Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

The Entropy as a State Function

The Entropy as a State Function

Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...

Entropy Change in Reversible Processes

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.

Entropy

Entropy

The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...

Entropy

Entropy

Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...

Entropy and the Second Law of Thermodynamics

Entropy and the Second Law of Thermodynamics

The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
The relation between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...

Entropy and the Second Law of Thermodynamics

Entropy and the Second Law of Thermodynamics

Consider an isolated system in which a hot object is placed in contact with a cold one. This is an irreversible process that eventually leads both objects to reach the same equilibrium temperature. It is crucial to note that the constituents of any substance exhibit increased disorder at higher temperatures. As a cold substance absorbs heat, its constituents become more disordered. The energy transfer from a hotter object to a cooler one increases the system's disorder or randomness. This...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same author

Timescapes of non-human experience.

Trends in cognitive sciences·2026

Same author

Evolving reservoir computers reveal bidirectional coupling between predictive power and emergent dynamics.

Patterns (New York, N.Y.)·2026

Same author

Mapping of Subjective Accounts into Interpreted Clusters (MOSAIC): Topic Modelling and LLM applied to Stroboscopic Phenomenology.

Neuroscience of consciousness·2026

Same author

The role of active inference in conscious awareness.

PloS one·2025

Same author

On the minimal theory of consciousness implicit in active inference.

Physics of life reviews·2025

Same author

Hemispherotomy leads to persistent sleep-like slow waves in the isolated cortex of awake humans.

PLoS biology·2025

Search research articles

Related Experiment Video

Updated: Jun 14, 2026

New Framework for Understanding Cross-Brain Coherence in Functional Near-Infrared Spectroscopy (fNIRS) Hyperscanning Studies

New Framework for Understanding Cross-Brain Coherence in Functional Near-Infrared Spectroscopy (fNIRS) Hyperscanning Studies

Published on: October 6, 2023

Granger causality and transfer entropy are equivalent for Gaussian variables.

Lionel Barnett¹, Adam B Barrett, Anil K Seth

¹Centre for Computational Neuroscience and Robotics, School of Informatics, University of Sussex, Brighton BN1 9QJ, United Kingdom. L.C.Barnett@sussex.ac.uk

Physical Review Letters

|April 7, 2010

Summary

Granger causality and transfer entropy are equivalent for Gaussian variables, bridging econometrics and information theory for causal inference. This finding links predictive modeling with information transfer in data-driven analyses.

More Related Videos

Application of Granger Causality Analysis of the Directed Functional Connection in Alzheimer's Disease and Mild Cognitive Impairment

Application of Granger Causality Analysis of the Directed Functional Connection in Alzheimer's Disease and Mild Cognitive Impairment

Published on: August 7, 2017

Related Experiment Videos

Last Updated: Jun 14, 2026

New Framework for Understanding Cross-Brain Coherence in Functional Near-Infrared Spectroscopy (fNIRS) Hyperscanning Studies

New Framework for Understanding Cross-Brain Coherence in Functional Near-Infrared Spectroscopy (fNIRS) Hyperscanning Studies

Published on: October 6, 2023

Application of Granger Causality Analysis of the Directed Functional Connection in Alzheimer's Disease and Mild Cognitive Impairment

Application of Granger Causality Analysis of the Directed Functional Connection in Alzheimer's Disease and Mild Cognitive Impairment

Published on: August 7, 2017

Area of Science:

Neuroscience
Econometrics
Information Theory

Background:

Granger causality, rooted in econometrics, assesses causal influence via predictive vector autoregression.
Transfer entropy, an information-theoretic measure, quantifies time-directed information transfer between processes.
Both methods are widely applied, particularly in neuroscience, for causal inference.

Purpose of the Study:

To formally describe the relationship between Granger causality and transfer entropy.
To bridge autoregressive and information-theoretic approaches to data-driven causal inference.

Main Methods:

The study analyzes the mathematical relationship between Granger causality and transfer entropy.
Focuses on the case of Gaussian variables to establish formal equivalence.

Main Results:

Granger causality and transfer entropy are shown to be entirely equivalent for Gaussian variables.
This equivalence provides a formal link between these two distinct causal inference frameworks.

Conclusions:

The findings unify predictive modeling (Granger causality) and information transfer (transfer entropy) for causal inference.
This work offers a foundational understanding for applying these methods in fields like neuroscience and econometrics.