Scaling expansions:

universal tool for classification of complex systems

Jan Korbel, Rudolf Hanel and Stefan Thurner

• 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?

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 $$k=2,3,\dots$$

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 $$d_k^{(l)}$$: $$d_0^{(l)} = 1/c_0^{(l)}$$ $$d_k^{(l)} = - c_k^{(l)}/c_0^{(l)}$$

Process S(W)
Random walk 0 1 0
Aging random walk 0 2 0
Riemann random walk 0 1 1
Magnetic coins 0 1 -1
Random networks 0 1/2 0
Random walk cascade 0 0 1

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

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

By Jan Korbel

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