Thermodynamics

Thermodynamics relates energy, entropy, temperature, pressure, volume, work, and heat through state functions and process laws.

Diffusion is the spreading of heat or material by microscopic random motion. A concentration gradient produces flux down that gradient; with uniform diffusion coefficient $D$, conservation gives the diffusion equation $C_t=D{\nabla }^{2}C$. In a moving carrier, advection must also be included. Physical diffusion is a transport mechanism, distinct from the general probabilistic class of Markov diffusion.

The fractional mass density decrease per temperature increase at fixed pressure. In the Boussinesq approximation, a temperature excess $\theta$ gives reduced gravity $g^{\prime }=g{\alpha }_T\theta$. It converts a heat budget into a buoyancy budget in natural ventilation.

For a fluid at fixed pressure, the volumetric thermal expansion coefficient obeys $\rho (T_0+\theta )\simeq \rho (T_0)(1-\alpha \theta )$. Under the Boussinesq approximation it determines the buoyancy proportional to $g\alpha \theta$, while mass density is otherwise held constant.

Stochastic thermodynamics assigns work, heat, and entropy production to individual fluctuating trajectories while recovering ordinary thermodynamics after ensemble averaging.

Local detailed balance states that the logarithm of the probability ratio between a trajectory and its time reverse equals the entropy delivered to the environment in units of $k_B$. For one heat bath, $\log (P_F/P_B)=\Delta Q/(k_{B}T)$ when positive $\Delta Q$ flows into the bath.

For an equilibrium process, a path and its physically time-reversed path have equal probability after including their equilibrium starting weights. Conditional on their endpoints, this meansThis is the path version of detailed balance. A time-even order parameter is read backward without changing sign, while a transport flux changes sign. The Onsager--Machlup path probability implements this reversal in a coarse-grained stochastic description.

Entropy production is the sum of the entropy changes of a system and its environment. Under local detailed balance it is represented by a forward-to-reversed path-probability logarithm and has nonnegative expectation.

A fluctuation theorem relates probabilities of positive and negative entropy production. In a steady-state large-deviation form, the rate function often obeys $I(s)-I(-s)=-s$ in units where $s$ is the entropy-production rate.

For a Langevin particle with drag coefficient $\zeta$ and white-force covariance amplitude ${\sigma }^2$, detailed balance at temperature $T$ requires ${\sigma }^2=2\zeta k_{B}T$.

The Boltzmann constant converts temperature into an energy scale.

At thermal equilibrium, the Boltzmann distribution assigns a state of energy $E_i$ probability $P_i=Z^{-1}{e}^{-\beta E_i}$, where $\beta =(k_{B}T{)}^{-1}$ and the partition function $Z$ normalizes the probabilities.

Temperature is the intensive variable thermodynamically conjugate to entropy; at equilibrium it determines the direction of heat transfer.

Inverse temperature is $\beta =1/(k_{B}T)$, where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature. A canonical ensemble weights an energy $E$ by its Boltzmann factor $e^{-\beta E}$. Calling couplings dimensionless means absorbing this factor $\beta$ into their energy coefficients. The alternative convention $k_B=1$ makes inverse temperature $1/T$; physical and dimensionless couplings must not be mixed.

In a thermal Euclidean worldline path integral with $\hslash =1$, $\beta$ is also the circumference of the imaginary-time circle.

Systems are in thermal equilibrium when no net heat flows between them; their temperatures are equal.

In static thermal equilibrium, the local temperature times the stationary lapse is constant. Energy measured locally redshifts to conserved Killing energy by that same lapse; requiring a single equilibrium Boltzmann exponent gives the relation. The Killing vector field normalization fixes the constant but not the ratio of temperatures at different locations.

Boltzmann suppression is the exponential reduction of an equilibrium population when creating it costs energy much greater than the temperature. For a nonrelativistic species with negligible chemical potential, the leading factor is $e^{-m/T}$, multiplied by its momentum phase-space volume. It concerns the equilibrium abundance; freeze-out can preserve a larger population than the instantaneous equilibrium value at later times.

A nonrelativistic species with $g$ states and negligible chemical potential has equilibrium density $g(mT/(2\pi ){)}^{3/2}{e}^{-m/T}$. If particle-antiparticle annihilation maintains equilibrium after the species becomes massive, its eventual freeze-out population can be much smaller than a relativistically decoupled population. The nonrelativistic annihilation rate and any asymmetry determine the final abundance.

Momentum-exchange collisions maintain a common temperature and an equilibrium-shaped momentum probability distribution. Particle abundances and chemical potentials need not have their chemical equilibrium values: elastic scattering alone conserves particle number. Thus kinetic equilibrium does not by itself imply either zero chemical potential or conserved comoving entropy.

A heat bath, or thermal reservoir, is an idealized system so large that exchanging a finite amount of heat does not appreciably change its temperature.

If two otherwise isolated systems freely exchange energy while $E_1+E_2$ remains fixed, their total entropy isAt an interior maximum, $\partial S_1/\partial E_1=\partial S_2/\partial E_2$. Since thermodynamic temperature satisfies $1/T=\partial S/\partial E$, the systems have equal temperatures at thermal equilibrium.

Entropy is an extensive state function satisfying $dS=\delta Q_{\mathrm{rev}}/T$ for a reversible transfer of heat.

Specific entropy is entropy per unit mass. For a perfect gas with constant specific heat capacity $c_v$ at constant volume and specific-heat ratio $\gamma$, differences obeywhere $p$ is pressure and $\rho$ is mass density. Thus a homentropic flow has constant $p/{\rho }^{\gamma }$.

For a macrostate containing $\Omega$ equally likely microstates, the Boltzmann entropy is $S=k_B\log \Omega$.

For a homogeneous equilibrium system with energy density $\rho$, pressure $P$, chemical potential $\mu$, and number density $n$, the entropy density is

For an extensive equilibrium system, the thermodynamic Euler relation gives $E=TS-PV+\mu N$. At zero chemical potential, divide by $V$ to obtain $s=(\rho +P)/T$. Equivalently, $dP=(\rho +P)dT/T$ makesThe first law of thermodynamics identifies this as $dS$; the thermodynamic Euler relation fixes the entropy normalization. For adiabatic expansion satisfying the cosmological perfect-fluid continuity equation, the numerator on the right vanishes, proving cosmological entropy conservation.

For a dilute nonrelativistic gas with $x=m/T\gg 1$ and zero chemical potential, the Bose-Einstein distribution approaches the Maxwell-Boltzmann distribution. Expanding $E=m+p^2/(2m)+\cdots$ in the number density integral gives $n\sim g(mT/(2\pi ){)}^{3/2}{e}^{-x}$. Since $\rho =mn+3nT/2+\cdots$ and $P=nT$, the entropy density at zero chemical potential gives $s=n(x+5/2+\cdots )$ and hence the displayed leading expression. This includes the internal multiplicity $g$; a single real scalar field has $g=1$.

The second law forbids cyclic devices whose sole effect is complete conversion of heat from one reservoir into work or unassisted heat transfer from cold to hot. For a closed system it implies nondecrease of total entropy.

A thermodynamically reversible process can be reversed without a net change of the system and surroundings. Its entropy change equals reversible heat transfer divided by temperature. A reversible adiabatic process conserves entropy; adiabaticity alone does not forbid internal entropy production in an irreversible evolution.

No cyclic device can have as its sole effect the transfer of heat from a colder reservoir to a hotter reservoir.

No cyclic device can have as its sole effect the extraction of heat from one reservoir and its complete conversion into work.

No heat engine operating between two fixed reservoirs is more efficient than a reversible engine, and all reversible engines between those reservoirs have the same efficiency.

Absolute thermodynamic temperature can be defined so that a reversible engine exchanging heats $Q_h$ and $Q_c$ with reservoirs at $T_h$ and $T_c$ satisfiesA choice of one reference temperature fixes the scale.

A Carnot cycle consists of two reversible isothermal changes, one at each reservoir temperature, joined by two reversible adiabatic processes. It attains the maximum thermal efficiency permitted between those reservoirs, $1-T_c/T_h$.

An isothermal process is a thermodynamic change at constant temperature. In a reversible isothermal step, the working substance remains in thermal equilibrium with a reservoir while exchanging heat.

Heat is energy transferred because of a temperature difference.

For a homogeneous material, thermal inertia is $\Gamma =\sqrt{k\rho c_p}$, where $k$ is thermal conductivity, $\rho$ is density and $c_p$ is specific heat capacity. It measures the resistance of the surface temperature to periodic heating and controls the phase lag relevant to the Yarkovsky effect.

A body loses internal energy by emitting thermal radiation faster than it absorbs radiation or receives other heating. In a gravitating gas this can remove pressure support and allow further contraction.

Hot electrons can transfer energy to Cosmic microwave background photons through Compton scattering. For a fixed electron fraction $x_e$ and electron temperature well above the radiation temperature, the optically thin gas cooling time scales approximately as $(1+z{)}^{-4}/x_e$. Its volume loss scales with electron density, not with $n_H^2$, because the radiation density is independent of local gas density. The process tends toward the radiation temperature; it heats rather than cools electrons colder than that radiation.

Dust grains emit thermal radiation after absorbing energy from radiation or collisions. Collisions can transfer thermal energy from dense gas to grains, whose radiation then cools the gas. This channel requires dust and does not operate in strictly pristine hydrogen-helium material without prior enrichment.

Thermal electrons can transfer energy to lower-energy radiation photons through Compton scattering. In the nonrelativistic Thomson regime, the net cooling power per electron is $4{\sigma }_{T}c{U}_{\gamma }{k}_B(T_e-T_{\gamma })/(m_e{c}^2)$ for a blackbody photon bath. A sufficiently hot electron population therefore loses energy; a colder population is heated instead. The Cosmic microwave background radiation energy density grows as $(1+z{)}^4$, making this channel more important for high-redshift diffuse ionized gas.

An astrophysical cooling function packages radiative losses of a plasma into a temperature-dependent coefficient. In the hydrogen-density-squared convention, the volume energy loss is $n_H^2\Lambda (T)$, so the coefficient has units $\mathrm{erg}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. Its dependence on ionization state, chemical abundances and radiation field must be specified; a low-density collisional-equilibrium curve is not universal.

Radiation from rotational and vibrational transitions of molecules removes gas thermal energy. In primordial gas below the atomic threshold, molecular hydrogen is a principal coolant; hydrogen deuteride can matter in suitable chemical conditions. Molecular formation, photodissociation, optical depth and density-dependent level populations control the rate. This allows some small systems below the atomic virial temperature threshold to form stars.

Photoionization changes the ion population, often removing bound-state coolants at temperatures where collisional equilibrium would retain them. It can therefore suppress primordial line-cooling peaks. It also supplies photoionization heating. The net loss is cooling minus heating and depends on radiation spectrum, intensity, shielding and gas density. A universal density-independent function of temperature alone is not generally sufficient for irradiated gas.

For a nonrelativistic ideal gas with $n_{\mathrm{tot}}$ thermal particles per volume, divide its thermal energy $3n_{\mathrm{tot}}{k}_{B}T/2$ by the optically thin volume radiation loss. In the $n_H^2$ cooling convention this becomes $3\chi k_{B}T/(2n_H\Lambda )$, where $\chi =n_{\mathrm{tot}}/n_H$. Density scaling and ionization-dependent particle count must be kept consistent.

Equating the optically thin cooling time to the uniform-sphere free-fall time gives a critical gas density proportional to $T^2/{\Lambda }^2$. With hydrogen mass fraction $X$, particle ratio $\chi$ and gravitating gas fraction $f_g$, it is $[32G{m}_p/(3\pi X{f}_g)][3\chi k_{B}T/(2\Lambda ){]}^2$. At fixed temperature, gas above this density cools faster than free fall. A smaller gravitating gas fraction moves the boundary upward because gravity is faster at a fixed gas density.

The primordial atomic cooling curve describes hydrogen-helium gas without heavy elements. Near the hydrogen excitation threshold it rises sharply, then shows hydrogen and helium excitation/ionization features. Bound-state losses weaken when both elements become highly ionized. At sufficiently high temperatures thermal bremsstrahlung gives a slowly rising tail. Composition, density convention and ionization assumptions fix the numerical normalization.

Atomic line cooling removes gas kinetic energy by collisional excitation followed by photon escape. It is strongest where atoms or ions have bound Electrons and thermally accessible transitions. If photons are trapped, or if the relevant atoms are fully ionized, the simple optically thin line-cooling rate no longer gives the same loss.

Heavy elements provide many ionic transitions at temperatures where hydrogen and helium have lost their most effective line emitters. Metal-line cooling can therefore increase the plasma cooling function substantially and broaden its temperature features. Low-energy fine-structure transitions also permit cooling below the hydrogen atomic threshold when the relevant ions and Electrons are present.

Heat flux density is the rate of heat transfer through unit area. Its SI unit is watts per square metre, and Fourier's law gives its conductive contribution.

Thermal conduction transfers heat down a temperature gradient through microscopic interactions, without bulk transport of matter.

The temperature gradient is the gradient $\nabla T$ of the temperature field. It points in the direction of fastest temperature increase.

Fourier's law relates the conductive heat-flux density $\mathbf{q}$ to the temperature gradient by $\mathbf{q}=-k\nabla T$.

Thermal conductivity is the positive material coefficient $k$ in Fourier's law. It may depend on temperature, composition, and direction.

Thermal diffusivity is $\kappa =k/(\rho c_p)$, the ratio of thermal conductivity to volumetric heat capacity. It sets the diffusion length $\sqrt{\kappa t}$ over time $t$.

The Prandtl number is the ratio of kinematic viscosity to thermal diffusivity. It compares the rates at which momentum and temperature variations diffuse. In thermal-diffusion units the Boussinesq equations have momentum inertia multiplied by ${\mathrm{Pr}}^{-1}$; large, rather than small, Prandtl number supplies the usual instantaneous Stokes flow limit.

The thermal diffusion time across a distance $L$ is $L^2/\kappa$, where $\kappa$ is the thermal diffusivity. It is the time required for a temperature disturbance to spread over that distance by thermal conduction.

With thermal diffusivity $\xi$, a Fourier mode of wavenumber $k$ decays by diffusion at this rate.

The Stefan–Boltzmann law gives the thermal-radiation flux emitted by a surface as $F=ϵ\sigma T^4$, where $ϵ$ is its emissivity and $\sigma$ is the Stefan–Boltzmann constant.

The Stefan-Boltzmann constant $\sigma$ is the proportionality constant in the blackbody flux $F=\sigma T^4$. The blackbody radiation pressure is $p_{\mathrm{rad}}=4\sigma T^4/(3c)$.

Emissivity is the ratio of the thermal radiation emitted by a surface to that emitted by an ideal black body at the same temperature.

A body is in radiative equilibrium when absorbed radiative power equals emitted radiative power. A surface absorbing flux $F$ and emitting as a grey body has equilibrium temperature $T=(F/(ϵ\sigma ){)}^{1/4}$.

Heat capacity is the heat required per unit temperature change under a specified constraint.

Along a reversible path at fixed volume, $dS=C_V(T)dT/T$. Hence entropy differences can be measured macroscopically fromwith an additional $L/T_c$ across a first-order transition of latent heat $L$ at $T_c$.

For a thermodynamically stable system, the entropy is concave in energy, soConsequently $\partial E/\partial T>0$: energy increases with temperature, and the corresponding heat capacity is positive.

Specific heat capacity is heat capacity per unit mass. The volumetric heat capacity is $\rho c_p$, and $k/(\rho c_p)$ is the thermal diffusivity when $k$ is the thermal conductivity.

The specific heat capacity at constant volume is the temperature derivative of specific internal energy at fixed density and composition: $c_V=(\partial u/\partial T{)}_{\rho }$. It is the corresponding extensive heat capacity at constant volume divided by material mass.

The specific heat capacity at constant pressure is $c_P=(\partial h/\partial T{)}_P$, where $h$ is specific enthalpy. It includes expansion work as well as the change in internal energy.

For pure blackbody radiation, $P=a{T}^4/3$ depends only on temperature. Constant pressure therefore fixes temperature, so a usual finite constant-pressure temperature derivative is unavailable. In the material gas-radiation limit the specific-heat ratio diverges, although all three stellar adiabatic exponents approach $4/3$. The radiation index $4/3$ is an adiabatic pressure-volume response, not a finite heat-capacity ratio.

The heat capacity at constant volume is $C_V=(\partial E/\partial T{)}_V$ when particle number and other conserved quantities are also fixed.

In the classical high-temperature limit, each independent harmonic mode contributes $k_B$ to the constant-volume heat capacity.

For a two-dimensional system of fixed area $A$, the heat capacity at constant area is $C_A=(\partial E/\partial T{)}_{N,A}$ when the particle number and other conserved quantities are also fixed. It is the two-dimensional analogue of heat capacity at constant volume.

The third law implies that the entropy of a system with a nondegenerate ground state approaches zero as its temperature approaches absolute zero. In particular, ordinary equilibrium heat capacities approach zero.

Pressure is normal force per unit area and is thermodynamically conjugate to volume.

In the local rest frame of a gas, the pressure tensor is the directional momentum flux $P_{ij}=\int p_i{v}_{j}f(\mathbf{p})d^{3}p$, where $f$ is the phase-space distribution function per unit momentum volume and $\mathbf{v}$ is particle velocity. For an isotropic distribution, $P_{ij}=P{\delta }_{ij}$, recovering the kinetic pressure of an isotropic gas. An anisotropic distribution can exert different normal pressures along different directions.

Gas pressure is the mechanical pressure contributed by material particles. For a classical ideal gas, $p_{\mathrm{gas}}=n{k}_{B}T$.

Radiation pressure is momentum flux carried by electromagnetic radiation. Isotropic blackbody radiation has $p_{\mathrm{rad}}=a_{\mathrm{rad}}{T}^4/3$.

An equation of state is a relation among thermodynamic state variables such as pressure, volume, temperature, energy, and particle number.

A perfect fluid in general relativity stellar model in which pressure depends only on energy density, $p=p(\rho )$. A positive derivative makes this relation locally invertible and turns outward-decreasing pressure into outward-decreasing density. This general relation need not have the constant ratio $p/\rho$ used for a constant cosmological fluid parameter.

A Clausius gas with covolume parameter $b$ obeys $P(V-Nb)=N{k}_{B}T$. It is the repulsive-volume part of a Van der Waals equation without the attractive term.

The van der Waals equation $(P+a{N}^2/V^2)(V-Nb)=N{k}_{B}T$ augments excluded volume $b$ with a mean attractive interaction $a$.

The Van der Waals equation becomes independent of its material parameters when pressure, temperature and molecular volume are divided by their critical values: $\bar{p}=8\bar{T}/(3\bar{v}-1)-3/{\bar{v}}^2$. This explains the common reduced isotherms within this model, without claiming exact agreement for all real fluids.

An isotropic ideal gas whose particles obey the ultrarelativistic dispersion $\epsilon =pc$ satisfies

An adiabatic equation of state relates pressure and volume along a process with no heat transfer. It commonly has the form $p{V}^{\gamma }=\text{constant}$.

Thermodynamic volume is the volume occupied by a thermodynamic system, treated as an extensive variable.

Internal energy is the energy stored in the microscopic degrees of freedom of a system.

Specific internal energy is internal energy per unit mass. For a perfect gas with constant specific-heat ratio, $e=p/[(\gamma -1)\rho ]$. Its material balance is $\rho De/Dt=-p\nabla \cdot \mathbf{u}+Q$, where $Q$ is heat supplied per unit volume per unit time.

A free energy is a thermodynamic potential whose decrease governs equilibrium under specified environmental constraints.

A free-energy functional assigns a free energy to an entire spatial order parameter configuration rather than only to one uniform value. In Landau-Ginzburg theory, a local derivative expansion gives $\mathcal{A}[m]=\int d^{D}x\{\kappa (\nabla m{)}^2/2+V(m)\}$, with appropriate boundary terms and regular backgrounds. The Landau approximation minimizes this functional; a fluctuating statistical field theory instead integrates its Boltzmann weight over configurations. These procedures need not give identical critical exponents.

At fixed temperature and volume, the Helmholtz free energy is $F=E-TS=-k_{B}T\log Z$ in the canonical ensemble.

The chemical potential $\mu =(\partial E/\partial N{)}_{S,V}$ is the energy cost of adding a particle under fixed entropy and volume.

For a composition free energy $F=\int [f(\varphi )+\kappa |\nabla \varphi {|}^2/2]$, the local chemical potential is the functional derivativeAt phase coexistence, exchanges between the bulk phases require equal chemical potentials. Mechanical equilibrium additionally requires equal pressure for a flat interface, or a Young–Laplace equation pressure jump for a curved interface.

At chemical equilibrium, the sum of chemical potentials weighted by each reaction's stoichiometric coefficients vanishes.

A chemical system is out of local chemical equilibrium when reaction adjustment cannot keep pace with transport, imposed changes or external driving. Comparing the chemical relaxation time with transport and irradiation times identifies plausible departures. Disequilibrium chemistry in an exoplanet atmosphere combines this principle with mixing, horizontal advection and atmospheric photochemistry.

For one particle species in chemical equilibrium under $2\varphi \leftrightarrow 3\varphi$, chemical balance gives $2{\mu }_{\varphi }=3{\mu }_{\varphi }$, hence ${\mu }_{\varphi }=0$. Elastic $2\varphi \leftrightarrow 2\varphi$ imposes no such constraint. In an expanding universe the number-changing reaction must remain fast compared with the Hubble parameter; after chemical freeze-out a nonzero chemical potential can develop even if elastic collisions maintain kinetic equilibrium.

The chemical relaxation time estimates how rapidly a perturbed abundance returns toward chemical equilibrium. Comparing it with the eddy mixing time identifies a chemical quench level.