For symmetric stress in static elastic equilibrium without body force, write $E=⟨\epsilon ⟩$ and $S=⟨\sigma ⟩$. The difference between microscopic and macroscopic work is $|\Omega {|}^{-1}{\int }_{\partial \Omega }(\sigma -S)n\cdot (u-Ex)$. It vanishes for affine boundary displacement, uniform boundary traction, or matching periodic displacement fluctuations and opposite tractions on paired faces.

The Voigt elastic approximation assumes uniform strain. Differentiating its averaged strain energy density gives the averaged local stress. For linear elasticity the predicted elastic stiffness tensor is $C_V=⟨C⟩$. The uniform strain is an admissible affine displacement trial field, so this gives an upper elastic energy bound for convex local energy.

The Reuss elastic approximation assumes uniform stress. The local strain is the derivative of the convex conjugate $W^{\ast }(S,x)$; its mean is $D⟨W^{\ast }⟩(S)$. Under invertible differentiable duality, this mean determines $S=D{W}_R(E)$. For positive linear elasticity, $C_R=⟨C^{-1}{⟩}^{-1}$. Uniform stress is generally not accompanied by a compatible local displacement field, so this approximation is distinguished from an exact heterogeneous solution.

Take a positive comparison elastic stiffness tensor $C^0$, excess energies $F_r(e)=W_r(e)-e:C^{0}e/2$, and a phasewise constant elastic polarization. The lower conjugate inequality majorizes the local energy by a quadratic-plus-linear function. Minimizing that function over compatible strains gives the displayed bound, where $A_{rs}=\int \int ({\chi }_r-c_r)\Gamma ({\chi }_s-c_s)$. Stationarity in ${\tau }^r$ gives $c_r{e}^r+{\sum }_s{A}_{rs}{\tau }^s=c_{r}E$, with $e^r=D{F}_{r\ast }({\tau }^r)$. Without this stationarity, dual strains must not be identified with the comparison solution's phase means.

Conductivity homogenization replaces rapidly varying positive coefficients in a divergence-form elliptic equation by a macroscopic effective conductivity. Periodic media admit cell problems; statistically homogeneous media can instead be described by ensemble averages. The small-scale assumptions and boundary regime must be specified.

For isotropic phases with planar interfaces normal to $n$, normal conductivity is the weighted harmonic mean $a_H=(\sum p_i/a_i{)}^{-1}$ and tangential conductivity is the weighted arithmetic mean $a_A=\sum p_i{a}_i$. Continuity of normal flux and tangential temperature gradient gives the two averages directly.

An affine trial temperature in the minimum-energy principle gives the upper tensor bound by the volume-averaged local conductivity. For an isotropic two-phase composite, $a^{\ast }\le p_1{a}_1+p_2{a}_2$. The same uniform-field variational idea supplies the familiar corresponding elastic upper bound.

For a scalar reference $b$, choose a trial polarization $q$ and solve $b\Delta v+\nabla \cdot q=0$ with zero Dirichlet data. The trial functional isA reference above all phase conductivities gives $J(u)\le {\mathcal{F}}_b(q)$; one below all of them gives $J(u)\ge {\mathcal{F}}_b(q)$. If the reference equals a phase conductivity, admissible polarization vanishes in that phase and the inverse is taken only on the contrast subspace.

A self-consistent conductivity approximation sets the homogeneous comparison effective conductivity equal to the predicted response. For a uniformly oriented two-dimensional polycrystal with principal conductivities $a,b>0$, the comparison formula gives $C_{\mathrm{SC}}^2=ab$, so its positive solution is $\sqrt{ab}$. This is an approximation defined by a fixed-point closure; exactness requires additional geometric or statistical hypotheses.

For real scalar multiplets with antisymmetric generators $t_a$, transforming a source $J$ in the same representation leaves $J\cdot \varphi$ invariant. With an invariant functional measure, local changes of integration variables imply the Ward identity for the quantum effective action. In the convention $Z=e^{iW}$, $\phi =W_J$, $\Gamma =-W+\int J\phi$, it isThe Legendre transform gives ${\Gamma }_{\phi }=J$ and ${\Gamma }_A{|}_{\phi }=-W_A{|}_J$. Its classical limit is $\Gamma =-S_0$, not $S_0$, for the source term $+J\varphi$. Differentiating twice in $\phi$ produces two contact terms and the momentum-space identity $p_{1\mu }{\tau }_{a,ij}^{\mu }=i(t_a{)}_{ik}{\tau }_{kj}-{\tau }_{ik}i(t_a{)}_{kj}$ with the appropriate shifted two-point momenta. Changing the overall convention for $\Gamma$ changes both two-point and mixed vertices together; changing just one violates the identity. A gauge anomaly would invalidate the assumed invariant-measure argument.

The quantum effective potential is the derivative-free part of the quantum effective action evaluated on a spatially constant background. The displayed sign uses $\Gamma =-W+\int J\phi$ for $Z=e^{iW}$ and the source term $+J\varphi$; the opposite effective-action convention gives $-\int V_{\mathrm{eff}}$. Its derivatives give zero-momentum one-particle-irreducible vertices. The value depends on a renormalization scheme and scale; an arbitrary additive vacuum-energy constant may be fixed separately.

Use the Minkowski generating functional convention $Z=e^W$, source insertion $iJ\varphi$, classical field $\phi =-i\delta W/\delta J$, and $i\Gamma =W-i\int J\phi$. With $G_r^c=i^{-r}{\delta }^{r}W/\delta J^r$, the response kernel is $\delta \phi /\delta J=i{G}_2^c$. Differentiating $\delta \Gamma /\delta \phi =-J$ once gives the displayed two-point vertex, where the vertex means a derivative of $i\Gamma$. Differentiating the inverse kernel gives the three-point vertex: the connected three-point function is amputated by one inverse two-point kernel on each leg. In the alternative convention $W_c=-i\log Z$, the factors of $i$ are redistributed.

With $W=-i\log Z$, $\phi =\delta W/\delta J$ and $\Gamma =W-\int J\phi$, variation gives $\delta \Gamma /\delta \phi =-J$. Differentiating through the source-to-field map givesThe inverse is taken on the regulated nonsingular fluctuation space. Since $W^{(2)}=i{C}_2$, the full connected propagator is determined by the inverse effective-action quadratic kernel. Factors of $i$ and the minus sign depend on the explicitly chosen Minkowski Legendre transform convention.

// Target: perturbative-quantum-field-theory.bigb

Write $S=\frac{1}{2}\varphi A\varphi$ and use the vacuum prescription $A\mapsto A+i0$. Completing the Gaussian path integral gives $Z[J]/Z[0]=\exp [-iJ{A}^{-1}J/2]$, hence $W=-J{A}^{-1}J/2$, $\phi =-A^{-1}J$ and $\Gamma =\phi A\phi /2=S[\phi ]$. The source-independent determinant cancels in vacuum normalization. Without that normalization, the equality holds up to a field-independent vacuum constant.

// Target: perturbative-quantum-field-theory.bigb

The ratio of turbulent momentum diffusivity to turbulent scalar diffusivity. It need not equal the analogous molecular diffusivity ratio, and it is unrelated to the Schmidt number used for quantum entanglement. For ${\kappa }_T={\nu }_T/2$ its value is two.

The covariance of vertical velocity and buoyancy fluctuations has dimensions length squared per time cubed. It supplies buoyancy production in the local turbulent kinetic energy balance. This local flux density differs dimensionally from the section-integrated buoyancy flux of a plume, which includes an area or width integration.

The ratio of turbulent momentum diffusivity to buoyancy or heat diffusivity. Their ratio determines how the gradient Richardson number is related to the energy-budget Flux Richardson number. It is generally a closure-dependent function of the flow, not necessarily one.

Molecular scalar diffusion removes density-fluctuation variance at the positive rate displayed here. With this convention the budget for one half of the variance has dissipation term $-\chi$; the full variance budget has $-2\chi$. In a stationary locally homogeneous shear flow the half-variance budget gives $\overline{{\rho }^{\prime }{w}^{\prime }}{\overline{\rho }}_z+\chi =0$. An integral-scale model writes $\chi =\overline{{\rho }^{\prime 2}}q/L_{\rho }$, absorbing any normalization constant into $L_{\rho }$.

In a narrow tube with unstable mean density gradient $q={\partial }_z\bar{\rho }>0$, width-limited turbulent eddies have velocity scale $u_{\ast }\sim W\sqrt{gq/{\rho }_0}$. A mixing-length diffusivity gives downward density flux $F_{\rho }=K{W}^2\sqrt{g/{\rho }_0}{q}^{3/2}$ per unit area. With upward coordinate $z$, conservation is ${\bar{\rho }}_t={\partial }_z{F}_{\rho }$ and $q_t=K{W}^2\sqrt{g/{\rho }_0}{\partial }_z^2(q^{3/2})$. The closure presupposes established high-Reynolds number turbulence and requires boundary fluxes and finite available potential energy to be respected.

A positive association between a community's demographic composition and its incidence rate does not establish what proportion of incidents involved people from that demographic group. The same aggregate totals can be generated by different offender compositions and mechanisms. Individual incident records with age, sex and role would be needed to identify such proportions; aggregate regression coefficients concern community-level association.