Neo-Gibbsian Statistical Energetics with Applications to Nonequilibrium Cells

Abstract

Generalization through novel interpretations of the inner logic of the century-old Gibbs' statistical thermodynamics is presented: i) Identifying $k_B\to 0$ as classical energetics, one directly derives a pair of thermodynamic variational formulae \[ F(T) = \min_{E\ge E_{min}}\Big\{E-TS(E) \Big\} \,\text{ and }\ S(E) = \min_{T>0}\left\{\frac{E}{T}-\frac{F(T)}{T} \right\}, \] that dictate all the more familiar $1/T=d S(E)/d E$, $E=d\{F(T)/T\}/d(1/T)$, and $S(E)=-d F(T)/d T$ in equilibrium, which is maintained by a duality symmetry with one-to-one relation between $T^{\text{eq}}(E)=\arg\min_T\{E/T-F(T)/T\}$ and $E^{\text{eq}}(T)=\arg\min_E\{E-TS(E)\}$. ii) In contradistinction, taking derivative of the statistical free energy w.r.t. $T$, a mesoscopic energetics with fluctuations emerges: This yields two information entropy functions which historically appeared 50 years postdate Gibbs' theory. iii) Combining the above pair of inequalities yields an irreversible thermodynamic potential $\psi(T,E) \equiv \{E-F(T)\}/T-S(E)\ge 0$ for nonequilibrium states. The second law of thermodynamics as a universal principle reflects $\psi\ge 0$ due to a disagreement between $E$ and $T$ as a dual pair. Our theory provides a new energetics of living cells which are nonequilibrium, complex entities under constant $T$, pressure $p$ and chemical potential $\mu$. $\psi$ provides a ``distance'' between statistical data from a large ensemble of cells and a set of intrinsic energetic parameters that encode the information within.