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Thermal electrical noise is the fluctuating electric potential difference or electric current produced by thermal fluctuations in a dissipative electrical circuit, even without an applied voltage. A resistor at thermal equilibrium is both a source of noise and a sink for electrical energy. Its electrical resistance describes the dissipation, and its absolute temperature sets the scale of the fluctuations. This connects a practical limit on signal-to-noise ratio to statistical mechanics: a warm, unbiased resistor is not electrically silent.

John Bertrand Johnson measured the effect using a vacuum-tube amplifier and thermocouple; his 1928 paper reported agreement with the Boltzmann constant. Harry Nyquist's companion paper calculated the effect from thermodynamics and statistical mechanics.^[ref][ref]

Thermal energy is a context-dependent name for the microscopic energy associated with a system's thermal state, rather than a separate conserved substance. The characteristic single-particle energy scale is $k_{B}T$, with Boltzmann constant $k_B$ and absolute temperature $T$. Comparing this scale with an excitation gap or energy barrier helps decide which microscopic processes are thermally accessible.

Under the classical equipartition theorem, each independent quadratic term in the Hamiltonian contributes $k_{B}T/2$ on average. Three translational terms therefore give $3k_{B}T/2$ per particle of a monatomic ideal gas. Quantum levels can invalidate that count: a degree of freedom with excitation spacing much larger than $k_{B}T$ can remain nearly frozen. A measured heat capacity tests how the stored internal energy changes with temperature; it does not simply count a fixed amount of heat stored in the sample.

Consider two rigid bodies forming an isolated system, with fixed composition and constant whole-body heat capacity at constant volume values $C_1=1\mathrm{kJ}{\mathrm{K}}^{-1}$ and $C_2=3\mathrm{kJ}{\mathrm{K}}^{-1}$. Their initial temperatures are $T_{1,0}=300\mathrm{K}$ and $T_{2,0}=400\mathrm{K}$. Assume each body remains internally uniform, their contact stores negligible energy, no phase transition occurs and no work is done or extracted. These are stipulated model values, not material measurements.

Because $d{U}_i=C_{i}d{T}_i$, write $U_i=a_i+C_i{T}_i$ with arbitrary fixed reference constants $a_i$. The conserved relative internal energy isThis is an energy reference convention, not a claim about actual heat capacities down to zero temperature. At thermal equilibrium the two temperatures equal $T_f$, so the first law of thermodynamics givesThe larger heat capacity pulls the common temperature closer to its initial value: body 1 warms by $75\mathrm{K}$, while body 2 cools by $25\mathrm{K}$.

To predict a timescale, stipulate a constant contact conductance $G=1/R_{\mathrm{th}}=0.1\mathrm{kJ}{\mathrm{s}}^{-1}{\mathrm{K}}^{-1}$, the reciprocal of the contact's thermal resistance. The rate of heat transfer into body 1 is $G(T_2-T_1)$, givingAdding proves conservation of $E$. Subtracting gives exponential decay of the temperature difference, with time constantand thereforeExact equality is approached as $t\to \infty$ in this ideal model. Changing $G$ changes the rate but not $T_f$: equilibrium and the speed of reaching it are separate predictions. Real bodies can have internal temperature gradients, changing heat capacities and contact losses, so the plotted $7.5\mathrm{s}$ is an assumed model timescale.

Thermal energy comparable with an exciton binding energy promotes dissociation and often washes out narrow bound-exciton resonances. Comparing $k_{B}T$ with binding energy is a scale estimate, not an exact disappearance criterion: densities, screening, oscillator strength and linewidths also matter.

For perfect transmission above threshold $E_n$, integrating $-df/dE$ gives $G_n=(e^2/h)f(E_n)$. Thermal occupation rounds a zero-temperature conductance step over an energy range of order $k_{B}T$.

A thermal membrane undulation is a shape fluctuation excited by thermal energy. In a planar membrane, a Fourier mode has stiffness $k_c{q}^4+\sigma q^2$, so tension suppresses long-wavelength excess area while bending suppresses short wavelengths.

A thermal neutron belongs to a population approximately in thermal equilibrium with its surroundings. Such a population has a distribution of kinetic energies, not one fixed energy. The scale $k_{B}T$ uses the Boltzmann constant and thermodynamic temperature; it is about $0.0259\mathrm{eV}$ at $300\mathrm{K}$. The mean translational kinetic energy for a classical three-dimensional equilibrium population is $3k_{B}T/2$. Not every slow neutron has reached this equilibrium.

The formal product over fermionic Matsubara frequencies is divergent before its normalization and time-slice prescription are fixed. The coherent-state thermal trace fixes the single-level factor to $1+e^{-\beta \xi }$, counting its empty and occupied states. A symmetrically regularized product can instead expose $2\cosh (\beta \xi /2)$; the additional factor $e^{-\beta \xi /2}$ restores the physical empty-state energy convention. Dropping that factor can incorrectly shift the grand potential and particle occupation.

For one nondegenerate thermalized particle, excited-state occupation relative to the ground state is set by the excitation gap, not by the absolute energy origin. In an infinite square well of width $L$, the first gap is $3{\pi }^2{\hslash }^2/(2m{L}^2)$. Predominant ground-state occupation requires this gap to be much larger than $k_{B}T$; an explicit tolerated occupation gives a quantitative temperature criterion. State quantum degeneracy adds its statistical weight.

For a stable nonrelativistic thermal relic, compare energy densities rather than number densities. If $m_Y=m_{\mathrm{nr}}/x^3$ and $Y_Y=C{x}^{3/2}{e}^{-x}$, its density relative to baryons is $(C{m}_{\mathrm{nr}}/m_B{Y}_B)x^{-3/2}{e}^{-x}$. Requiring this ratio below an allowed dark-matter-to-baryon ratio $R$ means $x+\frac{3}{2}\log x\ge \log [C{m}_{\mathrm{nr}}/(R{m}_B{Y}_B)]$. With fixed rate parameters this excludes the weakly interacting large-mass branch, while the sufficiently Boltzmann-suppressed smaller-mass branch can satisfy the density bound. Decay or a nonthermal initial abundance requires a different constraint.

A characteristic random particle speed. Here temperature $T_s$ is measured in energy units, and the factor-two convention makes a Maxwell-Boltzmann distribution proportional to $e^{-v^2/v_{\mathrm{th},s}^2}$. Other conventions, such as the one-dimensional standard deviation $\sqrt{T_s/m_s}$, differ by fixed factors.

Thermal transpiration is gas transport driven by an imposed temperature difference in a rarefied connection. Gas can move from cold to hot even when the two end pressures start equal. If the reservoirs are closed, redistribution builds a counteracting pressure difference until the net transfer stops. This final stationary state is not global thermal equilibrium: the two temperatures remain different. It explains why a heated pressure sensor can read a different pressure from the vessel to which it is connected. The correction depends on gas, connection geometry and molecular interaction with its surfaces. ^[ref]

The simple molecular limit is beautiful: equal opposing particle rates need not mean equal pressures. Consider effusion through a thin hole connecting two locally equilibrated reservoirs of the same gas, at $T_h>T_c$. Equating their one-way rates givesAt equal pressure, the hotter reservoir has lower number density; its faster molecules do not fully compensate for that reduction in outgoing particle rate. With $T_c=300\mathrm{K}$ and $T_h=351\mathrm{K}$, the ideal zero-transfer ratio is $\sqrt{351/300}\simeq 1.0817$. The hydrodynamic, zero-flow limit instead has equal pressures when other forces are absent. These are limiting models, not an interpolation formula for a glass capillary. ^[ref] In a real tube, incomplete exchange of molecular energy with the wall can alter the limiting ratio; a Knudsen number does not specify that exchange. ^[ref]

The electric potential difference corresponding to thermal energy $k_{B}T$ per elementary charge $q$.

Release of electrons from a heated material; one source of electrons in an X-ray tube.

Heating a metal gives some electrons enough energy to escape its surface. A heated diode cathode can therefore supply an electric current across a vacuum gap to a collecting diode anode. This makes temperature a control for the supply of electrons. Conversely, measuring that supply at several temperatures lets us estimate the surface work function: the energy barrier measured from the Fermi level to the nearby vacuum level. A Richardson plot turns the strong exponential function dependence into a slope.

Three questions must be kept separate: how many electrons cross the surface, how many traverse the gap, and how many the collecting diode anode records. The Richardson-Dushman law answers the first under specified surface assumptions. Space charge can obstruct the second, while incomplete collection can alter the third.

The coefficient $A$ has dimensions of electric current density divided by squared temperature. For ideal nonrelativistic particles with the free electron mass, two spin states, a planar uniform surface, and unit escape transmission, the thermionic tail-flux derivation giveswhere $e$ is the positive elementary charge and $h$ is the Planck constant. The two area units differ by a factor of $10^4$. A fitted $A$ need not equal $A_0$: reflection at the surface, the electronic band structure, and uncertain emitting area affect the inferred coefficient. A constant escape transmission $\tau$ would give $A=\tau A_0$ in this simple model; an energy-dependent transmission generally requires an integral instead.

The name joins Owen Willans Richardson with Saul Dushman. Dushman's 1923 paper develops a thermodynamic treatment of electron emission analogous to evaporation and evaluates the prefactor using quantum ideas.^[ref]

A thermodynamic system is in thermodynamic equilibrium when its macroscopic state has no spontaneous tendency to change through any exchange allowed by its constraints. For an isolated system, stable thermodynamic equilibrium maximizes entropy over the accessible states at fixed total energy and conserved material inventories. A wall can forbid an exchange, so always specify what can cross or move. Thermal, mechanical and chemical conditions answer different questions about that same constrained state. ^[ref]

For example, two compartments separated by a rigid, impermeable wall that transmits heat can reach thermal equilibrium while retaining different pressures and compositions. Allowing an unloaded piston to move adds a mechanical equilibrium condition. Allowing one chemical species to cross adds chemical potential equality for particle exchange; allowing a chemical reaction adds its chemical equilibrium condition. A fixed barrier can therefore support a constrained thermodynamic equilibrium that ceases to be an equilibrium when the barrier is removed.

Uniformity is not the definition of thermodynamic equilibrium. An isothermal atmosphere in hydrostatic equilibrium can have a common temperature while its pressure and mass density vary with height: the pressure gradient balances gravity. In an external field, the exchange condition must include the relevant potential energy; for charged matter this leads to the electrochemical potential. Relativistic gravitational corrections to thermal equilibrium are described separately by the Tolman temperature law.

A steady state only says that the macroscopic description is constant in time. A bar maintained between hot and cold thermal reservoirs has steady heat flux and produces entropy, so it is not in thermodynamic equilibrium. Even matching temperatures at one perfect-contact interface can coexist with heat transfer through the surrounding material; matching those two local values does not establish equilibrium of the whole apparatus. See thermal conduction for that spatial transport problem.

The useful simplification is that equilibrium conditions predict an endpoint without computing every collision or flow during relaxation. The first law of thermodynamics supplies the conserved quantities, and the Second law of thermodynamics supplies the direction of change. For a system held at fixed temperature and volume, the corresponding criterion is minimization of Helmholtz free energy; at fixed temperature and pressure, it is minimization of Gibbs free energy, with the material and exchange constraints still imposed.

A thermodynamic force conjugate to an internal variable is the energetic driving quantity paired with that variable's rate in the dissipation. Its sign and density normalization must be specified. With specific internal energy $U(F,\eta ,\xi )$, reference mass density ${\rho }_0$ and the convention used here,Equivalently, if the specific Helmholtz free energy is $\psi (F,T,\xi )$, then $f_{\xi }=-{\rho }_0(\partial \psi /\partial \xi {)}_{F,T}$. The negative sign makes positive force-rate product correspond to free-energy decrease at fixed external variables.

For an isothermal free energy per reference volume $W=E(\epsilon -\xi {)}^2/2$, the driving force is $f_{\xi }=E(\epsilon -\xi )$. Choosing $\dot{\xi }=f_{\xi }/{\eta }_v$ with ${\eta }_v>0$ yields $f_{\xi }\dot{\xi }=f_{\xi }^2/{\eta }_v\ge 0$. The internal variable relaxes toward the imposed strain, reducing stored elastic energy. Such a law supplies the mechanical part of a Clausius-Duhem inequality check; the thermal contributions still need their own consistent description.

The thermodynamic limit describes a sequence of larger systems, not a single enormous specimen: particle number $N$ and volume $V$ tend to infinity with number density $n=N/V$ fixed. Hold the chosen intensive and extensive thermodynamic quantities such as temperature $T$ and an applied magnetic field fixed along that sequence. In the canonical ensemble the controls are $T,N,V$; in the grand canonical ensemble one instead fixes $T$ and chemical potential, and the limiting number density is a result, possibly with a jump at phase coexistence. These are different specifications of a limiting experiment.

The useful limit is usually an intensive and extensive thermodynamic quantities normalization of the Helmholtz free energy, not a finite limit of the total extensive quantity itself:Here $Z_{N,V}$ is the canonical partition function, $k_B$ is the Boltzmann constant, $f$ is free-energy density, and $a$ is Helmholtz free energy per particle. Thus $F_{N,V}=Vf+o(V)$ when the limit exists; finite systems need not be exactly extensive quantities because of the surface correction to thermodynamic free energy.

Specify the interactions and the growing containers too. For a conventional fluid, use a fixed stable pair potential with a tempered pair potential tail, and containers whose boundary layer of any fixed thickness occupies a vanishing fraction of the volume; uniformly enlarged cubes work, a tube of fixed cross-section does not give the same bulk limit. For a finite-range Ising model, keep the lattice spacing and bond coupling fixed while enlarging all intended spatial directions. Ordinary bounded wall interactions or periodic boundary conditions can then give the same bulk free-energy density, although boundary conditions can select different spontaneous symmetry breaking states. This is a set of hypotheses, not a theorem that every interaction has a boundary-independent limit.^[ref]

The Curie–Weiss model below deliberately uses a different limiting prescription: its all-to-all coupling is divided by $N$. Without that factor its interaction energy would grow as $N^2$, rather than as an extensive quantity. The model has no spatial wall or correlation length to compare with a container; it is an exactly summable illustration of finite-size rounding, not a quantitative model of the fluid experiment below.

Thermodynamics studies energy transfers and the constraints on macroscopic changes. Its vocabulary connects internal energy, entropy, temperature, pressure and volume to heat and work. The first law of thermodynamics balances energy; the Second law of thermodynamics decides which changes can occur spontaneously and limits how much work a heat engine can deliver. These principles let us bound performance before choosing a material or a detailed machine design. Applications include power generation, propulsion and refrigerators. ^[ref]

Start by choosing a system boundary. A closed system can exchange energy but no matter; an isolated system exchanges neither. A state function, such as internal energy or entropy, depends on the equilibrium state. A process specifies how that state changes. Thus a body's temperature is a property it has, whereas heat records energy crossing its boundary during a process. A body contains internal energy, rather than a stored quantity of heat.

Two insulated bodies initially at different positive temperatures approach thermal equilibrium when brought into thermal contact. Their combined energy is conserved, but their combined entropy increases. They do not spontaneously separate again into a hotter body and a colder one. This familiar asymmetry makes thermodynamics interesting: conserving energy leaves many imaginable changes available, and entropy supplies a further constraint. The same distinction underlies both entropy maximization under thermal contact and the thermal efficiency limit in Carnot's theorem.

A thermodynamic system is the part of the physical world selected for an energy and matter balance. Draw its boundary before classifying an interaction as heat, work or transported matter. The boundary may move: a sealed piston encloses a closed system even while its volume changes. An isolated system additionally excludes all external energy transfers.^[ref]

Boundary choice changes the ledger. If an electrical heater warms water, choosing only the water counts incoming heat; including the heater counts electrical work entering through its leads. The combined internal energy records storage, without separate stored categories for the two transfer mechanisms.^[ref]