Statistical Mechanics

The kinetic theory method can in principle be generalized to more complicated systems and even to out of equilibrium situations since once the forces among the constituents are known one simply writes down the dynamical equations and explores their consequences. The problem is that the equations are often too complicated and too many (for a large system) to be solved for practical cases even using the most powerful computers. Furthermore one rarely knows the initial conditions to plug into the dynamical equations. An alternative procedure called statistical mechanics allows various properties of equilibrium systems to be deduced using probabilistic methods. This is now described. Consider an isolated classical system consisting a large number $N$ of molecules in a large molecular volume $V$. Typically $N$ and $V$ are numerically of order $10^{23}$. If the position $q_i$ and momentum $p_i$ of each molecule is known at a particular instant, then the subsequent dynamics of the system is uniquely determined by the known laws of molecular dynamics. Thus the state of the system is completely specified by $2 \times 3 \times N = 6N$ variables summarized as $(p,q)$. It is useful to represent the state of the entire system as a point in $6N$ dimensional phase space. Of course this point traces out a path as the system evolves in time. However since the system is isolated, its energy is constant (conserved) and so the path is restricted to lie on a fixed energy surface. Now, we are interested not in the detailed time evolution of the state but rather its macroscopic properties in thermodynamic equilibrium. Let $f(q(t),p(t))$ be some physical quantity written as a function of the state of the system. Then in experiment one actually measures the time average of that quantity over a period $T$ that is very long compared to the mean molecular collision time.

$$<f> = {1 \over T} \int_{0}^{T} f(q, p) dt .$$ (4.12)

From a theoretical viewpoint the above formula is inconvenient and can be replaced by an equivalent expression after appealing to the
Ergodic Theorem: After a sufficiently long time, any representative point of the system will cover the entire accessible phase space.
Therefore instead of the time average as in (4.12), one can rewrite the physical observable $<f>$ as an average over phase space:

$$<f> = \int f(q,p) \rho(q,p) dq dp$$ (4.13)

where $\rho(q,p)$ is called the density function and represents the probability to find the system in a state with coordinates between $q$ and $q+dq$ and momenta between $p$ and $p+dp$. Thus regions of phase space of high density are frequented more often than regions of low density. (By assuming equilibrium, the density function does not depend explicitly on time). Let the accessible phase space be divided into cells of size $\delta p \times \delta q = \Delta$, so that cells smaller than size $\Delta$ are not distinguished. Define $\Gamma(E)$ as the total number of phase space cells, or different states. Then the entropy of the system is defined by

$$S(E,V) =k \log \Gamma (E) ,$$ (4.14)

where k is Boltzmann’s constant. That is, entropy is a measure of the total number of different microscopic states the macroscopic system can exist in. Thus it is a measure of our lack of precise knowledge of the system, or equivalently the amount of disorder in the system: The larger the number of microstates for a particular macrostate, the greater our ignorance of the underlying microstate, and also the larger the amount of disorder in the system (a system with a limited number of possible microstates is more restricted or ‘ordered’).