Entropy in thermodynamics and information theory

Zentralfriedhof Vienna - Boltzmann

Entropy is a fundamental concept in both thermodynamics and information theory, reflecting disorder or randomness within a system. In thermodynamics, entropy is a measure of the number of specific ways in which a thermodynamic system may be arranged, often understood as a measure of disorder or randomness. The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, indicating that systems naturally progress from order to disorder. In information theory, entropy represents the amount of uncertainty or unpredictability in the content of a message, essentially measuring the information content.

Thermodynamics

In thermodynamics, entropy (symbolized as S) is a physical property that is a measure of a system's thermal energy per unit temperature that is unavailable for doing useful work. Because work is obtained from ordered molecular motion, the amount of entropy is also a measure of the molecular disorder, or randomness, of a system. The concept of entropy was introduced by Rudolf Clausius in 1850, who gave it a precise mathematical definition. According to the second law of thermodynamics, the entropy of an isolated system always increases or remains constant; it never decreases. This principle explains a number of physical phenomena, such as why ice melts in warm water, and forms the foundation of the arrow of time.

Information Theory

In information theory, entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. Introduced by Claude E. Shannon in 1948, entropy in this context is a measure of the unpredictability or information content. For a given message, entropy helps determine the minimum number of bits required to encode it. High entropy means the message contains a lot of information and is less predictable, while low entropy means the message is more predictable. This concept is crucial in data compression algorithms and in the field of cryptography, where it is used to gauge the effectiveness of cryptographic systems.

Mathematical Formulation

Thermodynamic Entropy

The change in entropy in a thermodynamic process can be calculated using the formula: \[\Delta S = \int \frac{dQ}{T}\] where \(\Delta S\) is the change in entropy, \(dQ\) is the infinitesimal amount of heat added to the system, and \(T\) is the absolute temperature at which the process occurs.

Information Theoretic Entropy

The entropy \(H\) of a discrete random variable \(X\) with possible values \(\{x_1, x_2, ..., x_n\}\) and probability mass function \(P(X)\) is given by: \[H(X) = -\sum_{i=1}^{n} P(x_i) \log_b P(x_i)\] where the base \(b\) of the logarithm determines the unit of entropy (bits if \(b = 2\), nats if \(b = e\), etc.).

Applications

Entropy has wide-ranging applications across various fields. In thermodynamics, it is used to determine the efficiency of engines and refrigerators. In information theory, it is applied in coding theory, cryptography, and in the analysis of communication systems. Entropy also plays a role in statistical mechanics, where it is related to the number of microstates corresponding to a macrostate, and in cosmology, where it is used to study the evolution of the universe.

See Also

• Second law of thermodynamics
• Information theory
• Statistical mechanics
• Cryptography