# Posts Tagged ‘average’

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 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 16, 2013

[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

## 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 »

Posted by peeterjoot on March 3, 2013

In A compilation of notes, so far, for ‘PHY452H1S Basic Statistical Mechanics’ I posted a link this compilation of statistical mechanics course notes.

That compilation now all of the following too (no further updates will be made to any of these) :

February 28, 2013 Rotation of diatomic molecules

February 28, 2013 Helmholtz free energy

February 26, 2013 Statistical and thermodynamic connection

February 24, 2013 Ideal gas

February 16, 2013 One dimensional well problem from Pathria chapter II

February 15, 2013 1D pendulum problem in phase space

February 14, 2013 Continuing review of thermodynamics

February 13, 2013 Lightning review of thermodynamics

February 11, 2013 Cartesian to spherical change of variables in 3d phase space

February 10, 2013 n SHO particle phase space volume

February 10, 2013 Change of variables in 2d phase space

February 10, 2013 Some problems from Kittel chapter 3

February 07, 2013 Midterm review, thermodynamics

February 06, 2013 Limit of unfair coin distribution, the hard way

February 05, 2013 Ideal gas and SHO phase space volume calculations

February 03, 2013 One dimensional random walk

February 02, 2013 1D SHO phase space

February 02, 2013 Application of the central limit theorem to a product of random vars

January 31, 2013 Liouville’s theorem questions on density and current

January 30, 2013 State counting

Posted in Math and Physics Learning. | Tagged: adiabatic, angular momentum, average, average energy, binomial distribution, Boltzmann's constant, canonical ensemble, Central limit theorem, change of variables, conjugate momentum, connection formulas, continuity equation, cyclic, cylindrical coordinates, diatomic molecule, diffusion, diffusion equation, dispersion, dissipation, energy, energy conservation, ensemble average, entropy, equilibrium, ergodic, Euclidean space, Fick's law, first law, Fluctuation-Dissipation theorem, fourier transform, free energy, Gamma function, Gaussian, Gaussian approximation, Generating function, geometric algebra, hamiltonian, harmonic oscillator, heat, heat capacity, Helmholtz free energy, hypersphere, ideal gas, ideal gas law, independent and identical random variables, indistinguishable states, integral approximation, irreversible, jacobian, Liouville's theorem, logarithm Taylor expansion, Maxwell distribution, Maxwell-Boltzmann distribution, mean, Minkowski space, moment, N dimension volume, N dimensional sphere, one dimensional well, particle in a box, Partition function, pendulum, phase space, phase space current, phase space density, phase space region, phase space volume, PHY452H1S, Poincare recurrence, point transformation, Poisson distribution, probability, probability distribution, product of random variables, propagator, quantized rotation, random walk, reversible, second moment, SHO, small angle, spatial average, specific heat, sphere packing, spherical coordinates, spherical volume element, spin hamiltonian, spin magnetization, state multiplicity, statistical entropy, statistical mechanics, Stirling approximation, temperature, thermal stability, thermodynamic, thermodynamics, trace, trajectory, unfair coin toss, variance, volume, WKB, work | 1 Comment »

Posted by peeterjoot on February 3, 2013

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## Question: One dimensional random walk

Random walk in 1D by unit steps. With the probability to go right of and a probability to go left of what are the first two moments of the final position of the particle?

## Answer

This was a problem from the first midterm. I ran out of time and didn’t take the answer as far as I figured I should have. Here’s a more casual bash at the problem.

First we need the probabilities.

**One step: **

Our distance (from the origin) can only be .

**Two steps: **

We now have three possibilities

**Three steps: **

We now have three possibilities

**General case**

The pattern is pretty clear, but we need a mapping from the binomial index to the the final distance. With an index , and a guess

where

So

and

Our probabilities are therefore

**First moment**

For the expectations let’s work with instead of , so that the expectation is

This gives us

**Second moment**

So the second moment is

From this we see that the variance is just this second term

Posted in Math and Physics Learning. | Tagged: average, binomial distribution, mean, PHY452H1S, random walk, second moment, statistical mechanics | Leave a Comment »