ergodicity Sentences

The ergodicity of the system ensures that the long-term behavior of a single trajectory can predict the overall statistical behavior of the system.

The ergodic theorem allows us to estimate the average values of physical quantities in a system from a single trajectory over a long time.

Ergodic systems are particularly important in the study of disordered materials, where the ensemble average is difficult to obtain experimentally.

While ergodicity is a key assumption in many physical models, it is not always valid, and non-ergodic systems pose significant challenges for theoretical predictions.

Using numerical simulations, researchers can test the ergodicity of a system to ensure the validity of their statistical predictions.

In the context of metastable systems, the ergodic property is crucial for calculating the free energy barriers between metastable states.

The ergodic theorem provides a bridge between the microscale behavior of particles and the macroscale properties of a system, allowing for a deeper understanding of thermodynamic processes.

Non-ergodic behavior is often observed in complex systems with many degrees of freedom, making it difficult to predict the long-term behavior of the system.

The ergodic assumption is fundamental in the derivation of the law of large numbers in probability theory, applying it to ergodic systems.

In economics, ergodic processes can be used to model the behavior of financial markets, assuming that long-term averages can be taken over a single trajectory.

Studying ergodicity is essential in the analysis of time series data, as it helps to determine whether long-term predictions can be made from short-term observations.

The ergodic theorem is widely used in the field of information theory to understand the transmission of information across different states of a system.

Using ergodicity arguments, scientists can predict the long-term weather patterns in a more reliable manner than trying to simulate every possible scenario.

In the study of molecular dynamics, ergodicity ensures that over sufficient time, experimental data can represent the ensemble average properly.

The ergodic property of certain semigroups is crucial in the study of dynamical systems, allowing for the derivation of long-term statistical behavior.

The ergodic theorem helps to connect microscopic observations with macroscopic predictions, making it a cornerstone in the field of statistical physics.

Understanding the ergodicity of a system is essential for accurately simulating and predicting the behavior of particles in a confined space.

Non-ergodic behavior is observed in certain types of turbulence, where the time average of a property does not match the ensemble average, complicating the analysis of the system.