Statistical Physics & Thermodynamics from Beginner to Expert

Explore the mathematical basics of statistics with coin flip experiments, learn to calculate probabilities and probability distributions, and decide whether to study the math now or jump into the physics.

Use coin flip experiments and tree diagrams to explore probability, showing that a fair coin has 0.5 for heads and that three flips yield 0.5^3 via tree paths.

Demonstrate event and counter event in a dice experiment, using independence and counting methods to compute probabilities, such as four favorable outcomes out of 216.

Explore how to compute the expectation value for coin tosses, dice rolls, and urn problems, using probability trees and dependent-event reasoning to determine fair game prices.

Explore urn problems with distinguishable objects, comparing with and without replacement and with or without order, and learn probability via factorials and the binomial coefficient.

Explore the binomial distribution as a model for repeated yes/no outcomes, derive the formula with binomial coefficients, and apply it to coin tosses and dice to compute exact probabilities.

Explore the binomial distribution’s probability function and its cumulative form, with 40 coins at 50% and an expectation of 20, plus variance, standard deviation, and 68%, 95%, 99.7%.

Explore how the binomial distribution converges to the Gaussian distribution as n grows, described by e^{-x^2}, with mu and sigma^2 = p(1-p). Use the normal pdf to compute probabilities.

Explore the Poisson distribution as a limit of binomial and normal models for large event counts with small success probability, featuring a single parameter where variance equals the mean.

Challenge your understanding of mathematics and physics with exercises from this section. Sit down with paper and pencil to solve them, and watch the solution videos if needed.

Investigate probabilities for three coin tosses and three dice, finding all-same outcomes with probability 1/4 for coins and 1/36 for dice, and two-of-a-kind with probability 5/12.

Summarize probability types with urn and dice examples, covering replacement vs without replacement and order effects, plus a German lottery and binomial coefficient methods.

Explore distribution functions using 100 dice, compute the binomial expectation 16.67 and standard deviation 3.73, and compare binomial results with normal and Poisson approximations under the 3-sigma rule.

Explore how statistical physics connects micro and macro levels, compare microcanonical and canonical ensembles, and compute the average energy and average coordinate when energy transfers define temperature.

Explore microstates and macro states to see why statistical physics describes systems with Avogadro-scale particle numbers, and how the partition function links collective properties to individual coordinates and momenta.

Show how macro states, defined by energy and volume, come from many micro states of particle positions and momenta, using phase space and a simple two-atom gas example.

Explore the harmonic oscillator in one dimension, define macro and micro states, and analyze phase-space energy surfaces, time averages, ensemble averages, and the micro canonical ensemble.

Describe the microcanonical ensemble with constant energy, equal a priori probabilities, and a delta-distributed probability over phase space, normalized by the accessible volume Ω.

Explore the canonical ensemble as a temperature-based generalization of the microcanonical ensemble, where two weakly interacting systems exchange heat while keeping particle numbers fixed.

Derive the canonical ensemble probability density for two weakly interacting subsystems, showing energy additivity yields a beta-based exponential form and introduces the partition function.

Explore the canonical ensemble by deriving normalization constants, defining partition functions Z1, Z2, and the total Z, and use derivative identities to compute energy expectations and connect beta to temperature.

Calculate the average kinetic energy of an atomic gas using the canonical ensemble and partition function. Define temperature statistically via beta = 1/(KB T).

Derive Maxwell's speed distribution for a gas by integrating over all other velocities and coordinates, yielding a v^2 factor multiplied by an exponential Boltzmann form.

Derive the parametric barometric formula by treating gravity as the potential energy in a gas, linking pressure to height through Boltzmann weighted probability and the partition function.

Demonstrating the equivalence of canonical and microcanonical ensembles in the thermodynamic limit, the lecture derives energy averages and fluctuations from the partition function and shows macroscopic fluctuations vanish.

Compare microcanonical and canonical ensembles: microcanonical fixes energy, canonical exchanges heat with a bath, yielding finite temperature and energy fluctuations described by the partition function.

Build a solid foundation in statistical physics by reviewing the basics, including the micro canonical ensemble and the canonical ensemble, and prepare to apply this knowledge to thermodynamics.

Examine the quantum harmonic oscillator, deriving its hamiltonian and discrete energy levels. Use the geometric series to compute the partition function, then extract the average energy and occupation numbers.

Explore the Liouville equation, linking probability density and current in phase space, and show how Hamiltonian dynamics conserve phase-space volume for oscillators.

Explore thermodynamics by applying theory to actual physics, introducing the three fundamental laws and thermodynamic potentials. Derive temperature, entropy, pressure, and volume from the inner energy via derivatives.

Explore the first law of thermodynamics, showing energy conservation as kinetic energy to potential energy to thermal energy, and how heat, work, and entropy relate to internal energy.

Explore thermodynamic work and the first law by linking internal energy, Hamiltonian, probability distribution, and partition function to heat and work, with a classical analogy to force, pressure, and volume.

Derive how pressure arises from the partition function by differentiating Z with respect to volume and using d ln Z/dV, connecting rho, Hamiltonian, and the first law to work.

Explore the second law of thermodynamics, entropy, and the relationship between heat, work, and the first law using the partition function and the hamiltonian.

Explore entropy as a measure of randomness, the second law’s nondecrease, and its calculation from logarithms of probability densities for continuous and discrete states, plus additivity for independent systems.

Explore the third law of thermodynamics by analyzing the canonical distribution at zero temperature, showing that only the ground state is occupied and entropy vanishes.

Explore entropy with a loaded die by analyzing a discrete probability distribution where the probability of six shifts by delta; derive entropy as a function of delta and discuss extremes.

Calculate the entropy of a loaded die using a discrete sum, showing how delta shifts probabilities and yields maximum entropy at delta zero and zero when outcomes are deterministic.

Explore how black holes fit into thermodynamics by linking event horizon area to entropy, showing the second law holds and that heavier black holes are colder.

Study internal energy U as a thermodynamic potential depending on entropy and volume, and obtain temperature and pressure from U via dU = T dS − P dV.

Study the Helmholtz free energy F, defined as U minus T S. Derive F from the partition function Z via F = -1/β ln Z.

Enthalpy H, defined as U plus P V, is a thermodynamic potential with dH = T dS + V dP, giving T = (∂H/∂S)_P and V = (∂H/∂P)_S.

Explore Gibbs free energy, defined as enthalpy minus temperature times entropy, as the fourth thermodynamic potential that measures the maximum work at constant temperature and pressure, and derive Maxwell relations.

Explore Maxwell relations by examining mixed second derivatives of thermodynamic potentials, linking internal energy, temperature, and pressure with entropy and volume, and deriving four Maxwell relations.

Explore the thermodynamics of a gas focusing on the ideal gas, its heat capacity, and isothermal cycles. Learn how to analyze engines, calculate efficiency, and briefly discuss real gas corrections.

Derive the ideal gas law from kinetic theory and the partition function, relate pressure, volume, and temperature, and contrast with real gas behavior.

This lecture introduces thermodynamic processes on the ideal gas PV diagram, detailing isothermal, isochoric, and isobaric paths and the isotropic process with constant entropy.

Maintain entropy constant in isentropic processes, yielding zero heat. Demonstrate that for an ideal gas, p v^gamma is constant, with gamma the adiabatic index.

Define heat capacity as delta q over delta t, compare cv and cp for constant volume and constant pressure, and illustrate with ideal gas enthalpy, internal energy, and gamma ratio.

Explore isothermal and isotropic compressibilities for ideal gases, deriving kappa_T = 1/p, kappa_iso = 3/5 kappa_T, and linking their ratio to the heat-capacity ratio gamma.

Two thermal expansion coefficients, isobaric and isochoric, describe how temperature changes affect volume or pressure; for an ideal gas, both equal 1 over temperature, and in general they may differ.

discover how to combine isothermal, isobaric, and isochoric processes into a closed thermodynamic cycle to form an engine, and relate heat, work, and entropy via the p-v diagram.

Explore the four archetypical thermodynamic processes in a cycle, and see how heat input powers engines or refrigerators, including why the Kano engine offers highest efficiency in a temperature window.

The Carnot cycle yields the highest efficiency between two temperatures; in the temperature-entropy diagram it forms a rectangle, with work equals delta s delta T and efficiency 1 minus Tcold/Thot.

Explore how real gases deviate from ideal behavior using the van der Waals framework, introducing volume exclusion and interparticle attractions, and predict isotherms, phase transitions, and the critical point.

Conclude the section by recognizing how thermodynamics of gases, thermodynamic processes, and cycles relate to engineering applications. Build motivation for next section as we explore real gases and phase transitions.

Explore how Landau theory uses symmetry to describe phase transitions by formulating the system's free energy and its derivatives, revealing entropy and heat capacity changes.

Investigate phase transitions as discontinuities in derivatives of the free energy, distinguishing first and second order transitions by entropy and heat capacity, with the order parameter and Landau theory.

Explore Landau theory as a symmetry-based, simplified framework to analyze phase transitions, the partition function, magnetization, and free energy landscapes, distinguishing first and second order transitions.

Explore how Landau theory classifies phase transitions using a symmetric free energy with even terms, showing a paramagnetic to ferromagnetic, second-order transition and a critical exponent beta of 1/2.

Explore how a negative quartic and a stabilizing sixth-order term in Landau free energy yield three minima and a discontinuous order parameter, signaling a first-order transition.

Explore lambda theory, which focuses on the symmetry of the system, and learn how it helps reveal first-order and second-order phase transitions.

Investigate the grand canonical ensemble for open systems, contrasting energy and particle exchange with microcanonical and canonical ensembles, and derive the Bose-Einstein and Fermi-Dirac distributions.

Describe the grand canonical ensemble for open systems with energy and particle exchange, introducing the chemical potential and grand canonical partition function, and relate it to the canonical ensemble.

Derive the grand canonical potential from entropy in the grand canonical ensemble, linking Ω to the grand partition function and μ, and compare with canonical ensemble and Helmholtz free energy.

Demonstrate how the grand canonical potential and partition function determine the average particle number and energy-level occupation in a non interacting quantum gas, highlighting fermions and bosons.

Explore the quantum distinction between bosons and fermions through a two-particle wave function, showing how indistinguishability and symmetry allow or forbid double occupation, leading to Bose-Einstein and Fermi-Dirac statistics.

Derive the Fermi-Dirac distribution for fermions like electrons, showing occupation between 0 and 1, with mu as the Fermi level, and how temperature smooths the step function.

Transform the infinite boson occupation sum into a geometric series, differentiate the denominator with respect to mu, and obtain the Bose-Einstein distribution with a minus sign relative to fermions.

Examine the Bose-Einstein distribution and how, at low temperatures, a Bose-Einstein condensate forms when all particles occupy the lowest energy level at mu.

Explore the classical limit where Fermi and Bose distributions reduce to the Boltzmann distribution, highlighting low occupations and the role of mu and beta in the approximation.

Explore the occupation statistics of quantum gases and distinguish fermions from bosons, showing how a tiny sign change in the distribution causes dramatically different occupations.

Explore the phase transition between paramagnetic and ferromagnetic phases using statistical physics tools, calculate the partition function and derivatives, and study the transition with Monte Carlo simulations.

Explore Zeeman energy and how magnetization depends on temperature by analyzing three models: two-state spin, finite-j magnets, and the classical limit, with moments interacting with an external field.

Develops the quantum Ising model of two-state, noninteracting spins in a magnetic field, deriving the magnetization as tanh(beta mu B) from the density matrix and Boltzmann statistics.

Learn to compute paramagnet magnetization via the partition function, showing z = z1^N and orientation sums yield hyperbolic functions in quantum and classical limits.

Derive magnetization from the partition function across quantum, easing, and classical limits; differentiate with respect to beta B to reveal the brain function and hyperbolic tangent behavior.

Explore how exchange interactions in the Heisenberg Hamiltonian cause magnetic moments to interact and align, producing magnetism at zero field via an effective field and ferromagnetic effects.

Apply the mean field approximation to ferromagnetism and derive an effective field that includes magnetization. Explore how Monte Carlo simulations reveal phase transitions and hysteresis compared to the mean-field results.

Explore phase transitions between ferromagnet and paramagnet using mean-field theory, identify the critical temperature and exponent, and introduce Monte Carlo simulations in Python.

Explore a more accurate treatment of magnetic phase transitions via numerical Monte Carlo simulations of interacting magnets, using Python to model orientations across temperatures with a pi demonstration.

Install Python and Jupyter Notebook via the Anaconda Individual Edition, with Python 3.9 and Anaconda Navigator. Learn to create and run notebooks, manage kernels, and restart or run all cells.

Explore a Monte Carlo approach to estimating pi by generating random points in a square, counting those inside a circle, and using the pi over four ratio to compute pi.

Estimate pi by generating random points in a square, count those inside the circle, and compute pi approximately as four times the inside-to-total point ratio, with vectorized arrays and plotting.

Explore Monte Carlo pi approximation by comparing loop and array methods with 100,000 random points, counting inside the circle, and note speed differences via time measurements.

Set up a lattice of magnetic moments with unit length, generate random orientations via spherical angles, and plot the initial paramagnetic state before Metropolis Monte Carlo energy minimization.

Define the exchange (Heisenberg) energy for a ferromagnetic system, sum nearest-neighbor spin dot products, apply periodic boundary conditions, and prepare for implementing the Monte Carlo Metropolis step.

Simulate the metropolis step of a Monte Carlo loop on a 40×40 spin grid by randomly reorienting a single moment and accepting only energy-decreasing changes.

Run the monte carlo algorithm for a spin system, tracking magnetization and energy, and learn how steps, energy updates, and system size affect convergence toward the global minimum.

Introduce finite temperatures to the Monte Carlo update, allowing energy-increasing moves via the Boltzmann factor to escape local energy minima and approach a near fair magnet.

Add Zeeman interaction to a spin system to model external magnetic field effects, compute energy as -μ p·B, update Metropolis steps, and observe how field direction aligns all magnetic moments.

Demonstrate how a Monte Carlo simulation, with exchange and DMI terms driven by broken inversion symmetry, generates spin spirals and domain walls from random magnetic moments.

Learn to compute magnetization versus temperature by looping over temperatures, store results, plot M versus T, and use the absolute value of Z to reflect two directions, noting finite-size effects.

Examine how magnetization varies with the magnetic field at fixed temperature, focusing on the z-component, s-shaped M versus B, and mean-field versus full neighbor interactions with j set to zero.

Explore paramagnetic magnetization under zero exchange (J=0) with Monte Carlo simulations, compare to the analytical coth(x) solution (1/tanh x), and discuss finite-size and mean-field limitations.

Conclude by mastering the phase transition in magnetic systems through Landau theory, mean field theory, and Monte Carlo algorithms, and consider leaving a review or exploring more courses.