# Posts Tagged ‘chemical potential’

Posted by peeterjoot on September 5, 2013

I’d intended to rework the exam problems over the summer and make that the last update to my stat mech notes. However, I ended up studying world events and some other non-mainstream ideas intensively over the summer, and never got around to that final update.

Since I’m starting a new course (condensed matter) soon, I’ll end up having to focus on that, and have now posted a final version of my notes as is.

Since the last update the following additions were made

September 05, 2013 Large volume fermi gas density

May 30, 2013 Bernoulli polynomials and numbers and Euler-MacLauren summation

May 09, 2013 Bose gas specific heat above condensation temperature

May 09, 2013 A dumb expansion of the Fermi-Dirac grand partition function

April 30, 2013 Ultra relativistic spin zero condensation temperature

April 30, 2013 Summary of statistical mechanics relations and helpful formulas

April 24, 2013 Low temperature Fermi gas chemical potential

Posted in Math and Physics Learning. | Tagged: average, Bernoulli number, Bernoulli polynomial, binomial distribution, Bose condensate, Bose gas, Bosons, canonical ensemble, Central limit theorem, cheat sheet, chemical potential, classical limit, density of states, ergodic, Euler-MacLauren summation, Fermi energy, Fermi gas, Fermi-Dirac, Fermions, fugacity, Generating function, grand canonical ensemble, grand canonical partition function, ground state, Hamilton's equations, Handy mathematics, ideal gas, large volume, Liouville's theorem, low temperature, Maxwell distribution, Microstates, number density, PHY452H1S, Quantum free particle in a box, Radius of gyration of a 3D polymer, random walk, specific heat, spin, statistical mechanics, statistics, Statistics mechanics, surface with binding sites, thermodynamics, ultra relativistic gas, Velocity, zeta function | Leave a Comment »

Posted by peeterjoot on September 5, 2013

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Here’s part of a problem from our final exam. I’d intended to redo the whole exam over the summer, but focused my summer study on world events instead. Perhaps I’ll end up eventually doing this, but for now I’ll just post this first part.

## Question: Large volume Fermi gas density (2013 final exam pr 1)

Write down the expression for the grand canonical partition function of an ideal three-dimensional Fermi gas with atoms having mass at a temperature and a chemical potential (or equivalently a fugacity ). Consider the high temperature “classical limit” of this ideal gas, where and one gets an effective Boltzmann distribution, and obtain the equation for the density of the particles

by converting momentum sums into integrals. Invert this relationship to find the chemical potential as a function of the density .

Hint: In the limit of a large volume :

## Answer

Since it was specified incorrectly in the original problem, let’s start off by verifing the expression for the number of particles (and hence the number density)

Moving on to the problem, we’ve seen that the Fermion grand canonical partition function can be written

so that our density is

In the high temperature classical limit, where we have

This is

where

Inverting for we have

or

Posted in Math and Physics Learning. | Tagged: chemical potential, classical limit, Fermi gas, large volume, number density, PHY452H1S, statistical mechanics, surface with binding sites | Leave a Comment »

Posted by peeterjoot on April 24, 2013

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## Question: Low temperature Fermi gas chemical potential

[1] section 8.1 equation (33) provides an implicit function for

or

In class, we assumed that was quadratic in as a mechanism to invert this non-linear equation. Without making this quadratic assumption find the lowest order, non-constant approximation for .

## Answer

To determine an approximate inversion, let’s start by multiplying eq. 1.0.2 by to non-dimensionalize things

or

If we are looking for an approximation in the neighborhood of , then the LHS factor is approximately one, whereas the fractional difference term is large (with a corresponding requirement for to be small. We must then have

or

This gives us the desired result

# References

[1] RK Pathria. *Statistical mechanics*. Butterworth Heinemann, Oxford, UK, 1996.

Posted in Math and Physics Learning. | Tagged: chemical potential, Fermi energy, Fermi gas, low temperature, PHY452H1S, statistical mechanics | Leave a Comment »

Posted by peeterjoot on April 22, 2013

Here’s my third update of my notes compilation for this course, including all of the following:

April 21, 2013 Fermi function expansion for thermodynamic quantities

April 20, 2013 Relativistic Fermi Gas

April 10, 2013 Non integral binomial coefficient

April 10, 2013 energy distribution around mean energy

April 09, 2013 Velocity volume element to momentum volume element

April 04, 2013 Phonon modes

April 03, 2013 BEC and phonons

April 03, 2013 Max entropy, fugacity, and Fermi gas

April 02, 2013 Bosons

April 02, 2013 Relativisitic density of states

March 28, 2013 Bosons

plus everything detailed in the description of my previous update and before.

Posted in Math and Physics Learning. | Tagged: average energy, average energy density, average number of particles, average occupancy, BEC temperature, binomial coefficient, binomial series, Boltzmann factor, Bose condensate, Bose condensation, Bose-Einstein condensate, Boson, canonical ensemble, chemical potential, configurations, constraint, delta function, density, density of states, distribution function, energy, energy density, ensemble, entropy, extreme relativistic gas, factorial, Fermi energy, Fermi gas, Fermi-Dirac function, Fermion, fugacity, Gamma function, Gibbs entropy, grand canonical ensemble, harmonic oscillator, Helium-4, jacobian, Lagrange multiplier, mean energy, microstate, momentum, momentum space volume element, neutrino gas, neutron star, nucleon, number density, occupation number, phonon, phonon modes, photon, photon gas, PHY452H1S, Polarization, pressure, probability, relativistic, relativistic gas, special relativity, specific heat, spherical coordinates, statistical mechanics, Taylor expansion, temperature, velocity space volume element, volume, zeta function | 1 Comment »

Posted by peeterjoot on April 2, 2013

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# Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

# Bosons

In order to maintain a conservation of particles in a Bose condensate as we decrease temperature, we are forced to change the chemical potential to compensate. This is illustrated in fig. 1.1.

Fig 1.1: Chemical potential in Bose condensation region

Bose condensatation occurs for . At this point our number density becomes (except at )

Except for , is well defined, and not described by this distribution. We are forced to say that

Introducing the density of states, our density is

where

We worked out last time that

or

This is plotted in fig. 1.2.

Fig 1.2: Density variation with temperature for Bosons

For , we have . This condensation temperature is

This is plotted in fig. 1.3.

Fig 1.3: Temperature vs pressure demarkation by T_BEC curve

There is a line for each density that marks the boundary temperature for which we have or do not have this condensation phenomina where states start filling up.

**Specific heat: **

so that

Compare this to the classical and Fermionic specific heat as plotted in fig. 1.4.

Fig 1.4: Specific heat for Bosons, Fermions, and classical ideal gases

One can measure the specific heat in this Bose condensation phenomina for materials such as Helium-4 (spin 0). However, it turns out that Helium-4 is actually quite far from an ideal Bose gas.

**Photon gas**

A system that is much closer to an ideal Bose gas is that of a gas of photons. To a large extent, photons do not interact with each other. This allows us to calculate black body phenomina and the low temperature (cosmic) background radiation in the universe.

An important distinction between a photon sea and some of these other systems is that the photon number is actually not fixed.

Photon numbers are not “conserved”.

If a photon interacts with an atom, it can impart energy and disappear. An excited atom can emit a photon and change its energy level. In a thermodynamic system we can generally expect that introducing heat will generate more photons, whereas a cold sink will tend to generate fewer photons.

We have a few special details that distinguish photons that we’ll have to consider.

- spin 1.
- massless, moving at the speed of light.
- have two polarization states.

Because we do not have a constraint on the number of particles, we essentially have no chemical potential, even in the grand canonical scheme.

Writing

Our number density, since we have no chemical potential, is of the form

Observe that the average number of photons in this system is temperature dependent. Because this chemical potential is not there, it can be quite easy to work out a number of the thermodynamic results.

**Photon average energy density**

We’ll now calculate the average energy density of the photons. The energy of a single photon is

so that the average energy density is

Mathematica tells us that this integral is

for an end result of

**Phonons and other systems**

There is a very similar phenomina in matter. We can discuss lattice vibrations in a solid. These are called phonon modes, and will have the same distribution function where the only difference is that the speed of light is replaced by the speed of the sound wave in the solid. Once we understand the photon system, we are able to look at other Bose distributions such as these phonon systems. We’ll touch on this very briefly next time.

Posted in Math and Physics Learning. | Tagged: average energy density, Bose condensation, chemical potential, density of states, energy density, Helium-4, phonon, photon gas, PHY452H1S, Polarization, specific heat, statistical mechanics | Leave a Comment »

Posted by peeterjoot on March 27, 2013

Here’s my second update of my notes compilation for this course, including all of the following:

March 27, 2013 Fermi gas

March 26, 2013 Fermi gas thermodynamics

March 26, 2013 Fermi gas thermodynamics

March 23, 2013 Relativisitic generalization of statistical mechanics

March 21, 2013 Kittel Zipper problem

March 18, 2013 Pathria chapter 4 diatomic molecule problem

March 17, 2013 Gibbs sum for a two level system

March 16, 2013 open system variance of N

March 16, 2013 probability forms of entropy

March 14, 2013 Grand Canonical/Fermion-Bosons

March 13, 2013 Quantum anharmonic oscillator

March 12, 2013 Grand canonical ensemble

March 11, 2013 Heat capacity of perturbed harmonic oscillator

March 10, 2013 Langevin small approximation

March 10, 2013 Addition of two one half spins

March 10, 2013 Midterm II reflection

March 07, 2013 Thermodynamic identities

March 06, 2013 Temperature

March 05, 2013 Interacting spin

plus everything detailed in the description of my first update and before.

Posted in Math and Physics Learning. | Tagged: addition of angular momentum, addition of spin, angular momentum, anharmonic oscillator, average, average diatomic separation, average dipole moment, average energy, average number of particles, average occupancy, binomial distribution, Boltzmann distribution, Boltzmann factor, Boson, canonical ensemble, Central limit theorem, chemical potential, classical harmonic oscillator, degeneracy pressure, delta function, density, density of states, diatomic molecule gas, differential form, eigenvalue, eigenvector, electric dipole, electric field interaction, electron, energy, energy eigenstate, energy eigenvalue, entropic force, entropy, equilibrium, Fermi distribution, Fermi energy, Fermi gas, Fermi temperature, Fermion, four momentum, four vector, free energy, fugacity, Gaussian approximation, Gibbs sum, grand canonical ensemble, grand canonical partition, grand partition function, graphene, hamiltonian, harmonic oscillator perturbation, heat capacity, high temperature limit, hole, ideal gas, integral approximation to sum, low temperature limit, magnetic field, magnetization, mean energy, microstate, moment of inertia, momentum, multiple paired spin, nuclear spin interaction, number of particles, number operator, occupancy, occupation number, occupation numbers, one form, orthonormal basis, partial derivative, particle in a box, Partition function, Pathria, pauli matrix, perturbation, PHY452H1S, Planck's constant, polymer, position mean value, pressure, probability, quantum anharmonic oscillator, random walk, relativistic gas, reservoir, singlet state, special relativity, specific heat, spherical harmonic, spin, spin hamiltonian, spin one half, spring constant, Statistics mechanics, subsystem, temperature, thermal average energy, thermal de Broglie wavelength, thermodynamic identity, trace, triplet states, two form, two variable Taylor expansion, variance, volume, zipper DNA model | 1 Comment »

Posted by peeterjoot on March 27, 2013

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# Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

# Fermi gas

**Review**

Continuing a discussion of [1] section 8.1 content.

We found

With no spin

Fig 1.1: Occupancy at low temperature limit

Fig 1.2: Volume integral over momentum up to Fermi energy limit

gives

This is for periodic boundary conditions \footnote{I filled in details in the last lecture using a particle in a box, whereas this periodic condition was intended. We see that both achieve the same result}, where

**Moving on**

with

this gives

Over all dimensions

so that

Again

## Example: Spin considerations

{example:basicStatMechLecture16:1}{

This gives us

and again

}

**High Temperatures**

Now we want to look at the at higher temperature range, where the occupancy may look like fig. 1.3

Fig 1.3: Occupancy at higher temperatures

so that for large we have

Mathematica (or integration by parts) tells us that

so we have

Introducing for the *thermal de Broglie wavelength*,

we have

Does it make any sense to have density as a function of temperature? An inappropriately extended to low temperatures plot of the density is found in fig. 1.4 for a few arbitrarily chosen numerical values of the chemical potential , where we see that it drops to zero with temperature. I suppose that makes sense if we are not holding volume constant.

Fig 1.4: Density as a function of temperature

We can write

or (taking (and/or volume?) as a constant) we have for large temperatures

The chemical potential is plotted in fig. 1.5, whereas this function is plotted in fig. 1.6. The contributions to from the term are dropped for the high temperature approximation.

Fig 1.5: Chemical potential over degenerate to classical range

Fig 1.6: High temp approximation of chemical potential, extended back to T = 0

**Pressure**

For a classical ideal gas as in fig. 1.7 we have

Fig 1.7: Ideal gas pressure vs volume

For a Fermi gas at we have

Specifically,

or

so that

We see that the pressure ends up deviating from the classical result at low temperatures, as sketched in fig. 1.8. This low temperature limit for the pressure is called the *degeneracy pressure*.

Fig 1.8: Fermi degeneracy pressure

# References

[1] RK Pathria. *Statistical mechanics*. Butterworth Heinemann, Oxford, UK, 1996.

Posted in Math and Physics Learning. | Tagged: chemical potential, degeneracy pressure, density, energy, Fermion, high temperature limit, ideal gas, integral approximation to sum, low temperature limit, PHY452H1S, pressure, statistical mechanics, thermal de Broglie wavelength, volume | Leave a Comment »

Posted by peeterjoot on March 26, 2013

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# Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

**Review**

Last time we found that the low temperature behaviour or the chemical potential was quadratic as in fig. 1.1.

Fig 1.1: Fermi gas chemical potential

**Specific heat**

where

**Low temperature **

The only change in the distribution fig. 1.2, that is of interest is over the step portion of the distribution, and over this range of interest is approximately constant as in fig. 1.3.

Fig 1.2: Fermi distribution

Fig 1.3: Fermi gas density of states

so that

Here we’ve made a change of variables , so that we have near cancelation of the factor

Here we’ve extended the integration range to since this doesn’t change much. FIXME: justify this to myself? Taking derivatives with respect to temperature we have

With , we have for

Using eq. 1.1.4 at the Fermi energy and

we have

Giving

or

This is illustrated in fig. 1.4.

Fig 1.4: Specific heat per Fermion

**Relativisitic gas**

- Relativisitic gas
- graphene
- massless Dirac Fermion
Fig 1.5: Relativisitic gas energy distribution

We can think of this state distribution in a condensed matter view, where we can have a hole to electron state transition by supplying energy to the system (i.e. shining light on the substrate). This can also be thought of in a relativisitic particle view where the same state transition can be thought of as a positron electron pair transition. A round trip transition will have to supply energy like as illustrated in fig. 1.6.

Fig 1.6: Hole to electron round trip transition energy requirement

**Graphene**

Consider graphene, a 2D system. We want to determine the density of states ,

We’ll find a density of states distribution like fig. 1.7.

Fig 1.7: Density of states for 2D linear energy momentum distribution

so that

Posted in Math and Physics Learning. | Tagged: chemical potential, density of states, electron, energy, Fermi distribution, Fermi gas, graphene, hole, PHY452H1S, relativisitic gas, specific heat, statistical mechanics | Leave a Comment »

Posted by peeterjoot on March 26, 2013

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# Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

# Fermi gas thermodynamics

**How about at other temperatures?**

We had

FIXME: references to earlier sections where these were derived.

We can define a density of states

where the liberty to informally switch the order of differentiation and integration has been used. This construction allows us to write a more general sum

This sum, evaluated using a continuum approximation, is

Using

where the roots of are , we have

In *2D* this would be

and in *1D*

**What happens when we have linear energy momentum relationships?**

Suppose that we have a linear energy momentum relationship like

An example of such a relationship is the high velocity relation between the energy and momentum of a particle

Another example is graphene, a carbon structure of the form fig. 1.3. The energy and momentum for such a structure is related in roughly as shown in fig. 1.4, where

Fig 1.3: Graphene bond structure

Fig 1.4: Graphene energy momentum dependence

**Continuing with the ***3D* case we have

FIXME: Is this (or how is this) related to the linear energy momentum relationships for Graphene like substances?

where as usual, and we write . For the low temperature asymptotic behavior see [1] appendix section E. For large it can be shown that this is

so that

Assuming a quadratic form for the chemical potential at low temperature as in fig. 1.5, we have

Fig 1.5: Assumed quadratic form for low temperature chemical potential

or

We have used a Taylor expansion for small , for an end result of

# References

[1] RK Pathria. *Statistical mechanics*. Butterworth Heinemann, Oxford, UK, 1996.

Posted in Math and Physics Learning. | Tagged: chemical potential, delta function, density, Fermi energy, Fermi gas, Fermi temperature, occupancy, PHY452H1S, statistical mechanics | Leave a Comment »

Posted by peeterjoot on March 16, 2013

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## Question: Variance of in open system ([1] pr 3.14)

Show that for an open system

## Answer

In terms of the grand partition function, we find the (scaled) average number of particles

Our second derivative provides us a scaled variance

Together this gives us the desired result

# References

[1] E.A. Jackson. *Equilibrium statistical mechanics*. Dover Pubns, 2000.

Posted in Math and Physics Learning. | Tagged: average, chemical potential, grand partition function, number of particles, PHY452H1S, statistical mechanics, variance | Leave a Comment »