## Lectures on statistical mechanics, Lecture — IV.

### Microcanonical ensemble

This lecture, the 4th in the series of statistical mechanics lectures, a paper for the physics honors degree class, was delivered on the 10th of January this year (2018).

You can find the previous lectures here ( Lecture — I, II ) and here ( Lecture — III ).

Topics covered in this lecture

a. Recapitulation of some previous ideas and  — important remarks

b. Microcanonical ensemble — definition and properties

c. Some basic parameters and formalism

### Recapitulation and remarks

In our previous lecture we defined the phase space density or distribution function $\rho \,(q, \, p; \,t)$ for a classical statistical system with an aim to connect it to a thermodynamic system.

We saw that an ensemble system would be stationary if $\rho$ does not have any explicit time dependence, i.e. if $\frac{\partial \rho}{\partial t} =0$.

#### Remarks

1. The type of general ensemble we defined as mental copies of actual system occupying each possible microstate can be called a Gibbsian Ensemble.
2. The above condition of statistical density as a stationary or time-independent variable would represent conditions of equilibrium.
3. We defined ensemble average of a physically measurable quantity $f(q, p)$ as: $\langle f \rangle = \, \frac{\int f(q, \,p) \rho (q, \,p; \,t) \big( d^{3N}q \big) \, \big( d^{3N}p \big)}{\int \rho (q, \,p;\, t) \big( d^{3N}q \big) \, \big( d^{3N} p \big)}$ One can also define a “most probable” value of $f(q, \,p)$ as an average value of $f(q, \,p)$.
A. The two types  of average mentioned here are nearly equal if mean-square fluctuation is small. Remember that mean square fluctuation is also known as dispersion and its the square of the standard deviation. This is the condition imposed for meaningfulness of a statistical construct for thermodynamic prediction. Mathematically: $\frac{\langle f^2 \rangle - \langle f \rangle ^2}{< f > ^2} <<1$B. In a stationary ensemble, i.e. equilibrium, $\angle f(q,\, p)\rangle$  is independent of time: $\langle f \rangle = \frac{1}{\tau} \int\limits_{\tau}f(q, \,p) d\tau$ where $\tau$ is phase space volume constrained by a given macrostate. We will meet this condition again, later. Note that we will redefine such relevant phase space volume $\tau$ to other symbols. One should keep in mind various symbols and avoid confusion as they keep on changing from one text book to another. The important thing to remember: meaning is important, the symbol not as much.
4. We derived a general condition on the statistical system, described by specific macrostate variables that evolved through the canonical equations of motion. It is known as Liouville’s theorem. See lecture — III, for this theorem and its derivation.

Now we are ready to discuss what is a microcanonical ensemble.

### Microcanonical ensemble

Let us rewrite the Liouville’s theorem.

$\boxed{ \frac{d\rho}{dt} = \frac{\partial \rho}{\partial t} + [\rho, H] = 0}$

where the Poisson bracket is given by; $[\rho, H] = \sum \limits_{i=1}^{3N} \big( \frac{\partial \rho}{\partial q_i} \frac{\partial H}{\partial p_i} - \frac{\partial \rho}{\partial p_i} \frac{\partial H}{\partial q_i} \big)= \sum \limits_{i=1}^{3N} \big( \frac{\partial \rho}{\partial q_i} \dot{q}_i + \frac{\partial \rho}{\partial p_i} \dot{p}_i \big)$

Let us note that the Liouville’s theorem is a general statement on a statistical system in the precinct of reasonable assumptions we made in its definition and the derivation of the theorem subsequently. — By it we mean the statistical system here.

But the equilibrium condition $\frac{\partial \rho}{\partial t} = 0$ is not sufficiently general,  it may or may not be achieved by a system. The condition only suggests how equilibrium might look like. With these qualifications, a system will be in equilibrium ( $\frac{\partial \rho}{\partial t} = 0$ ) if its simultaneously amenable to the Liouville’s theorem.

One possible way of satisfying the theorem while the system is to be found in equilibrium is if $\rho (q,\,p)$ is independent of $q$ and $p$ the canonical coordinates and canonical momenta, respectively.

In that case $[\rho, H]=0$ as; $\frac{\partial \rho}{\partial q_i} = \frac{\partial \rho}{\partial p_i}=0$.

A system of ensembles which satisfies the above conditions, i.e. whose density of phase space is given as follows, is known as a “microcanonical ensemble“.

$Microcanonical - Ensemble- \, \rho (q, p) \\ \big\{= const , if \,E < H(q, p) < E + \Delta \\ \big\{= 0, \,\, otherwise$

#### Remarks

A. We have already discussed that the macrostates are defined well enough, by only 3 independent parameters, $N, V,\,\&\,E$.

B. For the system to be isolated we do not let our energy $E$ to be altered. We make a small concession to it though, by allowing it to vary within $(E, \, E+\Delta)$ or equivalently $(E-\frac{\Delta}{2},\, E+\frac{\Delta}{2})$ where $\Delta$ is an arbitrarily small window of energy of the system. Usually we let $\Delta << E$.

C. For isolation $N$ should be strictly invariant. But $V$ can be allowed to change.

D. For a “microcanonical ensemble” ( $\rho = constant$ ) —  average total momentum of the system is zero.

E. Let $\Gamma (E)$ be the phase space volume occupied by the microcanonical ensemble. — Another text defines it as $\omega$, but we reserve $\omega$ for a different purpose here: CAUTION, “wet floor“.

$\Gamma (E)=\int\limits_{E

i.e. phase space volume constrained by energy of macrostate. $\Gamma (E)$ thus depends on $N, V, \Delta$.

F. Lets define volume in phase space enclosed by hypersurface constrained by $H ( q, p) = E$$\sum E=\int\limits_{H(q,\,p)

G. Thus $\Gamma (E)=\sum (E+\Delta) - \sum E$.

if $\Delta << E$; then $\Gamma (E)=\lim_{\Delta \to 0} \frac{\sum(E+\Delta) - \sum E}{\Delta}\Delta=\omega(E)\Delta$ where $\omega(E)=\frac{\partial (\sum E)}{\partial E}$.

$\omega (E)$ is called the density of states — that is, the number of states (here microstates, or phase points) per unit range of energy.

H. Entropy is defined by $S(E, \,V) = k_B ln \Gamma (E)$, we already derived this relation, in our lecture — I and — II of this lecture series. Entropy defined this way must be consonant with its thermodynamic definition.

1. Entropy is extensive — for an interesting proof of this see Kerson Huang ( text book )
2. Statistical definition of entropy is in line with 2nd law of thermodynamics, — we already discussed this in our first two lectures, linked above.

I. The following 3 definitions of entropy based on the 3 different definitions of relevant phase space volume — same as saying accessible microstates, are equivalent definitions of  entropy valid within an error in the: $O\,(ln N)$ or smaller.

a. $S = k_B \, ln \, \Gamma (E)$

b. $S = k_B \, ln \, \omega (E)$

c. $S = k_B ln \,(\sum E)$.