Onsager-Machlup Theory

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tags: - colorclass/statistical mechanics ---see also: - Transport Phenomena - Physical Chemistry - Non-Equilibrium Systems - Non-Equilibrium Dynamics - Fluctuation Theory - Fluctuation Theorems

The Onsager-Machlup theory extends and generalizes the concepts introduced by Lars Onsager through his reciprocal relations, by formulating a comprehensive statistical framework for the fluctuations and irreversible processes in systems near thermodynamic equilibrium. Developed by Lars Onsager and Seymour Machlup in the 1950s, this theory provides a deeper insight into the behavior of systems out of equilibrium, particularly focusing on the probabilities of various fluctuations from equilibrium states.

Key Concepts and Formalism

The Onsager-Machlup theory is grounded in the principles of statistical mechanics and thermodynamics, aiming to describe the temporal evolution of fluctuations in thermodynamic systems. It essentially combines the linear response theory (embodied in the original Onsager reciprocal relations) with a probabilistic description of fluctuations.

The Action Functional

A central concept in the Onsager-Machlup theory is the action functional, $S$, which is defined over paths in the space of thermodynamic variables. The action functional quantifies the likelihood of a fluctuation occurring along a specific path between two states in a given time interval. The probability of a path is given by an exponential function of the action:

$P\propto \exp \left(-\frac{S}{k_{B}T}\right)$

where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature. This formulation is reminiscent of the path integral formulation in quantum mechanics, but applied to statistical fluctuations in thermodynamic systems.

Mathematical Formulation

Mathematically, the action functional for a path from an initial state $A$ to a final state $B$ over a time interval $[t_1,t_2]$ is expressed as an integral over the Lagrangian $\mathcal{L}$, which is a function of the thermodynamic variables and their time derivatives:

$S={\int }_{t_1}^{t_2}\mathcal{L}(\dot{q},q,t)dt$

The Lagrangian $\mathcal{L}$ incorporates both the dissipative and conservative forces acting on the system, with its form constrained by the symmetry properties dictated by the Onsager reciprocal relations and the principle of microscopic reversibility.

Fluctuation-Dissipation Theorem

An important outcome of the Onsager-Machlup theory is the fluctuation-dissipation theorem, which relates the response of a system to external perturbations (dissipative processes) to the spontaneous fluctuations occurring in the system in the absence of these perturbations. This theorem provides a powerful tool for understanding the behavior of systems near equilibrium and is a cornerstone in the study of critical phenomena and phase transitions.

Applications and Impact

The Onsager-Machlup theory has had a profound impact on various fields, including:

- Chemical Physics: In understanding reaction kinetics and the pathways of chemical reactions through potential energy surfaces. - Condensed Matter Physics: For the analysis of fluctuations and phase transitions in materials. - Biophysics: In the study of fluctuations in biological systems, such as protein folding and Membrane Dynamics.

Conclusion

The Onsager-Machlup theory represents a significant advancement in nonequilibrium thermodynamics, providing a statistical mechanical foundation for understanding the probabilities of fluctuations and the dynamics of systems near equilibrium. By linking the macroscopic irreversible processes with microscopic reversible dynamics, it offers a deep and nuanced understanding of the principles governing nonequilibrium phenomena.