On the Topology of State-space, Linear vs. Nonlinear Theories and Dry Friction

The work reported here aimed to eliminate the gap between static- and sliding friction on the ground of non-equilibrium thermodynamics with dynamic (internal) degrees of freedom introduced into the state space of the system. Here a germ of a thermodynamic theory is sketched without acquiring generality. Here the simplest model is looked for to eliminate the gap between static and kinetic friction. It is not a complete idea at all but may be a guideline for further modeling. The fork where the train of thoughts stops at gives way to several kinds of possibilities, a number of which by the author’s opinion may have their own realm of application. Finally, an accurately accountable model is given that may be assumed friction if we pretend not knowing its origin and the law differs signficantly from the nearly exclusively used Coulomb’s law. This example supports the idea that friction cannot be described in a simple and uniform theory.


Introduction
The possible states of a non-equilibrium system can be assumed a topological space. Let a neighbourhood of a point (a state) be the set of those points that cannot be distinguished with the help of an observation (measurement). The intersection of two environments of a point x is obviously the set of those that are distinguished none of the two observations. Similarly, the elements of their union are the set of points not distinguished by at least one of the observations. The environments of the points together with their union and finite intersections define the open sets. The topology defined this way is very important but too general for practical purposes, so we are looking for more familiar ones. Classical thermodynamics has been satisfied with the topology of a finite linear space in any case and non-equilibrium thermodynamics most often remained in the vicinity; the fields of the variables used by the equilibrium theory have been introduced in classical irreversible thermodynamics, and similarly, fields of a finite number of coordinates are used in extended thermodynamics and the theories with dynamic (internal) variables. Rational thermodynamics goes further. The finite linear space is replaced by function space. Another question concerns linearity. The linear approximations are easier to deal with and, in many cases, result in usable outcome; but there are problems a linear theory cannot deal with. Such problems are those of chemical reactions, fluid turbulence, and dry friction the present paper fo cuses on. Nevertheless, sliding friction is frequent and it was the first irreversible process we were faced in the school, a very rare e ort has been aimed to develop its non-equilibrium thermodynamic theory. The reason may be guessed as Coulomb's (Amontons' 3 rd ) law is often taught as if it where the final solution, even the velocity dependence of the friction force is rarely mentioned in textbooks or in papers, looking away from its direction. On the other hand, the phenomenon cannot be accounted with a linear theory even in the Onsagerian sense. Coulomb's law displays a strong singularity at equilibrium, which is, at least, extremely rare in nature. On the ground of the thermodynamic theory of rheology, one can guess, that the singularity is virtual. The friction force does depend on the magnitude of the velocity and the function is continuous, but it starts so steeply and falls back at practically very low values that to observe it is difficult. If the two solids are separated with a thin layer of a non-Newtonian fluid the thermodynamic theory of the latter can be applied. This is not a nonsense; the independence of the friction force on the contact area is explained by rather widely accepted ideas with the plastic flow of the materials at the surface region. In the last decades, the interest has increased, and several theories were published and experimentally verified that explain friction and wear at different circumstances and the results are not like to each other nor to Coulomb's law. Applying Onsager's thermodynamics beyond the realm of linearity is like guessing. Not a free brainstorm, but severely limited by observations. Here a germ of a thermodynamic theory is presented always keeping an eye on the topology of the state space. Nevertheless, sliding friction is frequent [1,2] and it was the first irreversible process we were faced in the school and probable one of the most important from technological point of view, very rare effort [3] has been aimed to develop its non-equilibrium thermodynamic theory if ever, even though, non-equilibrium thermodynamics [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] has been applied to several phenomena with full success [18][19][20][21][22][23][24][25][26][27][28][29]. The reason may be guessed as Coulomb's law is often taught as if it where the final solution, even the velocity dependence of the magnitude of friction force is rarely mentioned in textbooks or in papers. On the other hand, the phenomenon cannot be accounted with a linear theory even in the Onsagerian sense. Coulomb's law is highly non-linear and displays a strong singularity at equilibrium as shown in Figure 1 which is, at least, extremely rare in nature. Starting from the fact that the static friction is larger than the kinetic one, we may assume that the friction force at very slow-motion increases with decreasing velocity. At the end of the 19 th century the low velocity meant rom 3mm/s down to 0.06mm/s [30] while nowadays from 1000nm/s down to 1nm/s, approximately 7 molecules per second. The experiments support the suspicion. Amontons' 3rd law may be replaced with a function free of the strong singularity just at equilibrium. On the ground of the thermodynamic theory of rheology, [18], one can guess, that the singularity is virtual. The friction force does depend on the magnitude of the velocity and the function is continuous, as shown in Figure 2, but it starts so steeply and falls back at practically very low velocities that it is difficult to observe.

Thermodynamic Theory
The assumed system is two rigid bodies in contact and sliding on each other. Motion is assumed tangential to the surface. The first law of thermodynamics reads where u is the internal energy, F  is the applied force necessary for steady slip with velocity v  , I q the heat flow, and t is the time. The second law reads in general form. Here s is the entropy, T the temperature both system and the environment, and s P is the entropy production in unite time.

Model in local equilibrium
If local equilibrium is supposed than the entropy of the system depends on the internal energy alone; and for the linear Onsager law The linear approximation is obviously not able to fulfill the needs. In nonlinear approximation L may depend on the velocity rather arbitrarily; the only requirement of the second law is gives back Coulomb's law for sliding but knows nothing on static friction, moreover, L has severe singularity at equilibrium. The symbol µ stands for the coefficient of sliding friction and N for the normal force pressing the surfaces one to the other. If a function the graph of which is something like the line in Figure 2 is taken, then a better formula is obtained. The disadvantages are that such a function is not very simple at all and the function must be measured or guessed. I incline to abandon the hypothesis of local equilibrium2.

Model with one dynamic variable.
The first approximation out of local equilibrium is based on a single dynamic variable α  invariant under time inversion. It is assumed to be a vector because the local equilibrium theory contains vectorial force and flux. Basing on the Morse lemma, [31], the entropy is given as where e is a convenient positive constant not specified yet.
The linear laws take the form 00 01 Here, the sliding surfaces are supposed isotropic. The thermo- The coefficient b can be determined from the static and dynamic coefficients of friction: which seems quite natural if friction is due to chemical bounds. A non-linear law similar to the above is giving account on the static friction also with creep and that the friction force is independent of the velocity if it is high enough. For some materials, decreasing force has been reported at very low velocities [33].

Further considerations
The model presented here gives account on continuous and delayed change of the friction force when the velocity changes abruptly, but the account is not good. This model does not display the overshot found experimentally. This fact makes plausible that the coefficient 11 ( ) L α  cannot be zero but at α = 1. It may be something like

Model with two dynamic variables
With two dynamic degrees of freedom, the entropy function reads (1 ) The content of them says that the two-dynamic degree of freedom contribute to the force independently from each other. For a stationary sleep the force, the velocity, and even 2 α can be given in parametric form with 1 α . The plot with representative parameters in Figure 6 shows that the logarithmic approximation fits well in the domain where the friction decreases with sleep velocity. And as for the overshot, the contribution of 1 α to the force is decreasing for large enough 1 α but that of

Conclusions
Here the skeleton of a thermodynamic theory has been sketched without acquiring generality. It is not a complete idea at all but may be a guideline for further development. The fork where the train of thoughts stops at gives way to several kinds of possibilities, a number of which by the author's opinion may have their own realm of application. I incline to believe that friction is a great family of processes and each kind needs its own theory, but the one presented here may be a good common approximation for each case. If the two solids are separated with a thin layer of a non-Newtonian fluid or if the contact area is surrounded by a thin fluidal layer (self-lubrication) then the thermodynamic theory of the former can be applied. [2,18]. This is in the case of hydrodynamic lubrication and even without any lubricant; the independence of the friction force of the contact area is explained by rather widely accepted ideas with the plastic flow of the materials at the surface region, etc. [1,2]. In the last decades, the interest has increased, and several theories basing different mechanisms were published and experimentally examined that explain friction and wear at different circumstances and the results are not like to each other nor to Coulomb's law [2]. Finally, here an accurately accountable model has been given that may be assumed friction if we pretend not knowing its origin and the law differs significantly from the nearly exclusively used Coulomb's law. This example supports the idea that friction can be described but with a simple and uniform theory. Assume that the roughness of the surfaces involves perpendicular motion and visco-elastic waves are generated. It is similar when a car goes on a rough road. The part of the driving force due to the energy loss in the shock absorber (visco-elastic loss in the bulks of the bodies) does not depend on the normal force but on the masses of the bodies and on the velocity. The function is rather complicated and at high velocities it is proportional to the velocity and independent of the masses or the normal force.
Without violating the generality, we may suppose that the coeffcients ik L ξ do not depend on any k X . If they do, introduce the auxiliary quantities where the coefficients 0 ik L ξ are independent of the X's. The same procedure leads to the statement, that the coefficients ik L ξ ′ are also independent of the X's.

Lemma 2
The ratio of any two 0 ik L ξ coefficients of eq. (36) is independent of the thermodynamic forces. Assume the set of the differential equations The trajectories are well defined and a manifold of curves, referred here as "linear trajectory", can be defined by the equations Now map any trajectory to the linear trajectory to which it fits best at equilibrium. The mapping is a bijection, Next map the points of the space of the dynamic variable to the one the entropy of which is the same and lies on the corresponding linear trajectory. The mapping in two dimensions is shown in Figure (7). This way we obtain a new representation of the dynamic variables the trajectories of which is defined by the equations of the linear trajectories, eq. (39). With the help of the latest result and combining the above equations for the trajectories, eqs. (38) and (39), moreover, introducing the function