This paper is aimed to establish a dynamical law of fluctuations, and to derive the critical exponents based on the standard model with fluctuations, leading to correct critical exponents in agreement with experimental results.
1. For a thermodynamic system, the PDP proposed in
which offers a complete description of associated phase transitions and transformation of the system from non-equilibrium states to equilibrium states. This dynamic law (1) also describes automatically irreversibility.
In view of (1), we developed a systematic theory in
for deriving explicit expressions of thermodynamic potentials, based on first principles, rather than on the mean-field theoretic expansions.
The dynamic law (1) with expression formulas for the thermodynamic potentials and the dynamic transition theory developed in
provide a complete theoretical understanding of phase transitions and critical phenomena for thermodynamic systems. This is the basic theory of the standard model for thermodynamical systems.
2. There is, however, a discrepancy between the theoretical exponents and their experimental values, as in the case of mean-field theoretic approach. We demonstrate in this paper that in reality, there is a critical fluctuation effect, and we show that the discrepancy just mentioned is due entirely to the spontaneous fluctuation.
To have an accurate account of the fluctuations, we need to derive its governing fundamental law, which have to stem from the thermodynamic potential and the dynamic law (1).
where is the fluctuation of the external force. 3. We derive two basic theorems on critical exponents. The first theorem is based on the dynamic law (1).
First Theorem of Critical-Exponents.
For a second-order phase transition, near the critical point, using the dynamical law (1) without fluctuations, we derive the theoretical critical exponents as given by
The second theorem takes into consideration of fluctuations.
Second Theorem of Critical-Exponents.
For a second-order phase transition, near the critical point, using the dynamical law (3) with fluctuations, the fluctuation critical exponents are given by
We now list three groups of exponent data for different thermodynamic systems:
1) experimental exponents,
2) theoretical exponents without taking into consideration of fluctuations, and
3) theoretical exponents using the standard model with fluctuations.
This table shows clearly the strong agreement of the results using the standard model of thermodynamics with fluctuations.
|exponent||magnetic system||PVT system||binary system||without fluctuation||with fluctuation|
4. We have shown that the theoretical values from the dynamic law (1) do reflect the nature under the ideal assumption that no fluctuations are present in the system. However, the fluctuations are inevitable, and are completely accounted for by the dynamic law of fluctuations (3). In a nutshell,
this in return validates the standard model of thermodynamics, which is derived based on first principles.
Ruikuan Liu, Tian Ma, Shouhong Wang and Jiayan Yang