pH/$T$ duality - wall properties and time evolution of plant cells
We examined the pH/$T$ (or $μ$/$T$) duality of acidic pH and temperature ($T$) for the growth of grass shoots in order to determine the equation of state (EoS) for living plants. By considering non-meristematic growth as a dynamic series of 'state transitions' (STs) in the extending primary wall, we identified the critical (read: optimum) exponents for this phenomenon, which exhibit a singular behaviour at a critical temperature, critical pH and critical chemical potential ($μ$) in the form of four power laws: $F(τ)_π\propto|τ|^{β-1}$, $F(π)_τ\propto|π|^{1-α}$, $G(τ)_μ\propto|τ|^{-2-α+2β}$ and $G(μ)_τ\propto|μ|^{2-α}$. The power-law exponents $α$ and $β$ are numbers, which are independent of pH (or $μ$) and $T$ that are known as critical exponents, while $π$ and $τ$ represent a reduced pH and reduced temperature, respectively. Various 'scaling' predictions were obtained - a convexity relation $α+ β\ge 2$ for practical pH-based analysis and a $β\equiv 2$ identity in microscopic representation. In the presented scenario, the magnitude that is decisive is the chemical potential of H$^+$ ions (protons), enforcing subsequent STs and growth. The EoS span areas of the biological, physical, chemical and Earth sciences cross the borders with the language (adapted formalism) of phase transitions.