# V. TIME REVERSIBILITY AWAY FROM EQUILIBRIUM – 0532 MODEL

At equilibrium the forward and backward trajectories for canonical oscillators, using any of the four ergodic sets of motion equations, are qualitatively much the same. No holes in the cross sections, good values for the even velocity moments, longtime averaged Lyapunov exponent the same for any initial condition. In short – deterministic, time-reversible, ergodic.

Away from equilibrium, thermodynamic dissipation can be investigated, still time-reversibly, by adding a localized temperature gradient \( ( d T / d q ) \, = \, [ \, \, \epsilon / \cosh ^ { 2 } ( q ) \, \, ] \) ] enabling heat transfer through a nonzero average current \( ( p ^ { 3 } / 2 ) : \)

\[ 1 - \epsilon < T < 1 + \epsilon = T ( q ) = 1 + \epsilon \operatorname { t a n h } ( q ) \longrightarrow \langle \; ( p ^ { 3 } / 2 ) \; \rangle < 0 \longrightarrow ( { \dot { S } } / k ) < 0 \ . \]

Here \( \epsilon \) is the maximum value of the temperature gradient, \( T ^ { \prime } ( 0 ) \) . The negative entropy change, causing the phase volume to shrink onto a strange attractor is due to the net heat loss from the oscillator to the coordinate-dependent 0532 thermostat temperature T(q) . From the standpoint of steady-state irreversible thermodynamics the overall heat loss is offset by an internal “entropy production” so that the net change of oscillator “entropy” vanishes. We remind the reader that Gibbs’ entropy is minus infinity for fractal attractors so that the irreversible-thermodynamics concept of nonequilibrium entropy is problematic. The artificial entropy change could be also be viewed as the result of ongoing coarse-graining ( which would artificilly increase Gibbs’ entropy ) at the level of the computational roundoff error ( in the sixteenth or seventeenth digit ).

The temperature gradient destroys the “global [ overall ] reversibility” of the motion equations. Although in principle reversible the chaotic instability of the dynamics, evidenced by a positive Lyapunov exponent, makes this “irreversibility” possible. This irreversibility is evidenced by a Lyapunov spectrum with a negative sum so that the longtime averaged distribution is a fractal strange attractor with a reduced information dimension rather than a smooth three-dimensional Gibbsian distribution.

Among the thermostats we have considered only the Nose´-Hoover equations show that a fractal attractor is not inevitable. In the Nose´-Hoover case a majority of initial conditions give rise to phase-space tori, orbits with no longtime tendency toward dissipation. All of the ergodic thermostats invariably produce small-gradient dissipation rather than tori so that their orbits exhibit what we call “global irreversibility”.

The equilibrium ( \( \epsilon \) = 0 and unit temperature \( \overrightharpoon { T } \)= 1 ) Lyapunov spectrum for the 0532 model, \( \{ \ \lambda \ \} = \{ \ + 0 . 1 4 4 , 0 , - 0 . 1 4 4 \ \} \) sums to zero corresponding to the three-dimensional

Gaussian distribution, \( f \propto e ^ { - q ^ { 2 } / 2 } e ^ { - p ^ { 2 } / 2 } e ^ { - \zeta ^ { 2 } / 2 } \). The time-averaged growth rates of infinites-imal one-, two-, and three-dimensional phase space volumes are given by

\[ \left\{ \begin{array} { l } { { \lambda _ { 1 } , } } \end{array} \lambda _ { 1 } + \lambda _ { 2 } , \ \lambda _ { 1 } + \lambda _ { 2 } + \lambda _ { 3 } } \end{array} \right\} \, . \]

In the nonequilibrium case with \( \epsilon \) = 0.50 the time-averaged spectrum becomes asymmet-ric, \( \{ ~ + 0 . 1 1 3 5 , 0 , - 0 . 1 4 5 4 ~ \} \) , corresponding to the time-averaged growth of a length or an area in phase space \( \simeq e ^ { + 0 . 1 1 3 5 t } \) but to shrinkage of an infinitesimal three-dimensional phase volume \( \otimes : \):

\[ ( \dot { \otimes } / \otimes ) = 0 . 1 1 3 5 - 0 . 1 4 5 4 = - 0 . 0 3 1 9 \longrightarrow D _ { K Y } = 2 + ( 0 . 1 1 3 5 / 0 . 1 4 5 4 ) = 2 . 7 8 \; . \]

Kaplan and Yorke’s linear interpolation predicts a strange-attractor dimension of 2.78. Cross sections of the equilibrium and nonequilibrium 0532 dynamics are shown in Figure 2 . Just as at equilibrium the nonequilibrium strange-attractor’s motion equations are time-reversible. Any forward-in-time sequence \( \{ \ + q , + p , + \zeta \ \} \) corresponds to a twin sequence \( \{ \ + q , - p , - \zeta \ \} \) with the order of the \( ( q , p , \zeta ) \) points reversed. Locally this reversed se-quence satisfies the same equations of motion with errors of order (\( d t ^ { 5 } / 1 2 0 ) \) for fourth-order Runge-Kutta integration. But any attempt to generate such a reversed sequence numer-ically fails because the Lyapunov spectrum of the reversed sequence would correspond to \( \{ ~ + 0 . 1 4 5 4 , 0 , - 0 . 1 1 3 5 ~ \} \) . The positive exponent sum indicates an unstable repellor with a diverging phase volume,\( ( \dot { \otimes } / \otimes ) = + \) 0.0319 . Any attempt to follow the repellor numerically will instead seek out the nearby attractor ( both are still ergodic, at least if \( \epsilon \) is small ) which, though unstable for a line or an area, is less so than the repellor. The repellor properties can ( only ) be observed by the expedient of storing and reversing a trajectory. The cross section associated with a stored ten-billion-point attractor trajectory is illustrated in Figure 3 . Note the lack of \( \pm p \) symmetry in the coloring of the local Lyapunov exponent, \( \lambda _ { 1 } ( t ) \) .

This instructive problem illustrates two general principles :\( : \ [ \ 1 \ ] \) the phase volume of the steady-state attractor is zero and singular everywhere despite the time-reversibility of the motion equations ; [ 2 ] any typical three-dimensional phase volume first expands and leaves the vicinity of the ( ergodic ) fractal repellor and then shrinks in order to join its mirror-image ( and likewise ergodic ) fractal attractor exponentially fast. Both these features correspond to the paucity of nonequilibrium states and to the irreversibility described by the Second Law of Thermodynamics.