Kullback-Leibler Potential for Non-Ergodic Replication Dynamics:An Information-Theoretic Second Law

TL;DR

This work reframes replication as a non-ergodic, block-invariant Markov process and introduces a Kullback–Leibler divergence–based potential $V(p)$ that serves as an informational Lyapunov function relative to the reachable steady set. The authors prove monotonic decay of $V$ under block-invariant dynamics and show convergence of within-block distributions to block invariants, with $V(p_n)\to 0$ when blocks are primitive; they further extend the framework to leakage via a leakage-tolerant $V_\delta$. Two concrete instantiations—image replication using blockwise Gaussian convolution and DNA replication with proofreading—illustrate monotone $V$ alongside rising entropy measures and decreasing step divergences, linking the formalism to an informational free-energy interpretation. The results yield a unified information-thermodynamic view of replication, transmission, and repair, with potential implications for understanding biological information processing and designing information-preserving protocols under coarse-grained constraints.

Abstract

This study aims to quantify and visualize the degradation of fidelity (information degradation) that inevitably accompanies the replication of information within the framework of information thermodynamics and to propose an information-theoretic formulation of the second law based on this phenomenon. While previous research in information thermodynamics has focused on the thermodynamic costs associated with information "erasure'' or "measurement'' through concepts such as Landauer's principle and mutual information, little systematic discussion has addressed the inherently irreversible nature of "replication'' itself and the accompanying degradation of information structure. In this study, we construct a mathematical model of information replication using a discrete Markov model and Gaussian convolution, and quantify changes in information at each replication step: Shannon entropy, cross-entropy, and the Kullback--Leibler divergence (KLD). The monotonic decrease of KLD exhibits a Lyapunov-like property, which can be interpreted as a potential analogous to the free energy in the process by which a nonequilibrium system converges to a particular steady state. Furthermore, we extend this framework to the potential applicability to biological information processes such as DNA replication, showing that the free energy required for degradation and repair can be expressed in terms of KLD. This contributes to building a unified information-thermodynamic framework for operations such as replication, transmission, and repair of information.

Figures (5)

• Figure 1:
• Figure 2: Ergodic (global) vs. non-ergodic (blockwise) Gaussian blurring of a $256\times256$ image ($\sigma=1.5$, $n_{\mathrm{steps}}=50$). At each step we report $H$\ref{['eq:H-image']}, $H_\times$\ref{['eq:Hcross-image']}, $D_{\mathrm{KL}}$\ref{['eq:KL-image']}, and $V$\ref{['eq:V-image']}. In the blockwise case, monotonicity of $V$ follows from Theorem \ref{['thm:V-Lyapunov']}. (Upper) $2\times2$ blocks, (Middle) $4\times4$ blocks, (Lower) $128\times128$ blocks. All values in this figure are in bits (base 2; see Appendix B). Per-panel mixing strength (see Defs. \ref{['eq:Doeblin-cond']}--\ref{['eq:Dobrushin-def']}). Upper: $\varepsilon=0.073$, $\delta=0.83$; Middle: $\varepsilon=0.051$, $\delta=0.88$; Lower: $\varepsilon=0.019$, $\delta=0.93$.
• Figure 3: DNA block-diagonal replication: time series of informational metrics (nats).Shannon entropy, Cross entropy, KLD, and KLD potential. We iterate a four-state Markov chain on $\{\mathrm{A},\mathrm{T},\mathrm{C},\mathrm{G}\}$ with a block-diagonal kernel $\widetilde{T}=\mathrm{diag}(T_1,T_2)$ for $N=50$ steps. Base rates: $\alpha=0.020$, $\beta=0.010$, $\gamma=0.015$, $\delta=0.015$. Proofreading channel $\mathcal{R}$ with $\alpha'=0.005$, $\beta'=0.003$, $\gamma'=\delta'=0.004$ is mixed at $\rho=0.30$, giving effective rates $\alpha_{\!e}=0.0155$, $\beta_{\!e}=0.0079$, $\gamma_{\!e}=\delta_{\!e}=0.0117$ and block invariants $\pi_1^\ast=(0.3376,\,0.6624)$, $\pi_2^\ast=(0.5,\,0.5)$. Initial distribution $p_0=(0.6,0.1,0.2,0.1)$ so that $w_1(p_0)=0.7$, $w_2(p_0)=0.3$. At each step $q_n=p_n\widetilde{T}$ and we record: top-left, Shannon entropy $H(q_n)$; top-right, cross-entropy $H(p_n,q_n)$; bottom-left, step divergence $D_{\mathrm{KL}}(p_n\|q_n)$; bottom-right, KLD potential $V(p_n)=w_1(p_0)\,D_{\mathrm{KL}}(p_n^{(1)}\|\pi_1^\ast)+w_2(p_0)\,D_{\mathrm{KL}}(p_n^{(2)}\|\pi_2^\ast)$. As replication proceeds, $H$ and $H_\times$ increase, while both $D_{\mathrm{KL}}(p_n\|q_n)$ and $V(p_n)$ decrease monotonically; the monotonicity of $V$ follows from Theorem \ref{['thm:V-Lyapunov']}.All metrics are reported in nats
• Figure 4: KLD potential landscape $V(p)=w_1 D_{\mathrm{KL}}(p^{(1)}\parallel\pi_1^\ast)+w_2 D_{\mathrm{KL}}(p^{(2)}\parallel\pi_2^\ast)$ (nats) over $(x,y)\in[0,1]^2$, where $x$ is the A fraction in the AT block and $y$ is the C fraction in the GC block. Proofreading mixture $\rho=0.30$; AT rates $\alpha=0.020$, $\beta=0.010$, $\alpha'=0.005$, $\beta'=0.003$ give $\alpha_{\mathrm{eff}}=0.0155$, $\beta_{\mathrm{eff}}=0.0079$ and $\pi_1^\ast=(\beta_{\mathrm{eff}}/(\alpha_{\mathrm{eff}}+\beta_{\mathrm{eff}}),,\alpha_{\mathrm{eff}}/(\alpha_{\mathrm{eff}}+\beta_{\mathrm{eff}}))\approx(0.338,,0.662)$. GC rates (asymmetric) $\gamma=0.014$, $\delta=0.021$, $\gamma'=0.004$, $\delta'=0.006$ give $\gamma_{\mathrm{eff}}=0.0110$, $\delta_{\mathrm{eff}}=0.0165$ and $\pi_2^\ast=(\delta_{\mathrm{eff}}/(\gamma_{\mathrm{eff}}+\delta_{\mathrm{eff}}),,\gamma_{\mathrm{eff}}/(\gamma_{\mathrm{eff}}+\delta_{\mathrm{eff}}))\approx(0.60,,0.40)$. Block weights $w_1=w_2=0.5$. The blue marker indicates $(x^\ast,y^\ast)=(\pi_1^\ast(\mathrm{A}),,\pi_2^\ast(\mathrm{C}))\approx(0.34,,0.60)$. Grid $101\times101$, colormap All values in nats.
• Figure :

Theorems & Definitions (12)

• Lemma 1: Preservation of block mass and conditional evolution
• proof
• Lemma 2: Block decomposition of KLD
• proof
• Theorem 1: KLD potential is Lyapunov under block invariance
• proof
• Proposition 1: Nonnegativity and telescoping sum
• proof
• Theorem 2: Primitivity of block kernels
• Theorem 3: Second law for non-ergodic replication: minimal form