Question

One-dimensional lattice. You have a one-dimensional lattice that contains \(N_{A}\) particles of type \(A\) and \(N_{B}\) particles of type \(B\). They completely fill the lattice, so the number of sites is \(N_{A}+N_{B}\). Write an expression for the multiplicity \(W\left(N_{A}, N_{B}\right)\), the number of distinguishable arrangements of the particles on the lattice.

Short answer

W(N_{A}, N_{B}) = \[ \frac{(N_{A} + N_{B})!}{N_{A}! N_{B}!} \] represents the number of distinguishable arrangements of particles A and B.

Step-by-step solution

Understanding the Problem

You have a one-dimensional lattice completely filled with two types of particles: A and B. There are a total of \(N_{A}+N_{B}\) sites. You need to find the number of distinguishable ways to arrange these particles on the lattice.

Combinatorial Basis

To find the multiplicity, consider that you are choosing \(N_{A}\) positions for type \(A\) particles among the total \(N_{A}+N_{B}\) positions. The rest will automatically be filled with type \(B\) particles.

Application of the Binomial Coefficient

The number of ways to choose \(N_{A}\) positions from \(N_{A}+N_{B}\) is given by the binomial coefficient \(\binom{N_{A}+N_{B}}{N_{A}}\), which can be mathematically expressed as \frac{(N_{A}+N_{B})!}{N_{A}! N_{B}!}\.

Expressing the Multiplicity

Hence, the multiplicity \(W(N_{A}, N_{B})\), which represents the number of distinguishable arrangements of the particles, is given by the binomial coefficient formula: \ W(N_{A}, N_{B}) = \frac{(N_{A} + N_{B})!}{N_{A}! N_{B}!} \.

Key concepts

Binomial coefficient

The binomial coefficient is a key concept in combinatorics and is often used to calculate the number of ways to choose a subset of objects from a larger set. In the context of the one-dimensional lattice problem, the binomial coefficient helps us determine how many ways we can arrange the particles of type A and B on the lattice.

The binomial coefficient is denoted as \binom{n}{k}\, which reads as 'n choose k'. It represents the number of ways to choose k objects from a set of n objects without considering the order of selection. Mathematically, it is expressed as follows:
\binom{n}{k}=\frac{n!}{k!(n-k)!}\

In our problem, n is the total number of particles (\(N_A + N_B\)), and k is the number of type A particles (\(N_A\)). Therefore, the number of ways to arrange these particles on the lattice can be calculated as:
\binom{N_A + N_B}{N_A} = \frac{(N_A + N_B)!}{N_A! N_B!}\

This formula accounts for all possible distinct ways of placing the A and B particles on the lattice.

Multiplicity in statistical thermodynamics

Multiplicity is a fundamental concept in statistical thermodynamics, representing the number of microstates corresponding to a particular macrostate. A macrostate is defined by macroscopic properties like the number of particles and their arrangement, while microstates are the specific detailed arrangements at the microscopic level.

In our one-dimensional lattice example, the macrostate is defined by having \(N_A\) particles of type A and \(N_B\) particles of type B. The multiplicity (denoted by \(W(N_A, N_B)\)) tells us how many different ways the particles can be arranged on the lattice while satisfying this condition.

The importance of multiplicity lies in its connection to the concept of entropy. Higher multiplicity corresponds to greater entropy, as more microstates are accessible for a given macrostate. The expression for multiplicity in this context is:
\W(N_A, N_B) = \frac{(N_A + N_B)!}{N_A! N_B!}\

This formula captures the combinatorial nature of the problem, accounting for the fact that particles of the same type are indistinguishable from each other.

Distinguishable arrangements

In combinatorics, distinguishable arrangements refer to the different ways in which a set of objects can be arranged when the position of each object matters.

For our one-dimensional lattice filled with particles of type A and B, every unique sequence of A's and B's on the lattice represents a distinguishable arrangement. These arrangements are crucial for understanding the system's multiplicity and determining the number of microstates.

Consider that each arrangement must place all \(N_A\) particles of type A and all \(N_B\) particles of type B in the lattice. The total number of particles is \(N_A + N_B\), and we need to consider all possible orderings of these particles. The formula:
\W(N_A, N_B) = \frac{(N_A + N_B)!}{N_A! N_B!}\

effectively counts the number of distinguishable arrangements, treating identical particles as indistinguishable within their type. Therefore, even if you swap two particles of the same type, the arrangement does not change. Thus, the number of distinguishable arrangements gives us insight into the combinatorial possibilities and helps in understanding the statistical behaviors of the system.