How to connect theory of electricity and thermodynamics? [closed]

Question

A more simple way of asking the same:

When we have a simple electrodiffusion, say, through ion channels positive ions rush-in into the intracellular space of a neuron, in one packet, about $10^3$ ions per channel. Even, for all channels, in the order of $10^4$ channels on the membrane, the number of ions appearing (suddenly, in psec time) on the surface is only $10^7$. This charge appears as current at the beginning of the axon. This current moves (discharges) without external potential. The huge difference in the charge density (and concentration) on the membrane's surface and the no charge at the beginning of the axon can explain why a current flows. (the other way round: ions escape into the volume and they repulse each other; so they will move in the direction of the drain). The Nernsnt-Planck equation could explain that the concentration gradient causes a potential gradient, i.e., explain why we see a current without voltage. However, the number of ions is not sufficient to apply thermodynamics. At the same time, the ions exist in a volume having thickness about tenths of nanometer, i.e., the density of the ions can be sufficient to behave as a macroscopic charged fluid in that limited volume. The N-P equation refers to large (on this scale) volume. Does an equivalent formulation exist for densities, or do some statistical restrictions apply for the case? (Evidence shows that current in pA scale flows, and behaves in a macroscopic way.) This case is nearly neither "net" macroscopic nor microscopic. Can we tell anything about it?

When we assume $10^7$ rushed-in ions for evoking one single action potential, it means $1.6*10^{-12}$ Cb charge. If that charge flows out in period of 1.6 msec, we arrive at that we should measure current in the range $10^{-9}$ A, in good agreement with the measured tens of pico-Ampere currents. If we assume, as usual, the resistance of the membrane as 10 MOhm, we can assume that a voltage about 10 mV evokes due to the sudden increase in the charge density (a voltage gradient) on the membrane's surface. So, the charge from the micro-world can be excellently mapped to the measured macroscopic current. On the other way round, due to Nernst-Plack, it could be expressed as a consequence of the sudden increase in the concentration (a concentration gradient) of the chemical ions on the surface. Provided, that thermodynamics can be applied to a volume, of area of $10^{-2}\ mm^2$ and thickness say up to $10*10^{-9}\ m$. With those numbers we arrive at that the ion density during an action potential is $10^{23}\ m^{-3}$, which is not far from the numbers where thermodynamics can be applied. Is there something against applying N-P in this neuro-physiological context?

When we suddenly switch an electric potential gradient to a steady-state volume filled with ions, according to the Stokes-Einstein formula, the ions gain a speed in the direction of the gradient. That means that its momentum and energy will change, and after a while, it reaches a new steady state, where also the concentration is changed. I can imagine that there is no problem from the point of view of conservation: the electric gradient means also force and energy; and together with the voltage generator, we have a closed system. However, it means also a step-like change in the speed's distribution, at least in the direction of the gradient. The distribution, at the same time, means temperature, at macroscopic level, and the voltage step means a temperature step as well. The distribution should be different from which the old temperature has been derived. What will be the macroscopic "temperature" in the volume? In the case of non-point-like balls, non-central collisions may result in changing the momentum in other direction. Maybe, this mechanism will distribute the step-like change in speed in the volume, and no more global temperature exists, only local ones? That is, the temperature also changes in the presence of an electric gradient (maybe its effect is similar to the case of gravitation)? (my guess is the latter) Can we explain the effect with assuming point-like balls? Is this question discussed somewhere?

Answer 1

So to start, it is important to acknowledge that a system with some kind of flux or current (even in steady state) is out of equilibrium, and so there is no guarantee that fundamental thermodynamic quantities such as temperature are "well-defined", except in a heuristic sense (i.e. you can attach a thermometer to a nonequilibrium system and get a reading, e.g. a mercury level, but there is no guarantee that your mercury reading will agree with, e.g., a reading from an IR thermometer.) Assuming that the external EMF is sufficiently weak (i.e. the average electric field is essentially constant over the mean-free path of an ion) then it is reasonable to invoke the assumption of local equilibrium: you partition the fluid into tiny (but not 'microscopic') unit cells, each of which contains enough particles (e.g. of order $10^{18}$ or so, but no more than $N_A\sim 10^{23}$) to qualify as a thermodynamic system. Under these assumptions, you would model the system as a thermodynamic fluid in local equilibrium, accounting for the equation of state of the solution (i.e. solvent + dissolved ions) in a constitutive relation which would resemble the Navier-Stokes equations with several additional parameters. Assuming that the fluid comoves with the ions (i.e. the ions aren't drifting through a gel or a jammed colloidal suspension), the drift velocity on its own will have no measurable effect on the local temperature. However, you may be interested to read about Onsager reciprocity: in effect, a thermodynamic flux (e.g. heat diffusion) will generically tend to correlate with or induce other bulk/thermodynamic fluxes (e.g. "charge" diffusion), and do so in an essentially symmetric way, and so fluxes are never entirely independent (except under very carefully contrived conditions.)