Classification of statistical complex systems

• Many examples of complex system are statistical (stochastic) systems
• Statistical complex systems near the thermodynamic limit $N \rightarrow \infty$            can be characterized by asymptotics of its sample space $W(N)$                       - space of all possible configurations
• Asymptotic behavior can be described by 
  • Coefficients of scaling expansion correspond to scaling exponents
  • Scaling exponents completely determine universality classes
• ​We can find corresponding extensive entropy 
  • generalization of (c,d)-entropy $^\star$

$^\star$ R.H., S.T. EPL 93 (2011) 20006

Scaling expansion

Rescaling the sample space

• How the sample space changes when we rescale its size $N \mapsto \lambda N$?
  • The ratio behaves like $\frac{W(\lambda N)}{W(N)} \sim \lambda^{c_0}$ for $N \rightarrow \infty$ 
  • the exponent $c_0$ can be extracted by $\frac{d}{d\lambda}|_{\lambda=1}$: $c_0 = \lim_{N \rightarrow \infty} \frac{N W'(N)}{W(N)}$
  • For the leading term we have $W(N) \sim N^{c_0}$.
• Is it only possible scaling? We have $\frac{W(\lambda N)}{W(N)} \frac{N^{c_0}}{(\lambda N)^{c_0}} \sim 1$
  • Let us use the other rescaling $N \mapsto N^\lambda$
  • The we get that $\frac{W(N^\lambda)}{W(N)} \frac{N^{c_0}}{N^{\lambda c_0}} \sim \lambda^{c_1}$
  • First correction is $W(N) \sim N^{c_0} (\log N)^{c_1}$
  • It is the same scaling like for $(c,d)$-entropy

• ​Can we go further?

• We define the set of rescalings $r_\lambda^{(n)}(x) := \exp^{(n)}(\lambda \log^{(n)}(x)$ )  
  • ​ $f^{(n)}(x) = \underbrace{f(f(\dots(f(x))\dots))}_{n \ times}$
  • ​ $r_\lambda^{(0)}(x) = \lambda x$,  $r_\lambda^{(1)}(x) = x^\lambda$,   $r_\lambda^{(2)}(x) =  e^{\log(x)^\lambda}$, ...
  • They form a group: $r_\lambda^{(n)} \left(r_{\lambda'}^{(n)}\right) = r_{\lambda \lambda'}^{(n)}$,    $\left(r_\lambda^{(n)}\right)^{-1} = r_{1/\lambda}^{(n)}$,   $r_1^{(n)}(x) = x$
• We repeat the procedure:  $\frac{W(N^\lambda)}{W(N)} \frac{N^{c_0} (\log N)^{c_1} }{N^{\lambda c_0} (\log N^\lambda)^{c_1}} \sim 1$,
  • We take $N \mapsto r_\lambda^{(2)}(N)$
  • $\frac{W(r_\lambda^{(2)}(N))}{W(N)} \frac{N^{c_0} (\log N)^{c_1} }{r_\lambda^{(2)}(N)^{c_0} (\log r_\lambda^{(2)}(N))^{c_1}} \sim \lambda^{c_2}$,
  • Second correction is $W(N) \sim N^{c_0} (\log N)^{c_1} (\log \log N)^{c_2}$

Rescaling the sample space II

Rescaling the sample space III

• ​General correction is given by $\frac{W(r_\lambda^{(k)}(N))}{W(N)} \prod_{j=0}^{k-1} \left(\frac{\log^{(j)} N}{\log^{(j)}(r_\lambda^{(k)}(N))}\right)^{c_j}  \sim \lambda^{\bf c_k}$
• Possible issue: what if $c_0 = +\infty$? Then $W(N)$ grows faster than any $N^\alpha$
  • ​We replace $W(N) \mapsto \log W(N)$
  • The leading order scaling is $\frac{\log W(\lambda N)}{\log W(N)} \sim \lambda^{c_0}$ for $N \rightarrow \infty$
  • So we have $W(N) \sim \exp(N^{c_0})$
• If this is not enough, we replace $W(N) \mapsto \log^{(l)} W(N)$                                        so that we get finite $c_0$
• General expansion of $W(N)$ is $$W(N) \sim \exp^{(l)} \left(N^{c_0}(\log N)^{c_1} (\log \log N)^{c_2} \dots\right)$$

J.K., R.H., S.T. New J. Phys. 20 (2018) 093007

$$W^{(l)}(N) \equiv \log^{(l+1)}(W(x)) = \sum_{j=0}^n c_j^{(l)} \log^{(j+1)}(N) + \mathcal{O} (\phi_n (N))$$

Scaling Expansion

• Previous formula can be expressed in terms of                                            Poincaré asymptotic expansion

• Coefficients of the expansion are scaling exponents                                          and can be calculated from:

$$c^{(l)}_k = \lim_{N \rightarrow \infty}  \log^{(k)}(N) \left( \log^{(k-1)} \left(\dots\left( \log N \left(\frac{N W'(N)}{\prod_{i=0}^l \log^{(i)}(W(N))}-c^{(l)}_0\right)-c^{(l)}_1\right) \dots\right) - c^{(l)}_k\right)$$

Extensive entropy

• ​We can do the same procedure with entropy $S(W)$
• Leading order scaling: $\frac{S(\lambda W)}{S(W)} \sim \lambda^{d_0}$
• First correction $\frac{S(W^\lambda)}{S(W)} \frac{W^{d_0}}{W^{\lambda d_0}} \sim \lambda^{d_1}$
  • ​First two scalings correspond to $(c,d)$-entropy                                                                                 for $c= 1-d_0$ and $d = d_1$
• ​Scaling expansion of entropy $$S(W) \sim W^{d_0} (\log W)^{d_1} (\log \log W)^{d_2} \dots$$
• Requirement of extensivity  $S(W(N)) \sim N$ determines the relation between $c$ and $d$ :
  • $d_l = 1/c_0$,              $d_{l+k} = - c_k/c_0$     for $k = 1,2,\dots$

EXAMPLES

$\log W$

$(\log W)^2$

$(\log W)^{1/2}$

$\log \log W$

$d_0$

$d_1$

$d_2$

$\log W/\log \log W$

* H. Jensen et al. J. Phys. A: Math. Theor. 51 375002

$W(N) = 2^N$

      $W(N) \approx 2^{\sqrt{N}/2} \sim 2^{N^{1/2}}$                        

$W(N) \approx N^{N/2}  e^{2 \sqrt{N}} \sim e^{N \log N}$  

$W(N) = 2^{\binom{N}{2}} \sim 2^{N^2}$  

$W(N) = 2^{2^N}-1 \sim 2^{2^N}$   

Parameter space of $(c,d)$ - entropy

How does it change for one more scaling exponent?

R.H., S.T. EPL 93 (2011) 20006

Parameter space of $(d_0,d_1,d_2)$-entropy

To fulfill SK axiom 2 (maximality): $d_l > 0$, to fulfill SK axiom 3 (expandability): $d_0 < 1$

Perspectives

Fields of possible applications of scaling expansions:

• Non-equilibrium thermodynamics
• Information geometry$^\star$
• Critical phase transitions
• Information theory (Shannon-Khinchin axioms$^\dag$)
• Statistical inference (Shore-Johnson axioms$^\ddag$)
• Super-exponential processes 
• Processes with structures
• ...

$^\star$ J.K., R.H., S.T. Entropy 21(2) (2019) 112

$^\dag$ P. Tempesta, Proc. R. Soc. A 472 (2016) 2195

$^\ddag$ P.J., J.K. Phys. Rev. Lett. 122 (2019), 120601

References

• J.K., R.H., S.T. Classification of complex systems by their sample-space scaling exponents, New J. Phys. 20 (2018) 093007
•  H. Jensen, R. H. Pazuki, G. Pruessner, P. Tempesta, Statistical mechanics of exploding phase spaces: Ontic open systems, J. Phys. A: Math. Theor. 51 375002
• R.H., S.T. A comprehensive classification of complex statistical systems and an axiomatic derivation of their entropy and distribution functions,  EPL 93 (2011) 20006
• P. Tempesta, Formal groups and Z-entropies, Proc. R. Soc. A 472 (2016) 2195
• P.J., J.K. Maximum Entropy Principle in Statistical Inference: Case for Non-Shannonian Entropies,   Phys. Rev. Lett. 122 (2019), 120601
• J.K., R.H., S.T. Information Geometric Duality of ϕ-Deformed Exponential Families,         Entropy 21(2) (2019) 112

I am excited to discuss any possible application

of scaling expansions

during the welcome reception or later

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