2.4.2 Statistical mechanics

Thermodynamics is a somewhat misunderstood branch of physics. As its name suggests, it deals with heat, work, and temperature, and their relationship to energy, entropy, and the physical properties of matter. However, it is based on statistical mechanics, which combines two fundamentally different concepts of nature. Ever since physics discovered that things, initially thought to be continuous, are quantized, it has been dealing with matter in a kind of double consciousness. In some ways, we view gases and fluids as continuous, while in others, we view them as discrete. Although we know that electricity consists of elementary charges, we describe it essentially in terms of continuous quantities. We think about the continuous medium as comprising hard-shelled balls, but their distribution is continuous.

To handle discrete objects in a continuous distribution, Ludwig Boltzmann introduced the ingenious idea that we use use a continuous probability distribution in a phase space to form ensembles. That is, we assume that a well-defined $f(𝐫,𝐩,t)$ probability distribution function of the position vector $𝐫$ and the momentum vector $𝐩$ of individual particles exists at every single moment $t$, and the volume element $d^3𝐫d^3𝐩$ in the phase-space defines an ensemble; furthermore, the integral of the distribution over that ensemble gives the probability that the volume contains

particles which have identical positions and moments in that region, at an instant of time. From this point on, we assume that the statistical estimation is sufficiently good (the probability function is sufficiently ’dense’ or the volume element is sufficiently large). We consider that the probable number $dN$ is the actual number, and consider that the same forces affect all particles in the same way, as the laws of science describe it. Notice that the individual particles are independent, so they must not interact with each other; they are affected only by external forces. An internal force would change the infinitesimal volume in the considered infinitesimal period.

”The real strength of thermodynamics lies in the collective phenomena. Temperature, pressure, heat, and so forth, are terms from the world of ensembles.” [59] Notice that here we are on the boundary of the continuous and particle-like views of a system: we select the particles as if we were handling a continuum, but calculating their position and momentum individually, under the exertion of a jointly defined force. In thermodynamics, the continuous and the quantized views of nature are connected. The idea was to not handle the enormous amount of particles individually; instead, replace the concrete particles with a probability that they are at a given position (in the phase space). The abstract mathematical laws of probability and statistics have their conditions of applicability. The task of physics was to determine which conditions enable us to use those abstract laws for actual physical systems, also when combining the statistical and particle views of electrons. We must discuss whether changing the charge carrier and the medium, where charge is transmitted, changes the conditions of applicability. In other words, we must scrutinize how much the laws of electricity, which are based on the statistical behavior of electrons in solids as a medium, can describe the statistical behavior of ions in the special medium that biology represents.

A more usual form (with our contrain force included) of the equation is

Notice the explicit role of particles’ mass: even if there were no other fundamental differences between the solid-state-based and electrolyte-based electricity, the $\approx 50,000$ factor between the masses of the charge carriers, alone, would introduce drastic deviations in the equations describing them. Furthermore, the terms combine describing individual particles with a description of probabilistic groups of particles; practically, continuous and quantized matter.

The best-known case is when there is no long-range interaction between the colliding particles, see above; the case of gases is the textbook example of thermodynamics. The most problematic term, in general, is the one on the right side of Eq.(2.5) that describes the effect of collisions between particles, which is topped in biology by the effect of the limited volume (”the construction”). The above collisionless equation is relatively simple. If long-range interactions are present, they can be aggregated in some cases. Physically, in plasmas, the swift free electrons and in solids, the heavy ions at fixed positions provide such aggregated force fields, and one can describe the collisionless movement of the charge carriers (ions in plasmas and electrons in solids) in that field also in the presence of long-range interaction.

Correspondingly, there are two major branches of applying Eq.(2.4) for systems with collisions (the classical thermodynamics for particles without electric charge, such as gases) and without collisions (in plasmas, the swift and light electrons provide a strong and uniform electric field, and the electrons do not collide directly; furthermore, the long-distance positive charge of the freely moving ions prevents direct collisions). A common element in those two branches is the assumption that the cell in the phase space remains unchanged and the force that is exerted on the particles is identical and instant. Classical electricity, the case of electrons in solids, corresponds to the case of moving under the effect of an external force (electrical field) and suffering inelastic collisions with the ions that are fixed to the gridpoints. The gridpoints convert the lost energy to heat, so we can consider rest of the system collisionless. In this case, the particles form a kind of cloud, so they do not exert force on each other, at least in a statistical sense. The ions in electrolytes, however, represent a drastically different case. There exists a long-range interaction between the ions; moreover, the ions collide with the neutral particles and the neutral particles with each other. On the one side, the overwhelming majority of the particles is not ionized, so the neutral particles define the features of the medium, so the behavior of the electrolytes is similar to that of the non-interacting particles. However, in the processes studied by biology, the low proportion of ionized particles dominate, so the behavior of ions in electrolytes is somewhat similar to that of the electrons in solids. The ions in electrolytes represent a very different class. That is, to describe them mathematically, we must add one more term to Eq.(2.5) to consider their long-range interaction plus one more term for their movement in speed-dependent interaction with the viscous medium. When describing electrical phenomena, in the case of having electrons in solids, we can neglect the collisions (in the terminology below: the thermodynamic term): the cloud mediates the charge transfer ”instantly”, allowing a ”one electron in, one electron out” mode of the charge transfer. In the case of ions in electrolytes, the ions can collide with the neutral particles and each other, and the strong, long-distance Coulomb interaction contributes a force that makes (part of) particles non-independent, making the applicability of Eq.(2.4) questionable.

We see that for the case of ions moving in biological electrolytes, neither of the above assumptions is valid (not to mention that the system is not closed so various constraint forces may affect the ions’ movement; furthermore, the medium not only ”resists” ion movement due to collisions with neural molecules but it can be chemically and electrically active). The ions have a long-range Coulomb interaction with each other, so they can freely change their positions as well, that changes the cell size $d^3𝐫d^3𝐩$. (This way, the derivative of the momentum is also present in the corresponding equations.)

A key assumption of thermodynamics is that a force ”instantly” acts on each particle. External forces that exert an effect on the ions act instantly: the case of biological electrolytes represents a combination of instant collisions with long-range Coulomb interactions. The thermodynamic effect (the collision) of a particle arrives at the other particle later than its electric effect. The classical disciplinary physics is not prepared for such a case: it handles all processes at the same (instant) speed. This is one of the fundamental issues why the usual thermodynamics handling cannot successfully describe biological systems.

The essential points of Boltzmann’s general equation are that the force acting on the particles is instantaneous, the particles only come into contact by direct collisions, the volume is infinite, the number of particles is sufficiently large, and the system is closed. Only with these assumptions can we assume that the volume in the phase space does not change. In the case of neutral atoms or molecules in a large, closed volume, the conditions are met, and Boltzmann’s equations describe the observed behavior of the systems. Another approach can be derived (the so-called Vlasov equations) when there is a long-range interaction between particles (for example, the Coulomb interaction between ions of the same charge), which excludes direct particle collisions. The latter describes, for example, plasmas in which atoms are fully ionized and ions move through a cloud of free electrons. This model also partially describes electrons moving in the periodic force field of the ion lattice in a solid, but it is not considered a thermodynamic system in the former sense; that concept is reserved for particles that interact only by direct collisions.

The different counter-forces depend on the situation, and the ”construction” refers to the fact that the composition of forces can change drastically, even within a single biological process and within a small fraction of nanometer distances. The volumes and the number of particles are usually much smaller than those the thermodynamics used to work with; furthermore, the ”construction” changes the local physical state point by point. However, the crucial changes happen in small volumes (limited by physical effects) where the density is large enough (for a short period) so we can make statistically meaningful statements with the mentioned limitations. Handling ions in living matter at least requires much more attention. It represents a field (perhaps a sub-discipline or a new discipline) where a new idea of handling is required and which offers hope for describing biological processes.

Classical physics can only handle instantaneous interactions; in the Newtonian worldview, all interactions co-occur. The traditional approach to thermodynamics cannot be applied to biological systems. Erwin Schrödinger stated that ordinary scientific laws cannot describe living matter. At the same time, he expressed his conviction that no new force or unknown interaction is emerging; only a previously unknown regularity concept must be found, then the non-ordinary laws organically integrate into the fabric of science, together with the already known laws.

When applied to ions and especially to electrolytes, the conditions for the applicability of the Boltzmann equations are largely not met. Due to the low temperature compared to plasmas, the vast majority of molecules do not dissociate, so they interact only through direct collisions. A minimal number of ions, due to their identical charge, avoids direct collisions with each other through long-range interactions; they do, however, collide with neutral molecules and also the neutral molecules collide with each other. Ions play a leading role in biological processes and can therefore interact through both direct collisions and long-range forces. For this reason, the Boltzmann equations are certainly not applicable to the description of the processes taking place in electrolytes, which directly explains why thermodynamics cannot describe processes of life based on electrolytes.

Further limitations are that, in biological systems, statistical concepts must be applied to a very small number of particles in a closed space of finite volume, where coercive forces also occur, and processes beyond the experimenter’s control make it doubtful that we can interpret a closed system. The main problem, however, is that ion’s charge and mass are inseparable; furthermore, the speed of the electrostatic interaction of the particles with each other is instantaneous, in accordance with Boltzmann’s original assumption. However, the speed of the diffusional interaction, which is calculated from the change in the distribution of the particles, is by orders of magnitude slower. Of course, the particle’s motion simultaneously changes the strengths of the electrical and thermodynamic interactions. The diffusional effect, given that it is implemented through collisions, occurs only when the particle reaches the region where it changes the concentration, whereas the force field created by the other particles changes immediately. In biology, the ”construction” is also different. In addition to the usually discussed forces, some variable counter-forces are also present (such as mechanical obstacles represented by membranes, tube walls, and caps on ion channels).

In mathematics, the temporal gradients are usually described by partial derivatives of spatial or feature parameters. The definition of a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. In the case of ions, changing the concentration means simultaneously changing the electrical charge; that is, the concentration gradient and the electrical gradient depend on each other. In other words, one cannot calculate the partial derivatives of ions’ concentration and electrical potential, only their total derivative. As a consequence, equations that calculate the partial derivatives of the concentration and the electrical potential with respect to time are either incorrect or approximations. The discussions, for example, the one leading to Eq. (11.30) in [24], assume that the interaction speeds are identical and that the partial derivatives can be interpreted for ions. Both assumptions are wrong. However, the claim that ”while diffusion is like a hopping flea, electrodiffusion is like a flee that is hopping in a breeze” [24] (attributed to Hodgkin) is true; see also our Eq. (2.24).