Dual interpretations of the nonextensive statistical thermodynamics with parameter transformation

Abstract

In complex systems, due to the invalidity of ergodic property, the time average of observable quantity is not equivalent to its ensemble average, and then, the latter is nonmeasurable. In order to find measurable quantities, the prior probability is introduced, estimated in original Euclidean phase-space with the ergodicity. The prior probability is associated with the ensemble probability through two manners, by, respectively, evoking the second and third choices of energy constraints. Two different ensemble probabilities are inferred through maximum entropy procedure, of which one is predictable and another is not. It can be verified that these two probabilities are equivalent to each other. With the equivalence, the third energy is directly measurable while the second one is indirectly measurable. By doing this, a consistent formalism of statistical thermodynamics is then established, which consists of dual interpretations of ensemble probability and dual Legendre transformations. Lastly, the independence of prior probability leads to an interesting composition law for entropy and energy, with different parameters and even interactions, in which case the ordinary thermodynamic method is verified to be valid.

Appendices

Appendix

In order to improve the (78), let us rewrite down the distribution (25), namely,

$$\begin{aligned} p_{i} =\frac{1}{\bar{{Z}}_{q}^{(3)} }\left[ {1-(1-q){\beta }'(E_{i} -U_{q}^{(3)} )} \right] ^{\frac{1}{1-q}}. \end{aligned}$$

(A.1)

In light of (36), in limit of large particle number, there is $q\rightarrow 1$; therefore, according to(73) one has the approximate relation

$$\begin{aligned} p_{ij} \simeq p_{i} p_{j}. \end{aligned}$$

(A.2)

Let (A.2)/(73), we have

$$\begin{aligned} (p_{ij} )^{1-q}=(p_{i} )^{1-q_{1} }(p_{j} )^{1-q_{2} }. \end{aligned}$$

(A.3)

In views of the probability (A.1), (29) and the following relation

$$\begin{aligned} c_{q,(1,2)} =c_{q_{1} } c_{q_{2} } , \end{aligned}$$

(A.4)

we have

$$\begin{aligned}&\left[ {1-(1-q){\beta }'(1,2)(E_{ij} -U_{q,(1,2)}^{(3)} )} \right] \nonumber \\&\quad =\left[ {1-(1-q_{1} ){\beta }'(1)(E_{i} -U_{q_{1} ,1}^{(3)} )} \right] \left[ {1-(1-q_{2} ){\beta }'(2)(E_{j} -U_{q_{2} ,2}^{(3)} )} \right] . \end{aligned}$$

(A.5)

With the supposed equilibrium condition,

$$\begin{aligned} {\beta }'(1,2)={\beta }'(1)={\beta }'(2), \end{aligned}$$

(A.6)

we can get simultaneously from (A.5)

$$\begin{aligned} U_{q,(1,2)}^{(3)} =\frac{1-q_{1} }{1-q}U_{q_{1} ,1}^{(3)} +\frac{1-q_{2} }{1-q}U_{q_{2} ,2}^{(3)} , \end{aligned}$$

(A.7)

and

$$\begin{aligned} E_{ij} =\frac{1-q_{1} }{1-q}E_{i} +\frac{1-q_{2} }{1-q}E_{j} +\frac{(1-q_{1} )(1-q_{2} )}{1-q}{\beta }'(E_{i} -U_{q_{1} ,1}^{(3)} )(E_{j} -U_{q_{2} ,2}^{(3)} ).\quad \quad \end{aligned}$$

(A.8)

It can be easily found that the averaged value of (A.8) in the representation of (17) is just the (A.7). Then the (78) is verified. Apparently the deduced equilibrium condition (80) is consistent with the assumption (A.6).

Appendix

In order to improve the (86), let us rewrite down the distribution (19), namely

$$\begin{aligned} p_{i} =\frac{1}{Z_{q}^{(2)} }\left[ {1-(1-q){\beta }'E_{i} } \right] ^{\frac{1}{1-q}}. \end{aligned}$$

(B.1)

Similar to (A.3), we have

$$\begin{aligned} (p_{ij} )^{1-q}=(p_{i} )^{1-q_{1} }(p_{j} )^{1-q_{2} }. \end{aligned}$$

(B.2)

That is

$$\begin{aligned} \frac{\left[ {1-(1-q){\beta }'(1,2)E_{ij} } \right] }{[Z_{q,(1,2)}^{(2)} ]^{1-q}}=\frac{\left[ {1-(1-q_{1} ){\beta }'(1)E_{i} } \right] }{[Z_{q_{1} ,1}^{(2)} ]^{1-q_{1} }}\frac{\left[ {1-(1-q_{2} ){\beta }'(2)E_{j} } \right] }{[Z_{q_{2} ,2}^{(2)} ]^{1-q_{2} }}. \end{aligned}$$

(B.3)

In limit of large particle number, the ensemble partition function could be written as

$$\begin{aligned} Z_{q}^{(2)} =Z_{1}^{N}, \end{aligned}$$

(B.4)

where the $Z_{\mathrm {1}}$ is single particle partition function defined in the molecular dynamics [28], which is approximately irrelevant of the nonextensive parameter.

So according to the rule (74), we have

$$\begin{aligned} {[Z_{q,(1,2)}^{(2)} ]}^{1-q}=[Z_{q_{1} ,1}^{(2)} ]^{1-q_{1} }[Z_{q_{2} ,2}^{(2)} ]^{1-q_{2} }, \end{aligned}$$

(B.5)

which, with the same equilibrium condition (A.6), leads to

$$\begin{aligned} E_{ij} =\frac{1-q_{1} }{1-q}E_{i} +\frac{1-q_{2} }{1-q}E_{j} -\frac{(1-q_{1} )(1-q_{2} )}{(1-q)}{\beta }'E_{i} E_{j}. \end{aligned}$$

(B.6)

Then, in the representation of (16), one can obtain

$$\begin{aligned} U_{q,(1,2)}^{(2)} =\frac{1-q_{1} }{1-q}c_{q_{2} } U_{q_{1} ,1}^{(2)} +\frac{1-q_{2} }{1-q}c_{q_{1} } U_{q_{2} ,2}^{(2)} -\frac{(1-q_{1} )(1-q_{2} )}{(1-q)}{\beta }'U_{q_{1} ,1}^{(2)} U_{q_{2} ,2}^{(2)}. \end{aligned}$$

(B.7)

The (86) is proved then.