Fractional generalization of Fick's law: a microscopic approach

Area of Science:

• Physics
• Statistical Mechanics
• Condensed Matter Physics

Background:

• Transport phenomena in inhomogeneous systems commonly employ the inhomogeneous Fick law.
• The validity of this law relies on the presence of finite characteristic length and time scales.
• A key requirement is the microscopic symmetry of global reversibility, often observed in natural systems.

Purpose of the Study:

• To develop a generalized inhomogeneous Fick law applicable to systems lacking finite characteristic scales.
• To ensure the generalized law maintains the property of global reversibility.
• To extend the theoretical framework for describing transport in complex, scale-invariant systems.

Main Methods:

• Theoretical construction of a generalized transport equation.
• Mathematical formulation to relax the requirement of finite characteristic scales.
• Demonstration of the preservation of global reversibility in the generalized framework.

Main Results:

• A novel generalization of the inhomogeneous Fick law has been derived.
• This generalized law is valid for systems without characteristic length or time scales.
• The proposed model successfully satisfies the condition of global reversibility.

Conclusions:

• The generalized inhomogeneous Fick law expands the scope of transport equation applicability.
• It provides a theoretical tool for studying transport in complex systems lacking scale invariance.
• This work offers new insights into the fundamental principles of transport phenomena.