A refined classification theory for phase transitions in thermodynamics and statistical mechanics in terms of their orders is introduced and analyzed. The refined thermodynamic classification is based on two independent generalizations of Ehrenfests traditional classification scheme. The statistical mechanical classification theory is based on generalized limit theorems for sums of random variables from probability theory and the newly defined block ensemble limit. The block ensemble limit combines thermodynamic and scaling limits and is similar to the finite size scaling limit. The statistical classification scheme allows for the first time a derivation of finite size scaling without renormalization group methods. The classification distinguishes two fundamentally different types of phase transitions. Phase transitions of order λ>1 correspond to well known equilibrium phase transitions, while phase transitions with order λ<1 represent a new class of transitions termed anequilibrium transitions. The generalized order λ varies inversely with the strength of fluctuations. First order and second order transitions play a special role in both classification schemes. First order transitions represent a limiting case separating equilibrium and anequilibrium transitions. The special role or second order transitions is shown to be related to the breakdown of hyperscaling. For anequilibrium transitions the nature of the heat bath in the canonical ensemble becomes important.
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
aL,bL,…, 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
bth2 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
Tc and various amplitudes although "universal" ratios remain constant.
Based on the foundations of thermodynamics and the equilibrium conditions for the coexistence of two phases in a magnetic Ising-like system, we show, first, that there is a critical point where the isothermal susceptibility diverges and the specific heat may remain finite, and second, that near the critical point the entropy of the system, and therefore all free energies, do obey scaling. Although we limit ourselves to such a system, we elaborate about the possibilities of finding universality, as well as the precise values of the critical exponents using thermodynamics only.
The paper introduces a classification of phase transitions in which each transition is characterized through its generalized order and a slowly varying function. This characterization is shown to be applicable in statistical mechanics as well as in thermodynamics albeit for different mathematical reasons. By introducing the block ensemble limit the statistical classification is based on the theory of stable laws from probability theory. The block ensemble limit combines scaling limit and thermodynamic limit. The thermodynamic classification on the other hand is based on generalizing Ehrenfest's traditional classification scheme. Both schemes imply the validity of scaling at phase transitions without the need to invoke renormalizaton-group arguments. The statistical classification scheme allows derivation of a form of finite-size scaling for the distributions of statistical averages while the thermodynamic classification gives rise to multiscaling of thermodynamic potentials. The different nature of the two classification theories is also apparent from the fact that the generalized thermodynamic order is unbounded while the statistical order is restricted to values less than 2. This fact is found to be related to the breakdown of hyperscaling relations. Both classification theories predict the possible existence of phase transitions having orders less than unity. Such transitions are termed anequilibrium transitions.Systems near anequilibrium transitions cannot be described by conventional equilibrium thermodynamics or equilibrium statistical mechanics because of very strong fluctuations. Anequilibrium transitions are found to exist in statistical-mechanical model systems. The identification of the Lagrange parameter \ensuremath{\beta} in the canonical ensemble becomes invalid if a reservoir and a system of the same substance are in thermal contact and anequilibrium transitions are present. Based on the ergodic hypothesis and the theory of convolution semigroups it is shown that near anequilibrium transitions the equations of motion for macroscopic observables of infinite systems may involve modified time derivatives as generators of the macroscopic time evolution. The general solution to the modified equations of motion exhibits very slow dynamics as frequently observed in a nonequilibrium experiment.
We divide the free energy near the critical point into two parts. One is the regular part, the other is the singular part. The singular part is assumed to be a concrete possible form. The singular part in this form is different from Widom scaling hypothesis and can give out the scaling laws of the critical exponents about the thermodynamic quantities. Furthermore, the critical exponents will be right worked out. We proof the singular part in this form can satisfy the qualities of the scaling function mentioned out by Widom. As an assumption, we expect the free energy in this form will be significative for the future research. Note: If you have any suggestions when this paper is considered, connect us with E-mail:lanmanhuayu@yahoo.com.cn, please.:)