# Nonequilibrium Time Reversibility with Maps and Walks

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Hopf’s Equilibrium Baker Map E(x,y)

#### 2.2. From Equilibrium to Nonequilibrium with Time-Reversible Maps

`if(q < p)`

`qnew = (5/4)q - (3/4)p + 3d`

`pnew = -(3/4)q + (5/4)p - d`

`else if(q > p)`

`qnew = (5/4)q - (3/4)p - 3d`

`pnew = -(3/4)q + (5/4)p + d`

`end`

`d = sqrt(1/8)`. This “motion” is analogous to ordinary Hamiltonian mechanics, where the phase volume $dqdp$ is conserved by Hamilton’s motion equations. In nonequilibrium molecular dynamics the extraction of heat, corresponding to entropy loss, leads to a continuous loss of phase volume. A mapping analogy can be illustrated by constructing a compressible mapping, as shown in Figure 2 and Figure 3.

#### 2.3. The Time-Reversible Dissipative Baker Map

`qnew = +(11q/6) - (7p/6) + 14d`

`pnew = -(7q/6) + (11p/6) - 10d`

`d`is $\sqrt{(1/72)}$. Notice that the expanding map has a $(q,p)$ Jacobian determinant of $(121-49)/36=2$, signalling a doubling of area with each iteration. In the larger white region the map, likewise linear, contracts:

`qnew = +(11q/12) - (7p/12) - 7d`

`pnew = -(7q/12) + (11p/12) - d`

#### 2.4. Random Walk Analog of the Baker Map

`x`and

`y`. A new value

`xnew`can be chosen at random, while

`ynew`depends upon both a random number

`r`where $0<r<1$ and the current value of

`y`:

`if(r.lt.1/3)`

`ynew = (1+2y)/3`

`else if(r.gt.1/3)`

`ynew = y/3`

`end`

## 3. Results and Discussion

#### 3.1. Irreversibility through Shrinking Phase Volume

#### 3.1.1. Lyapunov Instability and Exponential Growth

#### 3.1.2. Compression with Expansion Leads to Irreversibility

#### 3.2. Poincaré Recurrence of the Baker Maps

#### 3.3. Characterizing Chaos in the Baker Map

#### 3.4. Results from the Random Walk Baker Map

#### 3.5. Nonuniform Convergence of the Information Dimensions

`random`${}_{-}$

`number(r)`. The dimensionality data are analyzed here using ${3}^{n}$ bins, with n varying from 0 to 10. The finest grid has ${3}^{10}=59,049$ bins of equal width $\delta =1/59049$. By combining the contents of 3, or 9, or 27, or…contiguous bins the entire set of 30 stepwise information dimensions for the ten binning choices can be obtained from a single run. The apparent information dimensions for the 300 problems (thirty iterations with ten bin sizes) are plotted as the ten lines shown in Figure 9.

#### 3.6. Massively-Parallel Implementations of the Confined Walk

## 4. Conclusions

`Prob`($\delta $)} which can then be analyzed for a bin-width dependent information dimension:

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Hopf’s deterministic Baker Map E$(x,y)$ maps the left/right sides of the unit square into the top/bottom halves with each iteration. The continued doubling in the x direction eventually reaches a fixed point. The Figure shows a series of 112 points generated with quadruple-precision arithmetic. Evidently Hopf could not imagine a computational implementation of his map! The diamond-shaped lower version of the map, $2\times 2$ rather than a unit square, produces a long periodic orbit and is, unlike Hopf’s, time-reversible.

**Figure 2.**The deterministic Baker Map B doubles an area $dqdp$ in the red region and halves an area in the white. The time-reversal Map T changes the sign of the vertical “momentum-like” variable p. The diamond-shaped domain of the map is $|\phantom{\rule{4pt}{0ex}}q\pm p\phantom{\rule{4pt}{0ex}}|<\sqrt{2}$. A counterclockwise circuit of the four states follows if B is replaced by B${}^{-1}$ as T and T${}^{-1}$ are identical.

**Figure 3.**One million iterations of the Baker Map and the Confined Walk are compared. A scaling and translation of the Baker Map solution at the left to the unit square replicates the solution of a stochastic confined random walk problem where x and y are stochastic variables. The walk confined to the unit interval $0<y<1$ is generated with a random number relating the “next” value of

`y`to the “last”. The latter is either

`y/3`or

`(1+2y)/3`, corresponding to steps to the bottom third or upper two thirds of the green million-iteration solution at the right of Figure 3. The $(q,p)$ Baker Map at the left and the $(x,y)$ Confined Walk at the right provide indistinguishable fractals when rotated 45 degrees and scaled by a factor of two, as shown in the figure. The Confined Walk shown there occupies a unit square, $0<(x,y)<1$. We show a $2\times 2$ version, with $|q\pm p|<\sqrt{2}\leftrightarrow |x,y|<1$, here to clarify the details of the fractal structure.

**Figure 4.**Comparison of the differences $dr=\sqrt{d{q}^{2}+d{p}^{2}}$ between single and double precision simulations (above) of the Baker Map and double and quadruple precision simulations (below) with all three trajectories started at the origin. The straight blue line has a slope corresponding to the largest Lyapunov exponent, ${\lambda}_{1}=0.63651$.

**Figure 5.**Excess of compressive over expansive iterations of the single-(red), double-(green), and quadruple-(black) precision Baker maps. The differences between them are visible between 20 and 1000 iterations of the maps. The final values of the excess are 1,000,742, 1,000,250, and 998,236 for the three sets of three million iterations beginning at $(0,0)$.

**Figure 6.**Additional iterations, (

**a**) 1 to 50 and (

**b**) 50 to 75 in double (black) and quadruple (red) precision for the two-dimensional $(q,p)$ Baker Map B with the initial point at the origin $(0,0)$. Lyapunov instability makes the difference between the two solutions visible after about 70 iterations, as can be seen in (

**b**).

**Figure 7.**Comparison of the y coordinate distribution of the Baker Map in the unit square converted from $(q,p)$ with the Confined Walk distribution obtained using the FORTRAN random-number generator

`random`${}_{-}$

`number(r)`. The Map and Walk data, one million points for each, have been collected here and displayed in $2187={3}^{7}$ bins of width ${3}^{-7}$.

**Figure 8.**Fractal probability densities for confined walks with bin sizes $\delta ={3}^{-10}$ and ${4}^{-8}$. For both bin sizes ${\int}_{0}^{1}\rho \left(y\right)dy=1.$ For ${10}^{8}$ iterations of the map were used for the point set that was analyzed with both these choices of binning.

**Figure 9.**The information dimensions ${D}_{I}$ of the developing random-walk fractal as ten functions of the number of iterations. The number of bins characterizing each curve varies from ${3}^{1}$ to ${3}^{10}$ for each of thirty iterations. Each point corresponds to an averaged ${D}_{I}$ from one million equally-spaced initial conditions on the unit interval $0<y<1$. In the limiting special cases (shown as red squares) that the number of iterations is equal to the logarithm, base-3, of the number of bins, the information dimension follows from Farmer’s analysis [5], $\left[\right(2/3)ln(2/3)+(1/3)ln(1/6\left)\right]/ln(1/3)=0.78967$. When the number of iterations approaches infinity ahead of the number of bins, the dimensionality is substantially lower, ${D}_{I}\simeq 0.{741}_{5}$ rather than the Kaplan-Yorke conjectured value (based on the Baker-Map Lyapunov exponents) 0.7337.

**Figure 10.**The information dimension ${D}_{I}$ of the developing random-walk fractal as a function of the numbers of bins and iterations. The number of bins varies from ${4}^{1}$ to ${4}^{8}$ for thirty iterations. As in Figure 7 each point corresponds to ${D}_{I}$ for one of the ten samples of one million initial conditions.

**Figure 11.**Cumulative densities and information dimensions are compared for bin widths (from bottom to top at the right) of ${(1/3)}^{5},\phantom{\rule{4pt}{0ex}}{(1/3)}^{7},\phantom{\rule{4pt}{0ex}}{(1/2)}^{11},\phantom{\rule{4pt}{0ex}}{(1/2)}^{8}$. The information dimensions for much narrower bins suggest different limiting dimensionalities for the two bin families.

**Figure 12.**The logarithms of the errors in ${D}_{I}\left(\delta \right)$ (due to insufficient iterations) are shown for two different bin widths, $\delta ={3}^{-6}=1/729$ at the left and $\delta ={3}^{-12}=$ 1/531,441 at the right. For both cases the slope is roughly $-1$ when plotted as a function of Logarithms of the number of iterations. Thus the error for ${D}_{I}$ varies inversely with the number of total iterations included. The data plotted were obtained from 100 successive samplings, corresponding to a parallel computation with 100 independent processors. The leftmost and rightmost points correspond to 1000 and ${10}^{10}$ iterations: $ln\left(1000\right)=6.908$ and $ln\left({10}^{10}\right)=23.026$.

**Figure 13.**Timing for ${10}^{10}$ iterations of the Confined Walk using ${3}^{18}$ bins run on varying numbers of processors. The time is split into the Confined Walk calculation itself (Run), a process which requires no communication between processes, and a communication step (Comm) where the collected bins are accumulated on the root process before being used to calculate ${D}_{I}$.

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**MDPI and ACS Style**

Hoover, W.G.; Hoover, C.G.; Smith, E.R.
Nonequilibrium Time Reversibility with Maps and Walks. *Entropy* **2022**, *24*, 78.
https://doi.org/10.3390/e24010078

**AMA Style**

Hoover WG, Hoover CG, Smith ER.
Nonequilibrium Time Reversibility with Maps and Walks. *Entropy*. 2022; 24(1):78.
https://doi.org/10.3390/e24010078

**Chicago/Turabian Style**

Hoover, William Graham, Carol Griswold Hoover, and Edward Ronald Smith.
2022. "Nonequilibrium Time Reversibility with Maps and Walks" *Entropy* 24, no. 1: 78.
https://doi.org/10.3390/e24010078