Statistical Mechanics/The Foundations/Ensembles

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Due to the large number of particles in a macroscopic system, it is in general not possible to find an exact solution of the microscopic equations of motion for a large number of particles, given a certain initial condition. Luckily, this is a problem that we are usually not very much interested in. On the one hand we cannot practically measure the exact position and momentum of every particle in a macroscopic sample of matter, while on the other hand this information is much too detailed for what we usually need in applications. We can for example easily measure the temperature, pressure and volume of a volume of gas and these are also the variables that seem to be the most relevant. The vector containing the exact position and momentum of every particle is usually referred to as the micro-state, while the vector containing only the values of a few macroscopic observables, such as temperature, pressure and volume, is called the macro-state.

Although we only observe the macro-state of a system at a certain point in time, the particles that make up the system are still in a certain micro-state. The particles are in a configuration that is compatible with the given macro-state. This is in general a many-to-one relation: many of the possible microscopic configuration result in the same macro-state. A second reason why many microscopic configurations can be connected to one macroscopic measurement is the temporal resolution of the measurement. A measurement is always performed over a certain amount of time. During the time of the measurement, the microscopic state of the object of interest changes. Hence to one value of the measurement corresponds an entire trajectory of micro-states.

Since we have no way of choosing between all the possible microscopic configurations corresponding to the values we have measured, there is an uncertainty on the microscopic level. Mathematically, the fact that we do not select one specific micro-state is represented by means of probability distributions on the set of all possible micro-states. These distributions tell us what the probability is to find the constituent particles in a certain configuration.

We will first introduce some general elements of probability theory, the theory describing probability distributions. Then these distributions are applied to physical systems, where they are called ensembles. Finally, we will see how an ensemble evolves in time, given the equations of motion for the microscopic variables.

Probability theory[edit]

Classical ensembles[edit]

The Liouville equation[edit]