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- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)/06%3A_Quantal_Ideal_Gases/6.03%3A_Quantum_Mechanics_of_Independent_Identical_ParticlesElementary quantum mechanics deals with this situation, and it tells us that there will be a certain number M of (possibly degenerate) energy eigenstates. (Usually M will be infinite, but there are ad...Elementary quantum mechanics deals with this situation, and it tells us that there will be a certain number M of (possibly degenerate) energy eigenstates. (Usually M will be infinite, but there are advantages to calling it M and maintaining the ability to take the limit M → ∞.) The rth energy eigenstate has energy ϵr and is represented by the wavefunction η r (x), where x denotes the arguments of the wavefunction: Thus for a spinless particle, x could stand for x, y, z or p x , p y…
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)/02%3A_Principles_of_Statistical_Mechanics/2.02%3A_Macroscopic_Description_of_a_Large_Equilibrium_SystemFinally, what condition is the system in? (In other words, what corresponds to the dynamical variables in a microscopic description?) Clearly the “point in phase space”, giving the positions and momen...Finally, what condition is the system in? (In other words, what corresponds to the dynamical variables in a microscopic description?) Clearly the “point in phase space”, giving the positions and momenta of each and every particle in the system, is far too precise to be an acceptable macroscopic description.
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)/07%3A_Harmonic_Lattice_Vibrations/7.01%3A_The_ProblemThe Hamiltonian for lattice vibrations, in the harmonic approximation, is \[ \mathcal{H}=\frac{1}{2} \sum_{i=1}^{3 N} m_{i} \dot{x}_{i}^{2}+\frac{1}{2} \sum_{i=1}^{3 N} \sum_{j=1}^{3 N} x_{i} A_{i j} ...The Hamiltonian for lattice vibrations, in the harmonic approximation, is H=123N∑i=1mi˙x2i+123N∑i=13N∑j=1xiAijxj. Notice that this Hamiltonian allows the possibility that atoms at different lattice sites might have different masses. Show that the Hamiltonian can be cast into the form by a linear change of variables. (Clue: As a first step, introduce the change of variable zi=√mixi.)
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)/05%3A_Classical_Ideal_Gases/5.04%3A_Specific_Heat_of_a_Hetero-nuclear_Diatomic_Ideal_Gas\[ \begin{aligned} \ln \zeta(T)=& 3 e^{-\theta / T}+5 e^{-3 \theta / T}+\mathcal{O}\left(e^{-6 \theta / T}\right) \\ &-\frac{1}{2}\left[3 e^{-\theta / T}+5 e^{-3 \theta / T}+\mathcal{O}\left(e^{-6 \th...lnζ(T)=3e−θ/T+5e−3θ/T+O(e−6θ/T)−12[3e−θ/T+5e−3θ/T+O(e−6θ/T)]2+13[3e−θ/T+5e−3θ/T+O(e−6θ/T)]3+O(e−4θ/T)
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)/08%3A_Interacting_Classical_Fluids
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)/01%3A_The_Properties_of_Matter_in_Bulk/1.02%3A_Outline_of_Bookgases of non-interacting particles) is interesting and often useful, but it clearly does not tell the full story. . . for example, the classical ideal gas can never condense into a liquid, so it canno...gases of non-interacting particles) is interesting and often useful, but it clearly does not tell the full story. . . for example, the classical ideal gas can never condense into a liquid, so it cannot show any of the fascinating and practical phenomena of phase transitions. The first five chapters (up to and including the chapter on classical ideal gases) are essential background to the rest of the book, and they must be treated in the sequence presented.
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)This is a book about statistical mechanics at the advanced undergraduate level. It assumes a background in classical mechanics through the concept of phase space, in quantum mechanics through the Paul...This is a book about statistical mechanics at the advanced undergraduate level. It assumes a background in classical mechanics through the concept of phase space, in quantum mechanics through the Pauli exclusion principle, and in mathematics through multivariate calculus.
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)/03%3A_Thermodynamics/3.04%3A_Multivariate_CalculusIf you know r and h for a cylinder you can readily calculate any quantity of interest—such as the area of the top, T(r, h), the area of the side S(r, h), and the volume V(r, h)—as shown on the left si...If you know r and h for a cylinder you can readily calculate any quantity of interest—such as the area of the top, T(r, h), the area of the side S(r, h), and the volume V(r, h)—as shown on the left side of the table below. dfdx=∂f∂x)ydxdx+∂f∂y)xdydx=∂f∂x)y+∂f∂y)xdydx with dx,dy on contour of g
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)/01%3A_The_Properties_of_Matter_in_Bulk/1.01%3A_What_is_Statistical_Mechanics_AboutTo make progress, we have to ask different questions, question like “How does the pressure change with volume?”, “How does the temperature change upon adding particles?”, “What is the mean distance be...To make progress, we have to ask different questions, question like “How does the pressure change with volume?”, “How does the temperature change upon adding particles?”, “What is the mean distance between atoms?”, or “What is the probability for finding two atoms separated by a given distance?”. Thus the challenge of statistical mechanics is two-fold: first find the questions, and only then find the answers.
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)/02%3A_Principles_of_Statistical_Mechanics/2.01%3A_Microscopic_Description_of_a_Classical_SystemThe dynamical variables are the position and velocity (or momentum) of each body. (Alternative dynamical variables are the position of the center of mass and the separation between the bodies, plus th...The dynamical variables are the position and velocity (or momentum) of each body. (Alternative dynamical variables are the position of the center of mass and the separation between the bodies, plus the total momentum of the system and the angular momentum of the two bodies about the center of mass.) You can see from this example that the mechanical parameters give you the knowledge to write down the Hamiltonian for the system, while the dynamical variables are the quantities that change accordi…
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Statistical_Mechanics_(Styer)/10%3A_Appendices/10.04%3A_D-_Volume_of_a_Sphere_in_d_DimensionsThat is, I will use the formula for d = 2 to find the formula for d = 3, the formula for d = 3 to find the formula for d = 4, and in general use the formula for d to find the formula for d + 1. Now we...That is, I will use the formula for d = 2 to find the formula for d = 3, the formula for d = 3 to find the formula for d = 4, and in general use the formula for d to find the formula for d + 1. Now we begin with a three-dimensional sphere of radius r 0 in (w, x, y) space and thicken it a bit into the fourth dimension (z) to form a thin four-dimensional pancake of four-dimensional volume dz V 3 (r 0 ).
