This section recalls the contents of section 2.2.4 from the viewpoint of thermodynamics. As we discussed, we can handle atomicity in different abstractions, as charge-less or mass-less points (but anyhow: ’material point’ as A. Einstein coined [102]). We derived laws for a single interaction, see Newton’s Law of Universal Gravitation and Coulomb’s law. However, ions require different thinking [22]: thermodynamics is about non-interacting particles. In the frame of classical science, the interactions are instant, and so the interaction speeds are the same. Evidence shows that this is not the case for electrodiffusion; they differ by about six orders of magnitude. The assumption of classic science that everything occurs at the same time evidently cannot be applied; correspondingly, the diffusion coefficient of electrodiffusion has yet to be discovered.
The ions as ’material points’ (having charge and mass) can be abstracted that they have a behavior that there are two underlying interactions and experience two different force fields that cannot be separated further; correspondingly they have less simple laws of forces and motion. Classical science describes one type of forces using laws of electricity (the points have charge but no mass), another type using laws of mechanics (the points have mass but no charge). We must recall that thermodynamics is about non-interacting particles (except their direct collisions as hard-ball points). When speaking about ions, the volume elements in the phase space are not independent: the energy and momentum of a particle depend on the other particles.
What truly sets ions’ thermodynamics apart in physics is the absence of a direct equivalent of the Maxwell-equations. By introducing time derivatives by Eqs. (2.13) and (2.16), we can provide their equivalent equations that describe the relation between concentration and potential for the case when the first time derivative of the position coordinate is not zero. The practical difficulty is that the diffusion speed is by several orders of magnitude smaller than that of the EM interaction. Furthermore, the applied electric field speeds up the ions to potential-assisted or -accelerated speeds, and classical physics is not prepared to handle mixing interaction speeds.
In classical physics, the EM interaction is instant, so the time derivatives of the electric and magnetic fields can change simultaneously. In the approximation we use, we consider the EM speed infinitely high – in the spirit of ’classical physics’ – and we consider the finite speed of ions using physical approximations, which are simplified representations of the actual physical processes. In our mathematical model, the electric field gradient acts instantly on the charge, but the effect of the concentration gradient reaches its position with some delay. In this unique field, the chemical concentration gradient and the electrical field generate each other at different pace, presenting a fascinating departure from traditional (’ordinary’) physics. Another speed-related difference is that we must introduce – among others– local and global conservation laws, similar to the case of general theory of relativity.
The macroscopic features (such as pressure, temperature, potential, and concentration) of systems of ’material points’ are interpreted as having statistical behavior and their laws are discussed by the scientific discipline thermodynamics. Its notions drastically differ from the ones of classical fields. Here the concepts such as ’temperature’ or ’pressure’ are a generalization: a homogeneous distribution means that classical physical substances (such as momentum and energy) have a well-established distribution instead of a single value or uniform values of parameters. At the same time (in infinitely large volumes) the macroscopic parameters ’concentration’ and ’potential’ (notice that they are based on the single-interaction abstractions ’mass’ and ’charge’, respectively) are simple densities, although to interpret them, a large number of particles must be considered. For a more careful experimenter, it is evident that this homogeneity is a dynamical one: particles’ movement changes it continuously and it is constant only as a statistical average.
There are quasi-closed volumes is biological matter and the processes happen at the boundaries of the microscopic and macroscopic worlds, and we must consider different interactions at different speeds. To describe the phenomena, which are neither purely microscopic nor macroscopic, more than one abstraction must be used. Still, they show the behavior of both worlds. Furthermore, they change their behavior during the course of the studied process. The inappropriate handling of mixing interaction speeds led to ’non-ordinary’ behavior and one could conclude ’non-ordinary’ laws when uses the appropriate approximation(s). We need a more careful handling (and more approximations) when we consider the interactions in a finite volume, with strongly different conditions on its boundaries. We need to conduct case studies and apply casual approximations to describe the phenomena, which are neither purely microscopic nor macroscopic and where more than one abstraction must be used. It is important to remember that we are dealing with a mixture of macroscopic and microscopic descriptions, and this understanding is a crucial aspect of our research.
Newton’s laws of motion apply also to ions in the cell, given that they are independent of how the exerting forces are generated. As we discuss in section 2.4.4, the ”special construction” of living matter leads to cases when ”conditional counterforces” may apply. Furthermore, there are constraints arising from the structure of the living matter, and they move the ’material point’ in a viscous media instead of vacuum; furthermore, the balanced states, as well as the processes from and to those states, are described by the interplay of those forces.
The distributions, however, can be calculated for charge-less or size-less ’material points’ only. The interference of those forces and that they affect two different features of the atomic particles led to unusual disciplinary consequences. For his discovery of the reciprocal relations in thermodynamics, Lars Onsager was awarded the 1968 Nobel Prize in Chemistry. The presentation speech referred to his result that ”Onsager’s reciprocal relations represent a further law making a thermodynamic study of irreversible processes possible”. In that sense, we provide mathematical equations of the fourth law of thermodynamics in section 2.4.5. The experimental verification [96] of that law mentions ”the well-known difficulty of carrying out these experiments”. By using our relations between the electrical and chemical diffusion, we can overcome that experimental difficulty. The significance of our Eq.(2.24) is, that one can derive the speed of electrodiffusion in electrolytes, which are otherwise not measurable (”hopping in a breeze” [24]: we would have to measure potential changes at distances of the size of the electrodes, with picosec resolution while the electrolytic electrodes cause nearly msec delays).
In our research, the key point is that life (including neural processes) is based (mainly) on thermo-electric processes. The contradictions and duality (mainly) arise from the enormously different interaction speeds of the electric and diffusion processes. In our approach, we divide ion movements into stages, based on the speed of the dominating electric interaction. We introduce diffusion (or potential-less), potential-assisted (based on the mutual repulsion only), and potential-accelerated (internal voltage on biological components accelerates the ions) speeds. In some cases, the diffusion and electric processes follow each other in separate phases, so in some phase, they can be better approximated as ”net” electrical system, combining ”fast” and ”slow” currents. We show that the processes can be staged in such a way that in addition to the dominant interaction, only one more significant interaction remains on the stage, and we can work out a physics-based approximation that a mathematical formalism can describe.