A dependence of behavior on the relative timing or ordering of concurrent activities. In the abstract example under software thread, two increments can leave $x=1$ or $x=2$ depending on their interleaving. A C data race is a language-defined conflict with undefined behavior, rather than a name for every race condition.

Humidity describes water vapor in a gas. Absolute and relative measures answer different questions; relative humidity compares vapor pressure with saturation at the current temperature. This matters biologically because evaporation and cooling depend on atmospheric drying conditions, not only air temperature.

Evaporation transfers molecules from a liquid surface into the surrounding vapour. It can occur below the boiling temperature, and competes with condensation from the vapour. Its net rate depends on the surrounding vapor pressure and transport conditions. Removing latent heat through evaporation can cool the remaining liquid.

Fever is a regulated elevation of body temperature associated with altered thermoregulatory control. Responses such as shivering and reduced skin heat loss can accompany movement toward the higher regulated level. It differs mechanistically from overheating caused by heat gain exceeding dissipation without the same regulatory change.

Vasodilation is widening of a blood vessel, often through reduced smooth-muscle contraction. It can increase local conductance and surface heat transfer. The net systemic effect depends on which vascular beds change and how cardiac output and other vessels respond.

Vasoconstriction is narrowing of a blood vessel through changes such as smooth-muscle contraction. It alters local resistance and can change flow distribution or heat transfer. A local narrowing does not imply that pressure or flow changes identically everywhere in the circulation.

Shivering is involuntary muscle activity that increases metabolic heat production. Its thermal role differs from purposeful movement even though both use muscle metabolism. The resulting temperature change depends on heat loss as well as the extra heat generated.

Sweating supports thermoregulation through evaporation of water, which carries away latent heat.

Twelve resting volunteers wore a water-perfused suit. Tubes covered the torso, upper arms and legs; the right forearm remained uncovered for laser Doppler blood-flow measurement. Six thermocouples estimated mean skin temperature; a sublingual thermocouple measured oral temperature. A dry-air capsule measured local sweat output from outgoing air moisture. ^[ref]

After two control scans near $34{}^{\circ }\mathrm{C}$ skin temperature, the suit cooled and then warmed the skin. Mean oral temperature was $36.6{}^{\circ }\mathrm{C}$ during both cooling and early warming; mean skin readings were $30.5$ and $38.2{}^{\circ }\mathrm{C}$. Cutaneous vascular conductance fell during cooling and rose during warming. The inference is that compensatory blood-flow responses can precede an appreciable internal temperature change. ^[ref]

Repeated baselines controlled comparison within participants; closed mouths reduced oral-probe cooling artifacts. A later control with reduced oral temperature was excluded. Oral temperature is a proxy, local flow is not whole-body heat transfer, and the fixed condition order leaves time effects possible. Brain imaging measured regional blood flow, not direct neuron firing or causal necessity. ^[ref]

Core body temperature refers to internal temperature, distinguished from skin temperature. Different measurement sites are proxies with their own limitations, not interchangeable readings of a perfectly uniform interior. ^[ref]

The Seebeck effect is the generation of a thermoelectric electromotive force by a temperature difference. In a thermocouple, two different homogeneous electrical conductors A and B connect a measuring junction at $T_h$ to two reference connections at a common $T_c$. With negligible electric current, the measured terminal voltage is determined by the difference of their Seebeck coefficients. A hot junction alone does not specify the reading: the reference temperature, the second material, and the terminal order matter.

For a motivating experiment, join one end of A to one end of B, insulating the wires elsewhere. Connect their free ends to identical meter leads at reference temperature $T_c$, for example in an ice-water heat bath. A meter with input electrical resistance much larger than the thermocouple's electrical resistance approximates an open circuit, meaning $I\approx 0$. Keep the measuring junction in a separately controlled heat bath at $T_h$ and wait for stable junction temperatures. Both reference connections must share $T_c$; identical leads with identical endpoint temperatures have cancelling thermoelectric contributions, allowing the meter to be elsewhere. ^[ref].

Define the reading as $V_{AB}=V_A-V_B$, with the positive input on the A reference terminal and the negative input on B. Use the Seebeck coefficient convention in which the voltage change along a homogeneous wire at zero electric current is $dV=-S(T)dT$. Following the path from A at $T_c$ to the common junction at $T_h$, then back along B to $T_c$, givesThis formula assumes homogeneous materials whose Seebeck coefficients depend only on the local temperature, and no additional uncompensated lead contributions. Here voltage is the electrical transport voltage associated with electrochemical potential, rather than a direct measurement of an isolated equilibrium contact electric potential.

Start with $T_h=T_c$, then increase $T_h$ while holding $T_c$ fixed and record $V_{AB}$ at each stable setting. In the ideal model the equal-temperature reading is zero. If $S_A-S_B>0$ throughout the interval, A reads positive when $T_h>T_c$. Exchanging meter inputs reverses the reading; exchanging the two junction temperatures reverses the oriented integral. Repeating the run with identical homogeneous materials gives zero because $S_A=S_B$. These controls test the stated model; a nonzero experimental baseline should be measured rather than silently removed by assuming it is zero.

For an explicitly toy linear example, stipulate constant Seebeck coefficients $S_A=25\mu V/K$ and $S_B=-15\mu V/K$. Choose reference temperature $T_c=20{}^{\circ }\mathrm{C}$ and measuring-junction temperature $T_h=70{}^{\circ }\mathrm{C}$. A temperature difference has the same numerical size in kelvins and on the degree Celsius scale, soSwapping the inputs gives $-2.00\mathrm{mV}$. Bringing both junction regions to the same temperature gives zero. With temperature-dependent Seebeck coefficients, shifting both junction temperatures equally while keeping their difference fixed can change the reading. No actual material pair is claimed to have the stipulated coefficients over this interval.

Real thermocouples need a calibrated voltage-versus-temperature relation: the constant-Seebeck coefficient approximation is useful only where the coefficient difference changes little. Wire inhomogeneity can make the result depend on where the temperature gradient lies, and reference connections at unequal or unknown temperatures introduce errors. An electronic reference correction uses a measured reference temperature in place of an ice bath. ^[ref]. Burns and colleagues provide the standardized reference functions for common letter-designated thermocouple types; the toy line above is not a replacement for them. ^[ref].

The historical experiment used a different detector. In his 1821–1823 investigations, Seebeck found that a closed electrical circuit of two dissimilar materials with junctions at different temperatures deflected a compass needle. The modern explanation is that thermoelectric electromotive force drives an electric current, whose magnetic field deflects the needle. The open-circuit experiment above detects voltage while drawing negligible electric current. The striking connection is that a thermal imbalance can become an electrical signal and then a magnetic response. ^[ref].

Two thermodynamic variables are conjugate when their product has units of energy and they occur as a paired term in a thermodynamic differential, such as pressure and volume or tension and length.

A real gas departs from the ideal gas law because interactions and molecular volume matter. Nonideal equations of state, including the Dieterici equation, model such departures.

A state function used to describe a thermodynamic system with a chosen set of natural variables. Internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy use different variables and serve different environmental constraints.

A statistical ensemble describes hypothetical copies of the same system under specified constraints, with a probability distribution over their microstates. An ensemble average is an expected value across those copies. The canonical ensemble fixes temperature and particle number while allowing energy exchange; the microcanonical ensemble fixes energy, and the grand canonical ensemble also allows particle exchange. These choices specify different physical preparations. ^[ref]

Maxwell relations turn pressure, volume and temperature measurements into information about entropy. For example, the pressure rise when heating a rigid sealed vessel determines the entropy change when expanding the same sample isothermally:The quantities on the two sides describe different experiments, yet both are mixed partial derivatives of one Helmholtz free energy. This is the useful surprise: a mechanical measurement can reveal a thermal property. Similarly, the coefficient of thermal expansion gives $(\partial S/\partial P{)}_{T,N}=-\alpha V$. ^[ref]

For a simple compressible system, hold particle number $N$ and composition fixed, and allow only pressure-volume work. Write $U$ for internal energy, $H=U+PV$ for enthalpy, $F=U-TS$ for Helmholtz free energy, and $G=H-TS$ for Gibbs free energy. Their natural-variable differentials areThus the first partial derivatives, with the subscript recording the variables kept fixed, areThe symmetry of second derivatives produces all four standard Maxwell relations, including their signs:For a brief derivation of the $F$ relation, differentiate $F_T=-S$ with respect to $V$ and $F_V=-P$ with respect to $T$: $-S_V=F_{TV}=F_{VT}=-P_T$. Each other row follows in the same way. Assume continuous second partial derivatives of the relevant thermodynamic potential in the region considered. At a phase transition, the equilibrium potential may lose this smoothness; apply the identities within each smooth thermodynamic phase, and handle latent heat and phase coexistence separately. If particle numbers vary, the potential differentials acquire chemical potential terms, so the fixed-$N$ constraints must remain explicit. ^[ref]

For an ideal gas with fixed $N$, the ideal gas law gives $P=N{k}_{B}T/V$. At fixed volume, heating changes the pressure at rate $N{k}_B/V$. Consequently an isothermal expansion at positive temperature hasFor one mole with $R=8.314\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ and $V_2=2V_1$, this is $\Delta S=R\ln 2\simeq 5.763\mathrm{J}{\mathrm{K}}^{-1}$. At $300\mathrm{K}$, a reversible isothermal expansion absorbs $Q=T\Delta S\simeq 1728.85\mathrm{J}$ and performs equal work, since its internal energy is unchanged. The entropy difference is a state function difference: the same equilibrium endpoints give the same $\Delta S$ in an insulated free expansion, although there $Q=0$ and the process is irreversible.

The same Maxwell relations let thermal measurements determine the difference between constant-pressure and constant-volume heat capacities. In a smooth single thermodynamic phase,using heat capacity at constant volume, heat capacity at constant pressure, coefficient of thermal expansion and isothermal compressibility. Along fixed pressure, $dV=\alpha VdT$, soThus measured expansion and compression responses determine a heat capacity difference without measuring both heat capacities separately. They do not determine the absolute heat capacities or absolute entropy by themselves; an independent thermal measurement supplies that missing information. For an ideal gas, $\alpha =1/T$ and ${\kappa }_T=1/P$ reduce the result to $C_P-C_V=PV/T=N{k}_B$, which is Mayer's relation.

When the necessary higher partial derivatives exist and commute, differentiating the Maxwell relations once more also givesIntegrating these slopes determines how a heat capacity changes with volume or pressure at fixed temperature. For an ideal gas, both right-hand sides vanish because the ideal gas law is linear in temperature at either fixed volume or fixed pressure. Consequently $C_V$ and $C_P$ depend only on temperature at fixed $N$, although they need not be constant in temperature. ^[ref]

Historically, James Clerk Maxwell's Theory of Heat, first published in 1871, presents these connections using the geometry of isothermal processes and reversible adiabatic processes. His fourth edition, page 167, explicitly relates expansion on heating at fixed pressure to heat emitted during an isothermal compression: the modern $G$ relation gives $\delta Q_{\mathrm{rev}}=TdS=-T\alpha VdP$. This explains why a substance with positive coefficient of thermal expansion emits heat when compressed isothermally, whereas one with negative coefficient absorbs it. ^{[ref] [ref]}

Compressibility measures how readily a sample's volume decreases when its pressure increases. For a simple compressible system with fixed particle number $N$ and composition, specify the constraint $X$:Here $P$ denotes pressure, also written $p$ elsewhere. The minus sign makes compressibility positive on a mechanically stable equilibrium branch. Its units are inverse pressure, $\mathrm{P}{\mathrm{a}}^{-1}$; its reciprocal $K_X=1/{\kappa }_X$ is the constrained bulk modulus, measured in pascal units. Holding temperature fixed gives isothermal compressibility; holding entropy fixed gives isentropic compressibility ${\kappa }_S$ along an isentropic process. The latter describes a reversible adiabatic compression, since an irreversible adiabatic process can increase entropy. ^[ref]

For an ideal gas, the ideal gas law $PV=N{k}_{B}T$ and the reversible ideal-gas adiabat give, at the same equilibrium state,Here $C_P$ and $C_V$ are the whole sample's heat capacity at constant pressure and heat capacity at constant volume, and $\gamma$ is the heat capacity ratio. Constant $\gamma$ is required for the finite-path formula $P{V}^{\gamma }=\mathrm{constant}$. Reversible adiabatic compression raises the temperature, making the pressure rise more steeply than in an isothermal compression. Thus keeping entropy fixed makes the gas harder to compress locally. For a classical monatomic gas at $100\mathrm{kPa}$, $\gamma =5/3$, so ${\kappa }_T=10^{-5}\mathrm{P}{\mathrm{a}}^{-1}$ and ${\kappa }_S=6\times 10^{-6}\mathrm{P}{\mathrm{a}}^{-1}$; the corresponding values of the bulk modulus are $100\mathrm{kPa}$ and $166.67\mathrm{kPa}$.

For a smooth, stable single thermodynamic phase at positive temperature, Maxwell relations also implywhere $\alpha$ is the coefficient of thermal expansion. The ideal gas result is one example of this broader link between mechanical and thermal responses. The equality ${\kappa }_T={\kappa }_S$ is possible when $\alpha =0$ and the other responses remain finite. These formulas compare equilibrium partial derivatives, rather than arbitrary fast or dissipative compression histories. ^[ref]

The laws organize the equilibrium and process constraints of thermodynamics: the zeroth law of thermodynamics makes thermal equilibrium transitive, the first law of thermodynamics expresses energy balance, the Second law of thermodynamics restricts entropy changes, and the third law of thermodynamics describes the zero-temperature limit. Their analogy with horizon mechanics becomes quantitative through Hawking temperature and Bekenstein-Hawking entropy.

A thermodynamic phase is a form of macroscopic equilibrium characterized by its structure, symmetries and thermodynamic response. Away from phase transitions, its free-energy density varies smoothly with the control parameters. Different phases can share a phase coexistence boundary. An ordered magnetic phase is distinguished from a disordered phase by its spontaneous order parameter; positive and negative ordered states are different pure thermodynamic phases related by spin inversion symmetry.

Free expansion occurs when a gas spreads into an initially evacuated region without driving an external load. The standard idealization uses a rigid, insulated vessel: initially a partition confines the gas to volume $V_1$, while the remaining compartment contains vacuum. Open a passage and wait for thermodynamic equilibrium; the same fixed amount of substance $n$ now occupies the larger accessible volume $V_2$. The outer vessel never changes size. This simple release separates three questions: what energy is conserved, whether temperature changes, and why entropy increases. ^[ref]

Choose the gas as the closed system, neglect heat exchange with the walls, and assume negligible changes in bulk kinetic energy and potential energy between the equilibrium endpoints. Insulation gives $Q_{\mathrm{in}}=0$; no moving outer boundary or external load receives work, so $W_{\mathrm{out}}=0$. The first law of thermodynamics therefore givesThe gas can develop internal motion during the release; this does not supply external work. The intermediate state need not have one equilibrium pressure or temperature, so there is no actual quasistatic process curve $p(V)$ to integrate. In particular, substituting the equilibrium ideal gas law into $\int pdV$ would describe a different process. ^[ref]

For an ideal gas, internal energy of an ideal gas depends only on temperature. With positive heat capacity at constant volume, $U(T_2)=U(T_1)$ implies $T_2=T_1$, and the ideal gas law gives $p_2/p_1=V_1/V_2$. Doubling the accessible volume thus halves the final pressure without changing the final temperature. This is an endpoint statement; it does not assert a spatially uniform, constant temperature during the rush. Insulation alone does not imply constant temperature: a reversible adiabatic expansion cools an ideal gas because it supplies work. ^[ref]

For a real gas, $U(T,V)$ can depend on volume as well as temperature, so unchanged internal energy need not mean unchanged temperature. For a fixed-composition simple compressible system in a single thermodynamic phase, equilibrium identities using a Maxwell relation giveHere $C_V$ is the whole sample's heat capacity at constant volume. The sign depends on the equation of state; for an ideal gas the numerator vanishes. For a finite free expansion, solve $U(T_2,V_2)=U(T_1,V_1)$, or integrate this slope along equilibrium states of equal internal energy; that mathematical curve does not describe the actual release. ^[ref]

Free expansion is useful because it tests how internal energy depends on number density while eliminating external expansion work. Its beautiful surprise is that an ideal gas can finish with the same temperature and internal energy but greater entropy: a thermometer alone cannot diagnose reversibility. See entropy change in free expansion for the calculation and temperature sensitivity of a free-expansion experiment for the experimental qualification. The continuous-flow throttling associated with the Joule-Thomson coefficient has a different balance: under its usual assumptions it preserves enthalpy, whereas the gas-only idealization here preserves internal energy. ^[ref]