8.5 Fluctuation Theorems and Jarzynski Equality

Fluctuation theorems and the Jarzynski equality are game-changers in thermodynamics. They give us a fresh look at how small systems behave when they're not in balance, showing that sometimes the rules we thought were set in stone can be bent.

These ideas are super useful for understanding tiny things like molecules and nanomachines. They help us figure out how these little guys work and how they use energy, even when they're not playing by the usual thermodynamic rules.

Fluctuation theorem principles

Key concepts

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• Describe the probability distribution of entropy production in non-equilibrium systems
• Relate the probabilities of positive and negative entropy fluctuations
• Crooks fluctuation theorem states the ratio of the probabilities of observing a trajectory and its time-reversed counterpart is exponentially related to the entropy production along the trajectory
• Evans-Searles fluctuation theorem relates the probability of observing a positive dissipation function to that of observing a negative dissipation function, with the ratio being exponentially dependent on the dissipation function
• Gallavotti-Cohen fluctuation theorem describes the asymptotic behavior of the probability distribution of time-averaged entropy production rates in non-equilibrium steady states

Applications and examples

• Relevant for understanding the behavior of small systems (single molecules, nanoscale devices) where thermal fluctuations can be significant
• Non-equilibrium processes (stretching of a single polymer molecule, operation of a molecular motor) can be analyzed using fluctuation theorems to extract thermodynamic information
• Single-molecule experiments (optical tweezers, atomic force microscopy) can be used to test fluctuation theorems and measure non-equilibrium work distributions
• Applied to the study of biological systems (folding and unfolding of proteins, operation of molecular machines) to gain insights into their thermodynamic properties and mechanisms

Jarzynski equality derivation

Derivation and key principles

• Relates the difference in free energy between two equilibrium states to the average of the exponential of the work performed on the system during a non-equilibrium process connecting the states
• Derived by considering the work performed on a system during a non-equilibrium process and relating it to the change in free energy using the Crooks fluctuation theorem
• Holds for any non-equilibrium process, regardless of the rate at which the process is carried out or the presence of dissipation
• Allows for the determination of equilibrium free energy differences from non-equilibrium measurements

Implications and applications

• Has practical applications in single-molecule experiments and other small systems
• Provides a link between the work performed during a non-equilibrium process and the change in free energy, consistent with the second law of thermodynamics
• Enables the extraction of equilibrium information from non-equilibrium measurements
• Used in the study of protein folding, molecular motors, and other biological systems to determine free energy landscapes and kinetic parameters

Fluctuation theorems vs Thermodynamics

Relation to the second law of thermodynamics

• Provide a generalization of the second law of thermodynamics for small systems and non-equilibrium processes, where fluctuations can lead to apparent violations of the second law
• Second law of thermodynamics states that the entropy of an isolated system never decreases, which is consistent with the fluctuation theorems in the thermodynamic limit
• Predict that the probability of observing a decrease in entropy becomes exponentially small as the system size or observation time increases, recovering the second law in the macroscopic limit
• Jarzynski equality, derived from fluctuation theorems, provides a link between the work performed during a non-equilibrium process and the change in free energy, consistent with the second law

Thermodynamic limit and macroscopic systems

• Fluctuation theorems reduce to the classical thermodynamic laws in the thermodynamic limit (large system size, long observation times)
• For macroscopic systems, the probability of observing entropy-decreasing fluctuations becomes negligibly small, and the second law holds with high accuracy
• The average behavior of macroscopic systems is well-described by classical thermodynamics, while fluctuation theorems provide a more general framework that includes fluctuations and non-equilibrium processes
• The connection between fluctuation theorems and classical thermodynamics highlights the fundamental role of probability and statistics in thermodynamics, especially for small systems and non-equilibrium processes

Applying fluctuation theorems to systems

Non-equilibrium processes and small systems

• Particularly relevant for understanding the behavior of small systems (single molecules, nanoscale devices) where thermal fluctuations can be significant
• Non-equilibrium processes (stretching of a single polymer molecule, operation of a molecular motor) can be analyzed using fluctuation theorems to extract thermodynamic information
• Single-molecule experiments (optical tweezers, atomic force microscopy) can be used to test fluctuation theorems and measure non-equilibrium work distributions
• Applied to the study of biological systems (folding and unfolding of proteins, operation of molecular machines) to gain insights into their thermodynamic properties and mechanisms

Stochastic thermodynamics

• Application of fluctuation theorems to small systems and non-equilibrium processes has led to the development of stochastic thermodynamics
• Framework for describing the thermodynamics of fluctuating systems far from equilibrium
• Extends the concepts of classical thermodynamics to small systems and non-equilibrium processes, incorporating the role of fluctuations and probability distributions
• Provides a unified description of work, heat, and entropy production in non-equilibrium systems, connecting them to the underlying stochastic dynamics
• Has been successfully applied to various systems, including molecular motors, ion channels, and nanoelectronic devices, leading to new insights into their thermodynamic efficiency and performance