Classification of complex systems by their sample-space scaling exponents

Jan Korbel, Rudolf Hanel and Stefan Thurner

Statistical Physics of Complex Systems, 7-11 May 2019, Nordita, Stockholm

this presentation is available online at

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 \(k = 1,2,\dots\)


Process S(W)
Random walk




Aging random walk




Magnetic coins *




Random network




Random walk cascade





\( \log W\)

\( (\log W)^2\)

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

\( \log \log W\)




\( \log W/\log \log W\)

* 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): \(d_0 < 1\)


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


  • 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

ENJOY THE welcome reception!