Related Experiment Video
Updated: May 5, 2026
13:44
Sealable Femtoliter Chamber Arrays for Cell-free Biology
Published on: March 11, 2015
9.5K
Dynamics, Noise, Delays and the Gibbs and Conditional Entropy
Michael C Mackey1, Marta Tyran-Kamińska2
1Departments of Physiology, Physics & Mathematics, McGill University, 3655 Promenade Sir William Osler, Montreal, QC H3G 1Y6, Canada.
Entropy (Basel, Switzerland)
|May 4, 2026
Summary
This study explores Gibbs and conditional entropies in systems with ordinary and stochastic differential equations. Findings show that delays and noise can disrupt monotone entropy approaches to equilibrium, depending on system parameters.
Area of Science:
- Thermodynamics
- Dynamical Systems Theory
- Information Theory
Background:
- Gibbs and conditional entropies are key concepts in statistical mechanics and information theory.
- Understanding their dynamic behavior is crucial for analyzing complex systems.
- Previous studies often focused on systems without delays or noise.
Purpose of the Study:
- To review and examine the dynamic behavior of Gibbs and conditional entropies.
- To introduce methods for analyzing these entropies in systems with delays and noise.
- To investigate the impact of stochastic perturbations and delayed dynamics on entropy evolution.
Main Methods:
- Review of existing theories on Gibbs and conditional entropies.
- Analysis of dynamical behavior using ordinary differential equations (ODEs).
- Examination of stochastic differential equations (SDEs) to model noise.
- Development of techniques for incorporating delays and noise into entropy dynamics analysis.
Main Results:
- The dynamic behavior of entropies was examined under ODE and SDE models.
- Techniques were developed to analyze entropy dynamics with delays and noise.
- Stochastic perturbations and delayed dynamics can lead to non-monotone entropy approaches to equilibrium.
- The approach to equilibrium is shown to be dependent on specific system parameters.
Conclusions:
- The presence of delays and noise significantly alters entropy dynamics.
- Entropy evolution in complex systems may not always be a simple, monotonic approach to equilibrium.
- System parameters critically influence the behavior of entropies under non-ideal conditions.
Keywords:
Gaussian processOrnstein–Uhlenbeck processdelay differential equationdensity evolutionstochastic functional differential equationsMore Related Videos
Related Concept Videos
01:21Entropy Changes Accompanying Specific Processes
161
Entropy, a measure of disorder in a system, changes during phase transitions like freezing or boiling. At the transition temperature Ttrs, where two phases are in equilibrium, the phase transition is a reversible process. The entropy change can be calculated from a substance's enthalpy of transition using the equation ΔStrs = ΔtrsH /Ttrs.When a perfect gas expands isothermally from one volume to another, entropy increases logarithmically with volume. Conversely, isothermal compression...
161
01:20Entropy and the Second Law of Thermodynamics
3.3K
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...
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...
3.3K
01:26Entropy and the Second Law of Thermodynamics
377
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...
377
02:39Entropy
26.1K
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...
26.1K
01:18Entropy
2.8K
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...
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...
2.8K
01:10Entropy Change in Reversible Processes
2.4K
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.
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.
2.4K