Detailed balance, nonequilibrium states, and dissipation in symbolic sequences

Symbolic sequences arising from the coarse graining of deterministic dynamical systems continuous in phase space are considered. The extent to which signatures of the time irreversibility and of the nonequilibrium constraints at the level of the original system, such as fluxes or dissipation, can be identified at the coarse-grained level is analyzed. The roles of the partition, of the time window, and of time averaging in distinguishing in a clear-cut way the equilibrium versus nonequilibrium character of the sequence are brought out.

Figure 1

Phase-space portrait of the harmonic oscillator [Eq. (15)] and its partitioning into four equal sectors of the invariant set (here the circumference of the circle $(p^2+q^2)/2=\text{constant}$).

Figure 2

Phase-space density plot in the $p-z$ plane generated by model system (26) starting with initial conditions $p=5, q=z=x=0$.

Figure 3

Entropy production as a function of the averaging window for model (26) ($\epsilon =0$, solid line) and for model (28) with $\epsilon =0.5$ (dashed line) and $\epsilon =2$ (dotted line). The system is monitored every one time unit and the number of realizations is ${10}^5$.

Figure 4

As in Fig. 2, but for the augmented model (28) with $\epsilon =0.5$.

Figure 5

(a) Probability of variable $z$ of model (28) with $\epsilon =0$ (dashed line) and $\epsilon =2$ (solid line); (b) mean value of variable $z$ (solid line) and difference of invariant probabilities between cells 1 and 4 (dotted line) as a function of the constraint parameter $\epsilon$; (c) contraction rate versus $\epsilon$. Number of realizations is ${10}^6$ and initial conditions as in Fig. 2.

Figure 6

Phase-space portrait of model (30) with $\epsilon =0.05$ and $r=0.2$. Other parameter values $s=10$ and $b=8/3$.

Figure 7

Entropy production as a function of the averaging window for model (30) with $r=0.2, \epsilon =0.05$ (solid line) and $\epsilon =0.07$ (dotted line). The system is monitored every one time unit and the number of realizations is ${10}^4$. Other parameter values as in Fig. 6.