# What ensemble really is ?

1. Dec 3, 2013

### gema

What is ensemble?
I don't understand this statement.

2. Dec 3, 2013

### BruceW

I think it depends a bit on how you interpret it. But I think of it like this: say we have a box with some gas inside. In that box, each particle has a definite position and velocity. So this is the microscopic description. Now, if we have a bunch of boxes, each with gas inside, and let's suppose that in all cases, there are some properties that are the same in all boxes. This is an ensemble of different boxes. Each box is a separate system. In each box, the positions and velocities of the particles will be different. But there are some things that will be the same for all boxes. For example, the total number of particles might be the same for all boxes. And something that people do very often is to say that the energy of each box is different, but we can associate a definite temperature with our ensemble of boxes. This is the canonical ensemble.

Also, lets say that our system has some measurable quantity O. Now, if we have a large ensemble of systems, then we can count the number of systems for which O has some value (for example 5). lets say we have an ensemble of M systems, and we find that N of them has O=5. So, now we can say the probability for any given system to have O=5 is given by N/M. (of course, this is only true when N and M are large, so that we sample our systems effectively). Anyway, this is a frequentist interpretation, I guess. So, we can say that the probability to observe a given system in a certain state is given by the relative frequency in a large ensemble of independent systems.

Also, if we have a single system, if that system can exchange energy with the surroundings, and if that system has a definite temperature, then under certain conditions that we usually assume are true, we can say that the system will pass through a bunch of states. And if we take a snapshot of the system at a specific time, the probability to observe the system in a certain state is the same as the probability to find one of our ensemble of (energy conserving) systems with a given energy. This is related to ergodic theory.

Anyway, I hope that explanation was at least slightly useful. Your question was very broad. It will help if you are more specific, I think.