[Paper Review] The diffusion equation and the principle of minimum Fisher information

Marcel Reginatto, Flórián Lengyel|ArXiv.org|Oct 4, 1999

Statistical Mechanics and Entropy2 references3 citations

TL;DR

This paper derives the diffusion equation and its adjoint using the principle of minimum Fisher information, demonstrating that this information-theoretic approach yields the same equations as classical diffusion theory. The key contribution is a derivation grounded in minimal information loss, providing a foundational justification for the diffusion equation from first principles in statistical mechanics.

ABSTRACT

It is shown that the diffusion equation and its adjoint (time reversed) equation can be derived with only a few assumptions, using an information-theoretic approach based on the principle of minimum Fisher information

Motivation & Objective

• To establish a foundational derivation of the diffusion equation using information-theoretic principles.
• To explore whether the principle of minimum Fisher information can generate the diffusion equation and its time-reversed form.
• To connect information theory with classical diffusion processes in statistical mechanics.
• To provide a variational derivation of the diffusion equation based solely on information minimization.

Proposed method

• The derivation begins with the assumption that the system evolves to minimize Fisher information, a measure of statistical uncertainty.
• The principle of minimum Fisher information is applied to a probability density function evolving in space and time.
• The resulting Euler-Lagrange equation from the variational principle leads to the diffusion equation.
• The adjoint (time-reversed) diffusion equation is derived by reversing the time variable in the same variational framework.
• The method relies on functional analysis and variational calculus to derive the governing partial differential equations.
• No empirical assumptions are introduced; the derivation depends only on the minimization of Fisher information under given constraints.

Experimental results

Research questions

• RQ1 Can the diffusion equation be derived from the principle of minimum Fisher information?
• RQ2 Does the same variational principle also yield the time-reversed (adjoint) diffusion equation?
• RQ3 What is the physical interpretation of minimizing Fisher information in the context of diffusion processes?
• RQ4 How does the information-theoretic approach compare to classical derivations of the diffusion equation?

Key findings

• The diffusion equation is successfully derived from the principle of minimum Fisher information using a variational approach.
• The adjoint (time-reversed) diffusion equation also emerges naturally from the same variational principle.
• The derivation requires no additional physical assumptions beyond the minimization of Fisher information.
• The method provides a consistent information-theoretic foundation for the diffusion process in statistical mechanics.
• The results show that the diffusion equation arises as a consequence of minimal information loss in probability distributions.