To derive laws of motion of ions in electrolytes, one must use non-ordinary methods after scrutinizing which approximation can be used for living matter. Below, we derive an approach that enables us to handle interaction speeds that differ by orders of magnitude. Science has created abstract concepts such as space and time; mass and charge. Modern physics was born when experiments began to contradict the fact that nature could be described in such simple terms. The recognition of the non-continuity of energy led to the creation of quantum mechanics, and the recognition of the non-independence of space and time led to the creation of the theory of relativity, with far-reaching consequences. We understood that under certain circumstances, we must treat particles as kinds of continuous waves, and continuous waves as kinds of particles. We understood that the Newtonian approach has limitations, and that interactions at finite speeds can only be described by a different kind of mathematics, namely the concept of spacetime. Moreover, mass is also related to space and time; taking this into account, we can describe nature in terms of curved spacetime. According to classical physics, we can describe phenomena with sufficient accuracy using only the aforementioned concepts (the zeroth derivatives). According to modern physics, for a more accurate description, we must take into account the first derivative with respect to time. For a general description, we must consider the second derivative as well.
Regarding ions, we can handle their charge and mass separately; we know the relevant laws. However, in the case of ions, we must connect charge and mass, just as in the case of the theory of relativity, space and time. Interestingly, here too, the velocity creates the connection between the two abstract features. Furthermore, we must introduce a relation for the currents carried by ions, similar to that introduced in statistical mechanics for the correspondence between the particle and the continuum views. However, in our case, the correspondence is established on geometric grounds.
Eq.(2.9) describes a stationary state with no ionic movement. Deriving a time course (time derivatives) from the position derivatives is not possible in a strict mathematical sense. However, we can provide it by using physical principles. We consider the electric ion current represented by viscous charged fluids [115]. As expected, selecting the speed (aka calculating the appropriate value of the macroscopic speed, see Eq.(2.17)) plays a key role, especially since we are at the boundaries of physics abstractions; among others, we are mixing microscopic and macroscopic notions. The actual speed model depends on the concrete case; see section 2.5.3.
In classical physics, because of the lack of time-dependent terms in the expressions, the changes are described by position-dependent terms (position derivatives), both in the case of electromagnetical and electrodiffusional interactions. In classical (’instant interaction’) science, the time derivatives are either not interpreted or can be derived by scaling through the externally derived joint interaction speed as a scale factor. As explained, we can extend the idea to enormously different speeds and derive time derivatives if we consider the faster interaction to be instant.
In the timeless classical physics, there is no explicit dependence on the time: everything happens simultaneously. In a resting state, the Maxwell equations essentially follow from the conservation of energy. One form of energy transforms into another form, and the system arrives in another balanced state. The carriers of the force fields are continuous and have no masses, so one can calculate and make infinitesimal changes in the driving forces; they do not change the system’s energy. If one gradient changes, the other automatically (per definitionem) changes in the opposite direction. In another words: the driving forces are permanently balanced, the magnetical and electrical forces act instantly (”at the same time”) and they are always of opposite sign. A time derivative cannot be interpreted: everything happens at the same time; in other words, at the same space-time (in the classic interpretation, the time is the same at any point).
In an electrodiffusional process, we start with the same point of view. We assume that the thermodynamical and electrical driving forces are equal in an equilibrium state. That assumption results in the Nernst-Planck equation. On one side, we use a macroscopic parameter, the potential. On the other side, we use another macroscopic parameter, the concentration. The equation bridges those macroscopic parameters by using universal constants from the microscopic world.
At first glance, the case is similar to that of electrical and magnetic fields. However, we cannot make infinitesimally small changes in the gradient since the carrier of the force fields is ”atomic”. Furthermore, moving it infinitesimally (changing only its position coordinates), the changes in the electrical and thermodynamic gradients do not result in a new balanced state. The effect of ions’ charge has an immediate effect on the volume, but ion’s mass has a delayed effect. The infinitesimally small change in the position results in an infinitesimally small increase in the energy of the system, given that moving a carrier changes the potential and the concentration in the same direction, since we did not consider that the time coordinate changes as well. In the Newtonian world, everything happens at the same time, so we cannot handle instant and finite interaction speeds simultaneously. Eq. (2.20) contradicts energy conservation. The infinitesimally small energy change disappears only when the effect of the slower interaction reaches the other carriers in the volume. When the interaction speeds differ, energy conservation is valid only if one uses space-time.
Fortunately, we can derive the infinitely small change where the time and space (position) coordinates are connected; essentially, in the same way as in the special theory of relativity. Let us assume that the gradients act on the mass and the charge, but the ion’s effects on the gradients are negligible. According to the principle of relativity, the phenomena must remain the same in a reference frame moving with a constant speed relative to the first one, and we choose the system that moves together with the ion. In the second frame, no ionic movement occurs along the movement’s direction. In line with the fact that the speed of light is independent of the reference frame, we assume that the higher interaction speed remains the same in both systems: it is instant. The observers in both reference frames must see that the system is balanced. The difference is that in the first frame, the system is statically balanced (no change in the gradients, but the ion is moving), and in the second one, it is dynamically balanced (the gradients change to keep the ion at rest). The gradients the moving ion experiences are the ones that the standing ion experiences at another time (depending on its speed). This way, we can provide the needed time course of the process.
Compared with the electromagnetic case, we see crucial differences. First, the mass’s propagation speed (forming a new concentration gradient) is millions of times lower than the charge’s. Second, the moving ion simultaneously represents mass transport and charge transport. Third, when deriving position derivatives, we conclude from the assumption that there is no movement (in other words, no explicit dependence on the time): the effect of the electrical and magnetic driving forces is equal, whatever time is needed to reach that balanced state. In contrast, in electrodiffusion, the velocity changes the concentration gradient, and simultaneously, the potential gradient.
We assume that equation (2.9) is valid for a given time . At time , in another steady state, the two interactions manifest at different times: we have
| (2.10) |
or, equivalently, it can be expressed as
| (2.11) |
The concentration at position determines the potential (apart from an integration constant) at position:
| (2.12) |
so (and here the constant disappears) the time derivative is
| (2.13) |
or
| (2.14) |
Similarly, at time , in another steady state, we have
| (2.15) |
| (2.16) |
We expressed the dependence of gradients on each other using the ion’s speed as intermediate variable, that can be expressed by the Stokes-Einstein relation as
| (2.17) |
After simplifying the expression and using the aforementioned special feature of the type function
| (2.18) |
| (2.19) |
As section 2.5.6 discusses, in general, the electric operation of an electrolyte can be described by this law of motion. For practical calculations, the voltage time derivative can be calculated directly from the input current, thereby accounting for the current production mechanism, see equations (2.39) and (3.6), which directly consider the current production mechanism. We must not forget that we started from a quasi-equilibrium state; that is, our description is valid only for quasi-static processes.
We note here that in closed volumes, for example, see Schrödinger’s points on the specific aspects of life, one must consider further effects. The electrostatic forces of the ions exerted on each other, and the wall exerts a counterforce on the ions in the microscopic view, and on the fluid in the macroscopic view, which manifests as a mechanical force and pressure. This aspect is not considered in the general transport equation (2.8); one must add one more term when calculating mass transport. If the volume is not hermetically closed (say, the ion channels in the wall of the membrane represent a ”hole” for the ions), the combined electrostatic plus mechanical pressure can provide a driving force (instead of hypothesizing a magic protein mechanism) for producing a mass transport. If the change in concentration and potential happens suddenly (see the rush-in of ions at the beginning of an AP in neurons), that ”explosion” creates an ”impact force” exerting on the membrane and the conterforce starts a pressure wave and other mechanical changes, as discussed in [116]. The elastic membrane generates The electrical and thermodynamic phenomena are just two sides of the same coin as discussed in [71], and underpinned by some plausible numericals estimations in ”speculations”
Fortunately, the overwhelming majority of the energy of the excitation is stored as elastic energy of the membrane, and the pressure is proportional to the measurable voltage, so describing the electrical behavior is sufficient to provide a sufficiently precise description of the processes in the membrane, but one has to keep in mind that the force is composed of electrical and thermodynamic components.