# Information length

Created by: Eun-jin Kim

The information length measures the total number of statistically different states that a system passes through in time in non-equilibrium/relaxation processes. It is a Lagrangian measure depending on the evolution of a system and is a useful index for understanding the information geometry underlying non-equilibrium processes.

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[1]The information length [1-7] quantifies the number of statistically different states that a system passes through in time . For a time-dependent Probability Density Function (PDF)  for a stochastic variable , the information length ${\cal L}(t)$ is the total information change between time $0$ and $t$, defined by

[2]\begin{eqnarray} {\cal{L}} (t) &=& \int_0^{t} \frac{dt_1}{\tau(t_1)} = \int_0^{t} dt_1 \sqrt{\int dx \frac {1} {p(x, t_1)} \left [\frac {\partial p(x,t_1)} {\partial t_1} \right]^2}. \label{eq2} \end{eqnarray}

Here  $\tau$ is the characteristic time scale over which the information changes, given by

\begin{eqnarray} \frac{1}{\tau^2} & = &\int dx \frac {1} {p(x,t)} \left [\frac {\partial p(x,t)} {\partial t} \right]^2. \label{eq1} \end{eqnarray}

Note that  $\tau$ is a dimensionless number measuring the clock time measured in unit of .

When the control parameters $z^{I}$ (i=1,2,3,…) determining the PDF are known, Eq. (2) can be written as the Fisher information metric as

\begin{eqnarray} &&\frac{1}{\tau^2} = g_{ij} \frac{d z^{i}}{dt}\frac{ dz^{j}}{dt}. \label{eq01} \end{eqnarray}

where

\begin{eqnarray} g_{ij}& =& \int dx \frac {1} {p(x,t)} \frac{\partial p}{\partial z^{i}} \frac{\partial p}{ \partial z^{j}} ,\label{eq3} \end{eqnarray}

In the relaxation problem where any initial PDF relaxes into the equilibrium PDF,  ${\cal L} (t \to \infty)$ was demonstrated to map out an attractor structure [1-5,7]. In particular,  the effect of different deterministic forces was demonstrated by the scaling of   again the peak position of a narrow initial PDF, with the minimum value of  occurring at the equilibrium point for a stable equilibrium [2-4]. In comparison, in the case of a chaotic attractor,  ${\cal L} (t \to \infty)$ exhibits a sensitive dependence on initial conditions like a Lyapunov exponent [5]. The information length was also applied to music data to quantify the information change associated with different classical music [6].

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## References

1. James Heseltine; Eun-Jin Kim; Novel mapping in non-equilibrium stochastic processes. Journal of Physics A: Mathematical and Theoretical 2016, 49, 175002, 10.1088/1751-8113/49/17/175002.
2. Eun-Jin Kim; Investigating Information Geometry in Classical and Quantum Systems through Information Length. Entropy 2018, 20, 574, 10.3390/e20080574.
3. Eun-Jin Kim; Rainer Hollerbach; Time-dependent probability density function in cubic stochastic processes. Physical Review E 2016, 94, 1, 10.1103/physreve.94.052118.
4. Eun-Jin Kim; Rainer Hollerbach; Geometric structure and information change in phase transitions. Physical Review E 2017, 95, 1, 10.1103/PhysRevE.95.062107.
5. S.B. Nicholson; Eun-Jin Kim; Investigation of the statistical distance to reach stationary distributions. Physics Letters A 2015, 379, 83-88, 10.1016/j.physleta.2014.11.003.
6. Schuyler Nicholson; Eun-Jin Kim; Structures in Sound: Analysis of Classical Music Using the Information Length. Entropy 2016, 18, 258, 10.3390/e18070258.
7. Eun-Jin Kim; Lucille-Marie Tenkès; Rainer Hollerbach; Ovidiu Radulescu; Far-From-Equilibrium Time Evolution between Two Gamma Distributions. Entropy 2017, 19, 511, 10.3390/e19100511.