Non-Markovian nonequilibrium information dynamics

We probe into the dynamics of interacting non-Markovian information systems. The stochastic dynamics of information has two aspects: the self-evolution and interaction. We show that self-evolution of a non-Markovian information system can be described by a Markov-type master equation with memory dependence. We also reveal that the interaction between systems can be fully embodied into the information dynamics of the composite information system. To characterize time irreversibility of the self-evolution and the interaction, we apply the landscape-flux theory to both stochastic and thermal information dynamics. The driving force of the nonequilibrium information dynamics can be decomposed into time-reversible (detailed balance preserving landscape part) and -irreversible (detailed balance breaking nonequilibrium flux part) parts. The time-irreversible part of the driving force fully depicts the time-irreversibility behavior in the stochastic dynamics. The time irreversibility of the interactions between systems reflected in nonequilibrium thermodynamics can be seen in the decomposition of the mutual information rate which corresponds to decomposition of the driving force. In particular, the time-irreversible part of mutual information rate reveals the underlying relationship among the entropy production rates of the information systems. We propose the finite memory approximation method and demonstrate that the above mentioned features can be found in a wide class of non-Markovian nonequilibrium information systems. Finally, we derive the lower and upper bounds for informational entities under concern with clear meanings.

Figure 1

Locations of two time-successive memories in a sequence of non-Markovian system $X$. (a) Locations of $({\chi }_{t-1},{\chi }_t)$ along the forward time arrow. (b) Locations of the time reversals $({\tilde{\chi }}_t,{\tilde{\chi }}_{t-1})$ along the backward time arrow.

Figure 2

Diagram of the two interacting information systems $X$ and $S$ in the example.

Figure 3

Geometrical interpretation for the probability flux of a non-Markovian system $X$ with memory. $X$ has memory length of 2 with state space $\mathcal{X}=\{0,1,2\}$. The joint probabilities $\mathrm{Pr}({\chi }_{t-1},{\chi }_t)=\mathrm{Pr}(x_{t-2},x_{t-1},x_t)$ are arranged into a tensor space of order 3, i.e., into the lattices of a cube. By fixing $x_{t-1}=0,1,2$, we obtain a series of matrices $\left[\mathrm{Pr}(x_{t-2},0,x_t)\right]$ (red), $\mathrm{Pr}(x_{t-2},1,x_t)]$ (green), and $\left[\mathrm{Pr}(x_{t-2},2,x_t)\right]$ (blue), with different colors corresponding to the planes shown in the cube. The probability flux which is also a third-order tensor is obtained by calculating the antisymmetrical part of each matrix, i.e., $\left[J_x\right]=\left[\mathrm{Pr}({\chi }_{t-1},{\chi }_t)\right]-{\left[\mathrm{Pr}({\chi }_{t-1},{\chi }_t)\right]}^{†}=\left[\mathrm{Pr}({\chi }_{t-1},{\chi }_t)\right]-\left[\mathrm{Pr}({\tilde{\chi }}_t,{\tilde{\chi }}_{t-1})\right]$.

Figure 4

The values of (a) $R\left({\mathfrak{X}}^{\left(m\right)}\right)$, (b) $R\left({\mathfrak{S}}^{\left(m\right)}\right)$, (c) $I({\mathfrak{X}}^{\left(m\right)},{\mathfrak{S}}^{\left(m\right)})$, and (d) $I_B({\mathfrak{X}}^{\left(m\right)},{\mathfrak{S}}^{\left(m\right)})$ from $m=1$ to 6.

Figure 5

Comparisons of (a) $R\left(X\right)$, (b) $R\left(S\right)$, (c) $I(X,S)$, and (d) $I_B(X,S)$ by using conventional method and exact values. Curved lines, conventional method. Horizontal lines, exact values with $m=2$.