--- title: "Thermal & Statistical Physics" description: "20 topics in Science, in the order they build on each other." canonical: https://lightmysky.com/learn/science/areas/thermal-and-statistical-physics source: https://lightmysky.com/learn/science/areas/thermal-and-statistical-physics.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Thermal & Statistical Physics 20 topics in Science, in the order they build on each other. Page: https://lightmysky.com/learn/science/areas/thermal-and-statistical-physics - [The Maxwell-Boltzmann Distribution of Molecular Speeds](https://lightmysky.com/learn/science/the-maxwell-boltzmann-distribution-of-molecular-speeds-mt_njr5r5YtuT): Molecules in a gas do not share one speed but a distribution with a peak, a mean and a longer tail at high speed. Raising the temperature flattens and widens the curve rather than shifting it rigidly. - [Equipartition and the Heat Capacities of Gases](https://lightmysky.com/learn/science/equipartition-and-the-heat-capacities-of-gases-mt_3OjJwVWlYD): Each way a molecule can store energy holds an average of half kT, so monatomic, diatomic and polyatomic gases have different heat capacities. Freezing out of modes at low temperature is the first hint that classical physics is incomplete. - [Thermodynamic State Variables and PV Diagrams](https://lightmysky.com/learn/science/thermodynamic-state-variables-and-pv-diagrams-mt_G3EYx84GZK): Pressure, volume and temperature describe the state of a system regardless of how it got there, and a path on a PV diagram shows a process. The area under the path is the work done. - [The First Law of Thermodynamics for Any Process](https://lightmysky.com/learn/science/the-first-law-of-thermodynamics-for-any-process-mt_LIWgeWSXHz): Internal energy changes only by heat added and work done, whatever route the system takes. Applying it means being careful about signs and about which quantities depend on the path. - [Work in Isothermal, Isobaric and Adiabatic Processes](https://lightmysky.com/learn/science/work-in-isothermal-isobaric-and-adiabatic-processes-mt_WzVxwN3svc): Each idealised process fixes one variable and gives its own expression for the work, with the adiabatic case following a steeper curve than the isothermal one. Which curve applies decides how much a compressed gas heats up. - [Heat Engines and the Second Law](https://lightmysky.com/learn/science/heat-engines-and-the-second-law-mt_M7CDvIi4Vs): An engine takes heat from a hot reservoir, does work and dumps the rest cold, and no arrangement of parts can skip the dumping. The second law states that limit in terms of what cannot be built. - [The Carnot Cycle and the Ceiling on Efficiency](https://lightmysky.com/learn/science/the-carnot-cycle-and-the-ceiling-on-efficiency-mt_aWH0BHM556): A cycle built from two isothermal and two adiabatic legs is the most efficient possible between two temperatures, and its efficiency depends on those temperatures alone. Every real engine is measured against it. - [Entropy as a State Function](https://lightmysky.com/learn/science/entropy-as-a-state-function-mt_bg_K5nU-GZ): Heat divided by temperature, summed round a reversible path, returns to zero, which means entropy is a property of the state. Real processes only ever increase the entropy of the whole system and its surroundings. - [Microstates, Multiplicity and Boltzmann's Entropy](https://lightmysky.com/learn/science/microstates-multiplicity-and-boltzmanns-entropy-mt_55I3VvYMc6): Counting the arrangements a system can take gives a multiplicity, and its logarithm times Boltzmann's constant is the entropy. The second law becomes a statement about which outcomes have overwhelmingly more arrangements. - [The Boltzmann Factor and the Partition Function](https://lightmysky.com/learn/science/the-boltzmann-factor-and-the-partition-function-mt_u9MSoTgJkF): The chance of a state falls exponentially with its energy divided by kT, and summing those factors over all states gives the partition function. Once it is known, the thermal properties follow by differentiation. - [Free Energy and the Direction of Spontaneous Change](https://lightmysky.com/learn/science/free-energy-and-the-direction-of-spontaneous-change-mt_ww43eqzupN): A system in contact with surroundings at fixed temperature settles where free energy is least, balancing low energy against high entropy. That is why some processes run even though they absorb heat. - [The Microcanonical and Canonical Ensembles](https://lightmysky.com/learn/science/the-microcanonical-and-canonical-ensembles-mt_NSCYLK-brt): An isolated system at fixed energy and a system held at fixed temperature are described by different ensembles, and each has its own way of assigning probabilities. For large systems the two agree on everything measurable. - [The Grand Canonical Ensemble and the Chemical Potential](https://lightmysky.com/learn/science/the-grand-canonical-ensemble-and-the-chemical-potential-mt_sdrQrWhBAF): Letting a system exchange particles as well as energy introduces a second multiplier, the chemical potential, which sets the free energy cost of one more particle. The grand partition function is what makes quantum gases tractable. - [Fermi-Dirac and Bose-Einstein Statistics](https://lightmysky.com/learn/science/fermi-dirac-and-bose-einstein-statistics-mt_rU5uPV4SB8): Identical quantum particles cannot be labelled, and whether they may share a state splits them into fermions and bosons with different occupation rules. Classical counting is what both reduce to when states are plentiful. - [The Free Electron Gas and the Fermi Energy](https://lightmysky.com/learn/science/the-free-electron-gas-and-the-fermi-energy-mt_4oTalBcKpm): Electrons in a metal fill states up to the Fermi energy even at absolute zero, which explains why they contribute so little to heat capacity. Only the few near the top of the stack can be excited. - [Ideal Quantum Gases and Bose-Einstein Condensation](https://lightmysky.com/learn/science/ideal-quantum-gases-and-bose-einstein-condensation-mt_wR2YnShTwJ): Occupation numbers from the grand ensemble reproduce the Bose and Fermi distributions, and below a critical temperature a finite fraction of bosons collects in the single lowest state. The condensate comes from statistics rather than from any interaction. - [Fluctuations, Linear Response and the Fluctuation-Dissipation Relation](https://lightmysky.com/learn/science/fluctuations-linear-response-and-the-fluctuation-dissipation-relation-mt_Ibsvdyx7uP): Equilibrium quantities fluctuate by an amount fixed by a derivative of the same free energy that gives their averages, so heat capacity measures the size of energy fluctuations. The same noise sets how strongly a system responds when it is pushed, which is why resistance and thermal noise are one measurement. - [Phase Transitions, Order Parameters and Broken Symmetry](https://lightmysky.com/learn/science/phase-transitions-order-parameters-and-broken-symmetry-mt_3DQjfNJ-Ef): A phase transition appears as a point where the free energy stops being smooth, and an order parameter measures which symmetry the ordered phase has given up. First-order and continuous transitions differ in whether that parameter jumps. - [The Ising Model and the Mean-Field Approximation](https://lightmysky.com/learn/science/the-ising-model-and-the-mean-field-approximation-mt_wPwyBTVI0C): Replacing the neighbours of a spin by their average turns an intractable interacting model into one self-consistent equation with a spontaneous solution below a critical temperature. The approximation gets the transition right and the exponents wrong, and knowing why is the point. - [Critical Exponents, Scaling and Universality](https://lightmysky.com/learn/science/critical-exponents-scaling-and-universality-mt_EaJOjyvjjj): Near a continuous transition the correlation length grows without bound and measured quantities follow power laws whose exponents depend on dimension and symmetry rather than on the material. A fluid at its critical point and a magnet at its own share the same numbers.