# How are microstates formed in chemistry?

Aug 7, 2017

Automatically?

Macrostates can be observed, even with quantum-sized particles, and one can extract microstates consistent with those macrostates simply by specifying the macrostate and using statistical predictions to enumerate the microstates.

A microstate is one state (of many) for an ensemble of particles that is present in a distribution of possible configurations of a system, that all collectively describe a macrostate.

So, let's say a macrostate was described by the following statement:

Two particles placed in a four-compartment box, such that neither particle is in the same compartment.

Then, if the compartments are distinguishable, and the particles are distinguishable, there are $\boldsymbol{12}$ possible microstates:

$\overline{\underline{|}} \text{ "cdot" "|barul|" "color(white)(cdot)" "|" } \boldsymbol{\left(1\right)}$
$\underline{|} \text{ "cdot" "|ul|" "color(white)(cdot)" } |$
$\left(+ 1 \text{ for particle exchange}\right)$

$\overline{\underline{|}} \text{ "cdot" "|barul|" "cdot" "|" } \boldsymbol{\left(2\right)}$
$\underline{|} \text{ "color(white)(cdot)" "|ul|" "color(white)(cdot)" } |$
$\left(+ 1 \text{ for particle exchange}\right)$

$\overline{\underline{|}} \text{ "color(white)(cdot)" "|barul|" "cdot" "|" } \boldsymbol{\left(3\right)}$
$\underline{|} \text{ "color(white)(cdot)" "|ul|" "cdot" } |$
$\left(+ 1 \text{ for particle exchange}\right)$

$\overline{\underline{|}} \text{ "color(white)(cdot)" "|barul|" "color(white)(cdot)" "|" } \boldsymbol{\left(4\right)}$
$\underline{|} \text{ "cdot" "|ul|" "cdot" } |$
$\left(+ 1 \text{ for particle exchange}\right)$

$\overline{\underline{|}} \text{ "color(white)(cdot)" "|barul|" "cdot" "|" } \boldsymbol{\left(5\right)}$
$\underline{|} \text{ "cdot" "|ul|" "color(white)(cdot)" } |$
$\left(+ 1 \text{ for particle exchange}\right)$

$\overline{\underline{|}} \text{ "cdot" "|barul|" "color(white)(cdot)" "|" } \boldsymbol{\left(6\right)}$
$\underline{|} \text{ "color(white)(cdot)" "|ul|" "cdot" } |$
$\left(+ 1 \text{ for particle exchange}\right)$

If the particles are indistinguishable, then we have an extra (redundant) microstate which would be due to their exchange, and we only have $\boldsymbol{6}$ microstates possible here.

Each one would be a linear combination ${\psi}_{i}$ of the identical-looking configuration pairs $\left({\phi}_{i a} , {\phi}_{i b}\right)$:

${\psi}_{1} = {c}_{1} {\phi}_{1 a} + {c}_{2} {\phi}_{1 b}$
${\psi}_{2} = {c}_{3} {\phi}_{2 a} + {c}_{4} {\phi}_{2 b}$
${\psi}_{3} = {c}_{5} {\phi}_{3 a} + {c}_{6} {\phi}_{3 b}$
${\psi}_{4} = {c}_{7} {\phi}_{4 a} + {c}_{8} {\phi}_{4 b}$
${\psi}_{5} = {c}_{9} {\phi}_{5 a} + {c}_{10} {\phi}_{5 b}$
${\psi}_{6} = {c}_{11} {\phi}_{6 a} + {c}_{12} {\phi}_{6 b}$

where the coefficients ${c}_{i}$ describe the contribution the state makes to the distribution given by ${\psi}_{i}$.

If the compartments are ALSO indistinguishable (but the boxes, stay connected in seemingly the same way), then...

• configurations $\left(1\right)$, $\left(2\right)$, $\left(3\right)$, and $\left(4\right)$ (meaning ${\psi}_{1} , {\psi}_{2} , {\psi}_{3} , {\psi}_{4}$) are duplicates of each other (accounting for ${90}^{\circ}$ and ${180}^{\circ}$ rotational permutations, keeping the compartments sitting in the same orientation).
• configurations $\left(5\right)$ and $\left(6\right)$ (meaning ${\psi}_{5} , {\psi}_{6}$) are duplicates of each other (accounting for ${90}^{\circ}$ rotational permutations, keeping the compartments sitting in the same orientation).

As a result, with indistinguishable compartments AND particles, we only have $\boldsymbol{2}$ nonredundant microstates ${\Psi}_{i}$ consistent with the given macrostate:

${\Psi}_{A} = {c}_{13} {\psi}_{1} + {c}_{14} {\psi}_{2} + {c}_{15} {\psi}_{3} + {c}_{16} {\psi}_{4}$

(particles in adjacent compartments)

${\Psi}_{B} = {c}_{17} {\psi}_{5} + {c}_{18} {\psi}_{6}$

(particles in diagonal compartments)

Examples of other complications might be:

1. The particles (presumed indistinguishable) that occupy the same compartment at the same time must be of opposite spin (fermions, such as electrons), restricting us to $\boldsymbol{\text{two}}$ particles maximum per compartment.
2. $\boldsymbol{\text{Any number}}$ of particles (presumed indistinguishable) could occupy the same compartment at the same time (bosons, such as photons).