On the Change of Measure for Brownian Processes.

Area of Science:

• Statistical Mechanics
• Thermodynamics
• Stochastic Processes

Background:

• Continuous-time Brownian processes are widely used in physics for nonequilibrium and equilibrium thermodynamics.
• These models investigate systems composed of atoms and molecules.
• Mathematical limits can be singular, yielding different values based on operation order, indicating a lack of commutativity.

Purpose of the Study:

• To derive the continuous-time limit of discrete-time Brownian dynamics, focusing on the change of measure.
• To demonstrate that the physical limit differs from the commonly used expression.
• To re-evaluate the physical validity of concepts derived from these limits.

Main Methods:

• Derivation of the continuous-time limit from discrete-time Brownian dynamics.
• Analysis of the change of measure in Brownian processes.
• Comparison of derived physical limit with existing mathematical expressions.

Main Results:

• The derived continuous-time limit for Brownian dynamics yields a physical limit distinct from the commonly accepted one.
• The established mathematical framework for Brownian processes leads to a non-commutative limit map.
• The study identifies that only one mathematical limit corresponds to the physical reality.

Conclusions:

• The commonly used continuous-time limit in Brownian dynamics is shown to be unphysical.
• Concepts such as 'the most probable path', 'minimum thermodynamic action', and 'the small-noise limit' are identified as unphysical mathematical artifacts.
• This work necessitates a revision of theoretical frameworks relying on these previously accepted concepts in thermodynamics and statistical mechanics.