The Biggest Ideas in the Universe | 20. Entropy and Information

Sean Carroll introduces the topic of entropy, emphasizing its relationship with information. He mentions that entropy is multifaceted and challenging to define, playing a crucial role in discussions of time's arrow and the second law of thermodynamics. Carroll also highlights the historical context, involving figures like Boltzmann, Carnot, and Clausius, and touches on the statistical nature of entropy.

Carroll delves into two definitions of entropy within classical statistical mechanics. He discusses the concept of coarse-graining, where macroscopic states encompass numerous microscopic arrangements. The Boltzmann entropy is introduced as a measure of ignorance about a system's microscopic state, quantified by the logarithm of the volume of the macro state in phase space.

The video explores the connection between entropy and probability distributions. Carroll contrasts Boltzmann's entropy, which relies on coarse-graining, with Gibbs' entropy, which is based on the probability distribution across phase space. The Gibbs entropy is defined as the negative integral of the probability distribution function multiplied by the logarithm of that probability, leading to a discussion on the relationship between entropy, knowledge, and information.

Carroll examines the second law of thermodynamics in relation to the Gibbs and Boltzmann entropies. He explains that while the Gibbs entropy implies a constant entropy for closed systems, the second law is more accurately described by the change in entropy being greater than or equal to the heat exchanged with the environment. This leads to the concept that entropy can increase even in open systems when considering the effects of a heat bath.

The discussion shifts to the entropy of the early universe, addressing misconceptions about its high or low entropy state. Carroll argues that the early universe's entropy was low due to gravitational effects, contrasting it with a box of gas in thermal equilibrium. He also touches on the maximum entropy the universe could have had and the implications for the past hypothesis, which suggests the universe began in a low-entropy state.

Carroll acknowledges the mystery of why the early universe had a low-entropy state. He mentions his own speculative work with Jennifer Chen, which attempted to provide a dynamical explanation involving baby universes and quantum nucleation. He calls for more research into this area, emphasizing the lack of a widely accepted theory to explain the initial low entropy of the universe.

Carroll presents historical objections to Boltzmann's statistical interpretation of the second law, including Loschmidt's reversibility objection and Zermelo's recurrence objection. He explains the content and implications of these objections and Boltzmann's responses, highlighting the philosophical debates surrounding the nature of entropy and the probabilistic nature of physical laws.

The video addresses Boltzmann's idea regarding fluctuations in an infinite universe and the concept of Boltzmann brains. Carroll explains the flaw in Boltzmann's reasoning, which was highlighted by Eddington, showing that it's more likely for a single observer (a brain) to arise from random fluctuations than a complex entity like a person or a planet.

Carroll argues that the thermodynamic arrow of time, as described by the second law of thermodynamics, underlies all other arrows of time, including the psychological, biological, and cosmological arrows. He discusses the past hypothesis and its role in providing a boundary condition that breaks the time-reversal symmetry, explaining the directionality of time and the possibility of free will.

The video connects the concept of entropy to the existence of life on Earth. Carroll explains how life relies on the increase of entropy, as illustrated by the process of photosynthesis and the use of ATP in cells. He emphasizes the importance of the sun providing low-entropy energy that is then expelled in a higher entropy form, allowing life to maintain its integrity.

Carroll explores Maxwell's thought experiment involving a demon controlling the entropy of a system, which seemed to contradict the second law. He discusses the resolution to this paradox, as proposed by Landauer and Bennett, which involves the inevitable increase of entropy when information is erased, thus upholding the second law of thermodynamics.

The video concludes with a discussion on Claude Shannon's work on information theory and its connection to entropy. Carroll explains Shannon's concept of information content, or surprisal, and how it relates to the probability of symbols in a communication system. He highlights the formal similarity between Shannon's entropy (information entropy) and Gibbs' entropy, noting the different interpretations in communication theory and physics.

Carroll introduces the concept of quantum entropy, or entanglement entropy, as developed by Von Neumann. He explains that even when the complete state of a quantum system is known, entropy can still arise when considering subsystems that are entangled. This entropy is inherent and does not depend on ignorance or coarse-graining, reflecting a fundamental aspect of quantum mechanics.