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# Information length

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.

^{[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].

^{[3]}^{[4]}^{[5]}^{[6]}^{[7]}

## References

- 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. - Eun-Jin Kim; Investigating Information Geometry in Classical and Quantum Systems through Information Length.
*Entropy***2018**,*20*, 574, 10.3390/e20080574. - 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. - Eun-Jin Kim; Rainer Hollerbach; Geometric structure and information change in phase transitions.
*Physical Review E***2017**,*95*, 1, 10.1103/PhysRevE.95.062107. - 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. - Schuyler Nicholson; Eun-Jin Kim; Structures in Sound: Analysis of Classical Music Using the Information Length.
*Entropy***2016**,*18*, 258, 10.3390/e18070258. - 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.

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