Science laws about separate interactions of masses and charges are based on abstractions, which enable and require approximations and omissions. While we understand that the speeds of electrical and gravitational interactions are finite, we can use the ’instant interaction’ approximation in classical physics. This is because one effect of the first particle reaches the second particle at the same time as the other effect, leading to the absence of a time-dependent term in the mathematical formulation (although in special cases the ’retarded interaction’ shall be discussed). However, this is not the case in electrodiffusion, where the mass transfer is significantly slower than the transfer speed of the electromagnetic field. To describe the interrelation of these two effects, we need to conduct case studies and apply casual approximations. Science uses the notion ’instant’ in the sense that one interaction is much faster than the process under study; we consider the faster interaction as instant.
From a physical point of view, ionic solutions are confined to a well-defined volume, with a limited interaction with the rest of the world. What makes the things more complicated, their volume has evidently finite size with bounding surface(s), so we must adapt the corresponding laws to the case of finite resources. At a microscopic level, on the one hand, we use the abstraction they consist of chargeless and sizeless simple balls with mass, have thermal (kinetic) energy, and collide with each other, as thermodynamics excellently describes it. On the other hand, we use another abstraction, which is massless and sizeless charged points with mutual repulsion. Combining those different concepts is not possible without conflicts and contradictions. At a microscopic level, in both abstractions, they attempt to distribute as equally as possible in a given volume. However, we can notice the difference that the equilibrium (nomen est omen) is dynamic for the thermodynamic and static for the electrostatic interaction. At a macroscopic level, we use the abstraction that the respective volume is filled with a continuous medium with uniformly distributed macroscopic parameters such as temperature, pressure, concentration, and potential.
One can parallelize describing how ions change their position with how Newton’s laws of motion relate an object’s motion to the forces exerting on it. The first and third laws are static ones, the second one is dynamic. We can translate the first law to ions that without external invasion, their volume at rest will remain at rest. The third law, for ions’ volume, essentially states that in a resting state at every points the electrostatic and thermodynamic forces are equal; this is expressed by the Nernst-Planck electrodiffusion equation (for the case of no transport). The second law, for mechanics, expresses the time course of the object: the position’s time derivative. Notice that in this case we make one abstraction that the object (the carrier) has one attribute, its mass. (Recall, how important was for the special theory of relativity that the accelerated mass and the gravitational mass were identical.)
For ions, we have two abstractions, and two attributes ’charge’ and ’mass’, and the two forces act on the two attributes which science classified to belong to different science disciplines. We cannot express easily how the electric and thermodynamic forces will change the object’s position because those forces act differently on different attributes. No time derivatives are known, only position derivatives. Due to this hiatus, physics (and consequently: physiology) cannot describe the electrochemical processes: the second law of motion for electrodiffusion is missing. As a consequence of the instant interaction, classical science has no mechanism for handling the case when two different force fields (gradients) having different propagation speeds act on an object and two different abstractions (charge and mass, belonging to different science disciplines) translate the force into acceleration. Here we use Boltzmann’s idea: we derive the particles’ speed from the continuum mass’s speed. Furthermore, our idea is to derive a pseudo-electrical field (see section 2.6.3) and potential to combine those two interactions and their respective disciplines.
When describing processes (i.e., dynamical systems), we must have one or more equations of motion (aka changing speeds): how the time gradient of the fundamental entities are changing in the function of the fundamental entities. In classical science, we have only one fundamental entity: the position and also the driving forces depend only on the position. The Newtonian laws of motion do not depend on a temporal gradient. In ’ordinary’ science, we have a single-speed, single-substance interaction abstraction; so we have one law of motion; an analytical solution is possible. In the Einsteinian world, speed explicitly appears when describing the interrelation of fundamental concepts mass, position, and time. Actually, a tightly related second entity (position and time) appears; and the speed connects them. In all cases, the law has the form of a differential equation; i.e., we can derive the fundamental entities by integration. In the case of the dual-speed interaction of ions, only numerical solution is possible.
In our ’non-ordinary’ science, we have a double-speed, double-substance interaction abstraction; correspondingly, we have two laws of motion. Furthermore, for ions we use a continuous and a particle-based approach, sometimes simultaneously. In our laws of motion (see Eq.(2.13) and Eq.(2.16)) we also have explicit speed dependence in describing the interrelation of concentration and potential. Actually, we are thinking in two entities (concentration and potential), but both of them are parametrized by position . In line with the Einsteinian case, the time shines up, and, again, the speed connects those entities. However, the different interaction speeds act on the coordinates differently, that is, the effects of changed entities cannot be separated. This is why we need ’non-ordinary’ laws. (Given that our thermodynamical speed is always by orders of magnitude lower than the electrical (limiting) speed, we neglect transforming the time.)
In all cases, the law has the form of a differential equation; i.e., we can derive the fundamental entities by integration. In ’ordinary’ science, we have a single-speed interaction abstraction; so we have one law of motion; an analytical solution is possible. In a non-ordinary case, we have a dual-speed interaction, and a numerical solution is likely the only option.