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We report on a systematic investigation of the critical fluctuations in the complex conductivity of epitaxially grown ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ films for $T\ensuremath{\lesssim}{T}_{c}$ using a two-coil inductive technique at zero applied field. We observe the static three-dimensional (3D) $\mathrm{XY}$ critical exponent in the superfluid density near ${T}_{c}.$ Linear scaling analysis close to the critical temperature yields a dynamic critical exponent of $z\ensuremath{\approx}2.0$ for small drive currents, but nonlinear effects are seen to be important. At ${T}_{c},$ a nonlinear scaling analysis also yields a 3D dynamic exponent of $z=2.0\ifmmode\pm\else\textpm\fi{}0.1.$

The thermodynamic behavior of a fluid near a vapor-liquid and, hence, asymmetric critical point is discussed within a general ``complete'' scaling theory incorporating pressure mixing in the nonlinear scaling fields as well as corrections to scaling. This theory allows for a Yang-Yang anomaly in which \mu_{\sigma}^{\prime\prime}(T), the second temperature derivative of the chemical potential along the phase boundary, diverges like the specific heat when T\to T_{\scriptsize c}; it also generates a leading singular term, |t|^{2\beta}, in the coexistence curve diameter, where t\equiv (T-T_{\scriptsize c}) /T_{\scriptsize c}. The behavior of various special loci, such as the critical isochore, the critical isotherm, the k-inflection loci, on which \chi^{(k)}\equiv \chi(\rho,T)/\rho^{k} (with \chi = \rho^{2} k_{\scriptsize B}TK_{T}) and C_{V}^{(k)}\equiv C_{V}(\rho,T)/\rho^{k} are maximal at fixed T, is carefully elucidated. These results are useful for analyzing simulations and experiments, since particular, nonuniversal values of k specify loci that approach the critical density most rapidly and reflect the pressure-mixing coefficient. Concrete illustrations are presented for the hard-core square-well fluid and for the restricted primitive model electrolyte. For comparison, a discussion of the classical (or Landau) theory is presented briefly and various interesting loci are determined explicitly and illustrated quantitatively for a van der Waals fluid., Comment: 21 pages in two-column format including 8 figures

Thermodynamic functions in the vicinity of an ordinary critical point are expected to obey asymptotic scaling laws as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and the ordering field, $h$, approach zero. However, the optimal scaling variables are the nonlinear scaling fields, ${g}_{t}=t+{b}_{t}{h}^{2}+{c}_{t}{t}^{2}+\ensuremath{\cdots}$ and ${g}_{h}=h(1+{c}_{h}t+\ensuremath{\cdots})$. The nonlinearities yield correction factors to the leading power-law (and scaling) variation of thermodynamic quantities, $L$, of the form ($1+{a}_{L}t+{b}_{L}{t}^{2}+\ensuremath{\cdots}$), etc., where the correction amplitudes ${a}_{L},{b}_{L},\dots{}$, are uniquely determined by the nonlinear scaling-field coefficients. It follows that "analytic" corrections to, e.g., the susceptibility, are directly related to those for the free energy and magnetization (in zero field). The term ${b}_{t}{h}^{2}$ also generates nonanalytic contributions such as an additive, energylike term, varying as ${|t|}^{1\ensuremath{-}\ensuremath{\alpha}}$ in the zero-field susceptibility, and factors like ($1+{c}_{L}{|h|}^{2\ensuremath{-}\frac{1}{\ensuremath{\Delta}}}$) on the critical isotherm, $t=0$. Irrelevant scaling fields yield further, in general distinct, nonanalytic corrections, and cause shifts in ${T}_{c}$ and various amplitudes although "universal" ratios remain constant.