Paramagnetic system: computing number of microstates

In summary: B is the Boltzmann constant. You will need to use Stirling's approximation to simplify the equations. Finally, for part c, you can use the fact that in thermal equilibrium the total energy and magnetization are conserved, so,E = N1E1 + N2E2 and m = (N1m1 + N2m2)/(N1+N2).
  • #1
mondeo2015
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Homework Statement


We are given a paramagnetic system of N distinguishable particles with 1/2 spin where we use N variables
s_k each binary with possible values of ±1 where the total energy of the system is known as:
[tex] \epsilon(s) = -\mu H \sum_{k=1}^{N} s_k [/tex] where [tex] \mu [/tex] is the magnetic moment of the spin and H is the applied magnetic field. We are requested to assume that N is greatly larger than 1 and we are reminded of the definition of [tex] arctanh(x) = \frac{1}{2} ln(\frac{1+x}{1-x}) [/tex]. We have the following three parts:
1. We are asked to find the number of microstates for a given energy E [tex]\Omega(E)[/tex] with as many justifiable simplifications as possible.
2. We are asked find [tex] \beta = \frac{1}{kT} [/tex] for a given E and we are asked to use this information to find single-spin magnetization of the system defined as [tex] m= \frac{1}{N}\sum_{k=1}^{N} s_k [/tex] in terms of [tex] \beta [/tex] and H the applied magnetic field. We are advised that m can be treated as a continuous variable where needed.
3. We are told that two systems [tex](N_1,H),(N_2,H)[/tex] are Temperaturs [tex] \beta_1 , \beta_2 [/tex] and initial magnetizations [tex] m_1,m_2 [/tex] respectively, are brought into contact where only heat can be exchanged and we are asked to use the preceding parts to show that the finial magnetization at equilibrium is given by [tex] m = \frac{N_1m_1 + N_2m_2}{N_1+N_2} [/tex], noticing the magnetic field is the same for both systems.

Homework Equations


The Stirling approximation [tex] N! \approx Nln(N)-N [/tex] as seen in Wikipedia
https://en.wikipedia.org/wiki/Stirling's_approximation

The Attempt at a Solution


I was thinking for part A the idea is to use the fact that it is the total number of N distinct solutions to an integer equation where the sum of the s' must be some integer m so this is basic combinatorics where it is finding the number of +1 valued s' (denoted l) which determines the sum which is l-(N-l) = 2l-N this is equal to m therefore l must satisfy l = (N+m)/2 so the total number of possibilities is [tex] N \choose \frac{N+m}{2} [/tex] then to get rid of factorials we probably should use the assumption that N is much greater than 1 so a form of Stirling's approximation must be valid and is to be used to simplify it. For part b. I must admit I am stumped as to how to relate this to temperature and for part c I have no clue all I think must happen is thermal equilibrium but how do we reach this nice expression given? This is where I am stuck and need help, I thank all helpers.
 
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  • #2
To find the number of microstates of N distinguishable objects with spin up or spin down you need to divide the N spins into two smaller groups of nu and nd. There are N distinguishable states so N! possible arrangements, but there are nu indistinguishable states so there are nu! arrangements of these. By simple combinatory analysis the total number of microstates is,
Ω = N!/ (nu!(N- nu)!).
The total energy is E = nu(-μH) + nd(μH) and from this,
nu = (1/2)(N-E/(μH)) = (N/2)(1-E/(NμH))
nd = N - (1/2)(N-E/(μH)) = (N/2)(1+E/(NμH))
In order to find β and magnetization I suggest you use the basic thermodynamical relations:
(∂S/∂E)H,N = 1/T and (∂S/∂H)E,N = m/T where the entropy is S = kBln(Ω)
 

What is a paramagnetic system?

A paramagnetic system is a material or substance that exhibits paramagnetism, which is a form of magnetism that is only present when an external magnetic field is applied to the material. In a paramagnetic system, the individual atoms or molecules have magnetic moments that align with the external field, resulting in a weak attraction towards the magnetic field.

How is the number of microstates in a paramagnetic system computed?

The number of microstates in a paramagnetic system is computed using statistical mechanics, specifically the Boltzmann formula. This formula takes into account the number of particles, the energy levels of the system, and the temperature to calculate the number of possible arrangements of particles within the system.

What factors affect the number of microstates in a paramagnetic system?

The number of microstates in a paramagnetic system is affected by the number of particles, the energy levels of the system, and the temperature. As the number of particles or energy levels increases, the number of microstates also increases. However, as the temperature increases, the number of microstates decreases.

Why is it important to calculate the number of microstates in a paramagnetic system?

Calculating the number of microstates in a paramagnetic system is important because it helps us understand the behavior of the system and predict its properties. It is also a fundamental aspect of statistical mechanics, which is a branch of physics that studies the behavior of large systems based on the behavior of individual particles.

What other applications does the concept of microstates have?

The concept of microstates is not limited to paramagnetic systems, but it is also used in other areas of physics, such as thermodynamics and quantum mechanics. In thermodynamics, microstates are used to describe the possible arrangements of particles in a system, while in quantum mechanics, they are used to describe the possible states of a system based on its energy levels.

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