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The integral uses $0\log 0=0$ when it is defined. For the Boltzmann equation, the Boltzmann H theorem makes it nonincreasing under suitable regularity and integrability. This functional is distinct from the microstate-counting Boltzmann entropy; the physical kinetic entropy has the opposite sign.

A product trial distribution with actual mean Ising spin $m$ givesIts entropy is the sum of independent-spin entropies, with $0\log 0=0$. Differentiation gives $k_{B}T\mathrm{arctanh}m=qJm+h$, the self-consistency equation. The variational mean parameter is distinct from the assumed field parameter of the Ising auxiliary mean-field free energy until the equilibrium condition is imposed.

A coarse-grained variable retains selected macroscopic information while averaging over unresolved microscopic states. At thermal equilibrium, the statistical multiplicity of those states contributes entropy to the effective Helmholtz free energy of the retained variable.

If relic particle number and reference-gas entropy are separately conserved in a comoving volume, their ratio $Y=n/s$ is constant. A decay or entropy injection can change it. The reference entropy is that of the specified thermal sector, not necessarily the entropy of the decoupled relic itself.

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.

For a finite regular polytrope with fixed index, the scale ${\alpha }^2\propto K{\rho }_c^{(1-n)/n}$ implies $M^{n-1}{R}^{3-n}\propto K^n$. Thus, for $n\ne 3$, $R\propto K^{n/(3-n)}{M}^{(1-n)/(3-n)}$. For $n=3/2$, $R\propto K{M}^{-1/3}$. Fixed composition makes the cold electron degeneracy pressure coefficient approximately fixed, but a convective ideal gas has $K$ controlled by entropy. A stellar sequence with changing entropy therefore need not follow the fixed-$K$ negative slope.

For $N$ equal weak shocks with individual pressure ratio $1+\delta$ and total ratio $P_s=(1+\delta {)}^N$, entropy production in a perfect-gas shock givesAt fixed $P_s>1$, this tends to zero as the compression is divided into increasingly many smaller weak shocks. A single finite normal shock wave instead gives the positive value $\log P_s-\gamma \log D(P_s)$. Gradual compression can consequently approach reversible isentropic flow while an abrupt compression generates finite entropy.

For a single-component equilibrium system with extensive internal energy $E(S,V,N)$, scaling its entropy, volume and particle number together gives $E(\lambda S,\lambda V,\lambda N)=\lambda E(S,V,N)$. The Euler theorem for homogeneous functions therefore gives $E=S{\partial }_{S}E+V{\partial }_{V}E+N{\partial }_{N}E$. The first law of thermodynamics identifies these derivatives as $T,-P,\mu$, respectively, proving $E=TS-PV+\mu N$. Dividing by volume yields the entropy density formula $s=(\rho +P-\mu n)/T$. This step fixes the entropy normalization rather than merely identifying its differential. Differentiating the Euler relation and subtracting $dE=TdS-PdV+\mu dN$ gives the Gibbs-Duhem equation $SdT-VdP+Nd\mu =0$. The argument requires extensivity; it is not an assumption about arbitrary nonadditive systems.

An extensive quantity scales in proportion to the size of a thermodynamic system: combining $n$ independent identical copies multiplies it by $n$. Energy, entropy and volume are examples.

For a single-phase ideal gas mixture with molar amounts $N_i$, common temperature $T$, and pressure $P$, extensivity and ideal mixing giveHere $N={\sum }_i{N}_i$, $x_i$ is the mole fraction, $\mathcal{R}$ is the molar gas constant, and $P^{\circ }$ is the standard pressure. Integrating $\partial {\mu }_i/\partial P=\mathcal{R}T/P$ and including the ideal mixing entropy proves ${\mu }_i={\mu }_i^{\circ }+\mathcal{R}T\log (x_{i}P/P^{\circ })$. Reactions can change $N$, so fractions alone do not set the extensive Gibbs free energy. Its composition Hessian is $\mathcal{R}T({\delta }_{ij}/N_i-1/N)$; it is positive semidefinite by the weighted Cauchy-Schwarz inequality.

If an energy per site $C$ multiplies the identity operator, blocking sends $C$ to $b^{D}C+c(u)$. For free energy per site $F=C+f(u)$, exact partition function invariance gives the displayed inhomogeneous equation with $g=b^{-D}c$. It records eliminated-mode entropy, vacuum contributions and measure normalization. Removing a suitable regular background leaves homogeneous singular scaling; additive logarithmic resonances can obstruct a strictly analytic subtraction.

An isentropic process keeps the entropy constant. Every reversible adiabatic process is isentropic, while an irreversible adiabatic process can increase entropy.

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.

In a homentropic flow, the cross-helicity conservation law givesConsequently cross-helicity density is conserved along each trajectory exactly when the right-hand side vanishes. In a steady state, mass conservation gives $\nabla \cdot \mathbf{u}=-\mathbf{u}\cdot \nabla \log \rho$, so the criterion becomesThe absence of an entropy source in a conservation law does not alone make its density an advected scalar: compression and transport along the magnetic field still change that density.