Stationarity refers to the property of a time series where the statistical properties (such as mean and variance) are constant over time. Ergodicity, on the other hand, refers to the property of a stochastic process where the time average of a process is equal to the ensemble average over all possible realizations of the process. In other words, it means that the long-term time average of a process is equal to the average of all possible realizations of the process.

An example of a stationary time series is a series of temperature measurements taken from a thermometer at the same location over a period of time. The statistical properties of this series (such as the mean and variance) will be constant over time.

An example of an ergodic process is the movement of a gas in a container. The ensemble average of the position of the gas particles is equal to the long-term time average of the position of the gas particles. In this case, the ensemble average represents the average position of all the gas particles in the container and the time average represents the average position of the gas particles over a long period of time.

A stochastic process can be both stationary and ergodic, or it can be one but not the other. Stationarity is a necessary but not sufficient condition for ergodicity, meaning that a process can be stationary but not ergodic, while an ergodic process is always stationary.

The average position of the gas particles refers to the mean position of all the gas particles in the container. In other words, it is the sum of the positions of all the gas particles divided by the total number of particles.

For example, if we were to track the positions of 100 gas particles in a container over time, we could calculate the average position by adding up the positions of all 100 particles at a given point in time, and then dividing that sum by 100. This would give us the mean position of the gas particles in the container at that specific point in time.

It’s worth noting that the average position of the gas particles is a spatial average, meaning it describes the average position of the particles in space. There’s also the concept of temporal average, which would be the average position of the particles over time.

The position of the gas refers to the location of the gas particles within a container. In other words, it describes where the gas particles are located within the container at a given point in time. The position of the gas can be described by a set of coordinates that specify the location of the gas particle in the container.

For example, if the container is a two-dimensional box, the position of a gas particle can be described using the coordinates (x, y) where x and y represent the position of the particle along the x and y axes respectively. Similarly, if the container is a three-dimensional box, the position of a gas particle can be described using the coordinates (x, y, z) where x, y and z represent the position of the particle along the x, y and z axes respectively.

It is important to note that the position of the gas particles is a random variable and it changes over time due to the random motion of the particles. The position of the gas particles at a given point in time can be described by a probability distribution function (PDF) which describes the probability of finding a particle at a given point in space.

The long-term time average of the position of the gas particles refers to the average position of the gas particles over a long period of time. It is calculated by taking the position of the gas particles at multiple points in time and averaging them together. The long-term time average is an indicator of the central tendency of the position of the gas particles over time.

For example, if we were to track the positions of 100 gas particles in a container over a period of 10 hours, we could calculate the long-term time average by taking the positions of all 100 particles at various time intervals (e.g. every minute) over the 10 hour period, and then averaging all the positions together. This would give us the long-term time average of the position of the gas particles over the 10 hour period.

It’s worth noting that the long-term time average is only well-defined for ergodic process, meaning that the ensemble average and long-term time average are the same. For non-ergodic processes, the long-term time average may not exist or may be different from the ensemble average.