  Brownian ratchet like stochastic theory for the electrochemical membrane
system of Hodgkin-Huxley (HH) is developed. The system is characterized by a
continuous variable $Q_m(t)$, representing mobile membrane charge density, and
a discrete variable $K_t$ representing ion channel conformational dynamics. A
Nernst-Planck-Nyquist-Johnson type equilibrium is obtained when multiple
conducting ions have a common reversal potential. Detailed balance yields a
previously unknown relation between the channel switching rates and membrane
capacitance, bypassing Eyring-type explicit treatment of gating charge
kinetics. From a molecular structural standpoint, membrane charge $Q_m$ is a
more natural dynamic variable than potential $V_m$; our formalism treats
$Q_m$-dependent conformational transition rates $\lambda_{ij}$ as intrinsic
parameters. Therefore in principle, $\lambda_{ij}$ vs. $V_m$ is experimental
protocol dependent,e.g., different from voltage or charge clamping
measurements. For constant membrane capacitance per unit area $C_m$ and
neglecting membrane potential induced by gating charges, $V_m=Q_m/C_m$, and
HH's formalism is recovered. The presence of two types of ions, with different
channels and reversal potentials, gives rise to a nonequilibrium steady state
with positive entropy production $e_p$. For rapidly fluctuating channels, an
expression for $e_p$ is obtained.
