Related Experiment Video
Updated: Feb 19, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
Published on: September 23, 2025
Generalized second law for a simple chaotic system
Hiroshi H Hasegawa1, Tomomi Nakamura1, Dean J Driebe2
1Department of Mathematical Sciences, Ibaraki University, Mito 310-8512, Japan.
Abstract:
The generalized second law (nonequilibrium maximum work formulation) is derived for a simple chaotic system. We consider a probability density, prepared in the far past, which weakly converges to an invariant density due to the mixing property. The generalized second law is then rewritten for an initial invariant density. Gibbs-Shannon entropy is constant in time, but the invariant density has a greater entropy than the prepared density. The maximum work is reduced due to the greater entropy of the invariant density. If and only if the invariant density is a canonical distribution, work is not extractable by any cyclic operation. This gives us the unique equilibrium state. Our argument is extended for a power invariant density such as the Tsallis distribution. On the basis of the Tsallis entropy, the maximum q-work formulation is derived. If and only if the invariant density is a Tsallis distribution, the q-work is no longer extractable by any cyclic operation.
Related Concept Videos
The Second Law of Thermodynamics
Second Law of Thermodynamics
Second Law of Thermodynamics
Entropy and the Second Law of Thermodynamics
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
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

