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
provides the dynamic law for statistical physics:
which offers a complete description of associated phase transitions and transformation of the system from nonequilibrium 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 meanfield theoretic expansions.
The dynamic law (1) with expression formulas for the thermodynamic potentials and the dynamic transition theory developed in
Tian Ma & Shouhong Wang, Phase Transition Dynamics, SpringerVerlag, xxii, 555pp., 2013
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 meanfield 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).
In fact, for an equilibrium state of a thermodynamic system, the fluctuation of is the deviation from :
Then the needed dynamic law for fluctuations is given by
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 CriticalExponents.
For a secondorder 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 CriticalExponents.
For a secondorder 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 
0.300.36  0.320.35  0.300.34  1/2  1/3  
4.24.8  4.65.0  4.05.0  3  3.06.0  
0.00.2  0.10.2  0.050.15  0  02/3  
1.21.4  1.21.3  1.21.4  1  15/3 
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,

the standard model (1), together with the dynamic law of fluctuation (3), offers correct information for critical exponents; and

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
Very profound posts Prof. Wang. Deeply appreciate all the works done by you and Prof. Ma!
I have two simple questions:
1. Does every (or most of) physical system have the four “theoretical critical exponents” defined in (4)?
2. To derive formula (5) of “theoretical critical exponents” with fluctuation, do we need any regularity assumptions on external force (\tilde{f}), or can \tilde{f} be a stochastic term (so that equation (3) becomes a nonautonomous system)?
Thank you — Xige
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