Nonequilibrium Information Landscape and Flux , Mutual Information Rate Decomposition and Entropy Production

We explored the dynamics of the two interacting information systems. We show that for 1 the Markovian marginal systems the driving force for information dynamics is determined by both 2 the information landscape and information flux. While the information landscape can be used to 3 construct the driving force to describe the equilibrium time reversible information system dynamics, 4 the information flux can be used to describe the nonequilibrium time-irreversible behaviours of 5 the information system dynamics. The information flux explicitly breaks the detailed balance and 6 is a direct measure of the degree of the nonequilibriumness or time irreversibility. We further 7 demonstrate that the mutual information rate between the two subsystems can be decomposed 8 into the equilibrium time-reversible and nonequilibrium time-irreversible parts respectively. This 9 decomposition of the mutual information rate (MIR) corresponds to the information landscape-flux 10 decomposition explicitly when the two subsystems behave as Markov chains. Finally, we uncover 11 the intimate relationship between the nonequilibrium thermodynamics in terms of the entropy 12 production rates and the time-irreversible part of the mutual information rate. We found that this 13 relationship and MIR decomposition still hold for the more general stationary and ergodic cases. We 14 demonstrate the above features with two examples of the bivariate Markov chains. 15


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There are growing interests in studying two interacting information systems in the fields of 19 control theory, information theory, communication theory, nonequilibrium physics, and biophysics 20 [1-9]. Significant progresses have been made recently towards the understanding of the information 21 system in terms of information thermodynamics [10][11][12][13]. However, the identification of the global 22 driving forces for the information system dynamics is still challenging. Here we aim to fill this 23 gap by quantifying the driving forces for the information system dynamics. Inspired by the recent probabilities are given by q x (x|x ) and q s (s|s ) (for x, x ∈ X and s, s ∈ S) respectively. Then we have 123 the following master equations (or the information system dynamics) for X and S respectively, and 125 p s (s; t + 1) = ∑ s q s (s|s )p s (s ; t), where p x (x; t) and p s (s; t) are the probabilities of observing X = x and S = s at time t respectively. 126 We consider that both Eqs.(3,4) have unique stationary solutions π x and π s which satisfy π x (x) = 127 ∑ x q x (x|x )π x (x ) and π s (s) = ∑ s q s (s|s )π s (s ) respectively. Also, we assume that when Z is in SS, 128 π x and π s are also achieved. The relations between π x , π s and π z read, 129 π x (x) = ∑ s π z (x, s), π s (s) = ∑ x π z (x, s).
In the rest of this paper, we let X T = {X (1) This is also the the probability of the time-reverse sequence C T = {C(1) = c, C(2) = c }, (T = 2).

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The difference between these two transition numbers measures the time-reversibility of the forward = π c (c )q c (c|c ) − π c (c)q c (c |c), for C = X, S, or Z.
Then, J c (c → c) is said to be the probability flux from c to c in SS. If J c (c → c) = 0 for arbitrary c and 5 of 16 The transition probability determines the evolution dynamics of the information system. We 146 can decompose the transition probabilities q c (c|c ) into two parts: the time-reversible part D c and 147 time-irreversible part B c , which read From this decomposition, we can see that the information system dynamics is determined by two 149 driving forces. One of the driving force is determined by the steady state probability distribution. This 150 part of the driving force is time reversible. The other driving force for the information dynamics is the 151 steady state probability flux which breaks the detailed balance and quantify the time irreversibility.

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Since the steady state probability distribution measures the weight of the information state, therefore it 153 can be used to quantify the information landscape. If we define the potential landscape for the information is expressed in term of the difference of the potential landscape. This 156 is analogous to the landscape-flux decomposition of Langevin dynamics in [15]. Notice that the 157 information landscape is directly related to the steady state probability distribution of the information 158 system. In general, the information landscape is at nonequilibrium since the detailed balance is often 159 broken for general cases. Only when the detailed balance is preserved, the nonequilibrium information 160 landscape is reduced to the equilibrium informational landscape. Even though the information 161 landscape is not at equilibrium in general, the driving force D c (c → c) is time reversible due to the 162 decomposition construction. The steady state probability flux measures the information flow in the 163 dynamics and therefore can be termed as the information flux. In fact, the nonzero information flux 164 explicitly breaks the detailed balance because of the net flow to or from the system. It is therefore a 165 direct measure of the degree of the nonequilibriumness or time irreversibility in terms of the detailed 166 balance breaking.

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Note that the decomposition for the discrete Markovian information process can be viewed as the 168 separation of the current corresponding to the 2B c (c → c)π c (c ) here and the activity corresponding 169 to the 2D c (c → c)π c (c ) in a previous study [22]. The landscape and flux decomposition here for the 170 reduced information dynamics is in the similar spirit as the whole state space decomposition with 171 the information system and the associated environments. When the detailed balance is broken, the 172 information landscape (defined as the negative logarithm of the steady state probability φ = − log π) 173 is not the same as the equilibrium landscape under the detailed balance.
As we can see in next section, D c and B c are useful for us to quantify time-reversible and 184 time-irreversible observables of C respectively.
We give the interpretation that the non-vanishing information flux J c fully measures the 186 time-irreversibility of the chain C in time T for T ≥ 2. Let C T be arbitrary sequence of C in SS, 187 and with no loss of generality we let T = 3. Similar to Eq. (6), the measure of time-irreversibility of 188 C T can be given by the difference between the probability of C T = {C(1), C(2), C(3)} and that of its Then by the relations given in Eq.(9), we have P(C T ) − P( C T ) = 0 holds for arbitrary C T if and only if 192 This conclusion can be made for arbitrary T > 3. Thus, non-vanishing J c can fully describe the 193 time-irreversibility of C for C = X, S, or Z.

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We show the relations between the fluxes of the whole system J z and of the subsystem J x as 195 following: Similarly, we have These relations indicate that the subsystem fluxes J x and J s can be seen as the coarse-grained levels of 198 total system flux J z by averaging over the other part of the system S and X respectively. We should 199 emphasize that, Non-vanishing J z does not mean X or S is time-irreversible and vice versa. Moreover,

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for the completeness and uniqueness of , It measures the correlation between X and S in unit time, or say, the efficient bits of information that X indicates that X and S are independent of each other. More explicitly, the corresponding probabilities 210 of these sequences can be evaluated by using Eqs.(2,3,4), we have By substituting these probabilities into Eq.(12) (see Appendix A), we have the exact expression of MIR 212 as 213 This means that the mutual information representing the correlations between the two interacting Then I B (X, S) measures the change of averaged conditional correlation between X and S when a 229 sequence of Z turns back in time, A negative I B (X, S) shows that the correlation between X and S becomes strong in the time-reversal

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The Entropy Production Rates (EPR) or energy dissipation (cost) rate at steady state is a quantitative 244 nonequilibriumness measure which characterizes the time-irreversibility of the underlying processes.

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The EPR of a stationary and ergodic process C (here C = Z, X, or S) can be given by the difference 246 between the averaged surprisal (negative logarithmic probability) of the backward sequences C T and 247 that of forward sequences C T in long time limit, i.e., where R c is said to be the EPR of C [19]; − log P(C T ) and − log P( C T ) are said to be the surprisal 249 of a forward and a backward sequence of C respectively. We see that C is time-reversible (i.e., 250 P(C T ) = P( C T ) for arbitrary C T for large T) if and only if R c = 0. And this is due to the form 251 of R c which is exactly a Kullback-Leibler divergence. When C is Markovian, then R c reduces into the 252 following form when Z, X or S is assigned to C respectively [17,20], where total and subsystem entropy productions R z , R x , and R s correspond to Z, X, and S respectively.

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Here, R z usually contains the detailed interaction information of the system (or subsystems) and 255 environments; R x and R s provide the coarse-grained information of time-irreversible observables 256 of X and Z respectively. Each non-vanishing EPR indicates that the corresponding Markov chain 257 is time-irreversible. Again, we emphasize that a non-vanishing R z does not mean X or S is 258 time-irreversible and vice versa. 259 We are interested in the connection between these EPRs and mutual information. We can associate 260 them with I B (X, S) by noting Eqs. (10,11,14). We have We note that I B (X, S) is intimately related to the EPRs. This builds up a bridge between these 262 EPRs and irreversible part of the mutual information. Moreover, we also have This indicates that the time-irreversible MIR contributes to the detailed EPRs. In other words, The

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As a concrete example, we consider a two-state system coupled to two information baths a and b.

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The states of the system are denoted by X = {x : x = 0, 1} respectively. Each bath sends an instruction with X ∈ X and S ∈ S to denote the BMC of the system and the demon.

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Consequentially, the transition probabilities of the system read The transition probabilities of the demon read 289 q s (s|s ) = P(s).
And the transition probabilities of the joint chain read 290 q z (x, s|x , s ) = P(s) s (x).
We have the corresponding steady state distributions or the information landscapes as, π s (s) = P(s), π z (x, s) = P(s)π x (x). We obtain the information fluxes as, Here, we use the notations s (x ) and s (x) (s, s = a or b) to denote the probabilities of the instructions 293 x or x from bath a or b briefly. We obtain the EPRs as We evaluate the MIR as The time-irreversible part of I(X, S) reads,

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In this work, we identify the driving forces for the information system dynamics. We show 298 that for marginal Markovian information systems, the information dynamics is determined by both 299 the information landscape and information flux. While the information landscape can be used to 300 construct the driving force for describing the time reversible behavior of the information dynamics, the 301 information flux can be used to describe the time irreversible behavior of the information dynamics.

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The information flux explicitly breaks the detailed balance and provides a quantitative measure of 303 the degree of the nonequilibriumness or time irreversibility. We further demonstrate that the mutual 304 information rate which represents the correlations can be decomposed into time reversible part and time 305 irreversible part originated from the landscape and flux decomposition of the information dynamics.

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Finally we uncover the intimate relationship between the difference of the entropy productions of 307 the whole system to those of the subsystems and the time irreversible part of the mutual information.

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This will help for understanding the non-equilibrium behaviour of the interacting information system 309 dynamics in stochastic environments. Furthermore, we verify that our conclusions on the mutual 310 information rate and entropy production rate decomposition can be made more general for the 311 stationary and ergodic processes. The following abbreviations are used in this manuscript: Here, we derive the exact form of Mutual Information Rate (MIR, Eq.(13)) in steady state by using 321 the cumulant-generating function.

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We write arbitrary time sequence of Z in time T in the form as following where Z(i) (for i ≥ 1) denotes the state at time i. The corresponding probability of Z T is in the following form We let the chain U = (X, S) to denote a process that X and S follow the same Markov dynamics in Z but are independent of each other. Then we have the transition probabilities of U read Then the probability of a time sequence of U, U T , with the same trajectory of Z T reads with π u (x, s) = π x (x)π s (s) being the stationary probability of U. We can see that Thus, our idea is to evaluate K(m, T) at first. We have where we realize that the last equality can be rewritten in the form of matrices multiplication.
We introduce the following matrices and vectors for Eq. (A.6) such that where Q Q Q z is the transition matrix of Z; π π π z is the stationary distribution of Z. It can be also verified that where 1 1 1 † is the vector of all 1's with appropriate dimension.

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Then K(m, T) can be rewritten in a compact form such that Then, we substitute Eq. (A.9) into Eq. (A.5) and have By noting Eq. (A.8) and T ≥ 2, we obtain Eq. (13) from Eq. (A.10) that For general cases, indeed, we do not expect that both X and S are Markovian. Even the joint chain 328 Z may be non-Markovian. This means that Eq.
(2) may fail to depict the dynamics of Z. Then the 329 landscape-flux decomposition needs to be generalized to this situation. Such decomposition was not 330 developed yet for the non-Markovian cases. This will be discussed in a separate work. However, when 331 Z is stationary and ergodic process (also assume that both X and S are stationary and ergodic), we 332 show that the MIR can be decomposed into two parts as is shown in Eq. (14) and interesting relation 333 between the MIR and EPRs can still be found in the same form of the last expression in Eq. (17). 334 We are interested in the correlation between the forward sequences of X and S which can be measured by log P(Z T ) P(X T )P(S T ) (Z T = (X T , S T )), then the MIR can be used to quantify the average rate of this correlation in the long time limit as shown in Eq. (12). Furthermore, we are interested in the averaged difference between the rate of the correlation of the backward processes and that of the forward processes. This comes the time-irreversible part of the MIR defined by quantifies the correlation between the backward sequences of X and S. Clearly, the time-irreversible part of MIR depicting the correlation of the forward processes of X and S is enhanced (I B (X, S) > 0) or weakened (I B (X, S) < 0) compared to that of the backward processes. The other important part of the MIR, namely the time-reversible part, shows the averaged rate of the correlation that remains in both forward and backward processes, Consequentially, the MIR I(X, S) is decomposed into two parts shown as I(X, S) = I D (X, S) + I B (X, S).

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In Markovian cases, each part of the MIR reduces into the form in Eq. (14) respectively.

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The relation between the time-irreversible part of the MIR and EPRs can be shown as follows, which is in the same form of Eq. (17). And due to the non-negativity of the EPRs, the inequalities in 337 (18) still hold for general cases.

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The transition probabilities of the BMC read where s (x) denotes the probability of the instruction x from bath s = a, b. We assume that there is a 355 unique stationary distribution of z, π z such that 356 π z (z) = ∑ z q z (z|z )π z (z ).
The stationary distribution of S and X then read The behavior of the demon can be seen as a Markovian process in steady state. The corresponding 358 transition probabilities of the system read 359 q s (s|s ) = 1 It can be verified that π s is the unique stationary distribution of S. However, the dynamics of the 360 system always behaves as a non-Markovian process in general.

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To characterize the time-irreversibility of Z, X, and S, we use the definition of EPR in Eq. (15) and 362 have 363          R z = 1 2 ∑ z,z J z (z → z) log q z (z|z ) q z (z |z) , R s = 1 2 ∑ s,s J s (s → s) log q s (s|s ) q s (s |s) = 0, R x = lim T→∞ 1 T ∑ X T P(X T ) log P(X T ) P( X T ) , where 364 P(X T ) = ∑ S T P(Z T = (X T , S T )).
To quantify the correlation between the system and demon, we use the definition of MIR in Eq.

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(12). 366 We are also interested in the time-irreversible part of MIR, I B (X, S) which influences the EPR of    , for large T, where Z T = (X T , S T ) is a typical sequence of Z (hence X T and S T are typical sequences of X and 371 S respectively). The convergence of this numerical simulation can be observed as T increases. To 372 confirm the result R x = R z − R s − 2I B (X, S), we use different typical sequences in calculating R(X) 373 and I B (X, S) respectively. R(z) and R(s) are calculated by using the corresponding analytical results 374 shown above.

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For numerical simulations, we randomly choose two groups of the parameters: the probabilities 376 of the instructions of the baths a and b , and probabilities of the demon choices d (see Tables A.1 and   377 A.2). We evaluate R(X), I(X, S), and I B (X, S) for all two groups. The values of numerical results are 378 listed in Table A.3.