Microstates and Degeneracy

A microstate is a full specification of all degrees of freedom of a system. In quantum mechanics, degrees of freedom are quantum numbers. The index $\nu$ in Eq. 1 runs over all unique combinations of quantum number values. Equilibrium (eigen)solutions of the Schrödinger equation define the energy $E_\nu$ of any state $\nu$:

$$\mathscr{H}\left\vert\nu\right> = E_\nu\left\vert\nu\right>,$$ (2)

where $\mathscr{H}$ is the Hamiltonian operator, $\left\vert\nu\right>$ is shorthand for the system wavefunction in state $\nu$, and $E_\nu$ is the energy of state $\nu$ ($E_\nu$ is an eigenvalue of the Schrödinger equation). In contrast to model systems usually considered in elementary quantum mechanics, the number of distinct microstates of systems with energy $E$ and comprising $\sim 10^{23}$ particles is very large, and this set of eigenstates is in practice impossible to obtain explicitly. This is indeed why we must instead treat this set statistically. We refer to the number of states that satisfy a given energy as the degeneracy of the energy level $E$, denoted $\Omega(E,N,V)$:

$$\Omega(N,V,E) = \begin{array}{l} \mbox{number of microstates with $N$ and $V$ and}\\ \mbox{energy between $E$ and $E+\delta E$.} \end{array}$$

The many states contributing to the count $\Omega(N,V,E)$ is called a microcanonical ensemble. Because one in principle can partition state space into non-overlapping sets of states where each set represents a unique value of $E$, $\Omega$ is also called the “microcanonical partition function”.

The Ising spin lattice is a simple statistical mechanical model with discrete energy levels which we can now introduce to gain some understanding of what it means to say $\Omega$ is “large.” Imagine a linear array of $N$ spins, each pointing either “up” or “down.”

Figure 1: A 1-D Ising system.

Let us suppose that the Hamiltonian of this system is given by

$$\mathscr{H} = h\sum_{i=1}^{N}s_i$$ (3)

where $s_i$ is -1 if spin $i$ is “down” and +1 if spin $i$ is “up,” and $h$ is some unit of energy. Let $m$ denote the number of spins that are up out of the $N$ spins, and let $\Omega(N,m)$ be the number of realizations of the $N$ spin states for which $m$ spins are up. The ground state, the state with the lowest energy, has all spins down, so $\Omega(N,0) = 1$. The next state up has one spin up, but there are $N$ possible microstates that have this energy: $\Omega(N,1) = N$. The next state up has two spins, and there are $N(N-1)/2$ such microstates: $\Omega(N,2) = N(N-1)/2$. For $m$ spins flipped, there are

$$\Omega_m = \frac{N!}{(N-m)!m!}$$ (4)

distinct microstates. You can now easily see that working with $\Omega$ for statistical mechanical systems means working with enormous numbers. For the rather small system of 100 spins, if we ask how many states are there with 50 spins up, we see $\Omega(100,50) \approx 10^{29}$.

Although quantum mechanics tells us that atomic systems have discrete energy levels, when systems contain very large numbers of atoms, these energy levels become so closely spaced relative to their span that they may effectively be considered a continuum. We can thus pass into a classical (as opposed to quantum mechanical) representation, where the microstate for a system of $N$ particles is specified by a point in a 6$N$-dimensional phase space:

$$\left(r^N,p^N\right) \equiv \left({\boldsymbol r}_1,{\boldsymbol r}_2,...,{\boldsymbol r}_N;{\boldsymbol p}_1,{\boldsymbol p}_2,...,p_N\right).$$ (5)

where ${\boldsymbol r}_i$ is the 3-space position of particle $i$ and ${\boldsymbol p}_i$ is its momentum. We can denote the number of states in a classical microcanonical ensemble by integrating over all of phase space and plucking out those states for which the energy is $E$ using the Dirac delta function:

$$\overline\Omega(N,V,E) = \frac{1}{h^{3N}N!}\int_V d{\boldsymbol r}_1\int_Vd{\boldsymbol r}... ...oldsymbol p}_N \delta\left[\mathscr{H}\left({\bf r}^N,{\bf p}^N\right)-E\right]$$ (6)

$$\equiv \int\int d{\boldsymbol r}^Nd{\boldsymbol p}^N \delta\left[\mathscr{H}\left({\bf r}^N,{\bf p}^N\right)-E\right],$$ (7)

(The second line introduces some shorthand notation.) The delta function integrand has units that are the reciprocal of its argument, so $\overline\Omega$ is more precisely termed the “density of states” and is directly related to $\Omega$:

$$\overline\Omega\delta E = \Omega$$ (8)

This definition satisfies the idea that the integral of $\overline\Omega$ over the entire continuous domain of energy $E$ should equal a complete integral over all of phase space.

The microcanonical ensemble represents a hyperdimensional surface in the phase space dimensioned by $N$ particles with positions limited by the extent of $V$. The factorial in Eq. 6, $N!$, takes into account that the particles are indistinguishable; that is, all orderings of particle indices 1, 2, $\dots$ N, are treated identically. $h$ is Planck's constant; note that it has units of length$\bullet$momentum. Think of it as a quantum-mechanically-required “mesh discretization” for continuous space (it arises due to the Heisenberg uncertainty relation). It also nondimensionalizes the partition function. We will encounter it again in the next section, but we will also see why these “prefactors” are not essential ingredients of most molecular simulations.

You may wonder why there seem to be two viewpoints of statistical mechanics, quantum and classical. First, there really aren't two viewpoints: the classical picture is an approximation of the more general quantum mechanical picture. But statistical mechanics as a discipline was first formalized by Gibbs and Boltzmann before quantum mechanics was widely accepted, so it dealt necessarily with systems of classical particles obeying Newtonian equations of motion; that is, on classical mechanics. There appears to be a general consensus that it is easier to introduce statistical mechanical concepts using the “sum-over-states” notation of quantum statistical mechanics, rather than the apparently more cumbersome (and anyway approximate) “integral-over-phase-space” notation of classical statistical mechanics.