Scaling expansions:

universal tool for classification of complex systems

• Complex systems cover an enormous variety of systems (physics, chemistry, biology, sociology, economy...)         
  • Different systems have similar statistics 
  • No united classification 
• However, statistical complex systems near the thermodynamic limit (\(N \rightarrow \infty \))   can be characterized by asymptotics of its sample space \(W(N)\)
• Asymptotic behavior can be described by scaling expansion with coefficients corresponding to scaling exponents
• Scaling exponents completely determine universality classes

Asymptotic behavior of \(W(N)\) for \(N \rightarrow \infty\) can be described by the    Poincaré asymptotic expansion

$$ f(W(N)) = \sum_{j=0}^n c_j \phi_{j}(N) + \mathcal{O}(\phi_n(N))$$

where \(f(W(N)) = \mathcal{O}(\phi_0(N)) \) and \(\phi_{j+1}(N) = \mathcal{O}(\phi_{j}(N)) \)

Question: What set of functions \( \phi_j(N) \)  describes scaling of \(W(N)\)?

                   Do the coefficients \( c_j \) have any connection to scaling?

Answer: Yes! 

                 We can define an expansion, where the coefficients

                 \(c_j\) are the scaling exponents of \(W(N)\)

Notation: \(f^{(n)}(x) = \underbrace{f(f(\dots(f(x))\dots))}_{n \ \mathrm{times}} \) 

Let us choose \(\phi_j(N) = \log^{(j+1)}(N)\), Then for some \(l\) we can write $$W^{(l)}(N) \equiv \log^{(l+1)}(W(x)) = \sum_{j=0}^n c_j^{(l)} \log^{(j+1)}(N) + \mathcal{O} (\phi_n (N))$$

The coefficients \(c_j^{(l)}\) are the scaling exponents of \(W\)!

We define the set of rescalings \(r_\lambda^{(n)}(x) := \exp^{(n)}(\lambda \log^{(n)}(x) \) )

So  \(r_\lambda^{(0)}(x) = \lambda x\),  \(r_\lambda^{(1)}(x) = x^\lambda\),   \(r_\lambda^{(2)}(x) =  e^{\log(x)^\lambda} \), ...

Properties: \(r_\lambda^{(n)} (r_{\lambda'}^{(n)}(x)) = r_{\lambda \lambda'}^{(n)}(x) \), \(r_1^{(n)}(x) = x\)

Leading order scaling: \( \lim_{N \rightarrow \infty}\frac{W^{(l)}(r_\lambda^{(0)}(N))}{W^{(l)}(N)} = \lambda^{\bf c_0^{(l)}} \)

First correction scaling: \( \lim_{N \rightarrow \infty} \frac{W^{(l)} (r_\lambda^{(1)}(N))}{W^{(l)}(N)} \left(\frac{r_\lambda^{(1)}(N)}{N}\right)^{-c_0^{(l)}} = \lambda^{\bf c_1^{(l)}}\)

General rescaling: \( \lim_{N \rightarrow \infty} \frac{W^{(l)}(r_\lambda^{(k)}(N))}{W^{(l)}(N)} \prod_{j=0}^{k-1} \left(\frac{\log^{(j)}(r_\lambda^{(k)}(N))}{\log^{(j)} N}\right)^{-c_j^{(l)}} = \lambda^{\bf c_k^{(l)}}\) 

Scaling expansion: \( W(N) \sim \exp^{(l)}\left(\prod_{j=0}^n (\log^{(j)}(N))^{c_j^{(l)}} \right)\)

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

Examples

Random walk

• Two possibilities:

step to the left/step to the right

• For \(N\) steps we have

     \(W(N) = 2^N\)

     possible configurations (paths)

• Scaling exponents:
  • ​\(l=1\)
  • \(c_0^{(1)}=1\)
  • \(c_k^{(1)}=0\) for \( k=1,2,\dots\)

Exponential growth:

Aging random walk

R.H., S.T. EPL 96 50003, Entropy 15 5324.

Sub-exponential growth (stretched exponential):

•  RW with correlations
• After 1 step,

      2 steps in the same direction,

      3 steps in the same direction....

•  Asymptotically we get

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

• Scaling exponents:
  • ​\(l=1\)
  • \(c_0^{(1)}=1/2\)
  • \(c_k^{(1)}=0\) for \( k=1,2,\dots\)

Riemann random walk

Sub-exponential growth:

•  another version of correlated RW
•  The walker can make decision only for steps which are prime numbers

•  Density of prime numbers:                

      \(\pi(N) \sim N/\log N\)

•  \(W(N) = 2^{\pi(N)} \sim 2^{N/\log N}\)
• Scaling exponents:
  • ​\(l=1\)
  • \(c_0^{(1)}=1\)
  • \(c_1^{(1)}=-1\)
  • \(c_k^{(1)}=0\) for \( k=2,3,\dots\)

Magnetic coins

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

Super-exponential growth:

• \(N\) coins - head or tail
• Coins are magnetic - any two can stick together (1 state)
•  \( W(N) \approx N^{N/2}  e^{2 \sqrt{N}} \sim e^{N \log N}\)  
• Scaling exponents:
  • ​\(l=1\)
  • \(c_0^{(1)}=1\)
  • \(c_1^{(1)}=1\)
  • \(c_k^{(1)}=0\) for \( k=2,3,\dots\)

Random networks

Super-exponential growth (compressed exponential):

• Undirected network with \(N\) nodes has \(\binom{N}{2}\) possible links
• Number of possible networks \( W(N) = 2^{\binom{N}{2}} \sim 2^{N^2}\)  
• Scaling exponents:
  • ​\(l=1\)
  • \(c_0^{(1)}=2\)
  • \(c_k^{(1)}=0\) for \( k=2,3,\dots\)

Random walk cascades

Super-exponential growth (double-exponential):

• Generalization of RW
• The walker can go to the left, to the right or split
• After each split, there are two independent walkers
• \(W(N) = 2^{2^N}-1 \sim 2^{2^N}\)                        
• Scaling exponents:
  • ​\(l=2\)
  • \(c_0^{(2)}=1\)
  • \(c_k^{(2)}=0\) for \( k=1,2,\dots\)

Applications

Scaling expansions of extensive entropy

• Scaling expansion can be also found for entropic functionals
• This allows us to consistently define thermodynamics of complex systems and avoid paradoxes from BG thermodynamics
• For microcanonical ensemble ( \(p_i = 1/W\) ) is the scaling expansion given by $$ S(W) \sim \prod_{j=0}^n (\log^{(l+j)} W)^{d_j^{(l)}}$$
• Extensive entropy: \( S(W(N))  \sim N\) for \(N \rightarrow \infty\)
• This gives us relation between \(c_k^{(l)} \) and \(d_k^{(l)} \): $$d_0^{(l)} = 1/c_0^{(l)}$$ $$d_k^{(l)} = - c_k^{(l)}/c_0^{(l)}$$

​

$$ S(W) \sim \prod_{j=0}^n (\log^{(j)} W)^{d_j}$$

Extensive entropy for example processes

\( \log W\)

\( (\log W)^2\)

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

\( \log \log W\)

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

\(d_0\)

\(d_1\)

\(d_2\)

\( \log W \cdot \log \log W\)

Scaling expansions of critical phenomena

• Scaling expansions can determine critical exponents of systems near phase transitions
• Let us have a critical point \(x_c\) where a relevant quantity \(F(x)\) diverges 
• Let us write a scaling expansion of \( F \)

       in terms of \( \frac{1}{x-x_c} \)

$$ \log F(x) = \sum_{j=0}^n [\log^{(j+1)}(1/(x-x_c))]^{\alpha_j} + \mathcal{O}(\phi_n) $$

• Therefore, we get $$ F(x) \propto (x-x_c)^{-\alpha_0} \log(1/(x-x_c))^{\alpha_1} \log \log(1/(x-x_c))^{\alpha_2}\dots $$
• Typically, only leading-order term is considered, or some other expansions, which do not reflect higher-order rescalings

Information geometry of scaling expansions

• Information geometry applies techniques of diffrerential geometry in statistics and probability theory
• The central quantity of the IG is the information metric \(g_{ij}(p)\)
• The metric is connected to Entropy \(S(P)\) trough Bregman divergence \(D(p||q) = S(p)-S(q) - \langle \nabla S(q),p-q\rangle \) as

• Example:  information metric for the first two scaling exponents

\( g_{ij}(p) = \frac{\partial^2 D(p||q)}{\partial q_i \partial q_j}|_{p=q} \)

J.K., R.H., S.T. arXiv:1812.09893

Other possible applications:

• Non-equilibrium thermodynamics
• Dynamics of systems with structures
• ...

We would like to discuss any possible applications, where

• super-exponential processes
• processes with structures
• higher-order scaling exponents

might appear and play a role