Info Theory & Thermodynamics Interrelation

• Interrelation of Information Theory and Thermodynamics is defined as the interdisciplinary framework linking abstract entropy measures (Shannon/von Neumann) with macroscopic energy and disorder.
• Fundamental principles like Landauer’s limit and mutual information impose constraints on energy dissipation in computation, communication, and control protocols.
• Modern developments recast thermodynamics as a resource theory, quantitatively relating state convertibility, work extraction, and non-equilibrium dynamics through entropic monotones.

The interrelation of information theory and thermodynamics constitutes a fundamental cross-disciplinary framework in modern physics, mathematics, and engineering. Both disciplines analyze the interplay between randomness, uncertainty, and physical resources, but approach the formalization from distinct perspectives: thermodynamics is rooted in macroscopic energy and entropy flows, while information theory quantifies abstract uncertainty and inference. Contemporary research establishes that not only are the mathematical structures of thermodynamics and information theory closely linked, but operational protocols in statistical mechanics, quantum computation, communication, and control are deeply constrained by information-theoretic laws.

1. Core Entropic Quantities and Their Physical Realizations

The foundational connection arises via entropy measures. In classical thermodynamics, Boltzmann and Clausius entropy quantify macroscopic disorder through

$S_\text{thermo} = k_B \ln \Omega,$

where $\Omega$ is the accessible number of microstates. Information theory replaces $\ln \Omega$ with Shannon entropy,

$H(p) = -\sum_i p_i \log p_i,$

for a probability distribution $p$. Quantum mechanics generalizes to the von Neumann entropy,

$S(\rho) = -\mathrm{Tr} [\rho \log \rho].$

Axiomatic treatments (Lieb–Yngvason) rigorously demonstrate that thermodynamic entropy and information entropy coincide when the order of state transformations is defined by majorization and adiabatic accessibility. The unique entropy function consistent with the physical axioms is, up to scale, the Shannon or von Neumann entropy itself (Weilenmann et al., 2015).

The use of relative entropy, particularly the Kullback–Leibler (KL) divergence $D(p\|q)$, is central in quantifying “distance” from equilibrium and operationally appears as the dissipated work or irreversibility in protocols. Quantum generalizations (Umegaki, sandwiched Rényi divergences) extend these insights to nonequilibrium and single-shot regimes, capturing out-of-equilibrium thermodynamic resources precisely (Sagawa, 2020).

2. The Second Law, Information Processing, and Fluctuation Relations

Landauer’s principle formalizes the minimal heat dissipation required for erasure of information: erasing one bit requires at least $k_B T \ln 2$ of energy expended as heat. This lower bound is derived directly from the non-negativity of KL divergence and the Clausius inequality (Parrondo, 2023, Altaner, 2017). Extensions of the second law to measurement, feedback, and information acquisition protocols reveal that the entropy produced in a process can be decomposed as

$\Delta S_\text{tot} \geq -I(X;M),$

where $I(X;M)$ quantifies the mutual information accrued through measurement—this term quantifies the “information fuel” available to Maxwell’s demon–type protocols (Parrondo, 2023). The Sagawa–Ueda generalizations of the Jarzynski and Crooks fluctuation theorems further encode this informational correction, showing that rare fluctuation events violating the standard second law are exponentially suppressed with an exponent set by trajectory-level informational gains (Goold et al., 2015, Parrondo, 2023).

In stochastic thermodynamics, the pathwise KL divergence between the forward and time-reversed process measures entropy production, and system/bath entropy production can be decomposed via information-theoretic projections and coarse-grainings (Tsuruyama, 31 Dec 2025, Sagawa, 2020).

3. Minimax Principles, Resource Theories, and Single-Shot Constraints

Modern developments recast thermodynamics as an information-theoretic resource theory. The allowed “free” operations are those that cannot generate non-equilibrium resources (e.g., thermal, unital, noisy, random-reversible channels) (Scandolo, 2019). The set of monotones governing state convertibility are f-divergences and Rényi entropies, with majorization providing necessary and sufficient conditions under unrestricted reversibility (Scandolo, 2019, Sagawa, 2020). Notably, “single-shot” or “one-shot” thermodynamics incorporates min/max entropies and smooth divergences, capturing finite-sample or strong-non-equilibrium constraints absent from standard ensemble descriptions (Sagawa, 2020).

In this formalism, the maximum work extractable, or minimal work required for transformation, is dictated by orderings in the space of entropic monotones, yielding explicit trade-offs between informational and energetic resources (Bera et al., 2017).

4. Quantitative Trade-Offs: Precision, Dissipation, and Communication

Precise thermodynamic trade-offs have been derived that relate information acquisition, parameter estimation, and communication channel capacity to energy dissipation and entropy production. The thermodynamic cost of acquiring $\Omega$0 nats of parameter information with error variance $\Omega$1 using a probe system at temperature $\Omega$2 and dissipating average work $\Omega$3 is strictly bounded:

$\Omega$4

for encoding device resolution $\Omega$5 (Micadei et al., 2012). Shannon’s channel capacity and the minimum entropy production rate of a communication process are jointly constrained: at high capacity, entropy production is monotonic and convex in capacity, and inverse multiplexing can minimize total dissipation (Tasnim et al., 2023).

For feedback engines and information-driven protocols, the maximum work extracted per cycle from a single bath is bounded by the mutual information acquired,

$\Omega$6

with further corrections for finite time and dynamical rates (Taghvaei et al., 2021, Parrondo, 2023).

5. Equilibrium, Nonequilibrium, and Generalized Thermodynamics

Information theory provides the basis for reconstructing thermodynamics even in the absence of equilibrium concepts. Maximum-entropy inference yields generalized Gibbs (or reference) ensembles as solutions to constrained minimization of relative entropy, with Lagrange multipliers interpreted as intensive thermodynamic variables (temperature, chemical potential, information pressure) (Tsuruyama, 31 Dec 2025). Out-of-equilibrium systems are characterized by path-space KL divergences, and total entropy production is split naturally into system and bath contributions using the formalism of observational entropy (Tsuruyama, 31 Dec 2025, Altaner, 2017).

The temperature of an information system, as reconstructed in purely information-theoretic terms, corresponds to the Lagrange multiplier conjugate to the observational entropy or information volume, and pressure is generalized from a geometric to an information-theoretic “volume” conjugate (Barzi et al., 2024, Tsuruyama, 31 Dec 2025).

6. Quantum and Beyond: Universality of Information Principles

Quantum thermodynamics extends these connections further: the allowed state transformations, work extraction bounds, validity of the second law, and the emergence of equilibrium ensembles are all expressible in terms of information-theoretic monotones (e.g., relative entropy, mutual information, majorization relations) (Goold et al., 2015, Bera et al., 2017, Bera et al., 2018). Furthermore, these structures are robust beyond quantum mechanics, to general probabilistic theories. “Sharp theories with purification” and unrestricted reversibility axioms guarantee the existence of majorization and resource-theoretic duality between thermodynamic purity and bipartite entanglement (Scandolo, 2019).

A crucial theoretical advance is the demonstration that, in quantum and general probabilistic setups, all fundamental thermodynamic inequalities and efficiency bounds (e.g., Carnot, Kelvin–Planck) are encoded as data-processing inequalities for relative entropy, and that work/heat definitions are derived from information conservation—rather than physical modeling or particular bath assumptions (Bera et al., 2017, Bera et al., 2018).

7. Complementarity, Limits, and Broader Implications

While a traditional identification “information is physical” treats information and entropy as essentially equivalent, nuanced analyses highlight the complementarity—rather than strict equivalence—between information-theoretic and thermodynamic entropies, especially when error, memory lifetimes, and dynamical stability are accounted for (Alicki, 2014). Configuration stability and the operational meaning of entropy depend explicitly on system ergodicity.

This interrelation underpins not only the ultimate costs of computation, communication, and control, but also the statistical structure of learning, inference, and biological adaptation. Information processing cannot evade thermodynamic limitations: every logical irreversibility or information acquisition has a quantifiable energetic cost, and every non-equilibrium transformation is bounded by informational distance from equilibrium.

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References:

• (Weilenmann et al., 2015) Axiomatic relation between thermodynamic and information-theoretic entropies
• (Tsuruyama, 31 Dec 2025) Thermodynamics Reconstructed from Information Theory:An Axiomatic Framework via Information-Volume Constraints and Path-Space KL Divergence
• (Parrondo, 2023) Thermodynamics of Information
• (Altaner, 2017) Nonequilibrium thermodynamics and information theory: Basic concepts and relaxing dynamics
• (Sagawa, 2020) Entropy, Divergence, and Majorization in Classical and Quantum Thermodynamics
• (Bera et al., 2017) Thermodynamics as a Consequence of Information Conservation
• (Scandolo, 2019) Information-theoretic foundations of thermodynamics in general probabilistic theories
• (Bera et al., 2018) Thermodynamics from information
• (Goold et al., 2015) The role of quantum information in thermodynamics --- a topical review
• (Micadei et al., 2012) Thermodynamic cost of acquiring information
• (Alicki, 2014) Information is not physical
• (Barzi et al., 2024) Reformulation of Classical Thermodynamics from Information Theory
• (Taghvaei et al., 2021) On the relation between information and power in stochastic thermodynamic engines
• (Tian et al., 2021) Thermodynamics of Encoding and Encoders
• (Tasnim et al., 2023) Entropy production in communication channels
• (Ito, 2017) Stochastic thermodynamic interpretation of information geometry