The process of self-organization in friction system reduces the production of entropy in tribo films and reduces the wear rate [8]. The fundamental reason for this is that a portion of the frictional energy, which, in the absence of self-organization would have been spent on wear, is instead spent on the formation of dissipative structures in the wake of a self-organization process.

This generally applies for different friction systems such as metal cutting [44–46], friction with current collection [47], plain bearings [48, 49]. Self-organization can only occur after a friction system loses thermodynamic stability [42]. The conditions outlined by Lyapunov’s stability theorem are sufficient but not necessary for this to happen [42, 43].

According to [42, 43], the system loses thermodynamic stability when the following holds true:

\(\frac{1}{2}\frac{\partial }{\partial t}\left({\delta }^{2}S\right) \le 0\) (1),

where: \({\delta }^{2}S –\) second entropy variation, *t* – time.

\(\frac{1}{2}\frac{\partial }{\partial t}\left({\delta }^{2}S\right) = \sum _{i}\delta {X}_{i}\delta {J}_{i}\) (2),

where: *X**i* and *J**i* are the corresponding thermodynamic forces and flows.

The right side of (2) shows the excess production of entropy - defined as the variance of thermodynamic flows and forces from the stationary state. According to [42, 43], δ2S is the Lyapunov function and it is non-positive:

$$\left({\delta }^{2}S\right) \le 0$$

3

The system will be stable if the below condition is satisfied:

$$\frac{1}{2}\frac{\partial }{\partial t}\left({\delta }^{2}S\right) \ge 0$$

4

As long as condition (4) is satisfied, the system is stable and self-organization is impossible. But if the derivative of the Lyapunov function is non-positive, i.e. if condition (1) is met, then the system has the opportunity to lose thermodynamic stability, thus opening the possibility for self-organization. The development of a friction process depends on constant external conditions such as the pressing force, relative slip speed, compositions, structures, and properties of materials involved, as well as gas or liquid medium (lubricant). Considering the entropy of the friction system as a function of time of test (*S(t)*), the deviation of entropy from the stationary state can be expressed as follows:

$$\delta S=\frac{\partial S}{\partial t}\delta t+\frac{1}{2}\frac{{\partial }^{2}S}{{\partial t}^{2}}{\left(\delta t\right)}^{2}$$

5

According to [42, 43], the second degree term is sufficient in the Taylor series expansion of (5). Entropy variations in (5) occur over time of the test. In this case, the first entropy variation will consist of the sum of entropy production and entropy flow:

$$\frac{\partial S}{\partial t}\delta t= \frac{{\partial }_{i}S}{\partial t}\delta t+\frac{{\partial }_{e}S}{\partial t}\delta t=P\delta t+I\delta t$$

6

Where P and I are the corresponding production and flow of entropy.

The second variation of entropy over time is characterized by the sum of the derivatives of entropy production and flow. If the external conditions remain constant, it can be assumed that the entropy flow will also remain unchanged over a given time period. Therefore, the time derivative of the entropy flow will be zero, and the second entropy variation will be characterized by the derivative of entropy production with respect to time:

$${\delta }^{2}S= \frac{\partial P}{\partial t}({\delta t)}^{2}$$

7

In this case the left side of equations (1) and (4) will correspond to the second time derivative of entropy production. The conditions (1) and (3) required for the potential loss of thermodynamic stability and the initiation of a self-organization process are that the first and second derivatives of entropy production must be simultaneously negative.

Let us consider these results for the condition of friction. Once the process of friction has started, entropy production (and as a consequence, wear rate) will sharply rise from a state of rest until a steady state is reached under constant external conditions, i. e. a localized peak. At a maximum peak of entropy production, the first derivative is zero and the second derivative is negative. If entropy production (as well as wear rate) begins to decrease along a convex curve, then both derivatives will become negative, satisfying conditions (1) and (3). If this is the case, then the system may lose thermodynamic stability and self-organization may occur. If entropy production (wear rate) begins to decrease along a concave curve then the second derivative remains positive and self-organization can not occur.

Thus, for a self-organization process to initiate in the first place, it is necessary for entropy production to reach a maximum localized peak and then begin to decrease along a convex curve. Self-organization is a probabilistic process: if conditions (1) and (3) are met, the system has the possibility of losing thermodynamic stability and undergoing self-organization [27, 42]. If self-organization does not occur, then the system will remain in a state of maximum entropy production and wear rate under the given conditions, until it reaches its natural wear limit. This concurs with [27, 50], where it is stated that a high level of energy dissipation is a necessary requirement for the self-organization process.

Measuring the production of entropy during friction is a rather difficult task. That is why in experimental works the parameters that characterize entropy production are usually used [42]. Considering that any change in wear rate is directly dependent on entropy production [8], it can be expected that entropy production and wear rate change with time in the same manner.

The entropy change under friction conditions (\(\frac{dS}{dt}\)) is as follows [44, 47, 48]:

\(\frac{dS}{dt}=\frac{{dS}_{e}}{dt}+\frac{{dS}_{i}}{dt}\mp \left|\frac{{dS}_{f}}{dt}\right|-\frac{{dS}_{s}}{dt}\) (8),

where:

\(\frac{{dS}_{s}}{dt}\) is the entropy change in a friction body due to the wear of its base,

\(\frac{{dS}_{e}}{dt}\) is the entropy flow without the wear,

\(\frac{{dS}_{i}}{dt}\) is the entropy production without friction surface transformations,

\(\frac{{dS}_{f}}{dt}\) is the portion of entropy production associated with friction surface transformations.

The term \(\frac{{dS}_{s}}{dt}\) on the right side of expression (8) has a negative sign due to the removal of wear particles from the main body along with their entropy. The negative sign in front of \(\left|\frac{{dS}_{f}}{dt}\right|\) indicates that surface transformations are accompanied by a negative production of entropy. Moreover, it also signifies the passage of self organization along with the formation of dissipative structures. A plus sign in front of \(\left|\frac{{dS}_{f}}{dt}\right|\) indicates that the surface transformations in tribo-films are accompanied by a positive production of entropy. It also implies that self organization with dissipative structure formation had not taken place. The sum: \(\frac{{dS}_{i}}{dt}\mp \left|\frac{{dS}_{f}}{dt}\right|\) is the total entropy production, which remains positive according to the second law of thermodynamics.

Under the stationary conditions, the Eq. (8) will be as follows:

$$\frac{{dS}_{s}}{dt}=\frac{{dS}_{e}}{dt}+\frac{{dS}_{i}}{dt}\mp \left|\frac{{dS}_{f}}{dt}\right|$$

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Under constant external conditions, the value of \(\frac{{dS}_{e}}{dt}\) undergoes negligible changes. Taking into account the additivity of entropy, the value of \(\frac{{dS}_{s}}{dt}\) will be proportional to the wear rate. In this case, according to (9), the wear rate has a direct relationship with entropy production. This makes it possible to evaluate the change in entropy production through the assessment of a change in the wear rate. It follows from (9) that the wear rate decreases in the aftermath of self-organization.The achieved reduction in the production of entropy and the wear rate could only be accomplished if the friction system had reached the maximum of entropy production prior to the commencement of a self-organization process. The principle of maximum entropy production has thus been realized in the friction system.