atomic_and_molecular

## Spectra and Selection Rules

Atomic spectra are the primary observable and were the evidence that forced quantum mechanics. A
transition between two levels emits or absorbs a photon at the frequency corresponding to their
energy difference, and the resulting line spectrum is a fingerprint identifying the element
unambiguously — which is how the composition of stars is known.


atomic_and_molecular

Not every pair of levels is connected. Selection rules follow from the symmetry of the interaction
between the atom and the electromagnetic field, and for the dominant electric dipole term they
require the angular momentum to change by one unit and the parity to change. Transitions violating
these are not strictly impossible but proceed through much weaker higher-order terms, with rates
smaller by many orders of magnitude.


atomic_and_molecular

Forbidden transitions are astrophysically important precisely because they are slow. In a
laboratory gas, an atom in a metastable state is collisionally de-excited long before it can radiate
through a forbidden channel. In a nebula at a density of a few particles per cubic centimetre,
collisions are so rare that the forbidden transition is the dominant decay path, and the resulting
emission lines are among the brightest features of the spectrum. Lines that cannot be produced in a
laboratory therefore dominate the observation, which confused astronomers for decades before the
explanation was found.


atomic_and_molecular

Spectra and transitions:
  line spectrum        the element's fingerprint
  emission             downward transition, photon out
  absorption           upward transition, photon removed from a continuum
  electric dipole      the dominant interaction term
  selection rules      angular momentum changes by one, parity changes
  forbidden            allowed only at higher order; rates smaller by many orders
  metastable state     no allowed decay; long lived
  nebular lines        forbidden transitions dominating at very low density
  oscillator strength  the dimensionless transition probability
  lifetime             inverse of the total decay rate; sets the natural linewidth
  natural broadening   the uncertainty-principle linewidth; Lorentzian
  Doppler broadening   thermal motion; Gaussian, usually dominant in a hot gas
  pressure broadening  collisions interrupting the emission; increases with
                       density
  Voigt profile        the convolution of Lorentzian and Gaussian; what is
                       actually fitted


atomic_and_molecular

Q: An emission line in a nebula corresponds to a transition that cannot be produced in a
laboratory discharge tube. This is because:
A. The nebula contains an unknown element
B. The transition is forbidden and slow, so in the laboratory collisions de-excite the atom first;
   at nebular densities collisions are rare enough that the slow radiative decay dominates
C. The line is a measurement artifact
D. The physics differs in space
Reasoning: Forbidden transitions have very long radiative lifetimes. Whether they are observed
depends on whether collisional de-excitation is faster, which depends on density. The nebula's
extremely low density is the whole explanation. Answer: B.


atomic_and_molecular

## Fine, Hyperfine, and Atoms in Fields

The hydrogenic energy levels are a first approximation, and the corrections to them are where
atomic physics becomes precise. Fine structure arises from relativistic effects and from the
coupling of the electron's spin to the magnetic field it experiences in its own rest frame as it
orbits the nucleus. It splits levels by an amount scaling with the fine structure constant squared
relative to the gross structure, and it grows rapidly with nuclear charge.


atomic_and_molecular

Hyperfine structure arises from the interaction of the electron with the nuclear magnetic moment,
and it is smaller again by roughly the electron-to-proton mass ratio. It is nonetheless the basis
of the most consequential transitions in physics: the ground state hyperfine splitting of caesium
defines the second, and the corresponding splitting in atomic hydrogen produces the twenty-one
centimetre line that maps neutral hydrogen throughout the galaxy.


atomic_and_molecular

External fields split levels further and the splitting is diagnostic. The Zeeman effect splits by
magnetic field, with the splitting proportional to the field and to the magnetic quantum number,
which makes spectral line splitting a remote magnetometer — solar magnetic fields are measured this
way. The Stark effect splits by electric field, quadratically for most atoms and linearly for
hydrogen owing to its degeneracy, and it is the mechanism behind field ionization and the pressure
shifts in dense plasmas.


atomic_and_molecular

Corrections and external fields:
  fine structure       relativistic plus spin-orbit; scales as alpha squared
  spin-orbit coupling  the electron's spin in the field of its own orbit
  Lamb shift           a QED effect splitting levels degenerate in Dirac theory;
                       the measurement that launched quantum electrodynamics
  hyperfine structure  electron coupling to the nuclear moment
  the caesium standard  a hyperfine transition defines the SI second
  twenty-one centimetre  hydrogen's hyperfine line; maps galactic neutral gas
  isotope shift        mass and nuclear volume effects; measures nuclear radii
  Zeeman effect        magnetic splitting, linear in field
  anomalous Zeeman     the case with spin; requires the Landé g-factor
  Paschen-Back         strong field, where spin-orbit coupling is broken
  Stark effect         electric field splitting; quadratic generally, linear in
                       hydrogen
  field ionization     strong fields lowering the barrier until the electron
                       escapes
  Rydberg atoms        highly excited, enormous polarizability, extreme
                       sensitivity to fields; used as electrometers and as a
                       platform for strongly interacting quantum systems


atomic_and_molecular

## Molecular Structure

Molecules add nuclear degrees of freedom to the electronic problem, and the separation of scales
between them is what makes molecular physics tractable. The Born-Oppenheimer approximation treats
the nuclei as stationary while solving for the electrons, then uses the resulting electronic energy
as a potential surface on which the nuclei move. The justification is the mass ratio: electrons
respond essentially instantaneously to nuclear motion.


atomic_and_molecular

The resulting energy hierarchy is wide. Electronic transitions lie in the visible and ultraviolet,
vibrational transitions in the infrared, rotational transitions in the microwave, and each is
separated from the next by roughly two orders of magnitude. This separation is why infrared
spectroscopy identifies functional groups — a vibrational frequency depends mainly on the local
bond and its reduced mass, so it transfers between molecules — and why microwave spectroscopy
determines molecular geometry, since rotational constants are inverse moments of inertia.


atomic_and_molecular

Bonding itself is described by molecular orbital theory, in which atomic orbitals combine into
bonding and antibonding combinations, and the bond order follows from how the electrons fill them.
The competing valence bond picture localizes electrons into pairs and is more chemically intuitive;
both are approximations to the same underlying problem and each is more convenient for different
questions.


atomic_and_molecular

Molecular structure:
  Born-Oppenheimer     separate nuclear and electronic motion by mass ratio
  potential energy surface  electronic energy as a function of nuclear positions
  the energy hierarchy  electronic, vibrational, rotational, separated by roughly
                       two orders of magnitude each
  molecular orbitals   atomic orbitals combining into bonding and antibonding
  bond order           net bonding electron pairs
  vibrational modes    3N minus 6 for a nonlinear molecule, minus 5 for linear
  harmonic approximation  good near the minimum; anharmonicity matters for
                       overtones and dissociation
  infrared activity    requires a changing dipole moment
  Raman activity       requires a changing polarizability; complementary to
                       infrared
  rotational constants  inverse moments of inertia; determine geometry
  selection rules      vibrational transitions change v by one in the harmonic
                       limit; rotational by one
  Franck-Condon        electronic transitions are vertical on the nuclear
                       coordinate; the intensity pattern reflects wavefunction
                       overlap
  dissociation         the potential surface flattening; the asymptote gives the
                       bond energy


atomic_and_molecular

## Laser Cooling, Trapping, and Precision Measurement

Laser cooling exploits the fact that a photon carries momentum. An atom moving toward a laser tuned
slightly below its resonance sees the light Doppler-shifted into resonance, absorbs preferentially
from that direction, and re-emits in a random direction, so the momentum transfer averages to a
force opposing motion. Six beams along three axes produce a viscous damping in all directions,
cooling a gas to the microkelvin range and below.


atomic_and_molecular

Trapping requires a position-dependent force in addition, supplied by a magnetic field gradient
that Zeeman-shifts the atoms into resonance more strongly the further they are from the centre.
Combined with the cooling beams this makes a magneto-optical trap, the workhorse of the field.
Further cooling by evaporation — removing the most energetic atoms and letting the rest
re-thermalize — reaches the nanokelvin regime where quantum degeneracy appears and a
Bose-Einstein condensate forms.


atomic_and_molecular

The payoff is precision. A cold atom moves slowly, so it can be interrogated for a long time, and
the linewidth of a measurement is inversely proportional to the interrogation time. Optical clocks
based on a narrow transition in a trapped ion or in neutral atoms held in an optical lattice reach
fractional frequency uncertainties near one part in ten to the eighteenth, which is sensitive
enough that the gravitational redshift over a height difference of a centimetre is measurable.


atomic_and_molecular

Cooling, trapping, and metrology:
  radiation pressure   photon momentum transfer; the basic force
  Doppler cooling      red-detuned beams; velocity-dependent damping
  Doppler limit        set by the random-recoil heating from spontaneous emission
  sub-Doppler          polarization gradient mechanisms beating the naive limit
  magneto-optical trap  cooling beams plus a field gradient
  optical molasses     damping without confinement
  optical tweezers     a focused beam trapping by the dipole force; single-atom
                       control
  optical lattice      a standing wave making a periodic potential; a clean
                       platform for condensed matter analogues
  evaporative cooling  remove the hot tail, re-thermalize; the route to
                       degeneracy
  Bose-Einstein condensate  macroscopic occupation of the ground state
  degenerate Fermi gas  the fermionic counterpart; Pauli blocking observable
  atomic clock         a transition frequency as the time standard
  optical clock        an optical rather than microwave transition; the higher
                       frequency buys precision directly
  the redshift measurement  optical clocks resolve gravitational time dilation
                       over centimetres of height
  atom interferometry  splitting and recombining matter waves; measures
                       accelerations, rotations, and the gravitational constant
  the broader point    atomic physics supplies the most precise measurements in
                       science, and those measurements are how the constancy of
                       fundamental constants, the existence of new forces, and
                       the limits of the standard model are tested


biophysics

## Scale and Physical Constraints

Biophysics applies physical reasoning to biological systems, and the first thing it establishes
is that the physics at cellular scale is genuinely different from the physics of everyday
experience. Quantities scale with different powers of length, so ratios that are negligible for a
large organism dominate for a small one, and intuitions built at human scale mislead
systematically.


biophysics

Surface area scales as length squared while volume scales as length cubed, so the surface to
volume ratio rises as size falls. A bacterium has enormous surface relative to its volume, which
means diffusive exchange across its membrane is fast enough to supply its entire interior, and it
also means it loses heat and water rapidly relative to its content. This single scaling
relationship explains why small organisms need no circulatory system, why large ones do, why
cells are the size they are, and why heat loss constrains the minimum size of a warm-blooded
animal.


biophysics

The Reynolds number for a swimming organism is the ratio of inertial to viscous forces, and it
falls with size and speed. A human swimming operates at a Reynolds number around a million, where
inertia dominates and coasting works. A bacterium operates around ten to the minus five, where
viscosity dominates absolutely and inertia is irrelevant — a bacterium that stops swimming halts
within a fraction of an atomic diameter. This is why bacterial propulsion uses rotating helical
flagella rather than the reciprocal motion that works at high Reynolds number, since any
reciprocal motion at low Reynolds number returns the organism exactly to where it started.


biophysics

Scale and constraints:
  surface to volume    rises as size falls; governs exchange, heat loss, and the
                       need for transport systems
  diffusion time       scales with the square of distance; adequate over
                       micrometres and hopeless over millimetres
  the diffusion limit  why cells are small and why multicellular organisms need
                       circulation
  Reynolds number      inertial over viscous; ~1e-5 for a bacterium
  low Reynolds number  viscosity dominates absolutely; coasting distance is
                       subatomic
  the scallop theorem  reciprocal motion produces no net displacement at low
                       Reynolds number; hence rotating flagella and travelling
                       waves
  thermal energy       kT is the reference scale; ~4.1 pN nm at room temperature
  energy comparison    a covalent bond is ~100 kT, a hydrogen bond ~2-10 kT, so
                       hydrogen bonds break and reform continuously while covalent
                       bonds do not
  molecular crowding   the cell interior is 20-30% macromolecule by volume, not
                       dilute solution; this changes reaction rates and binding
                       affinities substantially
  allometric scaling   metabolic rate scales as roughly mass to the three-quarter
                       power across many orders of magnitude


biophysics

Q: A bacterium stops rotating its flagella. The distance it coasts before stopping is closest to:
A. Several body lengths
B. Less than an atomic diameter
C. One body length
D. It does not stop
Reasoning: At a Reynolds number around ten to the minus five, viscous drag dominates inertia
completely, and the coasting distance is of order the size times the Reynolds number — far below
an atomic diameter. This is why bacterial motility requires continuous propulsion and why the
mechanisms are rotational or wave-like rather than reciprocal. Answer: B.


biophysics

## Molecular Forces and Thermal Noise

The forces holding biological structures together are almost all non-covalent and individually
weak, and this is a design feature rather than a limitation. A structure held by covalent bonds
would be permanent and unable to assemble, disassemble, or respond. Structures held by many weak
interactions are stable in aggregate and reversible, so they self-assemble, they can be
disassembled when required, and they respond to conditions.


biophysics

The energy scale is what makes this work. Thermal energy at room temperature is about four
piconewton-nanometres, and a hydrogen bond is a few times that, so hydrogen bonds break and reform
continuously on nanosecond timescales. A structure requiring ten simultaneous hydrogen bonds is
stable because the probability of all ten breaking at once is negligible, while any individual bond
is transient — giving both stability and the capacity to rearrange.


biophysics

Brownian motion is the constant thermal agitation that dominates molecular-scale dynamics, and
its magnitude is what makes biological mechanism unintuitive. A protein in water is buffeted
continuously and moves erratically, so a molecular machine does not push its way forward against
resistance; it uses chemical energy to bias which thermally-driven fluctuations are captured. The
Brownian ratchet framing captures this: thermal motion supplies the movement and the energy input
supplies the directionality by preventing backward steps.


biophysics

Molecular forces and noise:
  covalent bond        ~100 kT; effectively permanent at biological temperature
  ionic interaction    strong in vacuum, greatly weakened by water's high
                       dielectric constant
  hydrogen bond        2-10 kT; transient individually, stable in aggregate
  van der Waals        <1 kT each; significant only over large complementary
                       surfaces, which is why shape complementarity matters
  hydrophobic effect   entropy driven; the dominant folding and assembly force
  the aggregate principle  many weak bonds give stability with reversibility
  Brownian motion      thermal agitation; dominates molecular-scale dynamics
  diffusion coefficient  inversely related to size and viscosity
  mean square displacement  proportional to time for diffusion, to time squared
                       for directed motion; the distinction identifies the
                       mechanism from a trajectory
  Brownian ratchet     thermal motion supplies movement, energy input supplies
                       direction
  fluctuation-dissipation  the same molecular collisions produce both random
                       forces and viscous drag
  entropic forces      polymer elasticity is entropic; a stretched chain recoils
                       because coiled configurations are more numerous


biophysics

Entropic elasticity deserves a note because it explains a property with no mechanical analogue.
A polymer's resistance to being stretched does not come from bond stretching but from
configurational entropy: an extended chain has far fewer possible arrangements than a coiled one,
so extending it reduces entropy and the free energy rises. The resulting force increases with
temperature, which is the opposite of ordinary elastic materials and is directly observable in
rubber, which contracts when heated under load. DNA's mechanical behaviour under tension follows
the same physics and is described by the worm-like chain model, which is how single-molecule
stretching experiments are interpreted.


biophysics

## Molecular Machines

Molecular motors convert chemical energy into directed motion, and they operate under conditions
where thermal noise is comparable to the forces involved. Their efficiencies are frequently high —
some approach the thermodynamic limit — which is remarkable given that they work by biasing random
fluctuations rather than by applying deterministic force.


biophysics

Kinesin and dynein walk along microtubules carrying cargo, taking discrete steps of a fixed
length powered by ATP hydrolysis. Myosin moves along actin and is what produces muscle
contraction, with the sliding filament mechanism converting many small myosin steps into
macroscopic shortening. ATP synthase is a rotary motor driven by a proton gradient, and the
rotation drives conformational changes that synthesize ATP — a mechanism proposed before it could
be observed and subsequently visualized directly.


biophysics

The forces and step sizes involved are the scale at which single-molecule measurement operates.
A motor generates a few piconewtons and steps by a few nanometres, so the work per step is a few
tens of piconewton-nanometres, comparable to the energy released by hydrolysing one ATP molecule.
That correspondence — one chemical event to one mechanical step — is what establishes the coupling
and was demonstrated by directly observing individual steps.


biophysics

Molecular machines:
  kinesin              walks toward the microtubule plus end; 8 nm steps, a few pN
  dynein               walks toward the minus end; larger and more complex
  myosin               actin-based; muscle contraction by the sliding filament
                       mechanism
  ATP synthase         a rotary motor driven by the proton gradient; visualized
                       directly
  bacterial flagellar motor  rotary, proton or sodium driven, reversible;
                       switching direction is what produces the run-and-tumble
                       search
  processivity         how many steps before detaching; kinesin is processive
                       because two heads coordinate so one is always bound
  step size            a few nanometres; matched to the structural periodicity
  force generation     a few piconewtons; comparable to thermal forces
  energy per step      comparable to one ATP hydrolysis; the correspondence
                       establishing the coupling
  stall force          where the motor stops against load
  ribosome             translates mRNA; a machine with a decoding and a catalytic
                       function, and its catalytic core is RNA
  polymerases          copy nucleic acids; proofreading raises fidelity by orders
                       of magnitude at the cost of speed
  kinetic proofreading  a general mechanism achieving accuracy beyond
                       thermodynamic equilibrium limits by expending energy on
                       discarding near-correct intermediates


biophysics

Kinetic proofreading deserves emphasis because it resolves an apparent impossibility. The
discrimination achievable by binding energy alone is limited by the free energy difference between
correct and incorrect substrates, and observed biological error rates are far below that limit.
The resolution is that the process includes an irreversible energy-consuming step between initial
binding and commitment, giving incorrect substrates additional opportunities to dissociate. The
discrimination is applied twice, so the error rate is squared, at the cost of energy expended on
discarding intermediates. The same principle operates in translation, in DNA replication, and in
immune receptor discrimination.


biophysics

## Membranes, Transport, and Bioelectricity

Membranes are lipid bilayers whose structure follows from amphipathicity, and their physical
properties determine much of cellular function. The bilayer is a two-dimensional fluid in which
components diffuse laterally, its thickness is set by the lipid tail length, and its mechanical
properties — bending rigidity and area compressibility — determine what shapes it can adopt and
what deformations cost energy.


biophysics

Membrane curvature is generated and sensed by several mechanisms, and it is what shapes the
organelles and drives vesicle formation. Lipids with different head and tail cross-sections
prefer curved arrangements. Proteins with wedge shapes insert and impose curvature. Protein
scaffolds bind and hold a curved shape. And cytoskeletal forces pull. The energy required to bend
a membrane is what makes vesicle budding a process requiring machinery rather than occurring
spontaneously.


biophysics

Transport across membranes divides by whether it requires energy, and the physical framing is
the electrochemical gradient — the combination of the concentration difference and the electrical
potential difference. Passive transport moves substances down that gradient. Active transport
moves them against it and requires energy input, either directly from ATP or by coupling to
another substance's favourable gradient.


biophysics

Membrane potential arises from the combination of ion concentration gradients and selective
permeability, and understanding it correctly requires recognizing that very few ions are involved.
The membrane is a capacitor, and charging it to a hundred millivolts requires moving a number of
ions that is negligible relative to the cell's total content — which is why the potential can
change rapidly without measurably altering the concentrations, and why the gradients that
establish it persist through many action potentials.


biophysics

Membranes, transport, and bioelectricity:
  bilayer              self-assembling from amphipathicity; two-dimensional fluid
  bending rigidity     the energy cost of curvature; makes vesicle budding require
                       machinery
  curvature generation  lipid shape, protein insertion, scaffolding, cytoskeletal
                       force
  lateral diffusion    fast within the plane; flip-flop between leaflets is slow
                       and requires enzymes
  lipid rafts          proposed ordered domains; their existence and size in
                       living cells remains contested
  electrochemical gradient  concentration plus electrical; the driving force
  passive transport    down the gradient; channels and carriers
  active transport     against it; primary from ATP, secondary by coupling
  Nernst potential     the equilibrium potential for one ion
  Goldman equation     combines ions weighted by permeability
  membrane capacitance  ~1 microfarad per square centimetre; nearly universal
                       because bilayer thickness is nearly universal
  few ions move        charging the membrane capacitance requires a negligible
                       fraction of the ion content
  cable theory         signal propagation along a process; the length constant
                       determines how far a passive signal travels
  myelination          raises the length constant and lowers capacitance;
                       saltatory conduction follows


biophysics

Cable theory deserves a note because it explains why axons are structured as they are. A passive
electrical signal spreading along a process decays exponentially with a length constant determined
by the ratio of membrane resistance to axial resistance. A thin process has high axial resistance
and a short length constant, so a signal dies within a fraction of a millimetre. The two solutions
are regenerating the signal periodically, which is what voltage-gated channels do in an
unmyelinated axon, and raising the length constant by increasing the membrane resistance, which is
what myelin does — permitting the signal to spread passively between nodes and be regenerated only
there.


biophysics

[user] I'm using optical tweezers to measure the force generated by a single kinesin motor, and my
measurements are noisy and inconsistent.
[assistant] tool_call biophys.single_molecule({"technique": "optical_tweezers",
"target": "kinesin_stall_force", "trap_stiffness_pn_nm": 0.02, "bead_diameter_nm": 500,
"expected_force_pn": 5, "sampling_hz": 200, "drift_nm_per_min": 15,
"buffer_atp_um": 2000})
[tool] {"thermal_force_noise_pn": 1.2, "signal_to_noise": 4.2,
"issues": ["trap_too_soft_for_force_range", "drift_comparable_to_signal",
"bandwidth_insufficient_to_resolve_steps"], "recommendations": ["increase_trap_stiffness",
"differential_detection", "higher_bandwidth", "drift_correction"]}
[assistant] Three separate issues, and the thermal noise is not actually your limiting factor.


biophysics

Your trap stiffness of 0.02 piconewtons per nanometre is too soft for a five piconewton stall
force — the bead would need to be displaced 250 nanometres, which is well outside the linear
range of the detection and comparable to the trap's own dimensions. Raise the stiffness to around
0.1, which brings the displacement at stall to about 50 nanometres and keeps you in the linear
regime. The thermal position fluctuation also falls with stiffer trapping, since the variance is
inversely proportional to stiffness.


biophysics

Your drift of 15 nanometres per minute is comparable to a kinesin step of 8 nanometres, which
means slow drift is indistinguishable from stepping over the timescale of a measurement. Two
remedies: use differential detection with a second trapped bead as a reference, so common-mode
drift cancels, or use a dumbbell geometry with two traps where the measurement is between the two
beads rather than relative to the stage. Active stabilization referenced to a fiducial marker is
the alternative.


biophysics

Your sampling at 200 hertz cannot resolve individual steps. A kinesin step takes under a
millisecond and the dwell between steps at saturating ATP is on the order of ten milliseconds, so
you need at least a few kilohertz to see the step structure rather than an averaged velocity.


biophysics

One experimental suggestion: lower the ATP concentration substantially. At two millimolar you are
saturating, so the motor steps as fast as it mechanically can and the dwells are short. At a few
micromolar the ATP binding becomes rate-limiting, the dwells lengthen to tens or hundreds of
milliseconds, and the steps become clearly resolvable at your current bandwidth. This is the
standard approach to resolving stepping, and it also lets you separate the mechanical and chemical
parts of the cycle by varying the concentration.


biophysics

## Cell Mechanics and Techniques

Cells are mechanically active and mechanically responsive, and the recognition that mechanics is
a signalling modality alongside chemistry has reshaped cell biology. The cytoskeleton provides
structure and generates force: actin filaments with myosin motors produce contractility,
microtubules provide compression resistance and transport tracks, and intermediate filaments
provide tensile resilience.


biophysics

Cells sense mechanical properties of their environment and respond, which is mechanotransduction.
Stem cells differentiate toward different lineages depending on substrate stiffness alone — soft
substrates toward neural, stiff toward bone — with no difference in chemical signals. Cells migrate
up stiffness gradients. Endothelial cells align with flow. The mechanisms include force-sensitive
ion channels, proteins that unfold under tension to expose binding sites, and adhesion complexes
that grow under load.


biophysics

Techniques for measuring these have developed alongside the understanding. Single-molecule methods
manipulate and measure individual molecules directly. Optical tweezers trap a dielectric bead in a
focused laser and measure forces of piconewtons with nanometre resolution. Magnetic tweezers apply
force and torque to a magnetic bead, which suits twisting DNA. Atomic force microscopy images
surfaces and measures forces on individual molecular interactions.


biophysics

Cell mechanics and techniques:
  actin and myosin     contractility; the force generation system
  microtubules         compression resistance and transport tracks
  intermediate filaments  tensile resilience
  cortical tension     the contractile layer beneath the membrane; sets cell shape
  mechanotransduction  mechanical signals converted to biochemical responses
  substrate stiffness  directs stem cell differentiation with no chemical
                       difference
  durotaxis            migration up stiffness gradients
  force-sensitive channels  open under membrane tension; touch, hearing,
                       proprioception
  cryptic binding sites  proteins unfolding under tension to expose interaction
                       surfaces
  optical tweezers     piconewton forces, nanometre resolution
  magnetic tweezers    force and torque; DNA twisting
  atomic force microscopy  imaging and force spectroscopy
  FRET                 distance between two fluorophores over 1-10 nm; a
                       molecular ruler reporting conformational change
  single molecule fluorescence  observe individual molecules; reveals
                       heterogeneity that ensemble measurement averages away
  super-resolution     beyond the diffraction limit by localizing individual
                       emitters or by patterned illumination
  cryo-electron microscopy  structures without crystallization; transformative for
                       large complexes and membrane proteins


biophysics

Single-molecule measurement deserves the closing emphasis for what it revealed rather than for
its precision. Ensemble measurements report averages, and an average conceals whether a population
is homogeneous or is a mixture of distinct behaviours. Single-molecule observation showed that
enzymes have memory — an individual molecule's catalytic rate fluctuates over time on scales far
longer than a single turnover — that molecular motors take discrete steps rather than moving
smoothly, that protein folding proceeds through distinct pathways rather than one, and that
apparently uniform populations frequently contain distinct subpopulations. These are qualitative
findings about mechanism that no amount of precision in an ensemble measurement could have
produced, because the averaging destroyed exactly the information they depend on.


classical_mechanics

## Kinematics and Newtonian Dynamics

Newton's three laws organize all of classical dynamics. The first defines the inertial
frame: a body with no net force maintains constant velocity, so any frame in which
this holds is inertial, and the law is best read as a statement about which frames the
second law applies in rather than as a special case of it. The second, F = dp/dt,
relates net force to the rate of change of momentum, reducing to F = ma only when mass
is constant — a caveat that matters for rockets, conveyor belts, and any system
shedding or accumulating mass. The third states that forces come in equal and opposite
pairs acting on *different* bodies, and the emphasis on "different" is what prevents the
common error of concluding that all forces cancel and nothing ever accelerates.


classical_mechanics

Free-body diagrams are the practical core of problem-solving, and most errors are
bookkeeping rather than conceptual. Draw one body at a time, include every force acting
*on* it and none of the forces it exerts on others, choose axes that align with the
expected acceleration, and resolve into components. On an inclined plane, tilting the
axes to lie along and perpendicular to the surface reduces the problem to two scalar
equations with the acceleration entirely in one of them. Normal force is a constraint
force whose magnitude adjusts to whatever prevents interpenetration, so it is never
simply "mg" except in the special case of a horizontal surface with no vertical
acceleration.


classical_mechanics

Friction has two regimes governed by different rules. Static friction is a constraint
force taking whatever value up to mu_s N is needed to prevent relative sliding;
writing f = mu_s N when the object is not on the verge of slipping is a standard and
consequential error. Kinetic friction, once sliding begins, has the definite magnitude
mu_k N opposing relative motion, with mu_k typically somewhat less than mu_s, which is
why an object breaks loose and then accelerates. Neither depends on contact area to
first order, a genuinely counterintuitive result explained by real contact occurring at
microscopic asperities whose total area scales with load rather than with apparent
area.


classical_mechanics

Kinematic relations for constant acceleration:
  v      = v0 + a t
  x      = x0 + v0 t + (1/2) a t^2
  v^2    = v0^2 + 2 a (x - x0)              time-independent; use when t is unwanted
  x      = x0 + (1/2)(v0 + v) t             average-velocity form


classical_mechanics

Projectile motion is the canonical application: horizontal and vertical motions
decouple because gravity acts only vertically, so the horizontal velocity is constant
and the vertical follows the constant-acceleration relations with a = -g. Range on
level ground is v0^2 sin(2 theta)/g, maximized at 45 degrees, and complementary angles
give equal range with different flight times. Air resistance breaks the decoupling
entirely, since drag depends on total speed and couples the components; for a
quadratic drag law there is no closed-form trajectory and the optimal launch angle
drops well below 45 degrees.


classical_mechanics

Q: A block sits on an incline without sliding, at an angle below the critical angle.
The friction force on it is:
A. mu_s * N
B. mg sin(theta), directed up the slope
C. mu_k * N
D. Zero
Reasoning: The block is in equilibrium, so friction must exactly balance the component
of gravity along the slope, which is mg sin(theta). Static friction is a constraint
force taking whatever value is required up to its maximum of mu_s N; it equals that
maximum only at the verge of slipping. Answer: B.


classical_mechanics

Q: A rocket in deep space ejects mass. Its acceleration is correctly obtained from:
A. F = ma with m constant
B. F = dp/dt, accounting for the changing mass and exhaust momentum
C. Conservation of kinetic energy
D. The impulse-momentum theorem with constant mass
Reasoning: Mass is not constant, so F = ma does not apply. The correct treatment tracks
total momentum of rocket plus exhaust, yielding the Tsiolkovsky equation delta_v = v_e
ln(m_initial/m_final). Kinetic energy is not conserved because the chemical energy of
the propellant is converted. Answer: B.


classical_mechanics

## Conservation Laws

Conservation laws are the most powerful tools in mechanics because they relate initial
and final states without requiring the trajectory between them. Energy is conserved
when all forces are conservative — derivable from a potential — which excludes
friction and other dissipative forces. Momentum is conserved for any system with no net
external force, regardless of the internal forces' nature, which is why it survives
collisions where energy does not. Angular momentum is conserved when there is no net
external torque.


classical_mechanics

Noether's theorem explains why these three and not others: each continuous symmetry of
the action yields a conserved quantity. Time-translation symmetry gives energy
conservation, space-translation symmetry gives momentum, rotational symmetry gives
angular momentum. This is not a reformulation but an explanation, and it tells you
immediately when a conservation law fails — a time-dependent potential breaks
time-translation symmetry and energy is not conserved, which is exactly the case of a
driven oscillator.


classical_mechanics

Collisions are classified by what happens to kinetic energy. Momentum is conserved in
all of them. Elastic collisions conserve kinetic energy as well, and in one dimension
the relative velocity of approach equals the relative velocity of separation — a
relation often faster to apply than the quadratic energy equation. Perfectly inelastic
collisions have the bodies moving together afterward, which is the maximum kinetic
energy loss consistent with momentum conservation. The coefficient of restitution
interpolates between the two, being the ratio of separation to approach speed.


classical_mechanics

Energy bookkeeping reference:
  kinetic, translational   K = (1/2) m v^2
  kinetic, rotational      K = (1/2) I omega^2
  gravitational, local     U = m g h
  gravitational, general   U = -G M m / r          zero at infinity, hence negative
  elastic (spring)         U = (1/2) k x^2
  work by a force          W = integral F . dr     path-independent iff conservative
  power                    P = F . v = dE/dt
  work-energy theorem      W_net = delta K         always, conservative or not


classical_mechanics

The distinction between the local and general gravitational potentials repays
attention. U = mgh is the first-order expansion of -GMm/r near the surface, valid only
while the height change is small compared with the planet's radius. The general form
being negative and approaching zero at infinity is a choice of reference point, and it
makes the escape condition transparent: a bound orbit has negative total energy, a
parabolic escape trajectory has exactly zero, and a hyperbolic one has positive energy
with speed remaining at infinity.


classical_mechanics

The virial theorem relates time-averaged kinetic and potential energies for a bound
system with a power-law potential: 2 average(K) = -n average(U) for U proportional to
r^n. For gravity, n = -1, giving 2K = -U, so the total energy is E = K + U = -K. This
has a startling consequence used constantly in astrophysics: a self-gravitating system
that loses energy — by radiating — gets *hotter*, since K = -E increases as E becomes
more negative. Stars contract and heat as they radiate, and gravitationally bound
systems therefore have an effective negative heat capacity.


classical_mechanics

## Rotational Motion

Rotational dynamics parallels translational dynamics with each quantity replaced by its
angular analogue: mass by moment of inertia, force by torque, momentum by angular
momentum, velocity by angular velocity. The parallel is exact, so every translational
result has a rotational counterpart, and the equation tau = I alpha is the rotational
second law.


classical_mechanics

Moment of inertia depends on the axis, not just on the body, and this is the main
practical difference from mass. It is the integral of r^2 dm with r measured
perpendicular to the axis, so mass far from the axis counts quadratically more. The
parallel axis theorem, I = I_cm + M d^2, gives the moment about any axis parallel to
one through the centre of mass, and it shows that the centre-of-mass axis always
minimizes the moment of inertia among parallel axes.


classical_mechanics

Moments of inertia about the centre of mass:
  point mass at radius r          m r^2
  thin ring / hoop, axis normal   M R^2
  solid disc / cylinder           (1/2) M R^2
  solid sphere                    (2/5) M R^2
  thin spherical shell            (2/3) M R^2
  rod, axis through centre        (1/12) M L^2
  rod, axis through end           (1/3) M L^2
  parallel axis theorem           I = I_cm + M d^2


classical_mechanics

The classic demonstration is a race down an incline. Objects rolling without slipping
share their energy between translation and rotation, and the split depends only on the
shape through the dimensionless factor c in I = c M R^2 — not on mass or radius at all.
The acceleration is g sin(theta)/(1 + c), so a solid sphere with c = 2/5 accelerates
at 5/7 g sin(theta), a disc with c = 1/2 at 2/3, a hoop with c = 1 at only half. A
sphere beats a disc beats a hoop, always, regardless of their masses or sizes, and any
of them loses to a frictionless sliding block that need not spin up at all.


classical_mechanics

Angular momentum conservation produces the most visible effects in mechanics. A skater
pulling in their arms reduces I and therefore increases omega, with kinetic energy
increasing because the skater does work pulling mass inward against the centripetal
requirement. A gyroscope precesses rather than falling because the torque from gravity
is perpendicular to the angular momentum and therefore rotates its direction rather
than changing its magnitude, with precession rate Omega = tau / L — faster spin means
slower precession, which is why a rapidly spinning top stands nearly upright and a
slowing one begins to wobble and fall.


classical_mechanics

[user] I'm sizing a flywheel to store 500 kJ at 6000 rpm in a 400 mm diameter package.
What mass do I need and what's the hoop stress?
[assistant] tool_call mechanics.flywheel({"energy_j": 500000, "rpm": 6000,
"outer_radius_m": 0.2, "geometry": "solid_disc", "material": "4340_steel"})
[tool] {"omega_rad_s": 628.3, "required_I": 2.533, "mass_kg": 126.7,
"thickness_m": 0.128, "rim_speed_m_s": 125.7, "hoop_stress_mpa": 296,
"material_yield_mpa": 710, "safety_factor": 2.4}
[assistant] A solid steel disc of 400 mm diameter needs about 127 kg and would be
roughly 128 mm thick. That comes from E = (1/2) I omega^2 with omega = 628 rad/s,
requiring I = 2.53 kg m^2, and I = (1/2) M R^2 for a solid disc.


classical_mechanics

The stress number is the one that should drive your design. Rim speed is 126 m/s, and
the peak hoop stress in a spinning disc scales as rho times rim speed squared — it is
296 MPa here against a 710 MPa yield for 4340, so a safety factor of 2.4. That is
acceptable but not generous for a rotating part where failure is energetic.


classical_mechanics

The key scaling to internalize: stress depends on rim *speed*, not on radius or rpm
separately, so you cannot make the flywheel safer by scaling it up at the same tip
speed. Specific energy per unit mass is likewise set by the material's strength-to-
density ratio, not by geometry. If you need more margin, the levers are a
higher-strength alloy, a composite rim where carbon fibre's strength-to-density ratio
is several times steel's, or accepting a lower rpm and a heavier wheel.


classical_mechanics

Two practical points beyond the sizing. A solid disc is not the most mass-efficient
geometry — a thick-rim design puts mass where the r^2 weighting pays, giving more
inertia per kilogram, though it raises the stress concentration at the hub. And at 127
kg spinning at 6000 rpm you are storing the energy equivalent of roughly 120 grams of
TNT, so containment design and a burst shield are not optional.


classical_mechanics

## Oscillations

Simple harmonic motion arises whenever a restoring force is proportional to
displacement, F = -kx. The equation of motion m x'' + k x = 0 has solution x = A
cos(omega t + phi) with omega = sqrt(k/m). The defining property is that the period is
independent of amplitude, which is why pendulum clocks work and why the harmonic
approximation is so useful: any smooth potential near a minimum is locally quadratic,
so small oscillations about any stable equilibrium are harmonic with omega =
sqrt(U''(x0)/m).


classical_mechanics

Damping adds a velocity-dependent force, giving m x'' + b x' + k x = 0. The behaviour
depends on the damping ratio zeta = b/(2 sqrt(mk)). Underdamped (zeta < 1) oscillates
with exponentially decaying amplitude at a frequency slightly below the natural one.
Critically damped (zeta = 1) returns to equilibrium in the shortest time without
overshoot, which is the design target for door closers, vehicle suspensions, and
control loops. Overdamped (zeta > 1) returns without oscillating but more slowly,
because the large damping force itself impedes the approach — a fact that surprises
people who assume more damping is always faster.


classical_mechanics

Driven oscillators exhibit resonance. Forcing at frequency omega_d produces a
steady-state response whose amplitude peaks near the natural frequency, with the peak
height set by the quality factor Q = 1/(2 zeta). High Q gives a tall narrow resonance
and long ringing; low Q gives a broad shallow one that settles quickly. The response
also shifts phase, lagging the drive by 90 degrees exactly at resonance and
approaching 180 degrees well above it. Resonance is either the mechanism or the enemy
depending on the application: it is how radios select a station, how NMR works, and
how a wine glass shatters, and it is why bridges, buildings, and rotating machinery are
designed with their natural frequencies deliberately placed away from expected
excitation.


classical_mechanics

Oscillator quantities:
  natural frequency        omega_0 = sqrt(k/m)
  period                   T = 2 pi / omega_0
  damping ratio            zeta = b / (2 sqrt(mk))
  damped frequency         omega_d = omega_0 sqrt(1 - zeta^2)
  quality factor           Q = 1/(2 zeta) = omega_0 m / b
  energy decay             E(t) = E_0 exp(-omega_0 t / Q)
  resonant amplification   A_peak / A_static ~ Q for lightly damped systems
  simple pendulum          omega = sqrt(g/L), small angles only
  physical pendulum        omega = sqrt(m g d / I)
  torsional                omega = sqrt(kappa / I)


classical_mechanics

Coupled oscillators introduce normal modes: special patterns of motion in which every
part oscillates at a common frequency with fixed relative amplitudes. Any motion of the
system decomposes into a superposition of normal modes, which is an eigenvalue problem
in disguise — the mode shapes are eigenvectors of the stiffness matrix relative to the
mass matrix, and the squared frequencies are the eigenvalues. This is the bridge from
mechanics to linear algebra and, in the continuum limit, to Fourier analysis and the
wave equation.


classical_mechanics

Q: A damped oscillator has Q = 50. After how many oscillation periods does its energy
fall to about 1/e of its initial value?
A. 50   B. 8   C. 25   D. 314
Reasoning: Energy decays as exp(-omega_0 t / Q), so it falls to 1/e when omega_0 t = Q,
that is t = Q/omega_0. One period is T = 2 pi / omega_0, so the number of periods is
t/T = Q/(2 pi) = 50/6.28 ≈ 8. The useful rule is that Q is roughly 2 pi times the
number of oscillations in an energy decay time. Answer: B.


classical_mechanics

Q: A vehicle suspension is designed to be critically damped rather than overdamped
because:
A. It uses less energy
B. It returns to equilibrium fastest without overshoot
C. It has a higher resonant frequency
D. It is cheaper to manufacture
Reasoning: Critical damping is precisely the boundary case minimizing settling time
subject to no overshoot. Overdamped systems do not overshoot either but return more
slowly, since the damping force resists the return. Underdamped systems return quickly
but oscillate. Answer: B.


classical_mechanics

## Lagrangian and Hamiltonian Mechanics

The Lagrangian formulation replaces forces with energies. Define L = T - V, the kinetic
minus potential energy expressed in any convenient generalized coordinates, and the
equations of motion follow from the Euler-Lagrange equation, d/dt(dL/dq') - dL/dq = 0,
one for each coordinate. The advantages are substantial: constraint forces drop out
automatically because the coordinates are chosen to satisfy the constraints, the
formulation is coordinate-independent so polar or spherical coordinates require no
special treatment, and energies are scalars while forces are vectors that must be
resolved.


classical_mechanics

This comes from Hamilton's principle: the physical trajectory is the one that makes the
action, the time integral of L, stationary among all paths with fixed endpoints. That a
variational principle underlies mechanics is one of the deepest facts in physics, and
the same structure recurs in optics as Fermat's principle, in general relativity
through the Einstein-Hilbert action, and in quantum mechanics where Feynman's path
integral sums exp(iS/hbar) over all paths and the classical trajectory emerges as the
stationary-phase contribution.


classical_mechanics

Cyclic coordinates make conservation laws automatic. If L does not depend explicitly on
a coordinate q, then the corresponding conjugate momentum p = dL/dq' is conserved. A
central potential's Lagrangian has no explicit dependence on the azimuthal angle, so
angular momentum is conserved without further argument. If L has no explicit time
dependence, the Hamiltonian is conserved, and when the kinetic energy is quadratic in
the velocities and the constraints are time-independent, that Hamiltonian is the total
energy. This is Noether's theorem in its most usable form.


classical_mechanics

The Hamiltonian formulation performs a Legendre transform to generalized momenta,
giving H(q, p) = sum p q' - L and the symmetric pair of first-order equations q' =
dH/dp and p' = -dH/dq. Trading n second-order equations for 2n first-order ones is not
obviously progress, but it opens phase space as the natural arena. Liouville's theorem
states that phase-space volume is conserved under Hamiltonian flow, which is the
foundation of statistical mechanics. Canonical transformations preserve the structure
and lead to Hamilton-Jacobi theory and action-angle variables. And the Poisson bracket
{f, g} obeys algebraic relations that map directly onto the quantum commutator, making
the Hamiltonian formalism the natural bridge to quantum mechanics.


classical_mechanics

Worked comparison — bead on a rotating hoop, radius R, angular velocity Omega:
  Newtonian:   resolve gravity, normal force, and the rotating-frame pseudo-forces
               along the hoop; requires care with centrifugal and Coriolis terms
  Lagrangian:  T = (1/2) m R^2 (theta'^2 + Omega^2 sin^2 theta)
               V = -m g R cos theta
               L = T - V, then Euler-Lagrange in theta alone
  result:      R theta'' = R Omega^2 sin theta cos theta - g sin theta
  equilibria:  theta = 0 always; additionally cos theta = g/(R Omega^2) when
               Omega^2 > g/R — a pitchfork bifurcation at the critical rotation rate


classical_mechanics

That bifurcation is the physical content: below a critical spin rate the bead sits at
the bottom, and above it the bottom becomes unstable and two new stable equilibria
appear off-axis. The Lagrangian route delivers this in a few lines with no vector
resolution and no explicit treatment of the constraint force holding the bead on the
hoop, which is exactly the kind of problem where the formalism earns its abstraction.


classical_mechanics

## Chaos and Continuum Mechanics

Deterministic systems can be unpredictable. A chaotic system has sensitive dependence
on initial conditions: nearby trajectories diverge exponentially at a rate given by the
largest Lyapunov exponent, so an uncertainty of epsilon in the initial state grows to
order one after a time proportional to ln(1/epsilon). Because that dependence is
logarithmic, improving initial measurements by a factor of a thousand buys only a
constant additive increase in the prediction horizon. This is why weather forecasting
has a hard practical limit of a couple of weeks regardless of instrumentation, and it
is a statement about the dynamics rather than about the model's quality.


classical_mechanics

Chaos requires nonlinearity and at least three dimensions in a continuous autonomous
system, since the Poincare-Bendixson theorem confines two-dimensional flows to fixed
points and limit cycles. Discrete maps can be chaotic in one dimension, the logistic
map x -> r x(1 - x) being the canonical example, which undergoes a period-doubling
cascade to chaos as r increases, with the ratio of successive bifurcation intervals
approaching Feigenbaum's constant 4.669. That this constant is universal across
entirely different systems with a quadratic maximum was the discovery that made chaos a
subject rather than a collection of curiosities.


classical_mechanics

The double pendulum is the standard mechanical example: two coupled rigid pendulums,
four-dimensional phase space, fully deterministic, and chaotic for all but small
amplitudes. Its equations follow in a few lines from the Lagrangian and are hopeless to
derive by force balance. Strange attractors — the Lorenz attractor from a truncated
convection model being the famous case — are the geometric signature: trajectories are
bounded and never repeat, converging onto a fractal set of non-integer dimension.


classical_mechanics

Continuum mechanics treats matter as a continuous field rather than as particles, valid
when the length scales of interest greatly exceed the molecular mean free path. Stress
is a rank-two tensor mapping a surface normal to the traction vector across it, with
the diagonal entries normal stresses and the off-diagonal ones shear. Strain is the
symmetric gradient of the displacement field. For a linear elastic isotropic material
they are related by two constants, conventionally Young's modulus and Poisson's ratio,
which reduce to bulk and shear moduli for volumetric and shape changes respectively.


classical_mechanics

Continuum reference:
  Cauchy stress tensor        sigma_ij; symmetric by angular momentum balance
  small strain                eps_ij = (1/2)(du_i/dx_j + du_j/dx_i)
  Hooke's law, 1D             sigma = E eps
  Poisson's ratio             nu = -transverse strain / axial strain, ~0.3 metals
  shear modulus               G = E / (2(1 + nu))
  bulk modulus                K = E / (3(1 - 2 nu))
  wave speeds                 c_longitudinal = sqrt(E/rho), c_shear = sqrt(G/rho)
  Navier-Stokes               rho(dv/dt + v.grad v) = -grad p + mu lap v + f
  Reynolds number             Re = rho v L / mu; inertial over viscous forces


classical_mechanics

The Reynolds number is the single most important dimensionless group in fluid
mechanics because it determines the character of the flow rather than merely its
magnitude. Below roughly 2,000 in a pipe the flow is laminar, with smooth layers and a
parabolic velocity profile, and the pressure drop is linear in flow rate. Above roughly
4,000 it is turbulent, with chaotic eddies, a flatter profile, far greater mixing, and
a pressure drop closer to quadratic in flow rate. Between the two lies a transitional
regime sensitive to disturbances. Dynamic similarity means two flows with the same
Reynolds number behave alike regardless of scale, which is what makes wind-tunnel and
towing-tank testing of scale models meaningful, and what forces model tests to
compensate with altered fluid density, viscosity, or velocity to match the full-scale
number.


computational_physics

## Simulation Methodology

Computational physics is a third mode of inquiry alongside theory and experiment, and its
distinctive characteristic is that a simulation is an experiment on a model rather than on
nature. This means its results are only as good as the model, and that the errors divide into
two kinds requiring different responses: model error, where the equations do not describe the
system adequately, and numerical error, where the equations are solved imprecisely. Reducing
numerical error in a poor model produces a precise wrong answer.


computational_physics

Every simulation makes choices about scale, and the choices determine what phenomena can appear.
A molecular dynamics simulation of a nanosecond cannot exhibit a process taking microseconds. A
grid with a cell size of a metre cannot resolve a feature of a centimetre. The physics not
resolved must either be absent, which is a modelling decision to be justified, or represented by
a subgrid model, which introduces parameters and assumptions. Recognizing which phenomena the
chosen scales admit is a prerequisite to interpreting any result.


computational_physics

Reproducibility in simulation requires more than in experiment because the entire computation
is in principle deterministic. Recording the code version, the input parameters, the random
seeds, the compiler and its flags, and the library versions is what permits a result to be
regenerated — and aggressive floating-point optimization changes results, so compiler flags are
part of the specification rather than an implementation detail. Archiving these alongside the
output is inexpensive and is frequently omitted.


computational_physics

Methodology:
  model error          the equations do not describe the system; not reduced by
                       better numerics
  numerical error      the equations are solved imprecisely; reduced by refinement
  verification         solving the equations correctly
  validation           solving the correct equations; requires comparison with
                       experiment
  resolved scales      what the discretization can represent
  subgrid model        represents unresolved physics; introduces parameters
  dimensionless groups  identify the governing regimes and permit scaling
  nondimensionalization  reduces the parameter count and improves conditioning
  convergence study    refine and confirm the answer stops changing; the single
                       most informative check
  method of manufactured solutions  construct an exact solution, derive the
                       required source, verify the code reproduces it at the
                       expected order
  reproducibility      code version, parameters, seeds, compiler, flags, libraries
  ensembles            repeat with varied initial conditions or parameters; a
                       single run tells you nothing about sensitivity


computational_physics

Q: A simulation's results change substantially when the compiler optimization level is raised
from -O0 to -O3. The most likely explanation is:
A. A compiler bug
B. Floating-point reassociation and fused operations change the rounding, which an unstable
   algorithm or a chaotic system amplifies
C. The code is incorrect
D. The hardware is faulty
Reasoning: Aggressive optimization reorders floating-point operations and may fuse multiply-add
pairs, both of which change rounding. For a well-conditioned stable computation the difference is
at the round-off level; for a chaotic system or an unstable algorithm it amplifies. This is why
floating-point flags are part of a simulation's specification, and it also indicates that
sensitivity should be quantified. Answer: B.


computational_physics

## Molecular Dynamics

Molecular dynamics integrates Newton's equations for a system of interacting particles, and its
appeal is that it produces a trajectory from which any observable can be computed. The
computational cost is dominated by evaluating the forces, which for pairwise interactions is
quadratic in particle count unless the interaction is truncated.


computational_physics

Short-range interactions are truncated at a cutoff distance beyond which the force is
negligible, and neighbour lists track which pairs are close enough to matter, reducing the cost
to linear. Long-range interactions, particularly electrostatics, cannot be truncated without
introducing serious artifacts because the sum converges conditionally, and Ewald summation splits
it into a rapidly converging real-space part and a reciprocal-space part evaluated by fast
Fourier transform.


computational_physics

The integrator must be symplectic, which is not a preference but a requirement for meaningful
long-time behaviour. A general-purpose high-order integrator has local errors that accumulate as
secular energy drift, so the simulated system slowly heats or cools and the thermodynamics
becomes meaningless. Symplectic integrators preserve phase space volume, so energy error
oscillates within a bound over arbitrarily long runs. Velocity Verlet is second order,
symplectic, time-reversible, and requires one force evaluation per step, which is why it is
near-universal.


computational_physics

Molecular dynamics:
  force field          the interaction model; bonded and non-bonded terms with
                       fitted parameters
  cutoff               truncate short-range interactions; use a switching function
                       to avoid a force discontinuity
  neighbour list       track nearby pairs; rebuilt periodically
  Ewald / PME          long-range electrostatics; real and reciprocal space
  periodic boundaries  eliminate surfaces; introduce artificial periodicity that
                       must be checked against the correlation length
  velocity Verlet      symplectic, time-reversible, one force evaluation per step
  timestep             bounded by the fastest motion, typically bond vibration;
                       ~1 fs, or ~2 fs with constrained hydrogen bonds
  constraints          SHAKE and LINCS remove the fastest vibrations, permitting a
                       larger timestep
  thermostat           couples to a heat bath; Nose-Hoover and Langevin sample the
                       canonical ensemble correctly, while naive velocity rescaling
                       does not
  barostat             constant pressure
  ensembles            microcanonical, canonical, isothermal-isobaric
  equilibration        discard the initial transient; the length required must be
                       demonstrated rather than assumed
  timescale gap        accessible times are microseconds while many processes take
                       milliseconds or longer; the field's central limitation
  enhanced sampling    umbrella sampling, metadynamics, replica exchange; bias the
                       simulation to cross barriers and correct afterward


computational_physics

The timescale gap is the field's defining constraint. The timestep is bounded by the fastest
motion in the system, and bond vibrations require a step of about one femtosecond, so reaching a
millisecond requires a trillion steps. Many processes of interest — protein folding, ligand
binding, conformational change — occur on exactly those timescales. Enhanced sampling methods
address this by biasing the simulation toward crossing barriers and then correcting the
statistics, which recovers equilibrium properties at enormously reduced cost while requiring the
relevant reaction coordinate to be identified in advance.


computational_physics

## Monte Carlo Methods

Monte Carlo methods use random sampling to compute quantities that are difficult to evaluate
directly, and their defining advantage is that the error scales as the inverse square root of the
sample count independent of dimension. Deterministic quadrature error scales as the sample count
to the power of the accuracy order divided by the dimension, so it degrades rapidly with
dimension while Monte Carlo does not. The crossover is around four to eight dimensions, above
which sampling is the only practical approach.


computational_physics

Markov chain Monte Carlo samples from a distribution known only up to a normalizing constant by
constructing a chain whose stationary distribution is the target. The Metropolis algorithm
proposes a move and accepts it with a probability depending on the ratio of target densities,
which is exactly why the normalizing constant is unnecessary — it cancels in the ratio. This is
what makes statistical mechanics computationally tractable, since the partition function is the
intractable quantity.


computational_physics

Detailed balance is the design principle: a chain satisfying it has the target as a stationary
distribution, and constructing the acceptance probability to enforce it is how any target
distribution is sampled. Ergodicity is the additional requirement that the chain can reach every
state, which fails when the target has well-separated modes and the proposal cannot cross between
them — producing a chain that samples one mode indefinitely while appearing converged.


computational_physics

Monte Carlo methods:
  error scaling        inverse square root of sample count, independent of
                       dimension
  crossover            around 4-8 dimensions; above this Monte Carlo dominates
                       quadrature
  importance sampling  sample from a distribution concentrating where the
                       integrand is large; reduces variance dramatically when the
                       proposal is well chosen and increases it when badly chosen
  Metropolis           propose and accept; needs the target only up to a constant
  detailed balance     the design condition ensuring the correct stationary
                       distribution
  ergodicity           the chain must reach every state; fails with separated
                       modes
  autocorrelation time  successive samples are correlated; the effective sample
                       count is far below the raw count and must be estimated
  burn-in              discard the initial non-stationary portion
  critical slowing down  autocorrelation diverges near a phase transition; cluster
                       algorithms address it by flipping correlated regions
                       together
  parallel tempering   run replicas at several temperatures and swap; crosses
                       barriers that a single chain cannot
  quantum Monte Carlo  path integral and diffusion methods for quantum systems
  sign problem         fermionic and real-time quantum Monte Carlo produce
                       oscillating weights whose cancellation destroys the
                       statistics; an unsolved obstacle limiting what can be
                       simulated


computational_physics

The sign problem deserves emphasis because it is a genuine barrier rather than a technical
difficulty. For fermionic systems and for real-time quantum dynamics, the quantity being sampled
has contributions of opposite sign that largely cancel, so the signal is a small difference of
large numbers and the statistical error grows exponentially with system size and inverse
temperature. No general solution exists, it is believed to be computationally hard in a formal
sense, and it is why certain regions of the quantum many-body phase diagram — including dense
quark matter and much of high-temperature superconductivity — remain computationally
inaccessible.


computational_physics

## Grids, N-body, and Field Simulation

Partial differential equations are solved by discretizing space and time, and the three main
spatial approaches suit different problems. Finite differences approximate derivatives on a
structured grid and are simple to implement and analyze at the cost of poor handling of complex
geometry. Finite elements expand the solution in local basis functions over an unstructured mesh,
handling arbitrary geometry with a rigorous variational error framework. Finite volumes enforce
conservation on control volumes exactly, which is why they dominate in fluid dynamics where
conservation is the point.


computational_physics

Explicit time stepping is bounded by the Courant condition requiring that information not travel
more than one cell per step, which for diffusion scales with the square of the cell size and
therefore becomes prohibitive under refinement. Implicit methods are unconditionally stable and
require solving a linear system per step, which is why diffusion is almost always treated
implicitly and advection frequently explicitly, with operator splitting handling problems
containing both.


computational_physics

N-body simulation computes gravitational or electrostatic interaction among many particles, and
the naive cost is quadratic. Tree methods group distant particles and approximate their combined
influence by a multipole expansion, reducing the cost to n log n with a controllable accuracy
parameter. Particle-mesh methods assign mass to a grid, solve for the potential by fast Fourier
transform, and interpolate the force back, giving linear scaling at the cost of resolution
limited by the grid. Hybrid methods use a mesh for long-range forces and direct summation for
close pairs.


computational_physics

Grids, N-body, and fields:
  finite difference    structured grid; simple, poor with complex geometry
  finite element       unstructured mesh, variational framework, complex geometry
  finite volume        conservation enforced exactly; fluids
  spectral method      global basis; exponential convergence for smooth periodic
                       problems, poor with discontinuities
  CFL condition        explicit stability limit; information travels at most one
                       cell per step
  implicit             unconditionally stable, requires a solve per step
  operator splitting   treat different terms with different methods
  adaptive mesh refinement  refine where the solution demands it; large savings
                       and substantial implementation complexity
  direct N-body        quadratic; exact, and limited to modest particle counts
  tree code            n log n; multipole approximation of distant groups
  particle mesh        linear; grid-limited resolution
  softening            a modified force at very small separation, preventing
                       unphysical accelerations from close encounters; a numerical
                       necessity with physical consequences that must be reported
  symplectic integration  required for long-term orbital and Hamiltonian dynamics
  lattice gauge theory  discretized quantum field theory; the primary tool for
                       non-perturbative strong interaction calculations


computational_physics

Gravitational softening deserves specific mention because it is a numerical device with
physical consequences that are frequently underreported. The gravitational force diverges at zero
separation, so a close pair of simulation particles receives an enormous acceleration that
neither the timestep nor the physics justifies — the particles represent a distribution of mass
rather than point masses. Softening modifies the force at small separations to remove the
divergence, and the choice of softening length sets the smallest scale on which the simulation's
dynamics is meaningful. Structures near or below that scale are numerical artifacts, and reporting
the softening length is necessary for a result to be interpretable.


computational_physics

[user] My N-body galaxy simulation shows small dense clumps forming that I don't think are
physical.
[assistant] tool_call sim.nbody_diagnose({"particles": 1e6, "softening_pc": 5,
"mean_separation_pc": 40, "clump_scale_pc": 8, "integrator": "leapfrog",
"timestep": "fixed", "two_body_relaxation_check": false})
[tool] {"clump_near_softening": true, "particles_per_clump": 12,
"relaxation_time_vs_runtime": 0.34, "verdict": "numerical_two_body_relaxation",
"recommendations": ["increase_particle_count", "check_clump_particle_count",
"adaptive_timestep", "report_resolution_limit"]}
[assistant] The clumps are almost certainly numerical. Two indicators point the same way.


computational_physics

Their scale is 8 parsecs against a softening length of 5, so they are barely above the
resolution limit — structures within a few softening lengths are not resolved and their
properties are set by the numerics rather than by the physics. And each clump contains about
twelve particles, which is far too few for its dynamics to be meaningful; a bound structure
needs at least of order a hundred particles before its properties are reliable, and preferably
more.


computational_physics

The mechanism is two-body relaxation. Your simulation particles are far more massive than real
stars, so gravitational scattering between them is much stronger than in the real system, and
the relaxation time scales inversely with particle number. Your relaxation time is 0.34 of your
runtime, which means the system has partially relaxed through an entirely artificial process —
real galaxies are collisionless on these timescales and your simulation is not.


computational_physics

Three responses, and the last one is the test that settles it.


computational_physics

Increase the particle count. Relaxation time scales roughly with N over log N, so more particles
push the artificial relaxation beyond the runtime. This is the direct fix and it is expensive.


computational_physics

Apply a resolution criterion to your analysis. Discard structures below some multiple of the
softening length — several softening lengths is a common threshold — and below a minimum
particle count. State both in any result, because they define what the simulation can claim.


computational_physics

Check whether your timestep is adequate in the dense regions. A fixed timestep sized for the mean
density under-resolves the orbits in a clump, which itself produces spurious heating or
collapse. Adaptive individual timesteps are standard for exactly this reason.


computational_physics

One diagnostic worth running: rerun at half and double the particle count with everything else
fixed. If the clumps' number and mass change systematically with resolution, they are numerical.
If they converge, they are physical. That test settles the question and is the standard evidence
to present.


computational_physics

## Validation and High-Performance Practice

Verification and validation are distinct and both are required. Verification asks whether the
equations are being solved correctly, and it is a mathematics and software question answered by
convergence studies, by the method of manufactured solutions, by comparison against analytic
solutions in limiting cases, and by conservation checks. Validation asks whether the correct
equations are being solved, and it is a physics question answered only by comparison against
experiment.


computational_physics

The convergence study is the single most informative verification test. Halving the
discretization should reduce the error by the factor the method's order predicts — a factor of
four for a second-order method. If it does not, either the implementation is not the order it
claims, which usually indicates a boundary condition implemented at lower order, or the solution
is under-resolved, or round-off dominates. Running this test costs a few extra runs and is what
separates a code believed to work from one demonstrated to.


computational_physics

High-performance practice has been reshaped by hardware trends. Arithmetic has become
inexpensive relative to memory access, so the binding constraint in most scientific codes is
memory bandwidth rather than floating-point throughput. This inverts traditional optimization
priorities: recomputing a value can be cheaper than storing and reloading it, and a
lower-arithmetic algorithm with poor memory access can lose to a higher-arithmetic one with good
locality.


computational_physics

Validation and performance:
  verification         solving the equations right; convergence, manufactured
                       solutions, conservation
  validation           solving the right equations; comparison with experiment
  convergence study    refine and confirm the predicted error reduction
  conservation checks  mass, momentum, energy to round-off where the scheme
                       should conserve them
  benchmark problems   standard cases with known solutions; the community's shared
                       verification set
  uncertainty quantification  propagate input uncertainty to the output; a
                       precise answer from uncertain inputs is not a precise
                       prediction
  sensitivity analysis  which inputs the result depends on; determines where
                       effort is worth spending
  memory bound         most scientific codes; bandwidth rather than FLOPS
  arithmetic intensity  operations per byte moved; the roofline model predicts
                       which bound applies
  cache blocking       restructure loops to reuse data while it is resident
  domain decomposition  partition the problem across processes; the standard
                       parallel strategy
  halo exchange        communicate boundary data between neighbouring subdomains;
                       overlap it with interior computation
  strong scaling       fixed problem, more processors; limited by Amdahl's law and
                       by communication growing relative to computation
  weak scaling         proportionally larger problem, more processors; the
                       realistic goal at large scale
  mixed precision      lower precision where accuracy permits; large speedups on
                       modern accelerators, and it must be validated rather than
                       assumed safe


computational_physics

The distinction between strong and weak scaling is worth stating precisely because reported
scaling results are frequently ambiguous. Strong scaling holds the problem size fixed and adds
processors, so the work per processor shrinks and communication eventually dominates — the
achievable speedup is bounded by the sequential fraction. Weak scaling grows the problem
proportionally with processors, holding work per processor constant, which is what large
simulations actually do and which can scale to very large machines. A code reported as scaling to
a hundred thousand cores is almost always weak scaling, and its strong scaling limit is typically
several orders of magnitude lower — a distinction that matters entirely when the question is
whether a fixed problem can be solved faster.


condensed_matter

## Crystal Structure and Bonding

Condensed matter physics studies the emergent behaviour of enormous numbers of interacting
atoms, and its recurring theme is that collective properties are not simply the sum of atomic
ones. Superconductivity, magnetism, and the very existence of electrical conductors and
insulators are properties of the assembly rather than of any constituent, which is why the
field's characteristic slogan is that more is different.


condensed_matter

A crystal is a periodic arrangement of atoms, described by a lattice specifying the repeating
translational pattern and a basis specifying what sits at each lattice point. Fourteen Bravais
lattices exhaust the possible three-dimensional periodicities, and the common metallic and
semiconductor structures are a small subset: face-centred cubic and hexagonal close-packed
achieve the densest possible sphere packing at 74 percent, body-centred cubic is slightly
less dense and common in transition metals, and the diamond structure — two interpenetrating
face-centred cubic lattices — is what silicon and germanium adopt.


condensed_matter

The reciprocal lattice is the Fourier transform of the direct lattice and is where most
physics is done, because waves in a periodic medium are naturally described by wavevector.
Diffraction occurs when the scattering vector equals a reciprocal lattice vector, which is
Bragg's law in reciprocal space, and this is how crystal structures are determined
experimentally. The Brillouin zone, the primitive cell of the reciprocal lattice, is the
region within which all distinct wave states lie, and electronic band structures and phonon
dispersions are always plotted along high-symmetry paths within it.


condensed_matter

Structure and bonding:
  Bravais lattices     14 distinct three-dimensional periodicities
  FCC / HCP            close packed, 74% filling, coordination 12
  BCC                  coordination 8; many transition metals
  diamond structure    tetrahedral coordination; Si, Ge, C
  basis                what decorates each lattice point
  reciprocal lattice   Fourier dual; where wave physics lives
  Brillouin zone       primitive cell of the reciprocal lattice
  Bragg condition      2d sin(theta) = n lambda
  ionic bonding        electron transfer; hard, brittle, high melting, insulating
                       until molten
  covalent bonding     shared pairs, directional; hard, often semiconducting
  metallic bonding     delocalized electron sea; ductile, conducting, opaque
  van der Waals        induced dipoles; weak, soft, low melting; holds layered
                       materials together
  hydrogen bonding     strong dipolar; water's anomalies, biomolecular structure


condensed_matter

Bond type predicts bulk properties with reasonable reliability, and the reasoning is
mechanical. Metallic bonding is non-directional, so planes of atoms can slide past one
another without breaking bonds, which is why metals are ductile. Covalent bonding is
directional, so displacement breaks bonds and the material fractures, which is why covalent
crystals are hard and brittle. Ionic crystals are brittle for a related reason: sliding brings
like charges into contact and the crystal cleaves. Van der Waals bonding between layers with
covalent bonding within them gives materials such as graphite that are simultaneously very
strong in plane and easily sheared between planes.


condensed_matter

Q: Metals are ductile while ionic crystals are brittle primarily because:
A. Metals have lower melting points
B. Metallic bonding is non-directional, so planes can slide without breaking bonds, while
   sliding an ionic lattice brings like charges together
C. Metals are less dense
D. Ionic crystals have stronger bonds
Reasoning: The mechanism is what happens under shear. The delocalized electron sea in a metal
maintains bonding regardless of relative atomic positions, permitting dislocation motion.
Displacing an ionic lattice by one lattice spacing aligns cations with cations, producing
strong repulsion and cleavage. Bond strength alone does not determine ductility. Answer: B.


condensed_matter

## Lattice Vibrations and Thermal Properties

Atoms in a crystal vibrate about their equilibrium positions, and the normal modes of that
vibration are called phonons when quantized. The dispersion relation between frequency and
wavevector has acoustic branches, in which neighbouring atoms move in phase and the frequency
goes to zero at long wavelength, and optical branches when the basis contains more than one
atom, in which atoms within a cell move against each other at finite frequency even at zero
wavevector.


condensed_matter

Phonons carry heat in insulators and contribute in metals, and their scattering determines
thermal conductivity. At low temperature, scattering from boundaries and defects dominates
and conductivity rises with temperature as more modes are populated; at high temperature,
phonon-phonon scattering dominates and conductivity falls as one over temperature. The peak
between them is characteristic, and high thermal conductivity requires both few defects and
weak anharmonicity — which is why isotopically pure diamond has the highest thermal
conductivity of any bulk material, exceeding copper by a factor of five despite having no
free electrons at all.


condensed_matter

Heat capacity was one of the early triumphs of quantum theory. Classical equipartition
predicts a constant molar heat capacity of 3R for a solid, the Dulong-Petit value, which is
observed at high temperature and fails badly at low temperature where the measured capacity
falls toward zero. Einstein's model, treating all modes as having the same frequency,
captured the qualitative behaviour; Debye's model, treating the modes as a continuum up to a
cutoff, gives the correct cubic temperature dependence at low temperature. The physical reason
is that modes with energy quanta exceeding the thermal energy cannot be excited and are frozen
out.


condensed_matter

Thermal properties:
  phonon                quantized lattice vibration
  acoustic branch       in-phase motion; frequency to zero at long wavelength
  optical branch        out-of-phase within the cell; finite frequency at zero k
  Debye temperature     the characteristic scale; above it, classical behaviour
  Dulong-Petit          3R molar heat capacity at high temperature
  Debye T^3 law         low-temperature heat capacity of a lattice
  electronic heat cap.  linear in T in a metal; only electrons near the Fermi
                        level participate
  thermal expansion     arises from bond potential anharmonicity; a perfectly
                        harmonic solid would not expand
  thermal conductivity  phonons in insulators, electrons in metals
  Wiedemann-Franz       the ratio of thermal to electrical conductivity in a metal
                        is proportional to temperature with a universal constant
  Umklapp scattering    phonon collisions that reverse momentum; what gives a
                        crystal finite thermal resistance at all


condensed_matter

Thermal expansion deserves note because its origin is easy to miss. If the interatomic
potential were exactly harmonic, increasing vibrational amplitude would not shift the average
separation, and the solid would not expand. Real potentials are asymmetric — steeply repulsive
at short range and gently attractive at long — so larger amplitude shifts the time-averaged
position outward. Expansion is therefore a direct consequence of anharmonicity, and materials
engineered for near-zero expansion achieve it by structural mechanisms that compensate rather
than by making bonds harmonic.


condensed_matter

## Electronic Structure and Semiconductors

The behaviour of electrons in a periodic potential is the central problem of the field, and
its resolution explains the existence of metals, insulators, and semiconductors. Bloch's
theorem states that electron states in a periodic potential take the form of a plane wave
modulated by a function with the lattice periodicity, which means an electron in a perfect
crystal propagates without scattering — resistance comes from deviations from periodicity,
not from the atoms themselves.


condensed_matter

Energy bands arise because the periodic potential opens gaps at Brillouin zone boundaries,
where the electron wavelength satisfies the Bragg condition and forward and backward waves
mix. Whether a material conducts depends on where the electrons fill. A partially filled band
means states are available at infinitesimally higher energy, so an applied field accelerates
electrons and the material conducts — a metal. A completely filled band separated by a gap
from the next means no states are accessible at small energy, so the material insulates. A
small gap means thermal excitation populates the upper band appreciably at room temperature —
a semiconductor.


condensed_matter

The semiconductor's usefulness comes from doping. Adding impurities with one more valence
electron than the host contributes electrons to the conduction band, giving n-type material;
impurities with one fewer contribute holes to the valence band, giving p-type. Carrier
concentration is thereby controlled over many orders of magnitude by adding parts per million
of impurity, and it is this controllability, together with silicon's stable native oxide, that
made silicon the basis of electronics rather than any superiority in intrinsic properties.


condensed_matter

Electronic structure:
  Bloch state           plane wave times a lattice-periodic function
  band gap              from Bragg reflection at zone boundaries
  metal                 partially filled band
  insulator             filled band, large gap
  semiconductor         filled band, small gap; Si 1.12 eV, Ge 0.66, GaAs 1.42
  direct vs indirect    whether the band extrema share a wavevector; direct gap
                        materials emit light efficiently, indirect ones do not,
                        which is why LEDs are GaAs-based and not silicon
  effective mass        the curvature of the band; determines mobility
  hole                  the absence of an electron in a nearly full band, behaving
                        as a positive carrier
  doping                n-type donors, p-type acceptors
  Fermi level           the chemical potential; midgap for intrinsic material,
                        shifted toward the relevant band by doping
  pn junction           built-in field from carrier diffusion; the basis of diodes,
                        transistors, and photovoltaics
  depletion region      carrier-free zone at the junction; widens under reverse bias


condensed_matter

The pn junction is the single structure from which most of semiconductor electronics is
built. Electrons diffuse from the n side and holes from the p side, leaving behind fixed
ionized dopants that create an electric field opposing further diffusion, and equilibrium is
reached when drift balances diffusion. Forward bias reduces the barrier and current flows
exponentially with voltage; reverse bias widens the depletion region and only a small leakage
current flows. Adding a third region gives a transistor, and controlling the depletion region
with an insulated gate gives the field-effect transistor that constitutes essentially all
modern digital logic.


condensed_matter

## Magnetism and Superconductivity

Magnetism in solids arises from electron spin and orbital angular momentum, and its
collective forms come from the exchange interaction — a consequence of the Pauli exclusion
principle rather than of magnetic dipole coupling, which is far too weak to explain ordering
at room temperature. Because identical fermions must have an antisymmetric total wavefunction,
the spatial and spin parts are correlated, and the resulting energy difference between
parallel and antiparallel spin configurations is electrostatic in origin and large.


condensed_matter

Diamagnetism is universal and weak, arising from induced currents opposing an applied field.
Paramagnetism occurs when unpaired moments align partially with a field and is disordered
without one. Ferromagnetism is spontaneous parallel alignment below the Curie temperature,
producing a moment with no applied field. Antiferromagnetism is spontaneous antiparallel
alignment with no net moment. Ferrimagnetism has antiparallel sublattices of unequal magnitude,
giving a net moment, and includes the magnetite that made lodestones and the ferrites used in
inductors.


condensed_matter

Domain structure explains why a ferromagnet is not always magnetized. The material divides
into regions of uniform magnetization oriented so as to reduce the external field energy, and
in the demagnetized state the domains cancel. Applying a field grows favourably oriented
domains at the expense of others, and this process is partly irreversible because domain walls
pin on defects, which is the origin of hysteresis. Hard magnetic materials have many pinning
sites and retain magnetization; soft materials have few and follow the applied field, which is
what transformer cores require.


condensed_matter

Magnetism and superconductivity:
  diamagnetism          universal, weak, opposes the field
  paramagnetism         unpaired moments, aligns partially, disordered without field
  ferromagnetism        spontaneous parallel order below the Curie temperature
  antiferromagnetism    spontaneous antiparallel order below the Neel temperature
  exchange interaction  electrostatic in origin via Pauli exclusion; the reason
                        ordering survives to high temperature
  domains               reduce external field energy; explain unmagnetized ferro-
                        magnets
  hysteresis            from domain wall pinning; area is the energy loss per cycle
  hard vs soft          permanent magnets versus transformer cores
  superconductivity     zero resistance below a critical temperature
  Meissner effect       expulsion of magnetic flux; not merely perfect conductivity
  Cooper pairs          electrons bound via phonon exchange; bosonic, condense
  BCS theory            explains conventional superconductors
  type I / type II      complete flux expulsion versus flux penetration in
                        quantized vortices; type II tolerates high fields and is
                        what practical magnets use
  high-Tc cuprates      above liquid nitrogen temperature; mechanism still not
                        settled after nearly four decades


condensed_matter

The Meissner effect is what makes superconductivity a distinct thermodynamic phase rather
than merely the limit of vanishing resistance. A hypothetical perfect conductor would trap
whatever flux was present when it became perfectly conducting; a superconductor actively
expels flux regardless of its history, which is a statement about the equilibrium state.
This is why the transition is a genuine phase transition with a well-defined free energy and
why the superconducting state can be described by a macroscopic quantum wavefunction — the
basis of flux quantization, the Josephson effect, and superconducting quantum interference
devices sensitive enough to measure the magnetic fields produced by the brain.


condensed_matter

## Defects, Mechanical Properties, and Low-Dimensional Systems

Real crystals contain defects, and mechanical properties are governed almost entirely by
them. Point defects — vacancies, interstitials, substitutional impurities — are
thermodynamically required at any finite temperature, since the entropy gain from disorder
outweighs the energy cost at low concentration. Line defects, dislocations, are the crucial
ones for mechanics. Planar defects include grain boundaries and stacking faults, and
volume defects include voids and precipitates.


condensed_matter

The theoretical shear strength of a perfect crystal, computed by displacing whole planes
simultaneously, exceeds the observed strength of real metals by two to three orders of
magnitude. The resolution is dislocations: plastic deformation occurs by the propagation of a
line defect through the crystal, breaking and reforming bonds a few at a time rather than all
at once, which requires far less stress. The analogy is moving a heavy rug by propagating a
ripple across it rather than dragging the whole thing.


condensed_matter

That mechanism immediately explains every strengthening technique in metallurgy, because all
of them impede dislocation motion. Work hardening increases dislocation density until they
obstruct each other. Grain refinement introduces boundaries that dislocations cannot cross,
giving strength increasing as the inverse square root of grain size. Solid solution
strengthening introduces size-mismatched substitutional atoms whose strain fields impede
motion. Precipitation hardening introduces second-phase particles that dislocations must cut
through or bow around. The universal tradeoff is that everything impeding dislocation motion
increases strength and decreases ductility, because ductility is dislocation motion.


condensed_matter

Defects and mechanical behaviour:
  vacancy               thermodynamically required; enables diffusion
  dislocation           line defect; carries plastic deformation
  Burgers vector        the lattice displacement a dislocation carries
  slip system           the plane and direction of easy dislocation motion; FCC has
                        many, which is why FCC metals are ductile
  work hardening        dislocation density impedes further motion
  Hall-Petch            strength scales as inverse square root of grain size
  solid solution        size mismatch strain fields impede dislocations
  precipitation         second-phase particles obstruct motion
  strength vs ductility the universal tradeoff; both come from dislocation
                        behaviour
  creep                 time-dependent deformation at high temperature; diffusion
                        mediated
  fatigue               failure under cyclic loading well below the yield stress;
                        crack nucleation and growth
  fracture toughness    resistance to crack propagation; brittle materials fail
                        from pre-existing flaws far below theoretical strength


condensed_matter

Low-dimensional materials have become a major subfield because confinement qualitatively
changes electronic behaviour rather than merely scaling it. Graphene, a single layer of
carbon, has a linear rather than quadratic dispersion near the Fermi level, so its charge
carriers behave as massless relativistic particles, giving extremely high mobility and an
anomalous quantum Hall effect. Transition metal dichalcogenides transition from indirect to
direct band gap in the monolayer limit, making them emissive. Quantum wells, wires, and dots
confine carriers in one, two, and three dimensions respectively, and the resulting discrete
energy levels are tunable by size — which is why quantum dot emission wavelength is set by
particle diameter and why they are used as tunable phosphors in displays.