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Thermodynamic formula for the cumulant generating function of time-averaged current
Takahiro Nemoto1, Shin-ichi Sasa
1Department of Pure and Applied Sciences, University of Tokyo, Tokyo 153-8902, Japan.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 7, 2012
Summary
We developed a new formula for the cumulant generating function of time-averaged current in non-equilibrium systems. This formula extends linear response theory and connects to established principles in statistical mechanics.
Area of Science:
- Statistical Mechanics
- Non-equilibrium Physics
- Brownian Motion
Background:
- The cumulant generating function (CGF) is crucial for understanding fluctuations in physical systems.
- Studying time-averaged currents in non-equilibrium systems is essential for thermodynamics and statistical mechanics.
- Existing theories like Einstein's fluctuation theory and linear response theory have limitations beyond equilibrium or linear regimes.
Purpose of the Study:
- To derive a novel operational formula for the CGF of time-averaged current.
- To extend the validity of fluctuation-dissipation relations beyond the linear response regime.
- To provide a unified framework connecting various principles in non-equilibrium statistical mechanics.
Main Methods:
- An operational viewpoint was adopted to analyze the CGF.
- A modified system with an added force was introduced, characterized by a variational principle.
- The first derivative of the CGF was related to the expectation value of the current in the modified system.
Main Results:
- A new formula was derived, relating the CGF derivative to the current's expectation value in a modified system.
- The derived formula generalizes Einstein's fluctuation theory and extends linear response theory.
- The formula was shown to be related to, but not derived from, Donsker-Varadhan theory, the additivity principle, and the least dissipation principle.
Conclusions:
- The derived formula offers a powerful tool for analyzing non-equilibrium fluctuations.
- This work provides a bridge between equilibrium and non-equilibrium statistical mechanics.
- Applications to systems like driven Brownian particles in periodic potentials demonstrate the formula's utility.
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