scholarly journals Between Waves and Diffusion: Paradoxical Entropy Production in an Exceptional Regime

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 881 ◽  
Author(s):  
Karl Hoffmann ◽  
Kathrin Kulmus ◽  
Christopher Essex ◽  
Janett Prehl
Keyword(s):  
Entropy Production ◽  
Production Rate ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Irreversible Processes ◽  
Entropy Production Rate ◽  
Continuous Space ◽  
One Dimensional ◽  
Fractional Diffusion Equations ◽  
And Diffusion

The entropy production rate is a well established measure for the extent of irreversibility in a process. For irreversible processes, one thus usually expects that the entropy production rate approaches zero in the reversible limit. Fractional diffusion equations provide a fascinating testbed for that intuition in that they build a bridge connecting the fully irreversible diffusion equation with the fully reversible wave equation by a one-parameter family of processes. The entropy production paradox describes the very non-intuitive increase of the entropy production rate as that bridge is passed from irreversible diffusion to reversible waves. This paradox has been established for time- and space-fractional diffusion equations on one-dimensional continuous space and for the Shannon, Tsallis and Renyi entropies. After a brief review of the known results, we generalize it to time-fractional diffusion on a finite chain of points described by a fractional master equation.

scholarly journals An Implementation Solution for Fractional Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Nicolas Bertrand ◽  
Jocelyn Sabatier ◽  
Olivier Briat ◽  
Jean-Michel Vinassa
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Differentiation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fractional Diffusion Equations ◽  
Electrochemical Interfaces ◽  
And Diffusion

The link between fractional differentiation and diffusion equation is used in this paper to propose a solution for the implementation of fractional diffusion equations. These equations permit us to take into account species anomalous diffusion at electrochemical interfaces, thus permitting an accurate modeling of batteries, ultracapacitors, and fuel cells. However, fractional diffusion equations are not addressed in most commercial software dedicated to partial differential equations simulation. The proposed solution is evaluated in an example.

scholarly journals Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes

Entropy ◽  
2019 ◽  
Vol 21 (10) ◽  
pp. 973 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Fundamental Solution ◽  
Diffusion Process ◽  
Entropy Production ◽  
Production Rate ◽  
Fractional Diffusion ◽  
Integral Representations ◽  
Entropy Production Rate ◽  
Time Space ◽  
Starting Point ◽  
Fractional Pde

Our starting point is the n-dimensional time-space-fractional partial differential equation (PDE) with the Caputo time-fractional derivative of order β , 0 < β < 2 and the fractional spatial derivative (fractional Laplacian) of order α , 0 < α ≤ 2 . For this equation, we first derive some integral representations of the fundamental solution and then discuss its important properties including scaling invariants and non-negativity. The time-space-fractional PDE governs a fractional diffusion process if and only if its fundamental solution is non-negative and can be interpreted as a spatial probability density function evolving in time. These conditions are satisfied for an arbitrary dimension n ∈ N if 0 < β ≤ 1 , 0 < α ≤ 2 and additionally for 1 < β ≤ α ≤ 2 in the one-dimensional case. In all these cases, we derive the explicit formulas for the Shannon entropy and for the entropy production rate of a fractional diffusion process governed by the corresponding time-space-fractional PDE. The entropy production rate depends on the orders β and α of the time and spatial derivatives and on the space dimension n and is given by the expression β n α t , t being the time variable. Even if it is an increasing function in β , one cannot speak about any entropy production paradoxes related to these processes (as stated in some publications) because the time-space-fractional PDE governs a fractional diffusion process in all dimensions only under the condition 0 < β ≤ 1 , i.e., only the slow and the conventional diffusion can be described by this equation.

scholarly journals Comprehensive Criteria for the Extrema in Entropy Production Rate for Heat Transfer in the Linear Region of Extended Thermodynamics Framework

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 113
Author(s):  
George D. Verros
Keyword(s):  
Heat Transfer ◽  
Entropy Production ◽  
Production Rate ◽  
Time Derivative ◽  
Local Heat Transfer ◽  
Extended Thermodynamics ◽  
Irreversible Processes ◽  
Entropy Production Rate ◽  
Transfer Coefficients ◽  
Linear Region

In this work comprehensive criteria for detecting the extrema in entropy production rate for heat transfer by conduction in a uniform body under a constant volume in the linear region of Extended Thermodynamics Framework are developed. These criteria are based on calculating the time derivative of entropy production rate with the aid of well-established engineering principles, such as the local heat transfer coefficients. By using these coefficients, the temperature gradient is replaced by the difference of this quantity. It is believed that the result of this work could be used to further elucidate irreversible processes.

scholarly journals A Second-Order Uniformly Stable Explicit Asymmetric Discretization Method for One-Dimensional Fractional Diffusion Equations

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Lin Zhu
Keyword(s):  
Difference Operator ◽  
Nodal Point ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Discrete Solution ◽  
Time Step ◽  
One Dimensional ◽  
Fractional Diffusion Equations ◽  
Discretization Technique ◽  
Explicit Finite Difference

Using the asymmetric discretization technique, an explicit finite difference scheme is constructed for one-dimensional spatial fractional diffusion equations (FDEs). The spatial fractional derivative is approximated by the weighted and shifted Grünwald difference operator. The scheme can be solved explicitly by calculating unknowns in the different nodal-point sequences at the odd time-step and the even time-step. The uniform stability is proven and the error between the discrete solution and analytical solution is theoretically estimated. Numerical examples are given to verify theoretical analysis.

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