统一熵：热力学熵与信息熵的统一框架

王彦明 | wangyanming@syinpo.com
摘要： 本文基于经典热力学与信息论，整合现有熵理论，构建统一熵框架，实现热力学熵与信息熵的形式化统一。通过公理化定义统一熵的数学表达与动力学演化方程，结合麦克斯韦妖、生命体代谢、量子计算等成熟场景验证框架自洽性；针对前沿物理领域，初步探讨与统一熵的潜在关联。本文给出可实验检验的定量预测，明确统一熵的适用边界与局限性，为信息热力学与复杂系统的熵理论研究提供自洽视角。
关键词： 统一熵；热力学熵；信息熵；互信息；兰道尔原理；信息热力学

1 引言

1.1 研究背景与问题

热力学熵与信息熵的数学结构高度相似，但长期分属不同学科领域。热力学熵表达式为 \(S = -k_B \sum p_i \ln p_i\)；信息熵表达式为 \(H = -\sum p_i \log_2 p_i\)。麦克斯韦妖思想实验揭示信息获取与擦除必然伴随物理熵变，兰道尔原理进一步明确：擦除1比特信息至少产生 \(k_B T \ln 2\) 的能量耗散。本文整合现有熵理论，构建自洽的统一熵框架。

1.2 研究目标与贡献

建立统一熵的公理化定义与数学表达，整合热力学熵与信息熵；验证与现有信息热力学理论的兼容性；给出可实验检验的定量预测。核心价值在于对现有熵理论的系统化整合与跨领域统一表述。

2 统一熵的理论框架

2.1 公理体系

存在性公理、可加性公理、等价性公理、熵增公理、开放系统公理。

2.2 统一熵的数学定义与分解

早期形式：\(\mathbb{S} = S_{\text{system}} + k_B\ln 2 \cdot H_M - k_B I\)。由于 \(S_{\text{system}} = k_B\ln 2 \cdot H_S\)，统一化形式为：

\[ \mathbb{S} = k_B\ln 2 \cdot \left( H_S + H_M - I_{S,M} \right) \]

其中转换系数 \(k_B\ln 2\) 由兰道尔原理确定。

2.3 动力学演化与自由能

统一熵的时间演化方程：

\[ \frac{d\mathbb{S}}{dt} = \frac{dS_{\text{sys}}}{dt} + \frac{dS_{\text{env}}}{dt} + k_B\ln 2 \cdot \frac{dH_M}{dt} + \sigma_{\text{comp}} \]

特殊情形涵盖孤立系统、封闭系统及开放系统。定义统一自由能 \(\mathbb{F} = U - T\mathbb{S}\)，恒温下 \(d\mathbb{F} \leq -dW\)。

图1 物理-信息复合系统 (示意图：系统、记忆与环境之间的熵流)
系统与环境通过能量交换、信息擦除产生熵产生率 \(\sigma_{\text{comp}}\)
图2 熵流方向 (系统熵变、环境熵变与信息熵变耦合)

3 统一熵在典型物理系统中的应用

3.1 麦克斯韦妖系统

统一熵表达式 \(\mathbb{S} = S_{\text{gas}} + k_B\ln 2(H_{\text{demon}} - I_{\text{gas,demon}})\)。三阶段(测量、利用信息、擦除)分析表明，测量和利用阶段统一熵不变，擦除阶段由于兰道尔原理产生熵增，整体满足熵增原理。

\table
阶段\(\Delta S_{\text{gas}}\)\(\Delta H_{\text{demon}}\)\(\Delta I\)\(\Delta \mathbb{S}\)
测量0\(+N\) bit\(+N\ln2\) nat0
利用信息\(-Nk_B\ln2\)0\(-N\ln2\) nat0
擦除0\(-N\) bit0\(\geq 0\)

3.2 生命体的信息热力学

生命体稳态条件：\(\frac{dS_{\text{physical}}}{dt} = -k_B\ln2 \frac{dH_{\text{brain}}}{dt} + k_B\frac{dI_{\text{body,brain}}}{dt}\)。学习过程额外能耗下限为 \(E_{\text{bit}} \ge k_BT\ln2\)。

3.3 量子计算的熵成本

量子统一熵 \(\mathbb{S}_Q = -k_B \text{Tr}(\widehat{\rho}_{\text{total}}\ln\widehat{\rho}_{\text{total}})\)，幺正变换不产生熵，测量产生熵 \(k_B H_{\text{Shannon}}\)，量子纠错每比特熵产生 \(\sigma \ge k_B\ln2\)，与兰道尔原理自洽。

图7 量子纠错中的熵变化 (测量与重置阶段的熵平衡)

4 与前沿物理的潜在关联

黑洞热力学形式化表述：\(\mathbb{S}_{\text{BH}} = S_{\text{BH}} + k_B\ln2(H_{\text{rad}} - I_{\text{BH,rad}})\)。量子场论纠缠熵可形式化为统一熵结构。此部分仅为理论形式兼容探讨。

5 讨论与理论局限

统一熵公理体系仍需完善微观动力学证明；对强非平衡态描述不完备；与前沿物理仅为形式兼容。

6 结论

本文整合现有熵理论构建自洽统一熵框架，实现热力学熵与信息熵的形式化统一，为信息热力学和复杂系统提供统一视角。

参考文献
  1. Sagawa T, Ueda M. Second Law of Thermodynamics with Discrete Quantum Feedback Control. Phys. Rev. Lett., 2008, 100, 080403.
  2. Seifert U. Stochastic thermodynamics, fluctuation theorems, and molecular machines. Rep. Prog. Phys., 2012, 75, 126001.
  3. Landauer R. Irreversibility and Heat Generation in the Computing Process. IBM J. Res. Dev., 1961, 5, 183-191.
  4. Bérut A, et al. Experimental verification of Landauer's principle linking information and thermodynamics. Nature, 2012, 483, 187-189.
  5. Jarzynski C. Nonequilibrium Equality for Free Energy Differences. Phys. Rev. Lett., 1997, 78, 2690-2693.
  6. Bennett C H. The thermodynamics of computation --- a review. Int. J. Theor. Phys., 1982, 21, 905-940.
  7. De Michielis M, Ferraro E. Energy and power scaling in quantum computers based on rotated surface codes with silicon flip-flop qubits. EPJ Quantum Technology, 2025, 12, 64.

Unified Entropy: A Unified Framework for Thermodynamic Entropy and Information Entropy

Yanming Wang | wangyanming@syinpo.com
Abstract: Based on classical thermodynamics and information theory, this paper integrates existing entropy theories to construct a unified entropy framework, achieving formal unification of thermodynamic entropy and information entropy. Through axiomatic definitions, mathematical expressions and dynamical evolution equations are established, and the framework’s self-consistency is verified via Maxwell's demon, organism metabolism, quantum computing, and complex systems. Quantitatively testable predictions are provided, clarifying applicable boundaries.
Keywords: unified entropy, thermodynamic entropy, information entropy, mutual information, Landauer's principle, information thermodynamics

1 Introduction

1.1 Background

Thermodynamic entropy \(S = -k_B\sum p_i \ln p_i\) and information entropy \(H = -\sum p_i \log_2 p_i\) share isomorphic structures. Maxwell's demon and Landauer’s principle (\(E \ge k_B T \ln 2\) per bit erased) reveal deep links. This work builds a unified entropy framework.

1.2 Objectives

Establish an axiomatic system, unify thermodynamic and information entropy, provide testable predictions, and integrate existing theories.

2 Theoretical Framework

2.1 Axioms

Existence, additivity, equivalence (1 bit ↔ \(k_B\ln2\) thermodynamic entropy), entropy increase for closed systems, and open system entropy production \(\sigma \ge 0\).

2.2 Mathematical Definition

Early form: \(\mathbb{S} = S_{\text{system}} + k_B\ln2 \cdot H_M - k_B I\). Using \(S_{\text{system}} = k_B\ln2 \cdot H_S\), unified entropy:

\[ \mathbb{S} = k_B\ln2 \left( H_S + H_M - I_{S,M} \right) \]

2.3 Dynamical Evolution

Complete time evolution equation:

\[ \frac{d\mathbb{S}}{dt} = \frac{dS_{\text{sys}}}{dt} + \frac{dS_{\text{env}}}{dt} + k_B\ln2 \cdot \frac{dH_M}{dt} + \sigma_{\text{comp}} \]

Unified free energy \(\mathbb{F}=U-T\mathbb{S}\) yields \(d\mathbb{F} \le -dW\) under isothermal conditions.

Figure 1 & 2 Physical-information composite system & entropy flow directions.

3 Applications

3.1 Maxwell's Demon

Unified entropy: \(\mathbb{S}=S_{\text{gas}}+k_B\ln2(H_{\text{demon}}-I_{\text{gas,demon}})\). Measurement and information-use stages yield \(\Delta\mathbb{S}=0\); erasure stage obeys Landauer’s bound, overall \(\Delta\mathbb{S}\ge0\).

Stage\(\Delta S_{\text{gas}}\)\(\Delta H_{\text{demon}}\)\(\Delta I\)\(\Delta \mathbb{S}\)
Measurement0\(+N\) bit\(+N\ln2\) nat0
Utilization\(-Nk_B\ln2\)0\(-N\ln2\) nat0
Erasure0\(-N\) bit0\(\ge 0\)

3.2 Organism Information Thermodynamics

Steady-state condition: \(\frac{dS_{\text{physical}}}{dt} = -k_B\ln2 \frac{dH_{\text{brain}}}{dt} + k_B\frac{dI}{dt}\). Learning cost \(\ge k_B T\ln2\).

3.3 Quantum Computing

Quantum unified entropy \(\mathbb{S}_Q = -k_B \text{Tr}(\widehat{\rho}_{\text{total}}\ln\widehat{\rho}_{\text{total}})\). Unitary gates are entropy-free, measurement produces entropy \(k_B H_{\text{Shannon}}\), and error correction per qubit satisfies \(\sigma \ge k_B\ln2\).

4 Potential Links with Frontier Physics

Black hole thermodynamics: \(\mathbb{S}_{\text{BH}} = S_{\text{BH}} + k_B\ln2(H_{\text{rad}} - I_{\text{BH,rad}})\). Formal compatibility only.

5 Discussion & Limitations

Microscopic proof of entropy increase principle is not fully derived; strong non-equilibrium needs extension; frontier links are formal.

6 Conclusion

This work integrates entropy theories into a unified framework, providing a coherent perspective for information thermodynamics and complex systems.

References
  1. Sagawa T, Ueda M. Phys. Rev. Lett., 2008, 100, 080403.
  2. Seifert U. Rep. Prog. Phys., 2012, 75, 126001.
  3. Landauer R. IBM J. Res. Dev., 1961, 5, 183-191.
  4. Bérut A, et al. Nature, 2012, 483, 187-189.
  5. Jarzynski C. Phys. Rev. Lett., 1997, 78, 2690-2693.
  6. Bennett C H. Int. J. Theor. Phys., 1982, 21, 905-940.
  7. De Michielis M, Ferraro E. EPJ Quantum Technology, 2025, 12, 64.