# Is there a link between the logistic differential equation and Fermi-Dirac statistics?

I was working out some statistical problems and I could not fail to notice that Fermi-Dirac distribution, $$f_{\rm Fermi-Dirac}(E)=\frac{N_{\rm sites}}{1+e^{\beta(E-\mu)}},$$ looks like the kind of distributions you get in epidemics (e.g. see South Korea total cases of coronavirus disease vs time). These are usually modeled by a logistic function in the SIS model (S for susceptible, I for infected) : $$f_{\rm logistic}(t)=\frac{N_{\rm population}}{1+e^{-b(t-t_0)}},$$ where $$t$$ is time and the rest are constants to the problem. This just comes out from a differential equation $$\partial_t f=b(1-f(t))f(t)$$, sometimes found in nuclear decay (e.g. decay products has to saturate at some point).

I know that Fermi-Dirac appears when you have fermion statistics, meaning that you have independent states that can either be empty or filled with just one particle. It does not have to be fermions, it can, for example, just be a finite number of traps/sites next to a reservoir of molecules, where one molecule can go get in or not in each trap (or simply a spin chain).

In epidemics, you don't have a thermodynamic equilibrium per se, you have a more dynamic system, but you have some similar conditions. The total number of people infected (active or not) is based on a model where each person can either be contaminated or not, there is no intermediate state. This gives me the feeling that the two might be related.

The issue here is that while the logistic function graphs a plot of number vs time, Fermi-Dirac is about number vs energy. Could there be a link between the two?

Energy and time are usually "conjugated variables", examples are the ill-defined uncertainty relation between the two in quantum mechanics, or the relativistic 4-momentum and 4-position $$0$$-th components. Also there are sometimes in physics some out of boundary replacements/mathematical tricks that lead to some useful relations between dynamics and equilibrium: like the imaginary energies to model decay in nuclear physics or dissipation, or the famous Wick rotation that gives a link between statistical mechanics and quantum mechanics (and between Lagrangians and Hamiltonians).

Does somebody know a if there is a mathematical link between quantum statistical mechanics and dynamic systems like those found in epidemics (or in classical master equations in general) that could provide a parallel between Fermi-Dirac and the logistic function?

Edit: the number of cases in South Korea has changed dramatically as there have been many waves since.

• It seems to me that you are trying to compare different things: $f_{fermi} \sim dN/dE$ probability density function, and $f_{logist} \sim N(t)$ -distribution(cumulative) function Commented Mar 20, 2020 at 22:26
• @AlekseyDruggist for Langmuir adsorption model, $N\sim f_FD$ Commented Mar 20, 2020 at 22:44
• I meant $f_{FD}$. By Langmuir adsorption model I mean a simple system of a gas at fixed temperature and a finite number of independent traps that can trap up to one atom of the gas. Commented Mar 20, 2020 at 22:58
• I see. You can also apparently add a system of noninteracting spins in a constant external magnetic field. But this is fundamentally not a Fermi distribution where different sites have fixed energy distribution Commented Mar 20, 2020 at 23:26

The similarity is mathematical: logistic regression models a dependence of a variable that can take only two values by equating the logarithm of its probability to be in one of the states to a linear function of parameters: $$\log P_{infected} \propto \sum_j\beta_jx_j.$$ Fermi-Dirac statistics is derived from similar arguments: the state is occupied or empty as a linear function of its position in respect to the chemical potential, $$\log P_{empty} \propto \beta(E-\mu)$$.
• "Note that the whole course of the pandemic will not be a logistic curve, since at some point the rate of infections will start to fall." It would not fall if you consider the infected as a cumulative number (you don't care if they get better, die or stay infected). Also are $x_j$ different times? Commented Apr 9, 2020 at 17:51