Fluctuation Theory and Critical Phenomena

Abstract

Singularities in equilibrium properties at critical points are related to the asymptotic forms of averages of products of $\mathrm{\nu (r)}$ , the deviation from the mean density at position r. Thus the forms of the singularities of the thermodynamic properties at the critical point can be obtained in a simple fashion from any theory that describes the forms of correlation functions like ${\mathrm{〈\nu (r}}_1{\mathrm{)\nu (r}}_2\mathrm{)\; 〉}$ for large r₁₂. As a particular example of such a theory we use the Landau–Lifshitz fluctuation theory to obtain the results $C_{\upsilon }{\sim }_{\kappa }^{\text{−1}}$ , $K_T{\sim }_{\kappa }^{\text{−2}}$ , and ${}_{\kappa }{\mathrm{\sim (T\hspace{0.17em}-\hspace{0.17em}T}}_c{)}^{\text{1/2}}$ for $T$ near $T_c$ . We demonstrate that the predictions of the fluctuation theory are consistent in the sense that the prediction of the form of $C_{\upsilon }$ from four‐particle correlations is the same as that obtained from a temperature integration of ${\mathrm{\partial C}}_{\upsilon }\mathrm{\hspace{0.17em}/\hspace{0.17em}\partial T}$ in terms of six‐particle corrections.