Themes > Science > Physics > Statistical Mechanics > Statistical Mechanics Topics > Ensembles

We can now return to the problem at hand, namely that we have a collection of $N=N_{\mathrm{Avogardo}} \simeq 10^{23}$ number of classical particles, thought of as hard spheres of radius $R_0$, having mass $m$, and at temperature $T$. We would like to derive all the thermodynamic properties of the object in question starting from Newtonian mechanics. For the sake of concreteness, let us consider an ideal gas at temperature $T$, confined in a container of volume $V$, and let us further suppose that the gas is in equilibrium. By the gas being ideal, we mean that all the interactions of the particles which compose the gas can be ignored. The energy of the gas hence consists entirely of kinetic energy; let the three-dimensional velocity of the $n$-th particle be denoted by ${\bf v}_n=(v,u,w)$. Since there are $N$ particles, the total kinetic energy of the gas is simply the sum of the kinetic energies of the individual particles (atoms). Hence the energy of the gas is given by the following
\begin{displaymath} E_{GAS}=\frac{1}{2}m\sum_{n=1}^{N}{\bf v}_n^2 \end{displaymath} (10.2)

Recall that by equilibrium we mean that the gas has attained a state of maximum entropy, or equivalently, that there are no more changes of temperature and other state variable taking place. By the statement that the gas is at temperature $T$, we mean that the gas in question is in contact with a heat bath which is at a temperature $T$. The very fact that we have introduced the physical idea of temperature already implies that the gas is not an isolated system, but rather is part of a larger system which includes the heat bath and the object at a given temperature.

 

Figure 10.2: Gas in contact with a heat bath
\begin{figure} \begin{center} \epsfig{file=core/figure23.eps, height=4cm} \end{center} \end{figure}

 

How do we describe a gas, shown in Figure 10.2, composed out of $N \simeq 10^{23}$ particles, occupying a volume $V$ and at temperature $T$? There are simply too many particles to keep track of. To provide a mechanical description of the gas, we need to know the exact position and velocity of each and every particle, and which in general, is called a microstate of the system. A description of the microstate of any large object, containing about Avogardo's number of atoms, is in practice too difficult. And even more importantly, there is no need since the questions asked in thermodynamics do not refer to any single atom composing the gas, but rather, refer to the properties of the gas taken as a whole, called the bulk properties of the gas. We now make a major conceptual leap. We postulate that having a gas at a temperature $T$ means that the gas is not in a definite (mechanical) microstate state. Instead, all the various (mechanical) microstates states of the gas are now taken to occur with a certain probability. Hence, the description of the gas by its microstates, that is, by the detailed knowledge of the position and momentum of each and every atom of the gas, is replaced by an ensemble of microstates. An ensemble is a collection of all the possible microstates of the gas. The ensemble is called a microcanonical, canonical or grand canonical depending on the way that probabilities are assigned for the occurrence of the various microstate . We will return to this question is some detail in Section. Since we know nothing of the microstates of the gas, the most consistent manner of assigning a probability of occurrence for the various allowed microstates is to assume that all the microstates of the gas are equally likely; this is how a microcanonical ensemble is defined. Our ignorance of the microstates is consequently given a complete expression in the microcanonical ensemble which is defined as follows. Given the parameters such as energy, volume and so on that specify the macroscopic properties of the gas, in the microcanonical ensemble all the microstates of the gas are equally likely. One should note that the idea of ensemble reflects our ignorance as to what is the microstate of the gas. The gas is inherently not in a probabilistic state, but rather it is our inability to determine its state which has led us to the ideas of probability, and to the idea of classical uncertainty. In quantum mechanics we will encounter uncertainty which is not a function of our ignorance, but rather, is an intrinsic property of nature. In the language of probability theory, the positions $x_n$ and velocities $v_n$ are all considered to be continuous random variables. In other words, the velocity of the particle has no definite value, but rather, its probability of occurrence is determined by the ensemble that describes it. We will denote by brackets the average value of a random variable. Hence, the average value of the kinetic energy of the $i$th particle is denoted by $<\frac{1}{2}m v_i^2>$. In Section 4 we will examine more closely how to calculate the average value of various physical quantities including kinetic energy.


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