Statistical Physics: relation to quanta and thermodynamics

Relate the average energy per mode to the energy density, cavity volume, and angular frequency omega, then divide by volume to obtain energy per unit volume.

We count the number of ways to distribute particles into energy levels, yielding w = n!/(n1! n2! ... nN!), under the constraint sum_j n_j = n.

The lecture introduces Stirling's approximation for factorials, explaining its use for large N by turning products into sums, integrating logs, and deriving N^N e^{-N} as an approximation.

Use Stirling's approximation to count particle arrangements in boxes, derive Omega as a function of temperature and angular frequency, and maximize dysfunction (disorder) under particle-number and total-energy constraints.

Maximize a function under two constraints by forming G = log W + α (constraint) − β (constraint), differentiate w.r.t J, and obtain J = exp(−α − β ε_J).

Explore Planck's mathematical trick by introducing the reduced Planck constant h-bar and assuming energy scales with frequency omega, deriving the energy as a function of omega for electromagnetic wave packets.

Derive the average energy per mode by substituting the single-mode energy and simplifying to a form with ħ, ω, and β; illustrate the classical limit ħ → 0.

Compare classical and quantum expressions for energy density per frequency, showing a parabolic omega dependency classically and discrete energy in the quantum case explained by Planck and Einstein.

Relate energy density per wavelength to frequency via lambda and omega, and show total energy per volume by integrating over all frequencies, scaling with temperature to the fourth power.

Learn how to represent a periodic function with period D as a Fourier series using cosine and sine terms, and compute the coefficients a0/2, ak, and bk via orthogonality integrals.

Using the function f(x)=x^2 on a periodic extension and applying the partial sums theorem, the lecture derives the series sum from n=1 to infinity of 1/n^4 as pi^4/90.

Compute final expression for the integral from zero to infinity of x^3/(e^x-1), yielding energy density u(T) and its frequency dependence as a constant times the temperature to the fourth power.

Derive the Stefan-Boltzmann law by normalizing the angular integral to four pi and converting to a double integral, yielding Q proportional to T^4 with h-bar and c.

Derive the Maxwell Boltzmann distribution by maximizing log w with energy constraints, using Lagrange multipliers and Stirling approximation to yield the exponential form.

Apply Planck's idea of quantized energy to solve the black body problem by counting electromagnetic modes in a cubic cavity and deriving the spectral energy density.

Einstein links total energy to potential plus kinetic energies of subsystems, derives average kinetic energy from momentum integration, and shows the relevant integral equals one over four H.

Einstein derives entropy from the second law using Lagrange dynamics, linking dissipative forces and heat to dQ/T and the Boltzmann constant via the state distribution.

This lecture analyzes Einstein’s thermodynamics math, rewriting the distribution and constants, defining E bar and E star, and deriving entropy relations from an energy average and normalization integral.

Learn Liouville's theorem and how canonical transformations preserve phase-space volume. The lecture derives the Jacobian equals one using Hamilton equations and generating functions, linking phase space to variational principles.

Preserve the system dynamics through canonical transformations and generating functions that rewrite the Lagrangian and action in new coordinates, while maintaining invariant variational principles and deriving Hamilton's equations.

Derive Hamilton equations from a lagrangian using the action principle and a formal variational method, obtaining q_j = ∂H/∂ p_j and p_j = −∂H/∂ q_j.

From Newton's second law, derive the variational principle for a single particle by rewriting the force as −dV/dx and showing the action S=∫(T−V) dt is stationary, i.e., Lagrangian governs dynamics.

The lecture proves Liouville's theorem by tracking a small square in phase space for a one-dimensional system, showing phase-space flow is incompressible.

Explore Einstein’s adiabatic-parameter approach to entropy, deriving it without Lagrange equations by treating interacting subsystems, linking heat and energy through a Boltzmann-constant–scaled distribution.

Derive the ideal gas law from statistical mechanics by linking entropy to energy and temperature, showing how P V = N k_B T (or n R T) emerges.

Einstein tests Planck's discrete oscillator view to explain emission and transformation of light, linking blackbody theory with light quanta and deriving Avogadro's number from Planck's law.

Relates entropy to probability via Boltzmann's constant, derives s0 = (R/n) ln(V/V0) for moving points, and shows low-density monochromatic radiation behaves as independent energy quanta with energy e0.

Einstein treats light as energy quanta, where absorbed quanta produce electron kinetic energy under energy conservation, linking emission and the photoelectric effect to ionization relations.

Demonstrates that entropy is a state function and a complete differential, derived from the second law for a Maxwell-distributed system using the Lagrangian framework and non-conservative work.

Explore entropy as a function of state through transitions between stationary states and deriving it from Lagrange equations, via average kinetic energy from Maxwell distribution and delta T.