection of some part of a phenomenon, whereas MEPP is the principle 
reflecting the dissipative properties that are observed in nature rather than in its 
model. We consider the investigation of nonequilibrium phase transitions in the 
homogeneous systems to be a possible way of MEPP falsification. For example, if 
the regime with the smallest entropy production (particularly, with the smallest 
generated heat of dissipation) proves to be the most probable (in the statistical 
sense) in the case of the fixed thermodynamic force (for example, the pressure or 
                                                          
8 Let us emphasize that MEPP states that the entropy production is not only above zero but is 
the maximum possible one.  
9 This will only bring unjustified discredit to the principle and result in a negative response 
thereto (see, e.g. [35, 36]).  

16  
temperature gradient) and several possible existing nonequilibrium phases 
(regimes), then we can disprove MEPP (or at least considerably narrow the range 
of its applicability). 
Certainly, microscopic methods of principle justification are important 
for understanding the scope of MEPP validity. However, their significance in this 
case should not be exaggerated. The role of the statistical method (physics) has 
always been of rather supplementary, subordinate nature: scientists tried to 
understand and illustrate macroscopic properties that were experimentally 
discovered and integrally generalized in the thermodynamic postulates (laws) 
using simple models (of ideal gas, Markovian processes, etc.). If results of the 
statistical investigation were not proved experimentally and/or contradicted 
thermodynamics, the used statistical model was considered to be too crude or 
erroneous, and the model was modified (and the opposite has never occurred in 
physics). It is therefore methodologically incorrect to state that the microscopic 
view on the world and the entropy substitutes (analytically derives etc.) MEPP. It 
is as crude to state that as to claim, for example, that Boltzmann’s H-theorem has 
become a proof of the second law of thermodynamics.  
4) The existing interpretation of MEPP from the viewpoint of Jaynes’ 
informational methodology should not be identified with the microscopic 
(statistical) interpretation. This is a special view on the foundation of the statistical 
physics, which has both its supporters and opponents (see e.g. [37]). The laconic 
brevity and the simplicity, which enables to formulate, for example, the 
foundation of the equilibrium statistical physics, can be reckoned among its 
benefits. The subjectivism of Jaynes’ method can be considered as a drawback. 
There are attempts to obtain MEPP (the principle of nonequilibrium physics) from 
the maximization of informational (trajectory) entropy [4-11].  This is a very 
interesting area of studies. However, we would like to raise a number of 
criticisms: 
♦ MEPP has still not been obtained in a sufficiently rigorous manner from 
the maximization of informational (trajectory) entropy [4-11]. Even if it is 
achieved, this will be done only with the involvement of additional important 
suppositions/assumptions. It is these suppositions that will prevent from 
concluding that MEPP is just a consequence; instead, they will indicate that MEPP 
is an independent statement.  
♦ The informational approach is a variety of microscopic approach. The 
incompleteness of statistical methods for justifying the statements (principles) 
based on experiments has already been mentioned in item 3. These can also be 
repeated for the considered approach. However, it also has particular 
disadvantages. Thus, Jaynes’ method is a procedure (often an effective one) of 
searching for the relations in the case of known or supposed constraints. If the 
result disagrees with the reality, then the constraints are replaced (or the priorities 
of the constraints are changed). The procedure itself (explicit form and properties 
of the informational entropy, existence and uniqueness of the solution, and the 
like) is postulated. As a result, this approach is useful as a simple algorithm of 

17 
obtaining the known (generally accepted) solution10.  If the solution is, however, 
unknown (or there are different opinions about the solution), then any sufficiently 
experienced scientist can use this method to obtain any desired result depending 
on his/her preferences11. In this regard, the use of a procedure (such as the 
informational one) rather than a physical law (objective by definition) for 
justifying MEPP (which is at the stage of its final formulation and understanding 
so far) provokes our objections. In other words, Jaynes’ mathematical procedure 
can be used for obtaining, depending on the selected physical and mathematical 
constraints, multiple other procedures (the value of such mathematical exercises 
becomes, nevertheless, rather doubtful for physics); however, if the maximum 
entropy production is considered to be a physical principle, then it will be 
fundamentally unachievable using Jaynes’ inference algorithm because MEPP 
itself is the key physical constraint (like the first/second law of thermodynamics or 
the charge conservation law). 
 
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10 Indeed, the researcher’s intention to mathematically make the most unprejudiced prediction in 
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11 If this cannot be achieved by selecting the constraints, then other kinds of informational 
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30]. 

18  
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Climatic Change 85, 267-269 (2007). 

19 
36. Goody, R.: Maximum entropy production in climatic theory. Journal of Atmospheric 
Science. 64, 2735-2739 (2007).  
37. Lavenda, B.H.: Statistical Physics. A Probabilistic Approach. John Wiley and Sons, Inc., 
New York, (1991). 
 
 
Figure Capture 
Fig1. An example of some transitions of the system from one state to another. For equilibrium 
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