Consider a lattice with \(N\) spin- 1 atoms with magnetic moment \(\mu\). Each atom can be in one of three spin states, \(S_{z}=-1,0,+1\). Let \(n_{-1}, n_{0}\), and \(n_{1}\) denote the respective number of atoms in each of those spin states. Find the total entropy and the configuration which maximizes the total entropy. What is the maximum entropy? (Assume that no magnetic field is present, so all atoms have the same energy. Also assume that atoms on different lattice sites cannot be exchanged, so they are distinguishable.)

Boltzmann's formula is a cornerstone in statistical mechanics. It helps us understand entropy, which is a measure of disorder in a system. Entropy, in this context, relates to the number of ways a system's microscopic states can be arranged while producing the same macroscopic state. Boltzmann's formula for entropy is given by:



where S is the entropy,
is upside down ykB is Boltzmann's constant, and Upper G /Lower (Divide by space) such as Nbottomboth EN, Hfactorials, is an element of matharea single characters that although it may appear full
We for singlechar_partial specific region intuition , this expression effectively counts the number of Microstates ,or different ways to arrange the atoms, subject to the constraints of their Spin states. Multiplying by the logarithm base e, most compoundinggives us a measure that links microscopicdetails with macroscopic observables.

Stirling's approximation is a useful mathematical technique when working with large numbers. It simplifies factorials, which can be quite large and unwieldy. Stirling's approximation states:

ln(Number)! approximately equal to Ifdeducted
Given a large value like N, it simplifies our calculations effectively down to:

Ns x lnH(ersionally) x N <=toleftlqHeightleftinterestingfeaturemn! s
When entropy calculations involve such large N, the factorial expressions in Boltzmann's formula can be daunting. Stirling's approximation simplifies this by converting factorials into logarithmic terms, making it easier to find entropy directly.

In this scenario, we are looking at a system with atoms that have three possible spin states: -1, 0, and +1. These states represent different orientations of the atoms' magnetic moments.

Each state is equally likely because there is no external magnetic field influencing them. Here are the general spin states concepts:

• -1 spin state: Magnetic moment points in one direction.
• 0 spin state: No magnetic moment (neutral).
• 1 state: Magnetic moment points in the opposite direction.

The total number of atoms is split among these states: n_{-1}, n_0, and n_1 . To maximize entropy, these atoms need to be distributed as equally as possible across these states.
Hence, for maximum entropy, the number of atoms in each spin state should be approximately

equal to (N/3).
This equal distribution balances the total number of ways we can arrange these atoms, producing a highly disordered (maximum entropy) state.