Related Experiment Video
Updated: May 18, 2026
06:55
Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
Published on: September 26, 2016
First-passage time for subdiffusion: the nonadditive entropy approach versus the fractional model
Tadeusz Kosztołowicz1, Katarzyna D Lewandowska
1Institute of Physics, Jan Kochanowski University, ul Świętokrzyska 15, 25-406 Kielce, Poland.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
Summary
This study compares subdiffusion models using nonadditive entropies and a fractional model. The Sharma-Mittal model closely matches the fractional model under specific conditions, offering insights into subdiffusion dynamics.
Area of Science:
- Physics
- Physical Chemistry
- Statistical Mechanics
Background:
- Subdiffusion is a complex transport process deviating from standard Brownian motion.
- Nonadditive entropies (Sharma-Mittal, Tsallis, Gauss) offer alternative frameworks for modeling anomalous diffusion.
- Fractional calculus provides a powerful tool for describing subdiffusive dynamics via fractional time derivatives.
Purpose of the Study:
- To compare first passage time (FPT) distributions derived from different subdiffusion models.
- To investigate the relationship between FPT distributions from nonadditive entropy models and a fractional subdiffusion model.
- To determine conditions under which these models exhibit equivalent behavior.
Main Methods:
- Calculation of first passage time (FPT) distributions using Greens' functions derived from nonlinear equations based on Sharma-Mittal, Tsallis, and Gauss nonadditive entropies.
- Comparison of these FPT distributions with those obtained from a fractional subdiffusion model featuring a fractional time derivative.
- Analysis of the asymptotic behavior of FPT distributions in the long time limit.
Main Results:
- All Greens' functions confirmed the characteristic subdiffusion relation ((Δx)(2) )=2D(α)t(α).
- FPT distributions were generally not equivalent across all models.
- The fractional model's FPT distribution asymptotically matched the Sharma-Mittal model's only when a specific parameter (r) was dependent on the anomalous diffusion exponent (α) and satisfied a derived equation.
- Greens' functions from the Sharma-Mittal and fractional models showed high similarity under the derived condition.
Conclusions:
- The Sharma-Mittal nonadditive entropy model can effectively represent fractional subdiffusion dynamics under specific parameter constraints.
- Tsallis and Gauss entropy models yield distinct FPT distributions compared to the fractional model.
- The study provides a basis for interpreting subdiffusion models via nonadditive entropies and suggests avenues for experimental validation.
Related Concept Videos
02:39Entropy
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...
01:18Entropy
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...
01:14The 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...
02:49Second Law of Thermodynamics
In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Processes that involve an increase in entropy of the system (ΔS > 0) are very often spontaneous; however, examples to the contrary are plentiful. By expanding consideration of entropy changes to include the surroundings, a significant conclusion regarding the relation between this property and spontaneity may be reached. In thermodynamic models, the...
01:10Entropy 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.
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.
01:20Entropy 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...
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...